Geometric Approximation Theory (Springer Monographs in Mathematics) 3030909506, 9783030909505

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Springer Monographs in Mathematics

Alexey R. Alimov Igor’ G. Tsar’kov

Geometric Approximation Theory

Springer Monographs in Mathematics Editors-in-Chief Minhyong Kim, School of Mathematics, International Centre for Mathematical Sciences, Edinburgh, United Kingdom; Korea Institute for Advanced Study, Seoul, South Korea Katrin Wendland, School of Mathematics, Trinity College Dublin, Dublin, Ireland Series Editors Sheldon Axler, Department of Mathematics, San Francisco State University, San Francisco, CA, USA Mark Braverman, Department of Mathematics, Princeton University, Princeton, NY, USA Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton, NY, USA Tadahisa Funaki, Department of Mathematics, University of Tokyo, Tokyo, Japan Isabelle Gallagher, Département de Mathématiques et Applications, Ecole Normale Supérieure, Paris, France Sinan Güntürk, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Claude Le Bris, CERMICS, Ecole des Ponts ParisTech, Marne la Vallée, France Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay, France Alberto A. Pinto, Department of Mathematics, University of Porto, Porto, Portugal Gabriella Pinzari, Department of Mathematics, University of Padova, Padova, Italy Ken Ribet, Department of Mathematics, University of California, Berkeley, CA, USA René Schilling, Institute for Mathematical Stochastics, Technical University Dresden, Dresden, Germany Panagiotis Souganidis, Department of Mathematics, University of Chicago, Chicago, IL, USA Endre Süli, Mathematical Institute, University of Oxford, Oxford, UK Shmuel Weinberger, Department of Mathematics, University of Chicago, Chicago, IL, USA Boris Zilber, Mathematical Institute, University of Oxford, Oxford, UK

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Alexey R. Alimov Igor’ G. Tsar’kov •

Geometric Approximation Theory

123

Alexey R. Alimov Faculty of Mechanics and Mathematics Moscow State University Moscow, Russia

Igor’ G. Tsar’kov Faculty of Mechanics and Mathematics Moscow State University Moscow, Russia

Steklov Mathematical Institute of Russian Academy of Sciences Moscow, Russia

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-030-90950-5 ISBN 978-3-030-90951-2 (eBook) https://doi.org/10.1007/978-3-030-90951-2 Mathematics Subject Classification: 41A65, 41A28, 41A46, 46B20, 54C60, 54C65 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to the memory of our teacher Prof. S. B. Stechkin

Preface

In the 19th century there arose a real need to deal with an important practical problems on the design of steam engines—here we speak about the famous Watt parallelogram, which was employed to convert the reciprocating motion of a piston into the rotational motion of the flywheel of the steam engine. The design of a steam engine looks surprisingly simple now, but it was only through the efforts of many engineers that it acquired its well-known form. It is in this context that in the middle of the 19th century Chebyshev introduced the important concept of best approximation (namely, the best uniform approximation) and made systematic use of it in practical applications. The initial impulses for Chebyshev’s engagement in approximation theory were set by the theory of hinge mechanisms1 (that are aimed at approximating any sufficiently smooth plane curve), which played an important role at that time of arising industrialisation. In his attempts to theoretically justify the construction of hinge mechanisms, Chebyshev was the first to pose the abstract problem of approximation of a given continuous function by a family of algebraic polynomials (or by a different system of functions) and introduced the Chebyshev (uniform) norm as a measure of proximity of continuous functions. The Chebyshev polynomials T n , which now bear his name (the symbol ‘T’ is derived from the continental transliterations of his name as ‘Tchebycheff’, ‘Tschebyscheff’) were also introduced by Chebyshev in a paper on hinge mechanisms presented to the St. Petersburg Academy in 1853. For the history of the origin of approximation theory, see Gusak [290], Steffens [542], Goncharov [283], and Butzer and Jongmans [163]. Subsequently, the concepts of best approximation and an element of best approximation (best approximant) were extended to general normed linear spaces and became the starting point of geometric approximation theory. The starting point of approximation theory is the concept of best approximation, that is, the distance from a given point x of a normed linear space X to a given nonempty set M
X, 1

For more on Chebyshev's mechanisms, see https://en.tcheb.ru.

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best approximation q ðx; MÞ :¼ inf k x  y k : y2M

By approximative properties one usually means concepts and properties defined in terms of best approximation or derived from it. The first of such concepts is that of an element of best approximation, or a nearest point. This is (for a given point x 2 X) a point y 2 M for which kx  yk ¼ qðx; MÞ. The metric projection of a point x 2 X onto a set M
X is the set PM x ¼ fy 2 M j q ðx; MÞ ¼ kx  ykg. In other words, PM x is the set of all nearest points in M for a given point x. A set M is called a set of existence (uniqueness) if, for each x 2 X, PM x is nonempty (empty or a singleton). Existence sets are also called proximinal sets. A set M is called a Chebyshev set if it is a set of existence and a set of uniqueness, that is, if for each x 2 X; PM x is a singleton. The fundamental concepts of approximation theory—the distance from a point to a set, a nearest point—are geometrically intuitive and, essentially, can be traced back a long way. A line in the Euclidean plane serves as an example of a Chebyshev set. The first significant results on Chebyshev sets were obtained in the theory of approximation of functions by P. L. Chebyshev, one of the founders of this theory. In 1859, he showed that (in modern terminology) in the space C½a; b the subspace Pn of polynomials of degree  n and the set Rn;m of rational functions with fixed n and m are Chebyshev sets (see Chaps. 2 and 11 below). The concepts of a Chebyshev set, an existence set, a uniqueness set, or a sun (see Sects. 1.1 and 10.2) and various variants thereof are considered as fundamental concepts of geometric approximation theory. Geometric approximation theory is mainly concerned with the relationship between various approximative properties of sets and their topological and geometrical properties (linearity, convexity, connectedness, etc.) under various conditions (strict convexity, smoothness, etc.) on the underlying normed space. Maturation of the geometric approximation theory, among the founders of which was S. B. Stechkin, N. V. Efimov and V. Klee, took place in the late 1940s and early 1950s. Among the precursors of geometric approximation theory we also mention P. L. Chebyshev, A. Haar, L. N. H. Bunt, T. Motzkin, A. N. Kolmogorov, S. I. Zukhovitskii, and M. G. Krein. In these and later years, much attention had been paid to problems of uniqueness, existence and stability of best approximation, which are central problems of approximation theory. In this period, many principal problems arising from the classical approximation theory were posed, and their solutions determined the contours of the modern geometric approximation theory. In this book we consider basic concepts and lines of research of the geometric theory of approximation. From the time of P. L. Chebyshev, the problems of existence and uniqueness of elements of best approximations have been considered as central problems of approximation theory. Theoretical studies were focused in properties and geometry of both concrete approximating sets like subspaces of polynomials and splines, classes of rational functions, etc., and abstract subsets of normed linear spaces. The foundations of the classical approximation theory were laid by pioneering studies of Chebyshev and his pupils. Later, in parallel with rapid

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development of approximation theory and considerable enlargement of approximating objects there arose a need for the study of approximative properties of abstract sets. The first steps in this direction was made by L. N. H. Bunt, T. Motzkin, N. Kritikos, H. Mann and by others, who began to study approximation problems in abstract finite-dimensional spaces. A new stage in the development of the subject began with N. V. Efimov and S. B. Stechkin, who were the first to deal with such problems in infinite-dimensional spaces and found first applications in the theory of approximation of functions. At the same time, V. Klee and his followers had also initiated extensive studies on this subject. The subsequent global rapid development of geometric theory of approximation has led to new concepts and methods and appearance of new research problems. For example, the study of problems on the characterization of elements of best approximation contributed to the emergence of new concepts like sun, Chebyshev sun, and Kolmogorov set. It became clear how the knowledge of approximative and structural properties of sets can be used in applications of geometric approximation theory in various fields involving approximation of functions, ill-posed problems, geometric topology, and in numerical methods. In Chap. 1, we give a brief introduction to geometric approximation theory. It is included for reference and selective reading. In this chapter, we also give main definitions, auxiliary results, examples, and some facts from functional analysis. In Chap. 2, we consider classical ideas and results that formed the basis of the theory of approximation by finite dimensional subspaces. In spaces CðQÞ, we give several results that either characterize or give sufficient conditions for existence of Chebyshev subspaces in CðQÞ. Among such condition, we menton de la Vallée Poussin’s estimates (see Sect. 2.1), the Haar characterization property (see Sect. 2.3), and Mairhuber’s theorem (see Sect. 2.5), which characterizes the metrizable compact sets Q such that the space CðQÞ contains nontrivial finite-dimensional Chebyshev subspaces. Many classical ‘least-square’ approximation problems are special cases of the general problem of best approximation in a Euclidean space by elements of a finite-dimensional subspace (or a convex set). In Chap. 3, we present two fundamental results on approximation by convex sets in the inner-product setting— the Kolmogorov criterion of best approximation and Phelps’s criterion of convexity of a Chebyshev set in a Euclidean space in terms of the Lipschitz continuity of the metric projection operator. Despite the fact that many problems of approximation theory in Euclidean spaces are well studied, it is in the Euclidean setting that the most famous unsolved problem of approximation theory is formulated: is any Chebyshev set convex in an infinite-dimensional Hilbert space? In Sect. 3.1, we give the particular case (which we call Deutsch’s lemma) of the well-known Kolmogorov criterion for a nearest element and prove Phelps’s theorem on the Lipschitz continuity of the metric projection onto Chebyshev sets (see Sect. 3.2). For more on approximation in Euclidean spaces, see, in particular, [5].

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The concept of compactness plays an important role in mathematics and applications. One direct extension of compact sets is the concept of boundedly compact sets (an intersection of such a set with any closed ball is compact). Further generalizations of this concept give rise to the important concept of approximative compactness introduced by Efimov and Stechkin in the 1950s. An approximatively compact set is always a set of existence. A number of results on proximally of abstract and concrete sets were obtained by employing the concept of approximative compactness and its variants. Moreover, approximative compactness and its generalizations turned out to be an important tool in proving existence theorems, solarity of Chebyshev sets, and existence of continuous eselections. In Chap. 4, we are mostly concerned with problems of existence of best approximation. In Sect. 4.1 we introduce boundedly compact and approximatively compact sets. In Sect. 4.2 we study basic properties of approximatively compact sets, recall the definition of uniformly convex spaces, and prove, in particular, that a convex closed nonempty subset of a complete uniformly convex space is approximatively compact, and hence, a set of existence. The important property of approximative s-compactness (with respect to a regular s-convergence) is discussed in Sect. 4.3. Using this concept, which is much more general than that of approximatively compact sets, it proves possible to establish the existence property for many classical and abstract objects like sets of rational functions, sets of exponential suns, and sets of splines with free knots in the space C½a; b (see Sect. 4.3.1). In Chap. 5, we consider the properties of elements of best approximation that distinguish them from other best approximant from a given approximating set. Much emphasis will be placed on characterization properties from which algorithms for construction of elements of best approximation can be derived. Some general theorems on characterization of an element of best approximation are given in Sect. 5.1. Discussion on suns and the Kolmogorov criterion for a nearest element, as well as local and global best approximations and unimodal sets (LG-sets) is given in Sect. 5.2. The Kolmogorov criterion of an element of best approximation in the space CðQÞ is formulated in Sect. 5.3. Continuity of the metric projection onto Chebyshev sets is discussed in Sect. 5.4. Some facts about differentiability of the distance function are given in Sect. 5.5. Relation of geometric approximation theory to geometric optics is discussed in Sect. 5.6. Convexity of Chebyshev sets and suns in studied in Chap. 6. In Sect. 6.1, we discuss the problem of convexity of suns. Convexity of Chebyshev sets in Rn is considered in Sect. 6.2. In 1966, V. Klee raised the conjecture that if in a Hilbert space there exists a nonconvex Chebyshev set, then this space also contains a Chebyshev set with convex bounded complement. This conjecture was solved in positive by Asplund. Moreover, Asplund showed that the existence of a Klee cavern is equivalent to the existence of a nonconvex Chebyshev set in Hilbert space. The Klee cavern and related question are discussed in Sect. 6.3. Johnson’s example of a nonconvex Chebyshev set in an incomplete pre-Hilbert space is recalled in Sect. 6.4.

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In Chap. 7, we study various structural connectedness-type properties of approximating sets (among which we consider Chebyshev sets, suns, moons, uniqueness sets, and so on). Classes of connectedness of sets are introduced and studied in Sect. 7.1. The problem of connectedness of suns is considered in Sect. 7.2. The main result here is that a sun in finite-dimensional normed linear space X is path-connected and locally path-connected. In Sect. 7.4, we present Klee’s example of a discrete Chebyshev set. Koshcheev’s example of a disconnected sun (in infinite-dimensional space) is constructed in Sect. 7.5. Several important properties of radial continuity of the metric projection are introduced and studied in Sect. 7.6. It is shown, in particular, that an approximatively compact Chebyshev sun is B-connected (that is, it has connected intersections with closed balls). For spans, segments, Menger-connected sets and monotone path-connected sets, see Sect. 7.7. Problems of continuous and semicontinuous selections of the metric projection, and their relations to solarity and proximinality of sets are examined in Sect. 7.8, where we also introduce and study the important property of Brosowski–Wegmannconnectedness. Solarity of the set of generalized rational fractions is studied in Sect. 7.10. Approximative properties of sets lying in a subspace are examined in Sect. 7.11. We finish this chapter by formulating some results on approximation by products (see Sect. 7.12). The problem of existence of Chebyshev subspaces is discussed in Chap. 8. In this chapter, we outline the known results on the existence of Chebyshev subspaces in finite- and infinite-dimensional spaces. We give some examples of infinite-dimensional Chebyshev subspaces of some infinite-dimensional spaces, formate and prove Garkavi’s theorem, which shows that there exists a Banach space that contains no nontrivial Chebyshev subspaces, and recall the well-known (presently unsolved) problem of whether there exists a separable Banach space which contains no nontrivial Chebyshev subspaces. For L1 -spaces, we give the classical M. G. Krein’s theorem to the effect that in L1 ½0; 1 (with Lebesgue measure) there are no finite-dimensional Chebyshev subspaces. Next, we formulate several extensions of Krein’s theorem, and in particular, provide Garkavi’s theorem (Theorem 8.4), which gives a necessary and sufficient condition that L1 ðQ; R; lÞ contain a Chebyshev subspace of dimension n\1. This result is followed by the result of P. Ørno, which says that there exists a separable reflexive Banach space without finite-dimensional Chebyshev subspaces. We conclude this section by two related results: the first one is the theorem by P. Ørno–Yu. A. Brudnyi–E. A. Gorin, which asserts that any Chebyshev set in L1 ½0; 1 is either a singleton or is infinite-dimensional. The second one is the new theorem, which says that any finite-dimensional sun in L1 ½0; 1 is convex. In Chap. 9, we consider the class of Efimov–Stechkin spaces (reflexive spaces with the Kadec–Klee property). This class is a natural class of spaces in which sets with “good structure” have “many” points of approximative compactness (points of stability of the metric projection operator). In particular, in such spaces any nonempty closed convex set is approximatively compact (and hence, an existence set); moreover, if the space is in addition rotund (strictly convex), then any

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nonempty closed convex set is a Chebyshev set. The Efimov–Stechkin spaces are characterized by the following important property: any hyperplane in such a space is approximatively compact. The class of complete uniformly convex spaces constitutes the well-known subclass of Efimov–Stechkin spaces. We start with several approximative–geometric properties that characterize the Efimov–Stechkin spaces (Theorem 9.1). Further in Sect. 9.2, we describe some classical properties of uniformly convex spaces and of their dual spaces (uniformly smooth spaces). In particular, in Sect. 9.3 and Sect. 9.4 we consider problems of uniqueness, strong uniqueness, and stability in such spaces. The problem of solarity of Chebyshev sets is considered in Chap. 10. In Sect. 10.1, we start with classical results on solarity of Chebyshev sets which are either boundedly compact or locally compact and have continuous metric projection. Several extensions of these results are discussed in Sect. 10.4. In Sect. 10.2, we examine relations between various classes of suns (suns, strict suns, a-suns, b-suns, c-suns, d-suns). Approximative properties of b-extensions of sets are considered in Sect. 10.2. The problem of solarity of Chebyshev sets is discussed in Sect. 10.3. In Sect. 10.4 we give some other results, in which the solarity of a set is proved under certain constraints on its geometric properties and continuity of the metric projection. We give a general theorem on solarity of monotone path-connected Chebyshev sets (see Sect. 10.4.1) and prove a theorem on solarity of boundedly compact P-acyclic sets (see Sect. 10.4.3). We also prove some other results on the relation of solarity with geometry of a set and stability of the metric projection operator. Rational approximation is studied in Chap. 11. Existence of best rational approximation is addressed in Sect. 11.1. In Sect. 11.2, we give de la Vallée Poussin’s estimate and Chebyshev’s characterization theorem of an element of best rational approximation. These results extend the classical Chebyshev’s and de la Vallée Poussin’s theorems from Sect. 11.2. In Sect. 11.3, we show that the set Rn;m of rational functions in Lp , 1  p\1, is a set of existence. However, by the classical Efimov–Stechkin’s theorem. the set Rn;m , m  1, is not a Chebyshev set in Lp , 1\p\1. In Sect. 11.4, we give theorems on characterization, existence, and uniqueness of best approximation by generalized rational fractions, when the numerator is taken from one linear subspace (or a convex set), and the denominator, from a different one. Results on characterization of best generalized rational approximation are given in Sect. 11.5. Uniqueness of general rational approximation is discussed in Sect. 11.6. Continuity properties of the best rational approximation operator are addressed in Sect. 11.7. In Chap. 12, we discuss the Haar cones, which is the natural extension of the Haar property from Sect. 2.3 to special nonlinear sets. These classes of sets are frequently encountered in various extreme problems. Properties of Haar cones, and uniqueness and strong uniqueness of best approximation by Haar cones are discussed in Sect. 12.1. The alternation theorem for Haar cones is given in Sect. 12.2. The related property of varisolvencity is considered in Chap. 12.3.

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Chapter 13 is devoted to approximation of vector-valued functions. In Sect. 13.1, we consider approximation of abstract (vector-valued) functions and study related interpolation and uniqueness problems. Uniqueness of best approximation in the mean for vector-valued functions is discussed in Sect. 13.2. The analogue of the classical Haar condition for vector-valued functions is studied in Sect. 13.3. Approximation of vector-valued functions by polynomials is considered in Sect. 13.4. Of special in interest is the problem of approximation in spaces of continuous bounded functions on sets which are not even locally compact. Here we consider one natural extension of the space of polynomials of degree  n, which, in general, is not a finite-dimensional space even for real-valued polynomials. Chapter 14 is concerned with the Jung constant. The Jung constant appears in many problems in various fields of mathematics. In Sect. 14.1 we give the definition and discuss some properties of the Jung constant JðXÞ :¼ rðMÞ supfdiam M jM
X; diam M\1g. In Sect. 14.2 we discuss the relations between the measure of nonconvexity of a space and the Jung constant. Relations between the Jung constant and fixed points of condensing and nonexpansive maps are discussed in Sect. 14.3. The problem of approximate solution of the equation f ðxÞ ¼ x is discussed in Sect. 14.4, in which the Jung constant in the Hilbert setting appears as a natural ingredient in the proofs. The Jung constant of the space ‘1n is considered in Sect. 14.5. In Sect. 14.6, we discuss the relation between the Jung constant (a geometric characteristic of a space) and the Jackson constant (an approximative characteristic of a space). The relative Jung constant is considered in Sect. 14.7. We formulate S. V. Berdyshev’s result, who found the relative Jung 1 constant of the space ‘ 1 n , n 2 N and described the extremal subsets of ‘ n , that is, 1 the sets M with J s ð‘ n Þ ¼ r M ðMÞ=diam M. Note that in his result, as distinct from the Jung constant problem JðXÞ, the exact value of the relative Jung constant J s ð‘ 1 n Þ is given for all n. The Jung constant of a pair of spaces is discussed in Sect. 14.8. In Sect. 14.9, we give remarks on intersections of convex sets and discuss the relation of this problem to the Jung constant. In Chap. 13 we present fundamental results on approximation of vector-valued functions. The Jung constant of a normed space space is defined as the radius of a smallest ball that can cover any subset of this space of diameter 1. If as a centre here one considers only points from the convex hull of the set, then the corresponding constant is called the relative Jung constant. Jung constants play an important role in the geometry of normed spaces, and also in approximation theory in relation to sharp constants in the Jackson inequality and the existence of fixed points of nonexpansive mappings. A survey of the results on the Jung and the relative Jung constants is given Chap. 14. As a particular example of approximation by set, we mention here the Chebyshev centre problem considered in Chap. 15. This problem has numerous applications in analysis, fixed-point theory, and in computational mathematics. We have tried to describe the current situation in this area as fully as possible. We also put forward some new results on existence and stability of Chebyshev centres.

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An important part of the theory of approximation of functions is related to the width theory, which considers optimal methods of approximation of classes of functions— here one compares classical means of approximation (algebraic polynomials, rational functions, trigonometric polynomials) with other means of approximation by a certain family of approximating sets described by the same number of parameters. In Chap. 16 we are focused on basic underlying geometric ideas in the width theory. Here we consider the inverse problem of approximation theory on the existence of a point with given best approximations from an increasing family of subsets. In Chap. 17, we give a survey on approximation by abstract sets and related problems on density characteristics of sets of uniqueness, existence, and stability. In Appendix A, results are given on Chebyshev systems of functions in the spaces C, C n and Lp . We define some classical abstract and concrete Chebyshev systems in the spaces C, C n and Lp , formulate extreme problems, of which some are still open, discuss structural formulas for extended polynomials, and formulate some conjectures in classical extreme problems. Radon, Helly, and Carathéodory theorems, which play an auxiliary role in our exposition, are recalled in Appendix B. Some open problems are formulated in Appendix C. Most results appear here for the first time in monograph form. The bibliographical notes should not be considered as being complete. The book was prepared with the financial support of the Ministry of Education and Science of the Russian Federation as part of the program of the Mathematical Center for Fundamental and Applied Mathematics under the agreement no. 075-15-2019-1621. The authors are grateful to all their colleagues from the Faculty of Mechanics and Mathematics at Moscow State University, from Steklov Mathematical Institute of Russian Academy of Sciences, and from Stechkin’s School for the insights they have shared with us. Special thanks are also due to M. V. Balashov, V. A. Berdyshev, S. V. Berdyshev, P. A. Borodin, S. Cobzas, N. A. Il’yasov, V. Ismailov, G. E. Ivanov, G. M. Ivanov, S. V. Konyagin, P.-L. Papini, K. S. Ryutin, E. M. Skorikov, V. M. Tikhomirov, and A. A. Vasil’eva for their helpful discussions and valuable suggestions, and to V. B. Demidovich and A. S. Kochurov for their help in preparation of Appendix A. The authors are also indebted to the referees for many valuable suggestions and corrections. The staff of Springer has been most understanding and patient in bringing this book to fruitition. In particular, we thank Donna Chernyk, Katherine Dolhon, David Kramer, Jayanthi Narayanaswamy and Boopalan Renu. We hope that this book will prove useful for researchers interested in advanced aspects of geometric approximation theory and its applications. Moscow, Russia 2022

Alexey R. Alimov Igor’ G. Tsar’kov

Contents

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3

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Existence. Compact, Boundedly Compact, Approximatively Compact, and s-Compact Sets. Continuity of the Metric Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Boundedly Compact and Approximatively Compact Sets . . . .

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Main 1.1 1.2 1.3

Notation, Definitions, Auxiliary Results, and Examples . . . Main Definitions of Geometric Approximation Theory . . . . Preliminaries and Some Facts from Functional Analysis . . . Elementary Results on Best Approximation. Strictly Convex Spaces. Approximation by Subspaces and Hyperplanes . . . .

Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Chebyshev’s and de la Vallée Poussin’s Theorems . . . . 2.2 Solarity and Alternant . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Haar’s Theorem. Strong Uniqueness of Best Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 A Short Note on Extremal Signatures . . . . . . . . . . . . . 2.5 Mairhuber’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Approximation of Continuous Functions by Finite-Dimensional Subspaces in the L1 -Metric . . . . 2.7 Remez’s Algorithm for Construction of a Polynomials of Near-Best Approximation . . . . . . . . . . . . . . . . . . . . Best Approximation in Euclidean Spaces . . . . . . . . . . . . . . . 3.1 Approximation by Convex Sets. Kolmogorov Criterion for a Nearest Element. Deutsch’s Lemma . . . . . . . . . . 3.2 Phelps’s Theorem on the Lipschitz Continuity of the Metric Projection onto Chebyshev Sets . . . . . . . 3.3 Best Least-Squares Polynomial Approximation. Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . .

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4.2 4.3

5

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Existence of Best Approximation . . . . . . . . . . . . . . . . . Approximative s-Compactness with Respect to Regular s-Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Applications in C½a; b . . . . . . . . . . . . . . . . . . 4.3.2 Applications in Lp . . . . . . . . . . . . . . . . . . . . .

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Characterization of Best Approximation and Solar Properties of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Characterization of an Element of Best Approximation . . . 5.2 Suns and the Kolmogorov Criterion for a Nearest Element. Local and Global Best Approximation. Unimodal Sets (LG-Sets) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Kolmogorov Criterion in the Space CðQÞ . . . . . . . . . . . . . 5.4 Continuity of the Metric Projection onto Chebyshev Sets . . 5.5 Differentiability of the Distance Function . . . . . . . . . . . . . . 5.6 Relation of Geometric Approximation Theory to Geometric Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convexity of Chebyshev Sets and Suns . . . . . . . . . . . . 6.1 Convexity of Suns . . . . . . . . . . . . . . . . . . . . . . 6.2 Convexity of Chebyshev Sets in Rn . . . . . . . . . . 6.2.1 Berdyshev–Klee–Vlasov’s proof . . . . . . 6.2.2 Asplund’s Proof . . . . . . . . . . . . . . . . . . 6.2.3 Konyagin’s Proof . . . . . . . . . . . . . . . . . 6.2.4 Vlasov’s Proof . . . . . . . . . . . . . . . . . . . 6.2.5 Brosowski’s Proof . . . . . . . . . . . . . . . . 6.3 The Klee Cavern . . . . . . . . . . . . . . . . . . . . . . . 6.4 Johnson’s Example of a Nonconvex Chebyshev Set in an Incomplete Pre-Hilbert Space . . . . . . .

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Connectedness and Approximative Properties of Sets. Stability of the Metric Projection and Its Relation to Other Approximative Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Classes of Connectedness of Sets . . . . . . . . . . . . . . . . . . . 7.2 Connectedness of Suns . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Dunham’s Example of a Disconnected Chebyshev Set with Isolated Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Klee’s Example of a Discrete Chebyshev Set . . . . . . . . . . . 7.5 Koshcheev’s Example of a Disconnected Sun . . . . . . . . . . 7.6 Radial Continuity of the Metric Projection. B-Connectedness of Approximatively Compact Chebyshev Suns . . . . . . . . . . 7.7 Spans, Segments. Menger Connectedness, and Monotone Path-Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7.8 7.9 7.10 7.11 7.12

xvii

7.7.1 The Banach–Mazur Hull . . . . . . . . . . . . . . . . . . . 7.7.2 Segments and Spans in Normed Linear Spaces . . 7.7.3 Monotone Path-Connectedness . . . . . . . . . . . . . . Continuous and Semicontinuous Selections of Metric Projection. Relation to Solarity and Proximinality of Sets . Suns, Unimodal Sets, Moons, and ORL-Continuity. Brosowski–Wegmann-connectedness . . . . . . . . . . . . . . . . Solarity of the Set of Generalized Rational Fractions . . . . Approximative Properties of Sets Lying in a Subspace . . . Approximation by Products . . . . . . . . . . . . . . . . . . . . . . .

8

Existence of Chebyshev Subspaces . . . . . . . . . . . . . . . . . . . 8.1 Chebyshev Subspaces in Finite-Dimensional Spaces . . 8.2 Chebyshev Subspaces in Infinite-Dimensional Spaces . 8.3 Finite-Dimensional Chebyshev Subspaces in L1 ðlÞ . . .

9

Efimov–Stechkin Spaces. Uniform Convexity and Uniform Smoothness. Uniqueness and Strong Uniqueness of Best Approximation in Uniformly Convex Spaces . . . . . . . . . . . 9.1 Efimov–Stechkin Spaces . . . . . . . . . . . . . . . . . . . . . . 9.2 Uniformly Convex Spaces . . . . . . . . . . . . . . . . . . . . . 9.3 Uniqueness of Best Approximation by Convex Closed Sets in Complete Uniformly Convex Spaces . . . . . . . 9.4 Strong Uniqueness in Uniformly Convex Spaces . . . . 9.5 Uniformly Smooth Spaces . . . . . . . . . . . . . . . . . . . . .

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10 Solarity of Chebyshev Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Solarity of Boundedly Compact Chebyshev Sets . . . . . . . 10.2 Relations Between Classes of Suns . . . . . . . . . . . . . . . . . 10.3 Solarity of Chebyshev Sets . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Solarity of Chebyshev Sets with Continuous Metric Projection . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Solarity and Structural Properties of Sets . . . . . . . . . . . . . 10.4.1 Solarity of Monotone Path-Connected Chebyshev Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Acyclicity and Cell-Likeness of Sets . . . . . . . . . . 10.4.3 Solarity of Boundedly Compact P-Acyclic Sets . . 11 Rational Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Existence of a Best Rational Approximation . . . . . . . . . . 11.2 Characterization of Best Rational Approximation in the Space C½a; b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Rational Lp -Approximation . . . . . . . . . . . . . . . . . . . . . . . 11.4 Existence of Best Approximation by Generalized Rational Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

11.5 11.6 11.7 11.8

Characterization of Best Generalized Rational Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniqueness of General Rational Approximation . . . . . . . Continuity of the Best Rational Approximation Operator Notes on Algorithms of Rational Approximations . . . . .

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12 Haar Cones and Varisolvency . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Properties of Haar Cones. Uniqueness and Strong Uniqueness of Best Approximation . . . . . . . . . . . . . . . . . . 12.2 Alternation Theorem for Haar Cones . . . . . . . . . . . . . . . . . 12.3 Varisolvency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Uniqueness of Best Approximation by Varisolvent Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Regular and Singular Points in Approximation by Varisolvent Sets . . . . . . . . . . . . . . . . . . . . . . . 13 Approximation of Vector-Valued Functions . . . . . . . . . . . 13.1 Approximation of Abstract Functions. Interpolation and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Uniqueness of Best Approximation in the Mean for Vector-Valued Functions . . . . . . . . . . . . . . . . . . . 13.3 On the Haar Condition for Systems of Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Approximation of Vector-Valued Functions by Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Some Applications of Vector-Valued Approximation . 14 The Jung Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Definition of the Jung Constant . . . . . . . . . . . . . . . . . 14.2 The Measure of Nonconvexity of a Space and the Jung Constant . . . . . . . . . . . . . . . . . . . . . . . . 14.3 The Jung Constant and Fixed Points of Condensing and Nonexpansive Maps . . . . . . . . . . . . . . . . . . . . . . 14.4 On an Approximate Solution of the Equation f ðxÞ ¼ x 14.5 On the Jung Constant of the Space ‘1n . . . . . . . . . . . . 14.6 The Jung Constant and the Jackson Constant . . . . . . . 14.7 The Relative Jung Constant . . . . . . . . . . . . . . . . . . . . 14.8 The Jung Constant of a Pair of Spaces . . . . . . . . . . . . 14.9 Some Remarks on Intersections of Convex Sets. Relation to the Jung Constant . . . . . . . . . . . . . . . . . .

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217 222 226 228

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15 Chebyshev Centre of a Set. The Problem of Simultaneous Approximation of a Class by a Singleton Set . . . . . . . . . . . . . . . . . 295 15.1 Chebyshev Centre of a Set . . . . . . . . . . . . . . . . . . . . . . . . . . 296 15.2 Chebyshev Centres and Spans . . . . . . . . . . . . . . . . . . . . . . . . 302

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xix

Chebyshev Centre in the Space CðQÞ . . . . . . . . . . . . . . . . Existence of a Chebyshev Centre in Normed Spaces . . . . . 15.4.1 Quasi-uniform Convexity and Existence of Chebyshev Centres . . . . . . . . . . . . . . . . . . . . . . 15.5 Uniqueness of a Chebyshev Centre . . . . . . . . . . . . . . . . . . 15.5.1 Uniqueness of a Chebyshev Centre of a Compact Set . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.2 Uniqueness of a Chebyshev Centre of a Bounded Set . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Stability of the Chebyshev-Centre Map . . . . . . . . . . . . . . . 15.6.1 Stability of the Chebyshev-Centre Map in Arbitrary Normed Spaces . . . . . . . . . . . . . . . . . 15.6.2 Quasi-uniform Convexity and Stability of the Chebyshev-Centre Map . . . . . . . . . . . . . . . . 15.6.3 Stability of the Chebyshev-Centre Map in Finite-Dimensional Polyhedral Spaces . . . . . . . . 15.6.4 Stability of the Chebyshev-Centre Map in CðQÞ-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.5 Stability of the Chebyshev-Centre Map in Hilbert and Uniformly Convex Spaces . . . . . . . 15.6.6 Stability of the Self-Chebyshev-Centre Map . . . . . 15.6.7 Upper Semicontinuity of the Chebyshev-Centre Map and the Chebyshev-Near-Centre Map . . . . . . 15.6.8 Lipschitz Selection of the Chebyshev-Centre Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.9 Discontinuity of the Chebyshev-Centre Map . . . . . 15.7 Characterization of a Chebyshev Centre. Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8 Chebyshev Centres That Are Not Farthest Points . . . . . . . . 15.9 Smooth and Continuous Selections of the Chebyshev-NearCentre Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.10 Algorithms and Applied Problems Connected with Chebyshev Centres . . . . . . . . . . . . . . . . . . . . . . . . . .

15.3 15.4

16 Width. Approximation by a Family of Sets . . . . . . . . . . 16.1 Problems in Recovery and Approximation Leading to Widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Definitions of Widths . . . . . . . . . . . . . . . . . . . . . . 16.3 Fundamental Properties of Widths . . . . . . . . . . . . . 16.4 Evaluation of Widths of ‘p -Ellipsoids . . . . . . . . . . 16.5 Dranishnikov–Shchepin Widths and Their Relation to the CE-Problem . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Bernstein Widths in the Spaces L1 ½0; 1 . . . . . . . . 16.7 Widths of Function Classes . . . . . . . . . . . . . . . . . .

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Contents

16.8 16.9

16.7.1 Definition of the Information Width . . . . . . . . . . . 16.7.2 Estimates for Information Kolmogorov Widths . . . 16.7.3 Some Exact Inequalities Between Widths. Projection Constants . . . . . . . . . . . . . . . . . . . . . . . 16.7.4 Some Order Estimates and Duality of Information Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7.5 Some Order Estimates for Information Kolmogorov Widths of Finite-Dimensional Balls . . . . . . . . . . . . 16.7.6 Order Estimates for Information Kolmogorov Widths of Function Classes . . . . . . . . . . . . . . . . . Relation Between the Jung Constant and Widths of Sets . . Sequence of Best Approximations . . . . . . . . . . . . . . . . . . .

17 Approximative Properties of Arbitrary Sets in Normed Linear Spaces. Almost Chebyshev Sets and Sets of Almost Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Approximative Properties of Arbitrary Sets . . . . . . . . . . . . 17.2 Sets in Strictly Convex Spaces . . . . . . . . . . . . . . . . . . . . . 17.3 Constructive Characteristics of Spaces . . . . . . . . . . . . . . . . 17.4 Sets in Locally Uniformly Convex Spaces . . . . . . . . . . . . . 17.5 Sets in Uniformly Convex Spaces . . . . . . . . . . . . . . . . . . . 17.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.7 Density and Category Properties of the Sets EðMÞ, ACðMÞ, and TðMÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.8 Category Properties of the Set UðMÞ . . . . . . . . . . . . . . . . . 17.9 Other Characteristics for the Size of Approximatively Defined Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.10 The Farthest-Point Problem . . . . . . . . . . . . . . . . . . . . . . . . 17.11 Classes of Small Sets ðZk Þ . . . . . . . . . . . . . . . . . . . . . . . . 17.12 Contingent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.13 Zajíček-Smallness of the Classes of Sets RðMÞ and R ðMÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.14 Zajíček-Smallness of the Classes of Sets Rk ðMÞ in Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.15 Almost Chebyshev Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.16 Almost Chebyshev Systems of Continuous Functions . . . . . A

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Chebyshev Systems of Functions in the Spaces C, C n , and Lp . . . . A.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Basic Facts About Structural Formulas for Extended Polynomials of Least Deviation from Zero . . . . . . . . . . . . . . . . A.3 Further Results and Inequalities for Derivatives . . . . . . . . . . . . A.4 Some Facts from Convex Analysis . . . . . . . . . . . . . . . . . . . . . . A.5 Tests for Least Deviation from Zero for Extended Polynomials from Chebyshev Spaces . . . . . . . . . . . . . . . . . . . .

459 462 463 466 467 468

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B

Radon, Helly, and Carathéodory Theorems. Decomposition Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 B.1 Radon, Helly, and Carathéodory Theorems . . . . . . . . . . . . . . . . 471 B.2 Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

C

Some Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

Chapter 1

Main Notation, Definitions, Auxiliary Results, and Examples

1.1 Main Definitions of Geometric Approximation Theory The best approximation (or the distance) of a given element x in a normed linear space X with norm  ·  from a given nonempty set M ⊂ X is, by definition, ρ(x, M) := inf  x − y. y ∈M

The concepts and properties defined in terms of best approximation (in particular, the existence, uniqueness, and stability properties of elements of best approximation) are called approximative properties. First and foremost is the concept of an element of best approximation, or a nearest point. This is (for a given x ∈ X) a point y ∈ M such that  x − y = ρ(x, M). For a given element x, the set of all nearest points (elements of best approximation or, briefly, best approximants) in M is denoted by PM x. In other words,   PM x := y ∈ M |  x − y = ρ(x, M) . The operator PM is called the best approximation operator (or the metric projection) onto a set M. Below, X is a real normed linear space, B(x, r) is the closed ball with centre x and radius r; ˚ r) is the open ball with centre x and radius r; B(x, S(x, r) is the sphere with centre x and radius r. For brevity, the unit ball and the unit sphere will be denoted by B := B(0, 1) and S = S(0, 1), respectively; B∗ is the unit ball of the dual space X ∗ . A set M   is called a set of existence (respectively, uniqueness) if PM x is nonempty (empty or a singleton) for each x ∈ X. Existence sets are also called proximinal sets. For a given   M ⊂ X, we set © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_1

1

2

1

Main Notation, Definitions, Auxiliary Results . . .

E(M) := {x ∈ X | PM x  }. If E(M) = X, then M is a set of existence. An existence set is always closed and nonempty. Indeed, if a cluster point of an existence set M were not contained in M, then this point would clearly fail to have a nearest point in M. The converse assertion clearly holds in every finite-dimensional space X: every nonempty closed subset of a finite-dimensional normed space is an existence set. A set M is called a Chebyshev set if it is a set of existence and a set of uniqueness, that is, if for each x ∈ X, the set PM x is a singleton. A line in the Euclidean plane serves as an example of a Chebyshev set. Every space contains trivial Chebyshev sets: the entire space and singletons. If Q denotes some property (for example, connectedness), then we say that a closed set M has one of the following properties under the given condition is: P-Q if for all x ∈ X, the set PM x is nonempty and has the property Q; P0 -Q if PM x has the property Q for all x ∈ X; B-Q if M ∩ B(x, r) has the property Q for all x ∈ X, r > 0; ˚ if M ∩ B(x, ˚ r) has the property Q for all x ∈ X, r > 0. B-Q For example, a closed subset of a finite-dimensional space is P-nonempty; that is, it is an existence set (a proximinal set). Let   M ⊂ X. A point x ∈ X \ M is called a solar point if there exists a point y ∈ PM x   (a luminosity point) such that   y ∈ PM (1 − λ)y + λx for all λ ≥ 0 (1.1) (geometrically, this means that there is a ‘solar’ ray emanating from y and passing through x such that y is a nearest point in M for every point on the ray). A point x ∈ X \ M is called a strict solar point if PM x   and condition (1.1) holds for every point y ∈ PM x. Further, if for every x ∈ X \ M, condition (1.1) is satisfied for every point y ∈ PM x, then a point x is called a strict protosolar point (in contrast to the case of strict solar points, in this setting, a nearest point y to x may fail to exist). A closed set M ⊂ X is called a sun (respectively, a strict sun) if every point x ∈ X \ M is a solar point (a strict solar point) for M (see Fig. 1.1). A set M ⊂ X is called a strict protosun if every point x ∈ X \ M is a strict protosolar point. A set M ⊂ X is called a protosun if for every x  M with PM x   there exists a luminosity point in M. As a rule, we always assume that a strict protosun is closed. Note that a (closed) strict protosun need not be an existence set. We also note that a convex set is always a strict protosun (a convex set of existence is always a strict sun). The concept of a ‘sun’ was introduced by Efimov and Stechkin in [222]. The term ‘strict protosun’ was introduced by the authors in [24] to avoid confusion in the definitions of a ‘strict sun of existence’ (a strict sun) and a ‘strict sun that is not necessarily an existence set’ (a strict protosun) — what is worse, such sets were simply called suns in different schools. We note that (closed) strict protosuns are exactly

1.1 Main Definitions of Geometric Approximation Theory

3

Fig. 1.1 A sun M that is not a strict sun in the space 2∞ (the 2∞ -unit ball is a ‘square’). Here, y0 is a luminosity point for x; the point y1 is a nearest point to x, but y1 is not a luminosity point.

closed Kolmogorov sets (these are the sets satisfying the Kolmogorov criterion for an element of best approximation; see Sect. 5.1). For suns, the Kolmogorov criterion asserts, in particular, that a point not lying in a sun can be strictly separated from it by an open convex support cone. ‘Suns’ have important characteristic features. For example, they can be characterized by separation properties of one kind or another: a ball can be separated from such a set by means of a larger ball or a support cone. These properties are akin to the well-known separation properties of convex sets by half-spaces (hyperplanes). Since a nonempty closed subset of a finite-dimensional space X is a set of existence, each strict protosun in X is a strict sun. This is not so in the infinitedimensional case — a strict protosun can be antiproximinal (that is, every point outside it has empty metric projection in it). Paying homage to P. L. Chebyshev as a founder of approximation theory, Efimov and Stechkin proposed the new term ‘Chebyshev set’, which was soon generally adopted. Chebyshev introduced the important concept of best approximation (in particular, best uniform approximation), made systematic use of it in practical applications, and developed its theoretical foundations. In studying best approximation in C[a, b] by the sets Pn of polynomials of degree at most n and the sets Rn,m := {p/q ∈ C[a, b] | p ∈ Pn, q ∈ Pm }

(1.2)

of rational fractions on [a, b], Chebyshev and his students pupil introduced the concept of alternance (the term ‘alternance’ was introduced much later by Akhiezer). Elaborating on Chebyshev’s alternance idea, Kirchberger [336], Borel [112], and Young (for polynomials) and Akhiezer [2] and Walsh [594] (for rational fractions) justified the uniqueness of best approximations and proved the existence theorem. For the historical background of approximation theory we refer the reader to the books by Gusak [283] and Steffens [530] and the survey papers by Goncharov [276], Butzer and Jongmans [156], Garkavi [262], Vlasov [592], and Balaganskii and Vlasov [61]. Thus, the sets Rm,n and Pn are Chebyshev sets in C[a, b]. Moreover, the elements of best uniform approximation in Rm,n and Pn can be characterized in terms of alternance. We shall see later that this characterization implies that the (nonlinear) set Rm,n is a Chebyshev sun in C[a, b].

4

1

Main Notation, Definitions, Auxiliary Results . . .

The set RV,W = {p/q | p ∈ V,

q ∈ W,

q(t) > 0,

t ∈ Q}

of generalized rational fractions is a strict protosun in C(Q) (see Sect. 7.10); here V, W are arbitrary convex subsets of a real or complex C(Q), where Q is a Hausdorff compact set (see Brosowski and Wegmann [143], Chong and Watson [165], Braess [132]). In contrast to the classical Rm,n -setting, a best generalized rational approximant from RV,W may not exist or may not be unique. Chebyshev sets and some generalizations thereof have been extensively studied. In the present book, we have not aimed at giving a detailed analysis of the available bibliography, but refer the reader to the surveys [592], [61], [24]. Below, we shall formulate and prove a number of general results of geometric approximation theory and provide an account of one of the most challenging problems (the Efimov–Stechkin– Klee problem) of geometric approximation theory — the problem of convexity1 of Chebyshev sets in Hilbert spaces: is every Chebyshev set convex in an infinite-dimensional Hilbert space?

1.2 Preliminaries and Some Facts from Functional Analysis In what follows, X is a real normed linear space with norm  · , ˚ r) := {y ∈ X |  x − y < r } is the open ball with centre x ∈ X and B(x, radius r > 0; B(x, r) := {y ∈ X |  x − y ≤ r } is the closed ball with centre x and radius r; S(x, r) := {y ∈ X |  x − y = r } is the sphere with centre x and radius r. We recall the definition of some classical normed spaces. By Rn we shall mean the standard real n-dimensional space with the norm  x2 =  xRn :=

n 

(x (k) )2

 1/2 ,

x = (x (1), . . . , x (n) ) ∈ Rn .

(1.3)

k=1

By  p (Γ), 1 ≤ p < ∞, we mean the space of all functions x : Γ → R that vanish on the complement of an at most countable set Γ and have the finite norm  1/p  |x(t)| p , x = x(t) ∈  p (Γ). (1.4)  x p := t ∈Γ

1 Let X be a linear space, C ⊂ X. A set C is convex if for every a, b ∈ C, the closed interval [a, b] is also contained in C.

1.2 Preliminaries and Some Facts from Functional Analysis

5

The space  p (Γ) with p = 2 is a Hilbert space (that is, a complete Euclidean√space2 with the inner product (x, y) := t ∈Γ x(t)y(t) and the induced norm  x2 = (x, x), x ∈  2 (Γ). p The classical sequence space  p and the n-dimensional vector space n are parp ticular cases of  (Γ). p The space n consists of all real vectors x = (x (1), x (2), . . . , x (n) ), x (i) ∈ R, with the norm n  1/p   x p := |x (k) | p , x = (x (1), . . . , x (n) ) ∈ Rn . k=1

The space  p , 1 ≤ p < ∞, consists of all infinite real sequences x = (x (1), x (2), . . . ) with finite norm ∞  1/p  |x (k) | p .  x p := k=1

The space  ∞ (Γ) consists of all functions

x : Γ → R that vanish at the complement of an at most countable set Γ. This space is equipped with the norm  x∞ := sup |x(t)|, t ∈Γ

x = x(t) ∈  ∞ (Γ).

(1.5)

The classical spaces n∞ and  ∞ are particular cases of the space  ∞ (Γ). The space C(Q), where Q is a compact space, is defined as the linear space of real continuous functions on Q equipped with the standard (Chebyshev or uniform) norm  f C(Q) := sup | f (t)| = max | f (t)|. t ∈Q

t ∈Q

It is easily checked that C(Q) is a normed linear space. The space c = c(N) is the vector space consisting of real sequences x = (x (n) ) with finite limit limn→∞ x (n) . The space c is endowed with the norm  x∞ := sup{|x (i) | | i ∈ N}. The space c0 = c0 (N) is the vector space consisting of real sequences x = (x (n) ) such that limn→∞ x (n) = 0. The space c0 is equipped with the norm  x∞ := max{|x (i) | | i ∈ N}. Let (Ω, μ) be a measure space. The set of all measurable functions (classes of functions) f : Ω → R with finite norm ∫  p1  f  L p := | f | p dμ , 1  p < ∞, Ω

2 Throughout, by a Euclidean space we mean an inner product space.

6

1

Main Notation, Definitions, Auxiliary Results . . .

is a vector space. Its quotient space over the linear manifold of all measurable functions that are equal on a null set is also a vector space. Being equipped with the norm  · , the latter space becomes a normed space. Besides, if μ is a σ-additive measure, then this space, which is denoted by L p = L p (Ω, μ), is complete (is a Banach space). All functions f ∈ L p and g ∈ L q satisfy the following well-known Hölder inequality: ∫  ∫   f g dμ  | f g|dμ   f  L p g L q ,  Ω

Ω

where p, q > 1 and p1 + q1 = 1. For p = 2, L p is a Hilbert space with the inner product ∫ ( f , g) := f g dμ. Ω

is not in general a normed space, because the For 0 < p < 1, the space p is not convex if dim L p > 1. But endowed with the metric ρ( f , g) := function  ·  L ∫ p p | f − g| dμ, f , g ∈ L , this space becomes a metric space. Ω Norms  ·  and  ·  on a linear space X are called equivalent if there exists a number C > 0 such that Lp

C −1  x   x  C x

for all x ∈ X.

On the linear space Rn , all norms are equivalent. As a corollary, the intersection of a finite-dimensional subspace with a closed ball is always compact. This, in turn, implies that a finite-dimensional subspace of a normed linear space is a set of existence. Indeed, consider an arbitrary point x0 ∈ X. The function ϕ(x) =  x − x0  is continuous, and by Weierstrass’s extreme value theorem has a minimum on the compact set L ∩ B(x0, (x0, L) + 1) at some point y0 . It is easily seen that  y0 − x0  = (x0, L). In what follows, by a linear manifold we shall mean a set that contains the linear hull of every pair of vectors from this set. A translation of this manifold by some vector is called an affine manifold. By a subspace we shall always mean a closed linear manifold in X. Its translation by a vector is called an affine subspace or a plane. A subspace Y is called a proper subspace of a space X if Y  X or Y  . The space of all continuous functionals with the standard operator norm  · X ∗ forms a linear space, which we denote by X ∗ . We recall the following classical result (Fig. 1.2). Lemma 1.1 (F. Riesz) Let L be a proper subspace of a normed linear space X. Then for every ε > 0, there exists a point xε ∈ S = S(0, 1) (called an ε-perpendicular) such that ρ(xε, L) ≥ 1 − ε. The following result is an extension of Riesz’s lemma (see, for example, [199, pp. 7–8]).

1.2 Preliminaries and Some Facts from Functional Analysis

7

x

1−ε

BX

Fig. 1.2 Riesz’s lemma.

Theorem 1.1 (C. A. Kottman) If X is an infinite-dimensional normed linear space, then on the unit sphere S of X there exists a sequence of points (sn ) such that sn − sm  > 1

for n  m.

The following result is frequently useful. Theorem 1.2 (completeness test) Let X = (X, ) be a metric space. Then X is complete if and only if the intersection of every nested sequence of closed sets (An ) ⊂ X such that diam An → 0, n → ∞, is nonempty. Let L be an affine manifold in some vector space over a subfield K of R (in particular, over Q or R). A function f : L → R is called convex over the field K if f (αx + βy)  α f (x) + β f (y) for all vectors x, y ∈ L and numbers α, β ∈ K, α, β > 0, α + β = 1. A function  : L → R is an affine function if f (αx + βy) = α f (x) + β f (y) for all x, y ∈ L and α, β ∈ K, α, β > 0, α + β = 1. An affine function is always convex. A liner combination of an affine function and a constant function is an affine function. The sum of a convex function and an affine function is a convex function. The problem of an extension of a linear functional frequently occurs in analysis. The principal role in this field of questions is played by the following classical theorem. Theorem 1.3 (Hahn–Banach) Let L be a vector space over a field K ⊂ R, let M0 be some plane, and let 0 be an affine function on M0 such that f  0 on M0 , where

8

1

Main Notation, Definitions, Auxiliary Results . . .

f : L → R is some convex function. Then there exists an affine function  : L → R such that f   on L and  ≡ 0 on M0 . Corollary 1.1 Let L be a linear space over a field K ⊂ R and let f : L → R be a convex function. Then for every point x0 ∈ L, there exists an affine function (x) supporting the graph of the function y = f (x) at the point (x0, f (x0 )); that is, f   on L and f (x0 ) = (x0 ). Remark 1.1 If in addition, f : L → R is a homogeneous function (that is, f (αx) = α f (x) for all x ∈ L and α  0), then the above function (x) is linear over the field K. In a linear space L over R, consider a convex set M  0 such that every line passing through the origin crosses this set in an interval for which the origin is an interior  point. Then t>0 t M = L. Consider on L the convex homogeneous nonnegative function fM (x) := inf{t > 0 | x ∈ t M }, which is known as the Minkowski functional of the set M (or the gauge of M). By definition, if fM (x) < 1, then x ∈ M, while if fM (x) > 1, then x  M. As such a set M one can take any open convex set containing 0. Then every point x ∈ X \ M can be separated from M by some linear functional x ∗ . Indeed, the Minkowski functional fM is a convex continuous function on X, and fM (x)  1. By Corollary 1.1, there is a linear functional  = x ∗ that supports the graph of fM at the point (x, fM (x)); that is, fM (y)  x ∗ (y) for all y ∈ M and 1  fM (x) = x ∗ (x). Since the Minkowski functional is bounded on M ( fM (x)  1 for x ∈ M), the linear functional x ∗ is bounded on some neighbourhood of the origin; that is, the norm of this functional is finite, and hence, x ∗ is a continuous linear functional. Geometrically, this means that the hyperplane L = {y ∈ X | x ∗ (y) = x ∗ (x)} contains x and is disjoint from the set M. Note that if a point x is at a positive distance from M (that is, fM (x) > 1), then the hyperplane L0 = {y ∈ X | x ∗ (y) = 1} strictly separates the set M and the point x (they lie in different half-planes with the boundary L0 ). Corollary 1.2 Every two distinct points x, y ∈ X can be strictly separated by an appropriate continuous linear functional x ∗ (that is, x ∗ (y) < x ∗ (x) for x ∗ ∈ X ∗ ). Corollary 1.3 Let V be a linear subspace of a normed linear space X. Then for every x ∈ X, there exists x ∗ ∈ X ∗ ,  xX ∗ = 1, such that ρ(x, V) = x ∗ (x),

x ∗ (V) = 0.

1.2 Preliminaries and Some Facts from Functional Analysis

9

Theorem 1.4 (Hahn–Banach) Let X = (X,  · ) be a nontrivial normed linear space over R. Then for each x ∈ X, there exists a functional x0∗ ∈ X ∗ such that  x0∗ X ∗ = 1 and for which the inequality |(x, x0∗ )| ≤  x  x0∗ X ∗ becomes an equality,

(x, x0∗ ) =  x  x0∗ X ∗ =  x.

Let X = (X,  · ) be a normed linear space over R (C) and let X ∗ be the dual space. The dual space to X ∗ is the space X ∗∗ = (X ∗ )∗ of all continuous linear functionals x ∗∗ on X ∗ ; this space is called the second dual of X. The norm of an element x ∗∗ ∈ X ∗∗ is defined as |x ∗∗ (x ∗ )| . sup  x ∗ X ∗ x ∗ 0 Remark 1.2 By duality, an element x ∈ X can be identified with a linear functional on X ∗ , which evaluates as (x, x ∗ ) = x ∗ (x), x ∗ ∈ X ∗ . By the Hahn–Banach theorem, the norm of x ∗∗ as an element of the second dual is equal to  x, because |(x, x ∗ )|   x  x ∗ X ∗ for all nonzero functionals x ∗ ∈ X ∗ and since this inequality becomes an equality on some nonzero functional x0∗ . So, there is a natural isometric embedding (called the natural embedding) of the space X into the space X ∗∗ . If this embedding sends X onto the entire X ∗∗ , then the space X is reflexive. If a space X is reflexive, then so is X ∗ . In what follows, we shall require the classical James theorem (see, for example, [198]), which characterizes the reflexive Banach spaces. Theorem 1.5 (R. James) The following properties of a Banach space X are equivalent: (a) X is reflexive; (b) X ∗ is reflexive; (c) each bounded sequence in X has a weakly convergent subsequence; (d) every nested sequence (Cn ) of closed bounded nonempty convex sets from X, Cn ⊃ Cn+1 , n ∈ N, has nonempty intersection; (e) each closed separable subspace of X is reflexive; (f) X is isomorphic to a reflexive space; (g) each bounded linear functional on X attains its norm;

10

1

Main Notation, Definitions, Auxiliary Results . . .

(h) the following assertion is not true: for all 0 < θ < 1, there exist a sequence (xn∗ ) from S ∗ and a sequence (xn ) from S such that xn∗ (x j ) ≥ θ for n ≤ j and xn∗ (x j ) = 0 for n > j; (i) every convex closed nonempty subset of X is a set of existence. A subset E of a convex set V is called its face if [a, b] ⊂ E whenever (a, b)∩E  . A point x ∈ V is an extreme point if {x} is a face of V; in other words, x is not an interior point of any interval lying in V. This is equivalent to saying that u = v whenever x = (u + v)/2, u, v ∈ E. The set of extreme points of a convex set V is denoted by ext V. Its role is revealed in the following fundamental result. Theorem 1.6 (Krein–Milman) Let K be a convex compact subset of a normed linear space X. Then the set of extreme points of this compact set is nonempty, and the closure of the convex hull of the extreme points of K coincides with K (that is, conv ext K = K). Remark 1.3 The set of extreme points of a convex compact set can be nonclosed (and hence, noncompact). In a space X, one can consider the weak topology (w-topology), which differs from the original strong (norm) topology. This topology is generated by the subbase consisting of open strips Ox ∗,α,β := {x ∈ X | x ∗ (x) ∈ (α, β)}, where x ∗ ∈ X ∗ and α, β ∈ R. So, an open set in the weak topology consists of unions of finite intersections of such strips. Of course, an open set in the weak topology is open in the strong topology. A set K ⊂ X is weakly compact if it is compact in the weak topology (that is, every weakly open cover of this set contains a finite subcover). The dual space X ∗ can be equipped with the weak-star topology (w ∗ -topology, for short). This topology is generated by the subbase consisting of open strips Ox,α,β := {x ∗ ∈ X ∗ | x ∗ (x) = (x, x ∗ ) = x(x ∗ ) ∈ (α, β)}, where x ∈ X and α, β ∈ R. So, an open set in the w ∗ -topology is a finite union of such strips. Of course, a w ∗ -open set is open in the strong topology. A set K ⊂ X is w ∗ -compact if it is compact in the w ∗ -topology (that is, every w ∗ -open cover of this set contains a finite subcover). Theorem 1.6 can be phrased in the following more general form. Theorem 1.7 (Krein–Milman) Let K be a convex weakly compact (respectively, a convex w ∗ -compact) subset of a normed linear space X (X ∗ ). Then the set of extreme points of K is nonempty, and the closure of the convex hull of the extreme points of K coincides with K.

1.3

Elementary Results on Best Approximation . . .

11

Example 1.1 The unit ball B of the space C[0, 1] has only two extreme points: f1 ≡ 1 and f2 ≡ −1 (it is easily checked that the remaining functions f ∈ B are interior points of some intervals whose endpoints lie in B). The closure of the convex hull of the extreme points is the interval [ f1, f2 ]  B. This shows that the space C[0, 1] is not a dual space, because the unit ball B of every dual space is w ∗ -compact (and hence B should coincide with conv ext B by the Krein–Milman theorem). Exercise 1.1 Indicate all extreme points of the unit balls of the spaces C[0, 1],  1 , L 1 [0, 1].

1.3 Elementary Results on Best Approximation. Strictly Convex Spaces. Approximation by Subspaces and Hyperplanes Let   M ⊂ X. The distance ρ( · , M) to the set M and the metric projection PM to M were defined in Sect. 1.1 (see p. 1). The following simple result is frequently useful. Proposition 1.1 Let   M ⊂ X. Then the distance function ρ( · , M) to M is 1-Lipschitz (and hence continuous). More precisely, if x, y ∈ X, M ⊂ X, then ρ(x, M) ≤  x − y + ρ(y, M), | ρ(x, M) − ρ(y, M)| ≤  x − y.

(1.6)

Proof Let x, y ∈ X, u ∈ M. By the triangle inequality and the definition of the distance function, we have ρ(x, M) ≤  x − u ≤  x − y +  y − u. Subtracting  x − y from both sides gives ρ(x, M) −  x − y ≤  y − u. Since u ∈ M is arbitrary and since the left-hand side of the previous inequality is independent of u, we have ρ(x, M) −  x − y ≤ ρ(y, M). By symmetry, ρ(y, M) −  x − y ≤ ρ(x, M), which gives − x − y ≤ ρ(x, M)− ρ(y, M) ≤  x − y, and so | ρ(x, M)− ρ(y, M)| ≤  x − y, the result required.  Proposition 1.2 The following equivalences hold: B(x, r) ⊂ B(x , r ) ⇐⇒  x − x  ≤ r − r; ˚ r) ⊂ B(x ˚ , r ) ⇐⇒  x − x  < r − r. B(x,

(1.7) (1.8)

Proof Indeed, if  x − x ≤ r − r, then for all y ∈ B(x, r), we have  x − y ≤  x − y +  x − x  ≤ r + r − r = r ; that is, y ∈ B(x , r ). Conversely, let B(x, r) ⊂ B(x , r ). We extend the line through the points x, x (if x = x , then we can take any of these lines) to its intersection with S(x, r) at the point y farthest from x . Then x ∈ [y, x ] and  x − x  =  y − x  −  y − x =  y − x  − r. Moreover, y ∈ B(x, r) ⊂ B(x , r ), and so  y− x  ≤ r , which gives  x− x  ≤ r −r. The proof of the second assertion is similar. 

12

1

Main Notation, Definitions, Auxiliary Results . . .

Definition 1.1 A space X is called strictly convex (or rotund) if its unit sphere does not contain nondegenerate intervals. The classic examples of strictly convex sets are the spaces L p and  p for 1 < p < ∞. It is also known that on every separable or reflexive space there exists an equivalent strictly convex norm. The space C[0, 1] and, more generally, C(K), card K ≥ 2, is not strictly convex. Proposition 1.3 The following conditions are equivalent: (1) the space X is strictly convex; (2) if x, y ∈ S, and  x + y = 2, then x = y; (3) if x, y ∈ X, and 2 x 2 + 2 y 2 −  x + y 2 = 0, then x = y; (4) if x, y ∈ X,  x + y =  x +  y, x  0, then y = λx for some λ ≥ 0.   = Proof (1) ⇔ (2). If the sphere contains a nondegenerate interval [x, y], then  x+y 2 1. Conversely, assume that the sphere does not contain nondegenerate intervals. Let x, y ∈ S, and  x + y = 2. If x  y, then there exists a point z ∈ [x, y] such that z < 1 (hence z  x+y 2 ). We can assume without loss of generality that   x+y z ∈  x, x+y . Hence = λz + (1 − λ)y for some λ ∈ (0, 1), and therefore, 2 2 x+y   1 = 2 ≤ λz + (1 − λ) y < 1. (3) ⇒ (2). Let x, y ∈ S,  x + y = 2. Then 2 x 2 + 2 y 2 −  x + y 2 = 0 and

x = y.

(2) ⇒ (3). Let 2 x 2 + 2 y 2 −  x + y 2 = 0. By the triangle inequality, 0 = 2 x 2 + 2 y 2 −  x + y 2 ≥  x 2 − 2 x  y +  y 2 ≥ ( x −  y)2 ≥ 0, and so  x =  y and  x+y 2  =  x. If x  0, then on dividing by  x and using condition (2), we get x = y. (4) ⇒ (2). Let x, y ∈ S,  x + y = 2. By condition (4), x = λy. Since x, y ∈ S, we have x = y. (2) ⇒ (4). Let x, y ∈ X,  x + y =  x +  y. We can assume without loss of generality that  x ≥  y > 0. We have   y  x    x ≥ 2 y ≥ y +  y  = y + x − x +  x  x (1.9)   y  ≥  x + y −  x 1 − =  x +  y −  x +  y = 2 y  x  y  y   (here we used the fact that x  ≤ 1, which gives  x + y −  x 1 − x  >0, inasmuch as  x + y >  x by the hypothesis). Dividing both parts of (1.9) by  y, we get  x y   + 2= ,  x  y

1.3

Elementary Results on Best Approximation . . .

and so yy  = is proved.

x x 

13

by condition (2), that is, y = λx for some λ > 0. Proposition 1.3 

Remark 1.4 The space C[0, 1] is not strictly convex. Indeed, the functions f (t) = 1 and g(t) = t, 0 ≤ t ≤ 1, both lie in the unit sphere S; this sphere also contains the interval λ f + (1 − λ)g, 0 ≤ λ ≤ 1, because the absolute value of the function λ f +(1−λ)g is 1 and is attained at the point t = 1. Nevertheless, the (nonrotund) norm in C[0, 1] can be replaced by an equivalent strictly convex norm that is arbitrarily close to the original norm. To this end, it suffices to put ∫ 1  1/2 | f | 2 dt ,  f  := max | f (t)| + ε t ∈[0,1]

0

where ε > 0 is sufficiently small. Remark 1.5 Every separable space can be renormed to be a strictly convex and even locally uniformly convex space.3 More generally, according to Troyansky’s theorem, every weakly compactly generated space (the class of such spaces contains the reflexive spaces and the separable spaces) can be equivalently renormed to be a locally uniformly convex space ((LUR)-space). Nevertheless, there exist Banach spaces that cannot be remormed to be a strictly convex space. An example of such a space is  ∞ (Γ), where Γ is any uncountable set. The following two results are clear. Proposition 1.4 Let M ⊂ X, x ∈ X. If y ∈ PM x and z ∈ [x, y], then y ∈ PM z. If in addition, X is strictly convex, then for all z ∈ (x, y], the metric projection PM z is the singleton {y}. Proof Indeed, using (1.6), we have ρ(z, M) ≤ z − y =  x − y −  x − z = ρ(x, M) −  x − z ≤ ρ(z, M), that is, z − y = ρ(z, M) and y ∈ PM z. Assume now that X is strictly convex and y1 ∈ PM z. If y1 does not lie on the line zx, then by Proposition 1.3,  x − y1  <  x − z + z − y1  =  x − z + z − y =  x − y = ρ(x, M), which is impossible. Hence y1 lies on the line zx, and hence, as is easily seen, y1 = y.  Proposition 1.5 Let M be a set of existence. Then M is nonempty, closed, and for all x ∈ X, the set PM x is closed. Proof Let x ∈ M, y ∈ PM x. Then  x − y = ρ(x, M) = 0, and hence x = y ∈ M. Next, if yn ∈ PM x, yn → y, then  x − y = lim  x − yn  = ρ(x, M). By the above,  y ∈ M, and hence y ∈ PM x. 3 See Definition 4.13 on p. 70 below.

14

1

Main Notation, Definitions, Auxiliary Results . . .

Consider some examples illustrating the concepts of the distance, the metric projection, and a Chebyshev set. Example 1.2 In the plane R2 with the Euclidean norm, consider the complement of the open unit ball M := R2 \ B˚ ( B˚ is the open unit ball). The set of nearest points from M to a point x ∈ X is as follows: if x ∈ M, then PM x = {x }, and if x ∈ B˚ \ {0}, then PM x = {x/ x  }; further, PM 0 = S (where S is the unit sphere). Example 1.3 In the plane R2 with the Euclidean norm, we consider as a set M the unit ball M := B = B(0, 1). It is clear that if x ∈ M, then PM x = {x }, while if x ∈ R2 \ M, then PM x = {x/ x  }; that is, M is a Chebyshev set. Example 1.4 In the plane R2 with the Euclidean norm, consider as a set M the open unit ball ˚ 1). If x ∈ S, then ρ(x, M) = 0, but since this point x does not lie in M, the set M M := B˚ = B(0, is not a set of existence. We also note that for no x  M are there any nearest points in M. Sets with this property are called antiproximinal sets. Example 1.5 Let M = P n be the subspace of algebraic polynomials of degree at most n on an interval [a, b] ⊂ R. By Chebyshev’s alternation theorem (see Chap. 2 below), there exists exactly one polynomial pn∗ of degree at most n that most closely approximates a given function f (t) in the uniform norm on [a, b]. So, M is a Chebyshev set in the space C[a, b], PM f = {pn∗ }. Note that in this case, the metric projection operator PM to the subspace M = P n is continuous on the space C[a, b] (for example, it is well known (see Theorem 5.10 below) that if M is boundedly compact, then PM ( · ) is continuous). However, the operator PM is not uniformly continuous on the unit ball of the space. Example 1.6 Let M = R n, m be the set of rational fractions on the interval [a, b] with fixed n and m (see (1.2)). By the classical Chebyshev uniqueness theorem (see Corollary 11.2 and Theorem 11.1 below), for a given continuous function f there exists exactly one rational function r ∗ ∈ R n, m that most closely approximates f in the uniform norm on [a, b]. So, M is a Chebyshev set in the space C[a, b]. However, as distinct from the case of the subspace of polynomials P n , the metric projection onto R n, m , m ≥ 1, has points of discontinuity (see, for example, Sect. 11.7 below and [137, Sect. 5.1.B]).

In general, the problem of finding an element of best approximation is a nontrivial problem. However, in the case of linear subspaces, the situation is geometrically quite clear. Proposition 1.6 Let L be a subspace of a normed linear space X. Then the metric projection operator is homogeneous and linear along L: PL (λx + y) = λPL (x) + y

for all x ∈ X, y ∈ L, λ ∈ R.

Moreover, if PL x = , then λPL (x) + y =  for all y ∈ L. Proof Let x ∈ X, y ∈ L, λ ∈ R. We have ρ(λx + y, L) := inf (λx + y) − z = |λ| inf  x − u = |λ| ρ(x, L), λ  0 (1.10) z ∈L

u ∈L

(in the next-to-last equality we used the fact that L is a subspace).

1.3

Elementary Results on Best Approximation . . .

15

For λ = 0, the assertion is trivial. Let λ  0. Then z ∈ PL (λx + y) ⇔ z − (λx + y) = ρ(λx + y, L)   z − y   (1.10)  − x  = |λ| ρ(x, L) ⇔ |λ|  λ  z − y   ⇔ − x = ρ(x, L) λ z−y ∈ PL x ⇔ z ∈ λPL x + y. ⇔ λ  Proposition 1.7 Let X be a normed linear space, x ∗ ∈ X ∗ \ {0}. Then the hyperplane Ker x ∗ := {x ∈ X | x ∗ (x) = 0} is a set of existence if and only if the functional x ∗ attains its norm (that is,  x ∗  = |x ∗ (x)| for some x ∈ S). Proof Assume that M := Ker x ∗ is a set of existence. Let us prove that the functional x ∗ attains its norm. We first show that there exists x ∈ X such that ρ(x, M) =  x = 1. Indeed, let z ∈ X \ M (such an element z always exists, because x ∗  0). Since M is a set of existence, we can find a point u ∈ M such that z − u = ρ(z, M). We set x :=

z−u z−u = . ρ(z, M) z − u

It is clear that  x = 1. By Proposition 1.6,  z−u   z  ρ(z, M) −u ρ(x, M) = ρ ,M = ρ + ,M = = 1. ρ(z, M) ρ(z, M) ρ(z, M) ρ(z, M) Let y ∈ X. Then y = x ∗ (y)/x ∗ (x)x + u for some u ∈ M. We have   x ∗ (y) |x ∗ (y)| |x ∗ (y)| |x ∗ (y)|   = ∗ ρ(z, M) = ∗ inf  x − z = inf  ∗ x + u − z ∗ z ∈M x (x) |x (x)| |x (x)| |x (x)| z ∈M (since M is a subspace, u ∈ M) = inf  y − z = ρ(y, M) ≤  y (because 0 ∈ M). z ∈M

So,

|x ∗ (y)| ≤ |x ∗ (x)|  y

for all y ∈ X.

Hence  x ∗  ≤ |x ∗ (x)| ≤  x ∗   x =  x ∗ , because  x = 1, which gives  x ∗  = |x ∗ (x)|.

16

1

Main Notation, Definitions, Auxiliary Results . . .

Conversely, assume that x ∗ ∈ X ∗ , x ∗  0, attains its norm on some x ∈ S; that is, = x ∗ (x). We set M := Ker x ∗ and consider u ∈ M. We have

 x∗ 

1=

 x∗  x ∗ (x) x ∗ (x − u)  x ∗   x − u = = ≤ =  x − u.  x∗   x∗   x∗   x∗ 

(1.11)

Taking the infimum in (1.11) over u ∈ M, we have 1 ≤ ρ(x, M) ≤  x − 0 =  x = 1. So, 0 ∈ PM x. Now consider y ∈ X. We have y = λx + u for some λ ∈ R and u ∈ M. By Proposition 1.6, we have PM y = PM (λx + u) = λPM x + u  . So M := Ker x ∗ is a set of existence.  Proposition 1.8 (The distance from a point to a hyperplane) Let x ∗ ∈ X ∗ , x ∗  0, c ∈ R, and let H := (x ∗ )−1 (c) = {x ∈ X | x ∗ (x) = c} be a closed hyperplane. Then for all x ∈ X, ρ(x, H) =

|x ∗ (x) − c| .  x∗ 

(1.12)

Proof For x ∈ H, (1.12) clearly holds. Let x  H. For y ∈ H, we have |x ∗ (x) − c| = |x ∗ (x) − x ∗ (y)| ≤  x ∗   x − y, which gives

|x ∗ (x) − c | ≤ ρ(x, H).  x∗ 

Next, let 0 < ε <  x ∗ . By the definition of the norm in X ∗ , there exists s ∈ S such that |x ∗ (s)| >  x ∗  − ε. We set  x ∗ (x) − c  s. y0 := x − x ∗ (s) Then y0 ∈ H and  x − y0  =

|x ∗ (x) − c| |x ∗ (x) − c| < , ∗ |x (s)|  x∗  − ε

which implies that ρ(x, H) ≤  x − y0 
0 (the case E  0 is trivial). n+2 of Necessity. Assume to the contrary that there is no Chebyshev alternant {ti }i=1 m length n + 2. Then there exists a partition {ui }i=0 , m  n + 1, of the interval [a, b] such that the sets   Hi := t ∈ [ui, ui−1 ] | |( f − p∗ )(t)| = E , i = 1, . . . , m, are nonempty. Note that the function f ∗ := f − p∗ preserves its sign on each such set, and these signs are interlacing. We can assume without loss of generality that at the points ui , i = 1, . . . , m − 1, the function f ∗ is zero and that f ∗ = E on Hm . On each interval [ui, ui−1 ] of the partition, consider the open set   E  Gi := t ∈ [ui, ui−1 ]  (−1)i−m f ∗ (t) > 2 (this set is open because the function f ∗ is continuous). Hence the remaining set Fi := [ui, ui−1 ] \ Gi is compact, and on it, | f ∗ | is smaller than E. By Weierstrass’s theorem, ri := maxt ∈Fi | f ∗ (t)| < E. Consider the polynomial p0 (t) = ε(t − u1 )(t − u2 ) · · · (t − um−1 ) ∈ Pn, where ε > 0 is so small that its uniform norm is smaller than d := mini=1,...,m−1 {E − ri, E2 }. By the construction, on the interval [ui, ui−1 ], the polynomial p0 has the same sign as f ∗ on Gi , and hence on this interval, the values of f ∗ − p0 vary in the range (−E, E). On the set Fi  t, we have |( f ∗ − p0 )(t)|  | f ∗ (t)| + |p0 (t)| < ri + d < E. So, the norm of f ∗ − p0 = f − (p∗ + p0 ) on [a, b] is strictly smaller than E; that is, the polynomial p∗ + p0 provides a better uniform approximation to the function f  than p∗ , contradicting the assumption. Theorem 2.1 is proved. The sufficiency in Theorem 2.1 is secured by the following result. Theorem 2.2 (Ch.-J. de la Vallée Poussin) Assume that for f ∈ C[a, b], there is a polynomial p ∈ Pn such that for r := f − p, there exists a tuple of points t1 < t2 < . . . < tn+2 from [a, b] such that r(ti ) r(ti+1 ) < 0,

i = 1, . . . , n + 2.

2.1 Chebyshev’s and de la Vallée Poussin’s Theorems

21

Then   En ( f ) ≥ min |r(ti )| | i = 1, . . . , n + 2 ,

(2.1)

where En ( f ) := ρ( f , Pn ) is the best approximant of f by polynomials of degree ≤ n. p − f  < |r(ti )|, Proof Assume that there exists a polynomial p ∈ Pn for which  i = 1, . . . , n+2. Writing the polynomial p−p ∈ Pn in the form p−p = ( f −p)−( f − p), we note that it alternates its sign at the points t1 < . . . < tn+2 , and hence it has at least (n + 1) zeros, which is impossible. Theorem 2.2 is proved.  For every (not necessarily Chebyshev) subspace of dimension n + 1, the number of points in de la Vallée Poussin’s estimate (2.1) cannot be reduced. Corollary 2.1 (P. L. Chebyshev) For every function f ∈ C[a, b] (a < b), a polynomial p∗ of best approximation is unique. Corollary 2.2 For all n ∈ Z+ , the set Pn is a Chebyshev set in the space C[a, b]. Proof (of Corollary 2.1) Assume that for some function f ∈ C[a, b], there are at least two different polynomials p and p∗ of best approximation from Pn ; that is, E = ρ( f , Pn ) := inf  f − r  =  f − p =  f − p∗ . r ∈P n

The polynomial q = function f , because

1 2 (p

+ p∗ ) is also a polynomial of best approximation to the

 1 1    f − q =  ( f − p) + ( f − p∗ ) 2 2 1 1 E E ≤  f − p +  f − p∗  = + = E. 2 2 2 2 By Theorem 2.1, for the polynomial q, there exists an increasing family of points n+2 ⊂ [a, b] for which {ti }i=1 |( f − q)(ti )| =  f − q = E, ( f − q)(ti ) = −( f − q)(ti+1 ),

i = 1, . . . , n + 2, i = 1, . . . , n + 1.

The polynomials p, p∗ have at most n equal values. Hence of the points t1, . . . , tn+2 , we have, for example, that t j is a point at which p and p∗ have different values. Then either | f (t j ) − p(t j )| or | f (t j ) − p∗ (t j )| is greater than | f (t j ) − q(t j )| = E, because q(t j ) lies strictly between p(t j ) and p∗ (t j ). Hence one of the polynomials p, p∗ is not a polynomial of best approximation, a contradiction. Corollary 2.1 is proved.  Proof (of Corollary 2.2) Indeed, the uniqueness of best approximation by the set Pn is secured by Corollary 2.1; the existence follows from the well-known fact that a boundedly compact set is necessarily a set of existence (see Sect. 4.1 below). 

22

2 Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems

There are many extensions of the Chebyshev alternation theorem (see, for example, Theorem 2.8 below). In particular, this theorem remains true if the subspace Pn is replaced by an arbitrary Chebyshev (Haar) system of functions. There are also some alternation theorems for spline approximation. Example 2.1 For the function f (t) = cos 2t, the polynomial p3∗ of best uniform approximation degree ≤ 3 in the uniform norm on the interval [0, 2π] is p3∗ ≡ 0 (the identically zero function). Indeed, we have ρ( f (t), P3 ) = maxt ∈[0,2π] | cos 2t − 0| ≤ 1, and that the error function f (t) − Q3 (t) = cos 2t assumes in succession the values +1 and −1 at the five (5 = 3 + 2) points 0, π/2, π, 3π/2, 2π. Example 2.2 For a continuous function f (t) on the interval [a, b] with minimal value m and maximal value M, the best uniform approximation of the form p0∗ ∈ P0 (by the constants) is given by the function p0∗ = (m + M)/2. Example 2.3 Let f (t) be a convex continuous function on [a, b], a < b, and let p1∗ be its polynomial of best uniform approximation of degree ≤ 1. Then the endpoints a and b belong to the alternant. Let t1, t2, t3 be the points of an alternant for f and p1∗ . We set r(t) := f (t) − p1 (t),

θ := min {ti | f (ti ) − p1 (ti ) = M }, i

where M := r . Since f (t) is continuous, we have f (θ) = M. Assume that θ > a. Then for sufficiently small ε > 0, the points θ − ε and θ + ε lie in [a, b]. Since f is convex, so is r(t) := f (t) − p1 (t) (r is the sum of the linear function p1 (t) and the convex function f ). Every convex function g satisfies the inequality



g (t1 + t2 )/2 ≤ g(t1 ) + g(t2 ) /2. So, since r is convex, we have M = r(θ) ≤

r(θ + ε) + r(θ − ε) M + r(θ − ε) M+M ≤ < = M. 2 2 2

This contradiction means that θ = a, that is, a is a point of the alternant. Similar arguments show that b is also a point of the Chebyshev alternant. For a convex differentiable function, the best linear polynomial can be constructed explicitly. Let c ∈ (a, b) be the third alternant point (in addition to the points a, b). By Chebyshev’s theorem, f (a) − (a0 + a1 a) = E,

f (c) − (a0 + a1 c) = −E,

f (b) − (a0 + a1 b) = E,

where E = E1 ( f ) := ρ( f , P1 ) is unknown. From the first and third equations, we get

a1 = f (b) − f (a) /(b − a). At the point c, the function f (t) − (a0 + a1 t) has en extremum; this point can be found from the equation f (c) − a1 = 0. Now from the first and second equations one finds a0 and E1 ( f ) = E. Geometrically, the solution of this problem is obtained as follows. We first draw the chord through the points (a, f (a)) and (b, f (b)) with slope a1 . Then we draw the tangent to the curve y = f (t) with the same slope. The secant line in the middle between the chord and the tangent line provides the best approximation among all lines to the function y = f (t) on the interval [a, b]. Example 2.4 Find a function and the corresponding polynomial of best uniform approximation of degree ≤ 6 for which there are 99 points of the Chebyshev alternant. Solution: On [a, b] we first fix a polynomial of degree p6 (t) and consider p+ (t) = p6 (t) + ε and p− (t) = p6 (t) − ε. Then inside the ‘corridor’ between p− and p+ , draw a continuous function f that oscillates about p6 and has 99 touch points (points of the alternant) with p6 .

2.2 Solarity and Alternant

23

Remark 2.1 If f is an odd (even) function, then the polynomial of best approximation on the interval [−1, 1] of any degree is odd (even). Indeed, let p∗n be the polynomial of best uniform approximation to f from the class Pn . We have | f (t) − p∗n (t)| ≤ En ( f ) := ρ( f , Pn ). Replacing t by −t and multiplying by −1 under the modulus sign, we have | − f (−t) − (−p∗n (−t))| ≤ En ( f ), which gives

| f (t) − (−p∗n (−t))| ≤ En ( f ).

Hence the polynomial −p∗n (−t) is also a polynomial of best uniform approximation. By the uniqueness theorem for best approximation for the class Pn , we have p∗n (t) = −p∗n (−t), the result required. A similar argument shows that if f is an odd (even) function with respect to the midpoint of the interval [a, b], then polynomial of best approximation is odd (even) with respect to the midpoint of [a, b]. Example 2.5 Let us find the polynomial of best approximation to the function f (t) = t 3 from the subspaces P2 and P1 on the interval [−1, 1]. By Remark 2.1, the polynomial of best approximation is odd, and hence it lies in P1 . It is easily seen that p ∗ (t) = 34 t, because the error function f (t) − p ∗ (t) = t 3 − 34 t has an alternant of length 4 at the points {−1; − 12 ; 12 ; 1}.

Note that the problem of the construction of an alternant can be highly nontrivial. With the help of Chebyshev’s and de la Vallée Poussin’s theorems, E. Ya. Remez designed an efficient method of construction of a polynomial of near-best uniform approximation and showed that these approximations converge rather fast (see, for example, [54], [216], [195, Sect. 3.8] and Sect. 2.7 below). Exercise 2.1 Let M be a finite-dimensional subspace in C[0, 1] of dimension n and let ϕ ∈ M be such that for some function f ∈ C[0, 1], the difference ϕ − f has an alternant of length n + 1. Does this imply that ϕ ∈ PM f ?

2.2 Solarity and Alternant We point out the following important characterization property of elements of best uniform approximation in terms of the alternant. Theorem 2.3 If, given a set M ⊂ C(Q), an element of best approximation is characterized in terms of the alternant, then M is a strict protosun. As a corollary (see Sect. 11.2), the set of rational functions Rn,m in C[a, b] and, more generally, any varisolvent set of existence (see Chap. 12) is a strict sun in C[a, b]. Proof (of Theorem 2.3) Let f  M, y ∗ ∈ PM f . Since in M an element of best approximation is characterized in terms of an alternant, it follows that for f − y ∗ ,

24

2 Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems

there exists an alternant t1 < . . . < tn . Let ϕ lie on the ray emanating from y ∗ and passing through f . Then ϕ − y ∗ = λ( f − y ∗ ) for some λ > 0. This equality shows that the family of points t1 < . . . < tn is an alternant for f − y ∗ , and hence y ∗ is an element of best approximation to ϕ. 

2.3 Haar’s Theorem. Strong Uniqueness of Best Approximation In the problem of best uniform approximation in the space C(Q) by elements of Chebyshev subspaces, the main tools are the above Chebyshev alternation (equioscillation) theorem and de la Vallée Poussin’s estimates, as well as Haar’s and Mairhuber’s theorems, which are given below. The space C[a, b] is not strictly convex, and hence a finite-dimensional subspace in C[a, b] is not necessarily a set of uniqueness. For example, if in the space C[−1, 1] one considers the subspace L generated by the functions t, t 2 , then for the function f0 ≡ 1, we have ρ( f0, L) =  f0 − αt 2  = 1

for any 0 ≤ α ≤ 1,

because 0 ∈ L, which gives ρ( f0, L) ≤ 1, and further, since ϕ(0) = 0 for all ϕ ∈ L, we have ϕ − f0  ≥ 1, which shows that ρ( f0, L) = 1 and so, αt 2 ∈ PL f0 for all 0 ≤ α ≤ 1. This suggests the problem of describing the finite-dimensional uniqueness subspaces in C[a, b]. Such subspaces are necessarily Chebyshev sets, because a finitedimensional subspace is always proximinal. Definition 2.1 A linearly independent system of continuous functions Φn := n on an interval [a, b] is called a Chebyshev system (or a Haar system) {ϕk (t)}k=1 if every nonzero polynomial p(t, α) = α1 ϕ1 (t) + . . . + αn ϕn (t) in this system has at most n − 1 zeros on [a, b]. An example of a Chebyshev system on an interval [a, b] is the system of power functions {1, t, . . . , t n−1 }, because every polynomial of degree n − 1 has at most n − 1 zeros. Another classical example of a Chebyshev system on the torus T = [0, 2π] (with identified points 0 and 2π) is the system of trigonometric polynomials {1, cos t, sin t, . . . , cos nt, sin nt}, because no trigonometric polynomial of order n has more than 2n zeros on the torus T. More examples can be found in [510], [432], [162], [163], and [195]. n be a linearly Theorem 2.4 (Chebyshev interpolation theorem) Let Φn = {ϕk (t)}k=1 independent system in C[a, b]. Then the following conditions are equivalent: (1) Φn is a Chebyshev system (a Haar system); (2) for every set of distinct points t1, . . . , tn ∈ [a, b], the Haar determinant    ϕ1 (t1 ) . . . ϕn (t1 )    Δ(t1, . . . , tn ) := . . . . . . . . . . . . . . . ϕ1 (tn ) . . . ϕn (tn )

2.3 Haar’s Theorem. Strong Uniqueness of Best Approximation

25

is nonzero. (3) for the system Φn , every interpolation problem is solvable; that is, for every set of distinct points t1, . . . , tn ∈ [a, b] and for all y1, . . . , yn ∈ R, there exists a unique polynomial p ∈ Ln := span(Φn ) such that p(tk ) = yk , k = 1, . . . , n (the interpolation property). Proof (of Theorem 2.4) (1)⇒(2). Assume that Δ(t1, . . . , tn ) = 0 for some t1, . . . , tn ∈ [a, b]. Then the system of equations n  αi ϕi (tk ) = 0, k = 1, . . . , n, i=1

α∗

has a nonzero solution = (α1∗, . . . , αn∗ ). The corresponding nontrivial polynomial n ∗ ∗ p(t, α ) = i=1 αi ϕi (t) has n zeros; that is, the system Φn is not a Chebyshev (Haar) system. (2)⇒(3). That the interpolation problem n  αi ϕi (tk ) = yk p(tk , α) = i=1

is solvable follows from the fact that the determinant Δ(t1, . . . , tn ) of this system is nonzero. n αi∗ ϕi (t) has n zeros t1, . . . , tn , (3)⇒(1). If a nontrivial polynomial p(t, α∗ ) = i=1 then Δ(t1, . . . , tn ) = 0, and the interpolation problem is not always solvable (by the theorem on solvability of a system of linear algebraic equations).  A system Φn satisfying condition (3) of Theorem 2.4 is called an interpolation system. It is easily checked that interpolation systems consisting of discontinuous functions exist on every set of cardinality c — it suffices to consider a one-to-one map of the interval to this set and consider the images of the interpolation system 1, t, t 2, . . . , t n on the interval under this mapping. The next classical theorem shows that on sets with nonempty interior there are no interpolation systems of size  2 consisting of continuous functions. Theorem 2.5 (A. Haar) If N > 1, if a set D ⊂ R N has nonempty interior (int D  ), and if n  0, then on D there are no continuous real interpolation systems (that is, n ) can be interpolated on every set of systems for which every set of values {yk }k=0 n knots {tk }k=0 ). n ⊂ Δ. Proof Consider a neighbourhood Δ ⊂ D. Let {tk }k=0 If { fk }, k = 0, . . . , n, is an interpolation system, then by Theorem 2.4, the system of equations n  ck fk (ti ) = yi (i = 0, . . . , n) k=0

is solvable for every tuple {yk }. Another appeal to Theorem 2.4 shows that det( fk (ti ))  0 for every {tk } ⊂ Δ. By the assumption, the functions fk are contin-

26

2 Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems

Fig. 2.1 A Haar system of length ≥ 1 cannot exist on a set with nonempty interior.

uous, and hence their determinant is continuous as a function of points {ti } in the domain Δ. Now we fix all the points tk in Δ, except two points, say t0 and t1 , which we continuously map to each other (see Fig. 2.1) so that all n + 1 points remain distinct and lie in Δ. The determinant changes continuously, but once the move is complete, it reverses sign (two rows of the determinant are swapped). Hence, for some intermediate set of points, this determinant must vanish. This contradiction proves the theorem.  n , of finite Theorem 2.6 (A. Haar) A subspace Ln := span(Φn ), Φn := {ϕk (t)}k=1 dimension n is a Chebyshev set in the space C[a, b] if and only if Φn is a Chebyshev (Haar) system; that is, if every nonzero function from Ln has at most n − 1 distinct zeros.

Proof (of Theorem 2.6) Sufficiency. We first assume that Φn is not a Chebyshev n αi∗ ϕi (t) system. Then by Theorem 2.4, some nontrivial polynomial p(t, α∗ ) = i=1 has n distinct zeros t1, . . . , tn . By the same theorem, the determinant Δ(t1, . . . , tn ) is zero. Hence there exists a tuple of numbers (a1∗, . . . , an∗ )  (0, . . . , 0) such that n 

a∗j p(t j , α) = 0

(2.2)

j=1

for every polynomial p(t, α) ∈ Ln := span Φn . Indeed, n  j=1

a∗j p(t j , α) =

n  j=1

a∗j

n  i=1

αi ϕi (t j ) =

n  i=1

αi

n 

(2.2)

a∗j ϕi (t j ) = 0.

j=1

Let g(t) ∈ C[a, b] be such that g(t j ) = sign a∗j , j = 1, . . . , n, gC = 1. We set

 1 f (t) := g(t) 1 − |p(t, α∗ )| , μ where μ := p(t, α∗ )C  0.

2.3 Haar’s Theorem. Strong Uniqueness of Best Approximation

27

It is clear that f (t) ∈ C[a, b], f (t j ) = g(t j ) = sign a∗j ,  f C = 1. We claim that ρ( f , Ln ) = 1. Since 0 ∈ Ln , we have ρ( f , Ln ) ≤  f C = 1. Assume that ρ( f , Ln ) < 1. Let p∗ (t) ∈ Ln be the polynomial of best uniform approximation to f . If j is such that a∗j  0, then | f (t j ) − p∗ (t j )| < 1 | f (t j )| = 1, and

  sign p∗ (t j ) = sign f (t j ) − ( f (t j ) − p∗ (t j )) = sign f (t j ) = sign a∗j .

Hence n  j=1

a∗j p∗ (t j ) =

n  j=1

a∗j sign p∗ (t j )|p∗ (t j )| =

n 

|a∗j | |p∗ (t j )| > 0,

j=1

which contradicts (2.2). This proves that ρ( f , Ln ) = 1. Now if ε ∈ R is such that |ε| < 1/μ, then

| f (t) − εp(t, α∗ )| ≤ |g(t)| 1 − μ−1 |p(t, α∗ )| + |ε| |p(t, α∗ )| ≤ 1 − μ−1 |p(t, α∗ )| + |ε| |p(t, α∗ )| = 1 − (μ−1 − |ε|)|p(t, α∗ )| ≤ 1 = ρ( f , Ln ). So a polynomial of best approximation p(t, α∗ ) is not unique. This proves the sufficiency in Haar’s theorem. To prove the necessity, we need the following auxiliary result. Let f ∈ C[a, b], and let p∗ (t) ∈ Ln := span(Φn ) be a polynomial of best approximation. The set    M( f , Φn ) := t ∈ [a, b]  | f (t) − p∗ (t)| = ρ( f , Ln ) is called the set of points of maximal deviation of the polynomial p∗ of best approximation to f . Lemma 2.1 Let f ∈ C[a, b], f  Ln := span(Φn ), and let Φn be a Chebyshev system.  Then the set M( f , Φn ) contains at least n + 1 points. Proof (of Lemma 2.1) Assume that M( f , Φn ) contains only m points t1, . . . , tm , m ≤ n. If m < n, then we augment this set of points by arbitrary points tm+1, . . . , tn . Using Theorem 2.4, we construct a polynomial p(t, α∗ ) such that p(tk , α∗ ) = sign( f (tk ) − p∗ (tk )), where p∗ (t) is the polynomial of best uniform approximation to f . At all points of the set M( f , Φn ), the polynomial p(t, α∗ ) has values of the same sign as r(t) := f (t) − p∗ (t). Let us show that this is impossible. We choose δ > 0 so as to have

28

2 Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems



 r(t ) − r(t

) < 1  f − p∗  = 1 r , 2 2 sign p(t, α∗ ) = sign p(tk , α∗ ) for all |t − t

| < δ and for all t ∈ [a, b] such that |t − tk | < δ for some k = 1, . . . , m. We set m 

 (ti − δ, ti + δ) [a, b], C = [a, b] \ A. A= i=1

Then the set A is open in [a, b], the set C is closed in the interval [a, b], and the functions f (t) − p∗ (t) and p(t, α∗ ) assume values of the same sign on A. Since C is closed and since C is disjoint from M( f , Φn ), we have max | f (t) − p∗ (t)| =  f − p∗  − ε = r  − ε t ∈C

(2.3)

for some ε ∈ (0, r ). Consider the polynomial p(t) := p∗ (t) +

ε p(t, α∗ ), 2μ

μ := p(t, α∗ ).

If t ∈ A, then in the representation

ε f (t) − p(t) = f (t) − p∗ (t) − p(t, α∗ ), 2μ the minuend and subtrahend have the same sign, and hence | f (t) − p(t)| <  f − p∗ ,

t ∈ A.

If t ∈ C, then | f (t) − p(t)| ≤ | f (t) − p∗ (t)| + So on the interval [a, b],

(2.3) ε ε |p(t, α∗ )| ≤  f − p∗  − ε + <  f − p∗ . 2μ 2

| f (t) − p(t)| <  f − p∗ ;

that is, p∗ is not a polynomial of best approximation to f . Lemma 2.1 is proved.  Let us continue the proof of the necessity in Haar’s theorem. Assume that Φn is not a Chebyshev system and that for some function f ∈ C[a, b], there are two polynomials p1, p2 ∈ Ln of best approximation. We have  p1 + p2  1 1   ≤  f − p1  +  f − p2  = ρ( f , Ln ), f − 2 2 2 and hence p = (p1 +p2 )/2 is also a polynomial of best approximation. By Lemma 2.1, there exist points t1, . . . , tn+1 such that

2.3 Haar’s Theorem. Strong Uniqueness of Best Approximation

29

| f (ti ) − p(ti )| = ρ( f , Ln ). For every point ti , we have



 1  ρ( f , Ln ) = | f (ti ) − p(ti )|  f (ti ) − p1 (ti ) + f (ti ) − p2 (ti )  2 1 1 ≤ | f (ti ) − p1 (ti )| + | f (ti ) − p2 (ti )| ≤ ρ( f , Ln ). 2 2 Hence for i = 1, . . . , n + 1, we have f (ti ) − p1 (ti ) = f (ti ) − p2 (ti ). So the polynomial p1 − p2 has n + 1 zeros, but this contradicts the fact that Φ is a Chebyshev system. Hence p1 ≡ p2 , completing the proof of Haar’s theorem.  It can be shown [285], [107] that Haar’s theorem (Theorem 2.6) also holds for every compact Hausdorff set. Namely, the following result holds, which gives an intrinsic description of Chebyshev subspaces of the space C(Q). Theorem 2.7 (A. Haar) A subspace G of finite dimension n is a Chebyshev set in the space C(Q), where Q is a compact Hausdorff space, if and only if G is a Haar subspace, that is, if and only if every nonzero function from G has at most n − 1 distinct zeros. The next theorem (see, for example, [432, Sect. 3.2], [3], [496]) is an analogue of the Chebyshev alternation theorem (Theorem 2.1) for the case of Haar subspaces (see Fig. 2.2). Theorem 2.8 (P. L. Chebyshev, Ch.-J. de la Vallée Poussin) Let V be a Chebyshev (Haar) subspace in C[a, b] of finite dimension n. Let g be an element of best approximation to f ∈ C[a, b] from V. Then there exist points a ≤ t1 < . . . < tn+1 ≤ b such that the error function r(t) := f (t) − g(t) satisfies |r(tm )| =  f − g, r(tk ) = −r(tk+1 ),

m = 1, 2, . . . , n + 1, k = 1, 2, . . . , n.

(2.4)

Remark 2.2 It is clear that the function g is uniquely defined by conditions (2.4). The following result was established by Kirchberger [337] in the particular case of algebraic polynomials. In the general case, an answer was independently obtained by Rice [494], Freud [246], and Maehly and Witzgall [413]. Theorem 2.9 (local Lipschitz continuity of the metric projection) Let V be a Chebyshev (Haar) linear subspace in C[a, b] of finite dimension n. Let g be an element of best approximation to f ∈ C[a, b] from V. Then there exists a number c = c( f , V) such that

30

2 Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems

Fig. 2.2 Alternation.

g −  g  ≤ c f −  f

∀ f ∈ C[a, b],

where  g is an element of best uniform approximation from V for  f. Remark 2.3 From the Hahn–Banach theorem, it follows that if V is a subspace and x ∈ X, then there exists a functional L ∈ X ∗ such that L = 1,

L|V = 0

(2.5)

and L(x) = ρ(x, V).

(2.6)

It is easily shown (see, for example, [432, Sect. 1.3]) that in fact, there exists a functional L ∈ X ∗ , L = 1, L|V = 0, for which inequality (2.6) becomes the equality L(x) = ρ(x, V).

(2.7)

Remark 2.4 If V is a Haar subspace, then one can easily construct a functional satisfying conditions (2.5). Namely, there exist a tuple of points t1 < . . . < tn+1 and numbers λ1, . . . , λn+1 such that the functional L( f ) = λm f (tm ) vanishes on the subspace V and has unit norm and such that L( f ) = ρ( f , V). This result can be proved from general considerations using Kolmogorov’s condition for a nearest element (see Sect. 5.3) or the decomposition lemma (see Sect. 6.2.3). We give a proof in the space C[a, b]. To this end, we choose n + 1 arbitrary points t1 < . . . < tn+1 from the interval [a, b] and consider the functional L(h) =

n+1 

λm h(tm ),

h ∈ C[a, b].

(2.8)

m=1

It is easily checked that L ≤ |λ1 | + . . . + |λn+1 |. Let hν , ν = 1, . . . , n, be a basis for the subspace V. We require that

2.3 Haar’s Theorem. Strong Uniqueness of Best Approximation n+1 

λm hν (tm ) = 0,

ν = 1, 2, . . . , n,

31

(2.9)

m=1 n+1 

|λm | = 1.

(2.10)

m=1

Since V is a Haar subspace, the coefficients λm in (2.9) are defined up to a constant; they are defined in a unique way from condition (2.10) and the assumption that λ1 has a fixed nonzero sign. Note that λm  0, m = 1, . . . , n + 1. The functional L thus constructed satisfies condition (2.5), whence by (2.6), we have |L( f )| ≤ ρ( f , V), f ∈ C[a, b]. Next, given f ∈ C[a, b], with any functional L defined by formula (2.8) one can associate in a unique way an element h ∈ V, which is determined from the condition h(tm ) + λ sign λm = f (tm ),

m = 1, . . . , n + 1.

(2.11)

Indeed, writing h as a linear combination in vectors of the basis {hν }, from (2.11) we get a linear system of equations with respect to the coefficients and the parameter λ. We claim that this system is uniquely solvable. To this end, we need to check that its determinant    h1 (t1 ) h2 (t1 ) . . . hn (t1 ) sign λ1    Δ := . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  h1 (tn+1 ) h2 (tn+1 ) . . . hn (tn+1 ) sign λn+1  is nonzero. We have λ1 · · · λn+1 Δ    λ1 h1 (t1 ) . . . λ1 hn (t1 ) λ1 sign λ1   = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . λn+1 h1 (tn+1 ) . . . λn+1 hn (tn+1 ) λn+1 sign λn+1  (adding to the last row all the above ones, we get)    λ1 h1 (t1 ) λ1 h2 (t1 ) . . . λ1 hn (t1 ) |λ1 |   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  =  |λn |  λn h1 (tn ) λn h2 (tn ) . . . λn hn (tn ) n+1  0 |λm |  0 ... 0   m=1  λ1 h1 (t1 ) λ1 h2 (t1 ) . . . λ1 hn (t1 )    =  . . . . . . . . . . . . . . . . . . . . . . . . . . .  λn h1 (tn ) λn h2 (tn ) . . . λn hn (tn )    h1 (t1 ) h2 (t1 ) . . . hn (t1 )    = λ1 · · · λn . . . . . . . . . . . . . . . . . . . . . = λ1 · · · λn Δ(t1, . . . , tn ),  h1 (tn ) h2 (tn ) . . . hn (tn ) where Δ(t1, . . . , tn ) is the Haar determinant. The last determinant is nonzero by Theorem 2.4, because V is a Haar subspace. It also follows that λn+1 = Δ(t1, . . . , tn )/Δ. A similar analysis shows that λi = (−1)n+1−i Δ(t1, . . . , ti−1, ti+1, . . . , tn+1 )/Δ,

32

2 Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems

i = 1, . . . , n. Note that Δ(t1, . . . , ti−1, ti+1, . . . , tn+1 ) preserves the sign (which is independent of the sign of i), and hence the signs of λi interlace. Next, using (2.11), we have λ sign λm = f (tm ) − h(tm ), where m = 1, . . . , n + 1, and hence λ=

n+1 

λ λm sign λm =

n+1 

m



λm f (tm ) − h(tm ) = L( f − h) = L( f ),

m

that is, λ = L( f ). Remark 2.5 The function h, as defined in (2.11), is an element of best uniform approximation to the function f from V on the mesh t1, . . . , tm , where V is the restriction of the subspace V to the mesh. This is easily checked using the Kolmogorov criterion for best approximation (see Sect. 5.3). Proof (of Theorem 2.9) Let f  V and t1 < t2 < . . . < tn+1 be points of a Chebyshev alternant for f − g. By Remark 2.4, we have g(tm ) = f (tm ) − ε(−1)m ρ( f , V),

m = 1, . . . , n + 1,

(2.12)

where ε = 1 or ε = −1 (the sign is fixed). By (2.11), there exists a unique element h ∈ V such that h(tm ) = f (tm ) − ελ sign λm,

m = 1, . . . , n + 1,

and moreover, by Remark 2.4, λ = L( f ), sign λm = ε(−1)m . Hence h = g and L( f ) = ρ( f , V); here L(h) := n+1 m=1 λm h(tm ) is the linear functional corresponding to this alternant. Next, L( f − g) = L( f ) =

n+1 

λm [ f (xm ) − g(xm )] =



λm (−1)m L( f ),

m=1

and so

n+1

m=1

λm (−1)m =

n+1

m=1

|λm | by (2.10), which gives λm = (−1)m |λm |.

(2.13)

Setting ϕ = g −  g , we have (−1)m ϕ(tm ) =     = (−1)m g(tm ) −  f (tm ) −  f (tm ) + (−1)m  g (tm )     (2.12) = (−1)m f (tm ) −  f (tm ) − L( f ) + (−1)m  g (tm ) f (tm ) −  ≤ f −  f  + ρ(  f , V) − ρ( f , V) ≤ 2 f −  f ,

(2.14)

m = 1, 2, . . . , n + 1. Further, ϕ ∈ V, and so another appeal to (2.13) shows that

2.3 Haar’s Theorem. Strong Uniqueness of Best Approximation

L(ϕ) =

n+1 

λm ϕ(xm ) =

m=1

n+1 

33

|λm |(−1)m ϕ(xm ) = 0.

(2.15)

m=1

Hence (2.10) −2 f −  f (1 − |λν |) = (2.14)

≤ −

n+1 

n+1 

|λm |(−2 f −  f )

m=1 mν

(2.15) |λm |(−1)m ϕ(tm ) = |λν |(−1)ν ϕ(tν ) ≤ 2 f −  f  · |λν |.

m=1 mν

Therefore, for every ν, we have   1 − |λν | |ϕ(xν )| ≤ max 2 f −  f −  f , f , 2 |λν | and so

1 − |λ |  ν , 1≤ν ≤n+1 |λν |

|ϕ(tm )| ≤ 2 f −  f  max

m = 1, 2, . . . , n + 1.

Now let (hν ), ν = 1, . . . , n be a basis for V. Since V is a Haar subspace, any nontrivial polynomial αν hν from V cannot have n zeros, and hence the Chebyshev norm on the mesh x1, . . . , xn is equivalent (with some constant  depending only on V and on the alternant) to the original norm  ·  on V. Hence the expression n      αν hν (t) ν=1

under n additional constraints n      αν hν (tm ) ≤ 1,

m = 1, 2, . . . , n,

ν=1

is bounded from above by the number . Hence ϕ ≤ c( f ) ·  f −  f , where

1 − |λ |  ν . 1≤ν ≤n+1 |λν |

c( f ) = 2 max This completes the proof of Theorem 2.9.



Marinov [422] established the local Lipschitz continuity of the metric projection operator onto a finite-dimensional Chebyshev (Haar) subspace M of C[a, b]. We give the necessary definitions. Let N ⊂ C[a, b]. If for every f ∈ N there exists r > 0 such that the Lipschitz constants K(ϕ) are uniformly bounded on (N ∩ B( f , r))  ϕ, then the operator PM is locally uniformly Lipschitz continuous on N. Here

34

2 Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems

K( f ) := sup

  P f − P ϕ   M M  ϕ ∈ C[a, b], ϕ  f .  f − ϕ

Theorem 2.10 (A. V. Marinov) Let M be a finite-dimensional Chebyshev (Haar) subspace of C[a, b]. Then the metric projection operator onto M is locally uniformly Lipschitz continuous on the set C[a, b] \ M. A. V. Marinov extended Theorem 2.10 to the case of the space C(Q) (Q is a metrizable compact set) and obtained a number of nontrivial results on the stability of the operator of near-best approximation onto a convex subset of a normed linear space in terms of moduli of continuity and smoothness of the space. Remark 2.6 (the lack of global Lipschitz continuity in C(Q)) S. B. Stechkin (see [251], and also Example 5.3 below) constructed, for all ε > 0, functions x, y ∈ C[−1, 1] such that  x − y < ε, but PL x − PL y ≥ 1, where L := span{1, t} is a (Chebyshev) subspace in C[−1, 1]. A similar result for an arbitrary finitedimensional Chebyshev subspace1 L ⊂ C(Q) (dim L ≥ 2, Q is an infinite compact set) was later established by A. K. Cline (see [170]). These results show the lack of global Lipschitz continuity (uniform continuity; see Theorem 2.11) of the metric projection operator when approximating by a Chebyshev subspace in C(Q) on an infinite compact set A. In the space n∞ , the metric projection onto a Chebyshev subspace is globally Lipschitz on the entire space (A. K. Cline [170, Theorem 4], V. I. Berdyshev [96, Theorem 2], and also M. Bartelt [79] and M. Finzel [236, Sect. 5]). Theorem 2.11 (R. B. Holmes, B. Kripke [294]) The metric projection onto a Chebyshev subspace of a normed space X is Lipschitz continuous if and only if it is uniformly continuous on X. p

Remark 2.7 In the finite-dimensional space n , 1 < p < ∞, the metric projection p onto a subspace Y ⊂ n is locally Lipschitz with a constant depending only on the parameter p (see [104, p. 54], [100]) and is globally Lipschitz for p > 2 (see [294, Theorem 4]). Holmes and Kripke [294, Theorem 4] proved a similar result on the local Lipschitz continuity of the metric projection onto a finite-dimensional subspace Y ⊂ L p in the space L p , 2 < p < ∞. For 1 < p < 2, the same does not hold (see [104]). Moreover, in [294], an example was constructed of a one-dimensional subspace in the space p , p > 2, with (globally) non-Lipschitz metric projection. Björnestål [104] showed that the metric projection onto the subspace of constants in L p [−1, 1] is not locally Lipschitz for 1 < p < 2 (for p > 2, local Lipschitz continuity holds by the above result of Holmes and Kripke [294]). Later, Al’brecht [6] showed that the metric projection onto the subspace of constants is non-Lipschitz in L p [0, 1] for all p ∈ (1, 2)∪(2, ∞) (for 1 < p < 2, this is a consequence of Björnestål’s results). Druzhinin [208] proved that in the space L p (E, Σ, μ), p ∈ (1, 2)∪(2, ∞), with atomless 1 By Mairhuber’s theorem (see Sect. 2.5 below) in C(Q), where Q is a metrizable compact set, there exists a Chebyshev subspace of every finite dimension n = 2, 3, . . . if and only if Q is homeomorphic to an infinite closed subset of the unit interval [0, 1].

2.3 Haar’s Theorem. Strong Uniqueness of Best Approximation

35

measure μ, the metric projection operator onto a one-dimensional subspace is not Lipschitz. Borodin, Druzhinin, and Chesnokova [119] established that the metric projection onto a finite-dimensional subspace Y of the space L p , p ∈ (1, 2) ∪ (2, ∞), satisfies the Lipschitz condition if and only if the supports of all functions from Y are concentrated on a finite number of atoms. The paper [119] also gives estimates for the Lipschitz constant of the metric projection to a one-dimensional subspace. See also Brown [149]. Definition 2.2 Let ∅  M ⊂ X. The metric projection PM satisfies the strong uniqueness property at a point x ∈ X if PM x = {y} is a singleton and there exists a number γ > 0 such that for all y ∈ M,

 y − y  ≤ γ  x − y  − ρ(x, M) . (2.16) The smallest of such constants γ ∗ is called the strong uniqueness constant. Such an element y is called a strongly unique element of best approximation for x. To explain the geometrical meaning of the above definition, we set z = x + (y − x). Inequality (2.16) (the inequality  y − y  ≤ γ y − z) means that the ball B(x, ρ(x, M)) near the point y looks like a cone, the constant γ ∗ characterizing the acuteness of this ‘cone’. The equivalent definition of the strong uniqueness property is as follows. For δ > 0, the set x−y  ρ(x, M)

δ PM x := {y ∈ M |  x − y ≤ ρ(x, M) + δ}

is called the δ-projection of an element x onto the set M (the set of near-best approximations). It is easily seen that the metric projection satisfies the strong uniqueness property at a point x if and only if there exists a constant η > 0 such that δ diam(PM x) ≤ ηδ

(0 < δ < ρ(x, M)).

It is worth pointing out that for L p -spaces (1 < p < ∞), the metric projection onto a proper subspace V ⊂ L p does not have the strong uniqueness property at any point (see [98]). We mention the following simple result on the stability of the metric projection [98]. Theorem 2.12 If the metric projection onto a set of existence M ⊂ X has the strong uniqueness property with constant γ > 0 at a point x0 , then it satisfies the following Lipschitz condition:  y − PM x0  ≤ 2γ x − x0 

∀x ∈ X,

∀y ∈ PM x.

Proof By the strong uniqueness property at the point x0 , the set PM x0 is a singleton (which we also denote by PM x0 ). For y ∈ PM x, inequality (2.16) gives

36

2 Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems



 y − PM x0  ≤ γ  x0 − y − ρ(x0, M)   ≤ γ ( x0 − x +  x − y − ρ(x0, M))   ≤ γ  x0 − x + | ρ(x, M) − ρ(x0, M)| . Applying inequality (1.6) (the Lipschitz continuity of the distance function), we complete the proof.  Remark 2.8 The strong uniqueness property of the polynomial of best uniform approximation to a real continuous functions from C[a, b] by algebraic polynomials of degree ≤ n was established by N. G. Chebotarev in 1943 (See [424]). Unfortunately, Chebotarev’s result was forgotten. In 1963, the strong uniqueness property was rediscovered by D. J. Newman and H. S. Shapiro in a more general setting of approximation of continuous real functions from C(Q) (where Q is a compact set) by finite-dimensional Chebyshev subspaces. Cheney [162] established strong uniqueness of the best rational fraction Rm,n in the space C[a, b]. The strong uniqueness property proved useful in justifying Remez-type algorithms and establishing estimates of their convergence rate [98], [451]. Theorem 2.13 (D. J. Newman, H. S. Shapiro, N. G. Chebotarev) Let L be a finitedimensional Haar (Chebyshev) subspace in the space C(Q). Then the (unique) element of best approximation to any f  L is a strongly unique element of best approximation. This result will be proved later in Sect.12.1 for Haar cones (see also [137, Sect. I.3.10]). Further results on strong uniqueness can be found in the survey by Kroó and Pinkus [376], as well as in the papers by Marinov [422], [424], Pokrovskii [479], and others. Exercise 2.2 Determine which of the following systems form a Haar system in the space C: (a) {1, t 2, t 4 } on the interval [0, 1]; (b) {1, t 2, t 4 } on the interval [−1, 1]; (c) {1, t 2, t 3 } on the interval [−1, 1]; (d) {1/(t + 1), 1/(t + 2), 1/(t + 3)} on the interval [0, 1]; (e) {1, e t , e2t } on the interval [0, 1].

2.4 A Short Note on Extremal Signatures The classical Haar condition for best approximation considered in this section had been developed significantly since it first appeared. The related more general problem of characterization of best approximations in C(Q)-spaces can be treated more effectively using the more advanced notion of extremal signatures, which plays a central role in the theory of Chebyshev approximation. Let V be an n-dimensional subspace of the space C(Q) on a compact Hausdorff space D; we equip C(Q) with the norm  f  = sup{| f (t)| | t ∈ Q}. A signature

2.5 Mairhuber’s Theorem

37

σ is a function on Q that has finite support and whose nonzero values are either +1 or −1. We say that a signature σ is extremal with respect to V if there exists a nonzero positive measure μ whose carrier is contained in the support of σ such ∫ that u(t)σ(t) dμ(t) = 0 for all u ∈ V. It can be shown by a convexity argument that this definition is equivalent to saying that there is no u ∈ V such that u(t)σ(t) > 0 for all x in the support of σ. The notion of an extremal signature plays a central role in the theory of Chebyshev approximation (see, for example, [510], [496]). Indeed, suppose f ∈ C(Q) is given, and for each u ∈ V, we define E f+ (u) = {t ∈ Q | f − u =  f − u}, E f− (u) = {t ∈ Q | f − u = − f − u}, E f ( f ) = E f+ (u) ∪ E f− (u). The best approximations of f out of V are characterized in the following theorem (T.J. Rivlin and H. S. Shapiro [510], [496]). Theorem 2.14 Let d = min{ f − u | u ∈ V }. Then  u ∈ V is an element of best approximation to f if and only if there is an extremal signature σ with support in E f ( u) such that ( f −  u)σ ≥ 0. Based on extremal signatures, lower bounds on the deviation of best Chebyshev approximations were obtained and effective numerical algorithms for computing best approximations were developed. The idea here is based on the following result by L. Collatz (see, for example, [510]). Theorem 2.15 Let σ be an extremal signature with support E. Suppose v ∈ V is such that ( f − v)σ ≥ 0. Then min{ f (t) − v(t) | t ∈ E } ≤ ρ( f , V) ≤  f − v. For more details, see [510], [496], [434], and [612].

2.5 Mairhuber’s Theorem A. Haar was already aware that the existence of a nontrivial Chebyshev subspace in the space C(Q), where dim Q > 1, places considerable restrictions on the structure of the compact set Q. A complete description of this situation was given by J. Mairhuber in 1956 and independently by Sieklucki [515]. Theorem 2.16 (J. Mairhuber) Let Q be a metrizable compact space. If C(Q) contains a real Chebyshev (Haar) subspace of dimension n ≥ 2, then Q is homeomorphic to a subset of the unit sphere S 1 .

38

2 Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems

More precisely, the set of integers n ≥ 1 with the property that C(Q) contains an n-dimensional Chebyshev subspace is as follows: (1) {1, . . . , k} if Q has only a finite number k of points; (2) {1, 2, 3, . . . } if Q is homeomorphic to an infinite closed subspace of the unit interval [0, 1]; (3) {1, 3, 5, . . . } if Q is homeomorphic to the unit circle; (4) {1} in all other cases. We explain Mairhuber’s theorem using an example of the ‘tripod’ T (Fig. 2.3).

Fig. 2.3 The tripod shows that a Chebyshev systsem of size ≥ 2 cannot exist on a domain of R, n≥ n from C[a, b], Consider an arbitrary linearly independent system Φn = {ϕk (x)}k=1 n  2. Given arbitrary distinct points x0 , x1, . . . , xn−1 ∈ T, there exists a homotopy τ(x, t) ∈ C(T × [0, 1], T) of the tripod T onto itself for which the points x0 and x1 are swapped, the remaining points x2, . . . , xn−1 are fixed, and the points τ(x0, t), τ(x1, t), . . . , τ(xn−1, t) are distinct for all t ∈ [0, 1]. The determinants (see Theorem 2.4)

Δ(τ(x0, 0), τ(x1, 0), . . . , τ(xn−1, 0)) = Δ(x0, x1, . . . , xn−1 ) and Δ(τ(x0, 1), τ(x1, 1), . . . , τ(xn, 1)) = Δ(x1, x0, . . . , xn−1 ) have opposite signs. The function ϕ(t) := Δ(τ(x1, t), τ(x2, t), . . . , τ(xn, t)) is continuous on the connected set T, and hence there exists t0 ∈ [0, 1] for which ϕ(t0 ) = 0. By Theorem 2.4, this means that Φn is not a Chebyshev (Haar) system. For history and references on Mairhuber’s theorem, see the book by Singer [519] (see also [107] and [144]). Later, Haar’s and Mairhuber’s theorems were extended by Blatter [107] and Brown [144] to the case of subspaces of C(Q) with unique continuous selection of the metric projection operator. Kamal [318] investigated the problem of existence and characterization of n-dimensional Chebyshev subspaces in L ∞ [a, b]-spaces in the space of bounded real functions on [a, b] and in some other spaces containing

2.6

Approximation of Continuous Functions . . .

39

discontinuous functions. A. K. Kamal also characterized the spaces that contain an n-dimensional Chebyshev subspace. Exercise 2.3 Identify the finite-dimensional Chebyshev subspaces in C[0, 1]2 .

2.6 Approximation of Continuous Functions by Finite-Dimensional Subspaces in the L 1 -Metric Let L = L[a, b] be the space of Lebesgue integrable functions on the interval [a, b]. It is easily checked that the one-dimensional subspace of constant functions in L is not a Chebyshev set. Moreover, according to the well-known Krein’s theorem (see Theorem 8.3), there are no nontrivial finite-dimensional Chebyshev subspaces in L[a, b]. Note that the space C[a, b] of continuous functions on the interval [a, b] is a dense linear manifold in L; that is, C[a, b] = L. Theorem 2.17 (D. Jackson) Let Φ := {ϕ1, ϕ2, . . . , ϕn } be a Chebyshev system of continuous functions on [a, b], and let Ln be the n-dimensional subspace spanned by the system Φ. Then for every function f ∈ C[a, b], there exists a unique polynomial ϕ∗ ∈ Ln such that  f − ϕ∗  L = ρ( f , Ln ) L . We need three auxiliary lemmas. Lemma 2.2 Let ψ1 , ψ2 be polynomials of best L 1 -approximation from a linear manifold M ⊂ C[a, b] for a given function f ∈ C[a, b]. Then



(2.17) f (x) − ψ1 (x) f (x) − ψ2 (x) ≥ 0 for any x ∈ [a, b]. Proof Let ψ1 and ψ2 be polynomials of best L 1 -approximation for f from M. Since M is linear, ψ = (ψ1 + ψ2 )/2 is also a polynomial of best approximation to f , and moreover,  ∫ b ∫ b ∫ b 1 | f − ψ| dt = | f − ψ1 | dt + | f − ψ2 | dt , 2 a a a that is, ∫ a

b

∫ | f − ψ1 + f − ψ2 | dt =

a

b

∫ | f − ψ1 | dt +

a

b

| f − ψ2 | dt.

The last equality (for continuous functions) holds only if the differences f − ψ1 and  f − ψ2 are of the same sign.

40

2 Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems

Lemma 2.3 Let ψ1 , ψ2 be distinct polynomials of best L 1 -approximation for f ∈ C[a, b] in a Chebyshev system Φ. We set ϕα (t) := αψ1 (t) + (1 − α)ψ2 (t), α ∈ (0, 1). Then the difference f − ϕα has at most (n − 1) zeros. Proof Assume that f (t) − ϕα (t) = 0 for some t. By definition, f (t) − ϕα (t) = α( f (t) − ψ1 (t)) + (1 − α)( f (t) − ψ2 (t)). By Lemma 2.2, the signs f (t) − ψ1 (t) and f (t) − ψ1 (t) are the same, and hence f (t) − ψ1 (t) = f (t) − ψ2 (t) = 0. So all the zeros of the function f − ϕα are zeros of the difference ψ1 − ψ2 . But this function has at most (n − 1) zeros, because Φ is a Chebyshev system.  Lemma 2.4 Let Φ := {ϕ 1n, ϕ2, . . . , ϕn } be a Chebyshev system on [a, b]. Consider ak ϕk (t) in this system such that the polynomials ϕ(t) = k=1 ∫ ϕ L =

b

a

|ϕ(t)| dt =: d

for some d. Then for every measurable set E from [a, b], we have ∫ |ϕ(t)| dt ≤ K d · mes E, E

where K depends only on the system Φ. Proof Being a continuous function of the coefficients ak , k = 1, . . . , n, the function ∫b n |ϕ(t)| dt attains its minimum on the finite-dimensional sphere k=1 ak2 = 1. This a minimum cannot be zero, because Φ is a Chebyshev system. Hence ∫ b  |ϕ(t)| dt ≥ C > 0 ∀ (a1, . . . , an ) : ak2 = 1. a

Now, if =

 n

k=1

∫

b

ak2  0 and

∫b a

|ϕ(t)| dt = d, then

|ϕ(t)| dt ≥ C > 0,

a

As a result, |ak | ≤ ≤ ∫ E

d , and hence C

|ϕ(t)| dt ≤

n  k=1

|ak |

∫ that is d =

|ϕ(t)| dt ≥ Cl.

∫ E

|ϕk (t)| dt

  d mes E ϕk ∞ ≤ K d mes E, C k=1 n

≤

a

b

2.6

Approximation of Continuous Functions . . .

41

where K is some constant depending only on ϕ1, ϕ2, . . . , ϕn .



Proof (of Theorem 2.17) We need to show that if f ∈ C[a, b], {ϕ1 (t), . . . , ϕn (t)} n is a Chebyshev system, then the polynomial ϕ(t) = k=1 ak ϕk (t) of best Lapproximation to f is unique. From Lemma 2.3, it follows that if R(t) = f (t) − ϕ(t) has at least n changes of sign, then ϕ is a unique polynomial of best approximation to f . Suppose that there were two polynomials ψ1 and ψ2 of best approximation to f . We set R(t) = f (t) − ϕα (t), where ϕα (t) is an arbitrary polynomial of the form ϕα (t) = αψ1 (t) + (1 − α)ψ2 (t),

α ∈ [0, 1].

We claim that the difference R(t) vanishes on a set of positive measure independent of α. This will lead to a contradiction to Lemma 2.3. We relabel the points of sign change for the difference R(t) as follows: a < t1 < . . . < tq < b,

q ≤ n − 1.

We increase, if required, the number of these points in the interval [b − δ, b], where b − δ > tq , to n − 1, t1 < . . . < tq < tq+1 < . . . < tn−1 < b. Since Φ is a Chebyshev system, there exists a polynomial F(t) in this system that vanishes only at these n − 1 points; moreover, it changes sign at these points. For  C or F(t) = −F(t)/  C , where example, one can take F(t) = F(t)/ F F    ϕ1 (t) . . . ϕn (t)       =  ϕ1 (t1 ) . . . ϕn (t1 )  . F(t)  ... ... ...     ϕ1 (tn−1 ) . . . ϕn (tn−1 )  Here the sign of F(t) is chosen so as to have sign F(t) = sign R(t) on the interval [a, b − δ] (see Fig. 2.4). Consider the difference R(t) − εF(t), ε > 0 and define three sets on [a, b]: U : |R(t)| > ε,

sign R = sign F,

V : |R(t)| ≤ ε,

sign R = sign F,

W : sign R(t)  sign F. By the construction of F, we have W ⊂ [b − δ, b] and R(t) − εF(t) = f (t) − ϕ (t), where ϕ  is some polynomial in the system Φ. Hence

42

2 Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems

R(t) F (t) a

t1

tq

F (t)

b−δ

b

R(t) Fig. 2.4 Proof of Theorem 2.17.

∫ ∫b

inasmuch as

a

b

a

∫ |R(t) − εF(t)| dt ≥

a

b

|R(t)| dt,

|R(t)| dt = ρ( f , Φ) L . Next, we have ∫ ∫ ∫ |R(t) − εF(t)| dt = |R(t)| dt − ε |F(t)| dt, U

U

U

because sign R(t) = sign F(t) and |R(t)| > ε, |F(t)| ≤ 1 for t ∈ U. Moreover, ∫ ∫ ∫ |R(t) − εF(t)| dt ≤ |R(t)| dt + ε |F(t)| dt. V

V

V

Therefore, ∫ ∫ ≤

U

∫

b

As a result, we have

∫ d=

b

a

a

W

∫ |R(t)| dt − ε

∫

U

∫

|F(t)| dt + ε

W

V ∪W

|F(t)| dt.

∫

U

V ∪W

∫

|R(t)| dt ≤ |R(t) − εF(t)| dt = + + a U W ∫ ∫ ∫ V |R(t)| dt − ε |F(t)| dt + |R(t)| dt + ε |F(t)| dt U V V ∫ ∫ + |R(t)| dt + ε |F(t)| dt =

∫

∫

a

∫

Adding

∫

b

|F(t)| dt ≤

V ∪W

|F(t)| dt.

|F(t)| dt to both sides and using Lemma 2.4, we get b

|F(t)| dt ≤ 2

∫



|F(t)| dt ≤ 2K d · mes(V ∪ W), K = K Φ .

V ∪W

2.6

Approximation of Continuous Functions . . .

43

Hence mes(V ∪ W) ≥ c > 0, where c is independent of d. Choosing ε < c/2, we have mes V ≥

c > 0, 2

but as ε → 0, the set V converges to a set on which R(t) = 0. So for all α ∈ [0, 1], the difference Rα = f (t) − ϕα (t) vanishes on the set V = V(α) of measure mes V ≥ c/2 > 0. But this contradicts Lemma 2.3. Theorem 2.17 is proved.  Remark 2.9 If the above functions are not assumed to be continuous, then equality (2.17) is satisfied only almost everywhere. Let us now give some advanced results on uniqueness of best approximation in general L 1 -spaces (for more details, see [476]). Let Q be a set, Σ a σ-field of subsets of Q, and ν a positive measure defined on Σ, i.e., ν(E) ≥ 0 for all E ∈ Σ. By L p (Q, ν) (we suppress Σ for brevity), 1 ≤ p < ∞, we denote the set of all real-valued ν-measurable functions f defined on Q for which | f | p is ν-integrable over Q. For each f ∈ L 1 (Q, ν), we consider its zero set Z( f ) := {t | f (t) = 0} and define N( f ) = Q \ Z( f ). The first result on L 1 -uniqueness in approximation of finite-dimensional subspaces L is a consequence of a characterization theorem of best L 1 -approximation (see [476, Theorem 2.1], [373], [519]). Theorem 2.18 Let f ∈ L 1 (Q, ν). Assume that g0 is a best approximant to f from L. Then  g ∈ L, g  g0 , is also a best approximant to f from L if and only if the following two conditions hold: g ) ≥ 0 ν-a.e. on ∫Q; (a) (∫f − g0 )( f −  (b)

Q

sign( f − g0 )( g − g0 ) dν =

Z( f −g0 )

| g − g0 | dν.

The following corollary to Theorem 2.18 (see [476, Corollary 2.5]) is sometimes easier to verify in practice. Corollary 2.3 Let g0 be a best approximant to f from L. Assume that ∫  ∫   |g| dν  sign( f − g0 ) g dν  < Q

Z( f −g0 )

for all g ∈ L, g  0. Then the best approximant to f from L is unique. Corollary 2.4 Let L be a finite-dimensional subspace of L 1 (Q, ν) and f ∈ L 1 (Q, ν) \ L. Then g0 is the strongly unique best approximant to f from L if and only if ∫  ∫   γ= |g| dν − sign( f − g0 ) g dν  g ∈ L, g L 1 = 1 Z( f −g0 )

Q

satisfies γ > 0. Furthermore, if γ > 0, then γ is the largest possible strong uniqueness constant.

44

2 Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems

2.7 Remez’s Algorithm for Construction of a Polynomials of Near-Best Approximation In the majority of cases, finding, for a given continuous function f (x), the polynomial n ck ϕk (x) is in general Pn∗ ( f , x) of its best approximation of the form Pn (x) = k=1 infeasible. So from a practical and theoretical point of view, it would be interesting to have an algorithm capable of constructing, for a given function f ∈ C[a, b], polynomials Pn ( f , x) that are arbitrarily close to the polynomial Pn∗ ( f , x) of its best uniform approximation. Of course, although this problem can be attacked via general minimization algorithms, the most powerful method is the one based on characteristic properties of polynomials of best uniform approximation and taking into account the specifics of the problem. There are several such algorithms (see, for example, Meinardus [432, pp. 105–130], Laurent [390, Sect. 3.7], Dzyadyk [216, Chap. I], Berdyshev [98, Sect. 8.1], Rivlin [495], DeVore and Lorentz [195], and Trefethen [547]). In this section, we mention only the so-called second Remez algorithm [430], which is also applicable in the general setting of approximation by real Chebyshev (Haar) systems. This algorithm is generally accepted and used in practice for approximate representation of continuous functions by polynomials in the Chebyshev norm. Consider the problem of uniform approximation of a continuous function f ∈ n aμ x μ . Given f and C[−1, 1] by algebraic polynomials P ∈ Pn , that is, P(x) = μ=0 P, we set n  δn (x) = δn (x, a) := f (x) − aμ x μ , μ=0

where a = (a0, a1, . . . , an ) ∈ Rn+1 . Remez’s algorithm is based on the Chebyshev alternation theorem and consists of the following sequence of operations. n+1 in the interval [−1, 1], Step 1. Start with n + 2 distinct points (xk )k=0 −1 ≤ x0 < x1 < . . . < xn+1 ≤ 1. Step 2. Find a0, a1, . . . , an, E from the linear system of equations δn (xk , a) = f (xk ) −

n 

aν xkν = (−1)k E,

k = 0, 1, . . . , n + 1.

(2.18)

ν=0

Because of (2.18), the function x → δn (x) has at least one zero zk (xk < zk < xk+1 ) in each of the intervals (xk , xk+1 ). Step 3. Determine the points z−1, . . . , zn+1 such that z−1 = −1,

δn (zk ) = 0, k = 0, 1, . . . , n,

zn+1 = 1.

Step 4. Select points  xk ∈ [zk−1, zk ], k = 0, 1, . . . , n + 1, such that   (sign δn (xk ))δn ( xk ) = max (sign δn (xk ))δn (x) . z k−1 ≤x ≤z k

2.7

Remez’s Algorithm for Construction of a Polynomials of Near-Best Approximation

45

Step 5. If δn ( · , a) > max0≤k ≤n+1 |δn ( xk , a)|, then there exists a point  x ∈ [−1, 1] x, a)| = δn ( · , a). In this case, put the point  x in place of some point at which |δn ( xn+1 so that the function x → δn (x, a) preserves the alternating signs on the  x0, . . . ,  xn+1 obtained in this way. new set of points  x0, . . . ,  Step 6. For a given tolerance ε > 0, check the condition    δn ( · , a) − |E |  < ε. If this condition is satisfied, then stop; otherwise, put xk :=  xk , k = 0, 1, . . . , n + 1, and go to Step 2. So at each iteration ν, Remez’s algorithm produces a polynomial Pν ∈ Pn of ‘near-best’ uniform approximation to the function f . The algorithm is not particularly sensitive to the choice of the initial points x0, . . . , xn+1 , but it is often convenient to take as knots the extreme points of the Chebyshev polynomial Tn+1 . Remez’s algorithm converges to the polynomial of best uniform approximation at the rate of a geometric progression. Namely, the following result holds (see, for example, [216, Chap. I], [466, pp. 13–15]). Theorem 2.19 Let f ∈ C[−1, 1] and let P∗ ∈ Pn be the polynomial of best uniform approximation to f from the class Pn . Then for the polynomial Pν ∈ Pn obtained at the νth iteration of Remez’s algorithm, we have Pν − P∗  ≤ C ρν, where 0 < ρ < 1, and the constant C is independent of ν. Example 2.6 (see [430]) √ We illustrate Remez’s algorithm on the following example. We approximate the function f (x) = 3 + 2x + 4x 2 by the class P3 on the interval [−1, 1]. At the νth iteration of the algorithm, we have δ3(ν) (x) = f (x) − (a0(ν) + a1(ν) x + a2(ν) x 2 + a3(ν) x 3 ). √ √ Starting from the extreme points {−1, − 2/2, 0, 2/2, 1} of the fourth-order Chebyshev polynomial (see, for example, [430, (1.1.32)]), Remez’s algorithm generates the following sequences of and the corresponding quantity E (see [430]): coefficients a(ν) k ν

a0(ν)

a1(ν)

a2(ν)

a3(ν)

E (ν)

1 2 3 4

1.75109137 1.74940896 1.74941135 1.74941135

0.52173410 0.52617035 0.52614796 0.52614796

0.88598318 0.88937920 0.88939426 0.88939426

−0.13976809 −0.14420434 −0.14418195 −0.14418195

−0.01904056 −0.02075418 −0.02077162 −0.02077162

So as a polynomial of near-best approximation, we can take P ∗ (x) = 1.74941135 + 0.52614796x + 0.88939426x 2 − 0.14418195x 3 (rounded to eight decimal places). The alternant points are x0 = 1, x1 = −0.72898482, x2 = −0.12747162, x3 = 0.58607094, x4 = 1. Correspondingly,

46

2 Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems

 f − P ∗  ≈ 2.077 × 10−2 .

Remark 2.10 Remez’s algorithm is considerably more convenient for most applications than approximation by finite sums of Taylor series — it is well known that a Taylor series converges very slowly and that there are constraints on the domain of its convergence and the smoothness of an approximated function. For example, the series 1 1 ln(1 + x) = x − x 2 + x 3 − . . . 2 3 does not converge for x > 1, whereas for x = 1, the tolerance 0.0001 is achieved with 104 terms of the Taylor series.

Chapter 3

Best Approximation in Euclidean Spaces

Many classical ‘least-squares’ approximation problems are special cases of the general problem of best approximation in a Euclidean space by elements of a finitedimensional subspace (or a convex set). We present two fundamental results on approximation by convex sets in the inner-product setting — the Kolmogorov criterion of best approximation and Phelps’s criterion for convexity of a Chebyshev set in a Euclidean space in terms of the Lipschitz continuity of the metric projection operator. Despite the fact that many problems of approximation theory in Euclidean spaces are well studied, it is in the Euclidean setting that the most famous unsolved problem of approximation theory is formulated: is every Chebyshev set convex in an infinite-dimensional Hilbert space? In this chapter, by a Euclidean space we mean a real linear space with given inner product. The case of finite-dimensional spaces will be mentioned separately. In Sect. 3.1, we give the particular case (which we call Deutsch’s lemma) of the wellknown Kolmogorov criterion for a nearest element, and we prove Phelps’s theorem on Lipschitz continuity of the metric projection onto Chebyshev sets in Sect. 3.2. For more on approximation in Euclidean spaces, see in particular [191].

3.1 Approximation by Convex Sets. Kolmogorov Criterion for a Nearest Element. Deutsch’s Lemma We first prove that every convex closed subset of a Hilbert (complete inner product) space is a Chebyshev set. The completeness of the space in this result is essential. Theorem 3.1 A closed convex nonempty subset M of a Hilbert space H is a Chebyshev set. Proof We shall require the well-known parallelogram identity, which characterizes Euclidean spaces:  x + y 2 +  x − y 2 = 2 x 2 + 2 y 2,

x, y ∈ H.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_3

(3.1) 47

48

3 Best Approximation in Euclidean Spaces

Given ε > 0 and an arbitrary x ∈ H, consider the closed convex set   Mε := y ∈ M |  x − y  ρ(x, M) + ε = M ∩ B(x, ρ(x, M) + ε). If y1, y2 ∈ Mε , then

y1 +y2 2

∈ M, and moreover,

 y1 − y2  2 = (y1 − x) − (y2 − x) 2 (3.1)

= 2 y1 − x 2 + 2 y2 − x 2 − (y1 − x) + (y2 − x) 2

 2(ρ(x, M) + ε)2 + 2(ρ(x, M) + ε)2 − (y1 − x) + (y2 − x) 2 2 y + y   1 2 − x = 4(ρ(x, M) + ε)2 − 4 2 ≤ 4(ρ(x, M) + ε)2 − 4ρ2 (x, M) = 4ε(2ρ(x, M) + ε).  Hence the diameter of the set Mε is at most 2 ε(2ρ(x, M) + ε). The family of sets {M1/n } is a nested sequence of closed sets, whose diameters tend to zero. Since H is complete, the completeness theorem  (Theorem 1.2) implies that the intersection of this sequence is a singleton y0 ∈ n∈N M1/n ⊂ M. Moreover,  y0 − x  ρ(x, M) +

1 → ρ(x, M), n

n → ∞,

and hence  y0 − x = ρ(x, M). So y0 is a nearest point to x, there being no other nearest points. The theorem is proved.  Corollary 3.1 For every subspace L of a Hilbert space H and a point x ∈ H \ L, there exists a unique nearest point y ∈ L to x. Moreover, (x − y)⊥L. Proof Since L is convex and closed, by the previous theorem there exists a unique point y ∈ L such that |y − x| = (x, L). Consider an arbitrary vector e parallel to the subspace L and consider the line  = {y + te | t ∈ R}. The point y ∈  is a nearest point to x in the set , and hence in the two-dimensional plane spanned by  and x, the vector x − y is perpendicular to the line . Hence (x − y) ⊥ e, and the required result follows.  Remark 3.1 In an incomplete Euclidean space, a convex closed set can fail to be proximinal. It suffices to consider the convex closed set  ∫ 1 

 M = f ∈ C[−1, 1]  f (t) dt = 0 0

in the space (C[−1, 1],  · 2 ) with the mean-square norm. We claim that M is a convex closed subset in (C[−1, 1],  · 2 ) with the meansquare norm, which is not an existence set. It is clear that M is nonempty (0 ∈ M) and is a linear manifold. Let ( fn ) be a sequence from M converging to some function f ∈ C[−1, 1]. By the Cauchy–Schwarz inequality, for every n ∈ N we have

3.1

Approximation by Convex Sets . . .

∫  

0

1

 ∫   f (t) dt  = 

49

∫

1

f (t) dt −

0

0

1

  fn (t) dt 

∫ 1 ∫ 1  ∫ 1   = ( f (t) − fn (t)) dt  ≤ | f (t) − fn (t)| dt ≤ | f (t) − fn (t)| dt 0 0 −1 ∫ 1 1/2 ∫ 1 1/2 √ ≤ 1 dt | f (t) − fn (t)| 2 dt = 2  f − fn 2 . −1

−1

∫1

Letting n → ∞, we get | 0 f (t) dt| = 0, that is, f ∈ M. Consider the function f ∈ C[−1, 1] defined by f (t) ≡ 1 for t ∈ [−1, 1]. Since ∫1 f (t) dt = 1, we have f  M. Given g ∈ M, we have 0 ∫  f − g22 =

1

−1 1

| f (t) − g(t)| 2 dt ∫

∫ =

|1 − g(t)| 2 dt +

−1

0

∫ =

1

0

0

|1 − g(t)| 2 dt

1 − 2g(t) + g (t) dt + 2

∫

=1+

1

∫ g (t) dt + 2

0

−1

0

∫

0

−1

|1 − g(t)| 2 dt

|1 − g(t)| 2 dt ≥ 1.

So g  M. The previous inequality becomes an equality if and only if ∫ 0 ∫ 1 |1 − g(t)| 2 dt = g 2 (t) dt = 0, −1

0

the latter equality being satisfied only if g|(−1,0) = 1 and g|(0,1) = 0. But this function g cannot be continuous, and so g  M. As a corollary, we have that  f − g2 > 1 for all g ∈ M. For all 0 < ε < 1, define the function gε : [−1, 1] → R by ⎧ ⎪ 1, ⎪ ⎨ ⎪ gε (t) = −t/ε, ⎪ ⎪ ⎪ 0, ⎩

−1 ≤ t ≤ −ε, ε < t < 0, 0 ≤ t ≤ 1.

Clearly, gε ∈ M. We have ∫ 1 ∫ 0 ∫ 1 t 2  +  f − gε 22 = | f (t) − gε (t)| 2 dt = dt + 1 dt  1 ε −1 −ε 0  t3  0 ε t2 =1+ . =1 + t + + 2 ε 3ε −ε 3 Since this inequality holds for all 0 < ε < 1, we get ρ( f , M) ≤ 1. As a result, ρ( f , M) = 1 <  f − g

∀g ∈ M, that is, PM f = .

50

3 Best Approximation in Euclidean Spaces

So the above closed convex set M is not proximinal. In this section, we also give Phelps’s theorem to the effect that a Chebyshev set in a Euclidean space is convex if and only if the metric projection onto it is 1-Lipschitz. We require the following lemma (Fig. 3.1), which in essence is a reformulation of the Kolmogorov criterion for a nearest element (Theorem 5.2) for Euclidean spaces (see, for example, [191, Sect. 4.1], [82, Theorem 3.16]). Lemma 3.1 (F. Deutsch) Let M be a convex subset of a Euclidean space (X,  · , · ) and let x ∈ X, y ∈ M. Then y ∈ PM x ⇐⇒ x − y, z − y ≤ 0 ∀z ∈ M. Proof Let x ∈ X, y ∈ M. Assume that the inequality of the lemma holds. If x = y, then there is nothing to prove. Now let x  y. For all z ∈ M, we have  x − y 2 = x − y, x − y = x − y, x − z + x − y, z − y ≤ x − y, x − z ≤  x − y  x − z (the last inequality was obtained using the Cauchy–Schwarz inequality). So  x−y ≤  x − z for all z ∈ M, and hence y ∈ PM x.

Fig. 3.1 Kolmogorov criterion for a nearest element for convex sets M in a Hilbert space (Deutsch’s lemma): y ∈ PM x if and only if x − y, z − y  ≤ 0 for all z ∈ M.

Conversely, assume to the contrary that x − y, z − y > 0 for some z ∈ M. Let λ be such that  2x − y, z − y

. 0 < λ < min 1, z − y 2 Hence 2x − y, z − y − λz − y 2 > 0. The set M is convex, and hence yλ := λz + (1 − λ)y ∈ M. As a corollary, we have    x − yλ  2 = x − yλ, x − yλ  = x − y − λ(z − y), x − y − λ(z − y)   =  x − y 2 − λ 2x − y, z − y − λz − y 2 <  x − y 2 .

3.2

Phelps’s Theorem on the Lipschitz Continuity . . .

51

So  x − yλ  <  x − y, which shows that y  PM x.



Exercise 3.1 Prove that the set M = {x ∈  2 | x (2n) = 0} is an infinite-dimensional Chebyshev subspace in  2 . What is PM x for every x ∈  2 ? Exercise 3.2 Let M be a convex Chebyshev set in a Euclidean space H. Verify the following strong uniqueness property: x − y  2 ≥ x − PM x  2 + y − PM x  2

x ∈ H, y ∈ M .

Hint. Use Deutsch’s lemma (Lemma 3.1). Exercise 3.3 Let M = {p ∈ C2 [a, b] | p ∈ P n, p ≥ 0}. Show that M is a convex Chebyshev set, that is a cone. What is PM x? Exercise 3.4 Let M = {x ∈  2 | x (i) ≥ 0, i ∈ N}. Show that M is a convex Chebyshev set, that is a cone. Verify that PM x = x+ := max{x, 0}. Exercise 3.5 Let M1 ⊃ M2 ⊃ · · · ⊃ Mn be Chebyshev subspaces in a Euclidean space. Show that PM1 PM2 . . . PM n = PM n = PM n PM n−1 . . . PM1 . Exercise 3.6 Let M and N be orthogonal Chebyshev subspaces. Show that M + N is a Chebyshev subspace and PM + N = PM + PN . Exercise 3.7 Let M be a convex Chebyshev set in a Euclidean space H. Show that for all x, y ∈ H, either  PM x − PM y  < x − y  or PM x − PM y = x − y.     1 (n) = 0 . Show that L is Exercise 3.8 Consider the following set in  2 : L = y ∈  2  ∞ n=1 2 n/2 y a Chebyshev hyperplane, and compute PL x and ρ(x, L). Exercise 3.9 Which of the following hyperplanes in the Euclidean space C2 [−1, 1] are (1) Chebyshev, (2) antiproximinal. For Chebyshev hyperplanes, find PL x. ∫1 (a) L = {y ∈ C2 [−1, 1] | −1 y(t) dt = 2}; ∫0 ∫1 (b) L = {y ∈ C2 [−1, 1] | −1 y(t) dt = 0 y(t) dt }; ∫1 (c) L = {y ∈ C2 [−1, 1] | −1 y(t) cos t dt = 0}; ∫0 ∫1 (d) L = {y ∈ C2 [−1, 1] | −1 e t y(t) dt + 0 (1 + t)y(t) dt = 0}.

3.2 Phelps’s Theorem on the Lipschitz Continuity of the Metric Projection onto Chebyshev Sets We can now formulate and prove a theorem to the effect that the metric projection onto a closed convex subset of a Euclidean space is 1-Lipschitz (Phelps [467]). Theorem 3.2 (R. Phelps) A Chebyshev set M in a Euclidean space X is convex if and only if ∀x, y ∈ X, PM x − PM y ≤  x − y that is, if and only if the metric projection operator is nonexpansive.

52

3 Best Approximation in Euclidean Spaces

Proof (of Theorem 3.2) Let M be a convex Chebyshev set and let x, y ∈ X. We can always assume that PM x  PM y. By Lemma 3.1, x − PM x, PM y − PM x ≤ 0

and

y − PM y, PM x − PM y ≤ 0;

x − PM x, PM y − PM x ≤ 0

and

PM y − y, PM y − PM x ≤ 0.

that is,

Adding these two inequalities, we get x − y + PM y − PM x, PM y − PM x ≤ 0, and hence PM y − PM x, PM y − PM x ≤ y − x, PM y − PM x. Next, PM y−PM x 2 = PM y − PM x, PM y − PM x ≤ y − x, PM y − PM x ≤  y − x PM y − PM x. So PM x − PM y ≤  x − y, the result required. To prove the converse result, we assume that the metric projection operator onto M is nonexpansive, but M is a nonconvex Chebyshev set. Since M is closed and nonconvex, M is not midpoint convex; that is, we can find x, y ∈ M such that z := (x + y)/2  M. It is clear that x  y. Let r :=  x − y/2 > 0. We have PM z − x = PM z − PM x ≤ z − x = r, PM z − y = PM z − PM y ≤ z − y = r, and so PM z ∈ B(x, r) ∩ B(y, r). Since X is strictly convex (as a Euclidean space) and since  x − y = 2r, if follows from Proposition 1.3 that B(x, r) ∩ B(y, r) = {(x + y)/2} = {z}. This implies that z = PM z ∈ M, which is impossible. This contradiction proves Theorem 3.2.  For more on approximation in Euclidean spaces, see Deutsch [191]. A number of results on Lipschitz continuity of the metric projection onto closed convex subsets of a Hilbert space were obtained by Balashov and Golubev [64], [65], [67]. For example, M. O. Golubev proved that the metric projection operator onto a strongly convex subset (in the sense of E. S. Polovinkin, M. V. Balashov, and G. E. Ivanov) of a Hilbert space is Lipschitz continuous with constant smaller than 1 on the complement of an open neighbourhood of this set. For more details, see [278].

3.3

Best Least-Squares Polynomial Approximation . . .

53

3.3 Best Least-Squares Polynomial Approximation. Orthogonal Polynomials Among the broad field of least-squares approximation, we consider one illustrative example. Our aim in this section is to minimize the integral ∫ b w(t) ( f (t) − pn (t))2 dx (3.2) a

over all polynomials pn (t) of degree ≤ n. The minimizer in this problem (the best least-squares polynomial) is denoted by p∗n (t). But we first recall some facts on orthogonal polynomials that play a central role in the solution of least-squares problems. The weighted inner product between two functions f (t) and g(t) (with respect to ∫b the weight w(t)) is defined by ( f , g)w = a f (t)g(t)w(t) dt. The weighted L 2 -norm  is obtained from the weighted inner product by  f 2,w = ( f , f )w . Given a weight w(t), orthogonal (or orthonormal) polynomials are constructed using the classical Gram–Schmidt orthogonalization process. We now give several examples of orthogonal polynomials and a very brief summary of some of their properties. Legendre polynomials. The Legendre polynomials are a family of polynomials that are orthogonal with respect to the weight w(t) ≡ 1 on the interval [−1, 1]. Legendre polynomials are the most commonly used of classical orthogonal polynomials and the only ones for which the condition of their orthogonality on [−1, 1] is satisfied in its ‘pure form’, i.e., through the equality to zero of the inner product: ∫1 ( f , g) = −1 f (t) g(t) dx, namely, (Pn, Pn ) =

2 . 2n + 1

(3.3)

Instead of calculating these polynomials one by one from the recurrence relation, they can be obtained directly from the explicit Rodrigues formula Pn (t) =

 1 dn  2 (t − 1)n , 2n n! dx n

n ≥ 0.

(3.4)

Legendre polynomials satisfy the recurrence relation (n + 1)Pn+1 (t) − (2n + 1)xPn (t) + nPn−1 (t) = 0,

n ≥ 1,

where the first two polynomials are as follows: P0 (t) = 1, P1 (t) = t. For n = 1, 2, 3, . . . , this formula gives P2 (t) = 12 (3t 2 − 1), P3 (t) = 12 (5t 3 − 3t), P4 (t) =  1 2n+1 4 − 30t 2 + 3), P (t) = 1 (63t 5 − 70t 3 + 15t). From (3.3) we have P = (35t 5 n 8 8 2 Pn . The recurrence relation (3.4) implies that the Legendre polynomial Pn is an even function for even k and an odd function for odd k.

54

3 Best Approximation in Euclidean Spaces

Chebyshev polynomials. The Chebyshev polynomials Tn (t) are orthogonal with respect to the weight w(t) = (1 − t 2 )−1/2 on the interval [−1, 1]. Chebyshev polynomials satisfy the well-known recurrence relation Tn+1 (t) = 2tTn (t) − Tn−1 (t),

n ≥ 1,

together with T0 (t) = 1 and T1 (t) = t, and they satisfy the orthogonality conditions ∫ 1 Tn (x)Tm (x) dx = 0 (n  m), √ −1 1 − x2 (3.5) ∫ 1 ∫ 1 (Tn (x))2 (T0 (x))2 π n ≥ 1, dx = , dx = π. √ √ 2 −1 −1 1 − x2 1 − x2 Laguerre polynomials. The Laguerre polynomials Ln (t) are orthogonal on [0, ∞) with the weight function p(t) = e−t . The recurrence relation for Laguerre polynomials is given by Ln (t) − (2n − 1 − t)Ln−1 (t) + (n − 1)2 Ln−2 (t) = 0, and for small n, we have L0 (t) = 1, L1 (t) = 1 − t, L2 (t) = t 2 − 4t + 2, L3 (t) = −t 3 + 9t 2 − 18t + 6. The Rodrigues formula for the Laguerre polynomials is as follows: Ln (t) =

et d n n −t (t e ), n! dt n

n ≥ 0.

The normalization condition is Ln  = 1. A more general form of the Laguerre polynomials is obtained when the weight is taken as e−t t α , α > −1 on [0, ∞). Hermite polynomials. The Hermite polynomials Hn (t) are orthogonal on R with 2 respect to the weight w(t) = e−t . They can be explicitly written as d n e−t , dt n 2

Hn (t) = (−1)n et

2

n ≥ 0.

The Rodrigues formula for the Hermite polynomials is Hn (t) =

[n/2]  k=0

(−1)k n! (2t)n−2k , k!(n − 2k)!

where [t] denotes the greatest integer that is ≤ t. The recurrence relation for Hermite polynomials reads as Hn+1 (t) − 2xHn (t) + 2nHn−1 (t) = 0,

n ≥ 1,

3.3

Best Least-Squares Polynomial Approximation . . .

55

together with H0 (t) = 1, H1 (t) = 2t. From this formula, for n = 1, 2, . . . , we have H2 (t) = 4t 2 −2, H3 (t) = 8t 3 −12t, H4 (t) = 16t 4 −48t 2 +12, H5 (t) = 32t 5 −160t 3 +120t. The Hermite polynomials satisfy the orthogonality relation ∫ ∞ √ 2 e−t Hn (t) Hm (t) dt = 2n n! π δnm . −∞

For more details on orthogonal polynomials, see, for example, [162], [496], [495], and [535]. Let us return to problem (3.2). Assume that  (Pk ) is an orthogonal family of polynomials with respect to w(x), and let pn (x) = nj=0 c j P j (x). Then ∫ f −

2 pn 2,w

b

= a

n  2  w(x) f (x) − c j P j (x) dt, j=0

and hence n n    2 = f− c j P j (x), f − c j P j (x) 0 ≤  f − pn 2,w j=0

= ( f , f )2 − 2 2 =  f 2,w −2

2 =  f 2,w −

n  j=0 n 

c j ( f , P j )w + c j ( f , P j )w +

j=0 n 

( f , P j )2w

j=0

j=0

2 P j 2,w

+

w

n  n 

ci c j (Pi, P j )w i=0 j=0 n  2 c2j P j 2,w j=0

n   ( f , P j )w j=0

P j 2,w

2 − c j P j 2,w .

(3.6)

The last expression is minimal if and only if ( f , P j )w − c j P j 2,w = 0, P j 2,w that is, if and only if cj =

( f , P j )w 2 P j 2,w

∀ 0 ≤ j ≤ n,

.

(3.7)

Hence a unique least-squares approximation exists and is given by p∗n (t) =

n  ( f , P j )w j=0

2 P j 2,w

P j (t) =

n 

c j P j (t).

(3.8)

j=0

If the polynomials (P j (t)) are also normalized so that (P j = P j , P j 2, j = 1), then c j = ( f , P j ) and

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3 Best Approximation in Euclidean Spaces

p∗n (t) =

n n   ( f , P j )wP j (t) = c jP j (t). j=0

(3.9)

j=0

The above method of finding a best least-squares polynomial is nothing other than a reformulation of the extremal property of the partial sums of Fourier series. Let us now consider some examples of finding best least-squares polynomials. Example 3.1 Find the best L 2 linear approximation to f (t) = 1 − t 2 with weight w(t) = (1 − t 2 )−1/2 on [−1, 1]. Solution. In this case, (Tk (t)) is the appropriate orthogonal system, and hence the best leastsquares polynomial from P1 reads as p ∗ (t) = c∫0T0 (t) + c1T1 (t). From (3.7) and (3.5), we have ∫ 1

1

(1−t 2 )−1/2 (1−t 2 )

(1−t 2 )−1/2 (1−t 2 )t

( f ,T0 ) ( f ,T1 ) c0 = (T = −1 , c1 = (T = −1 . Substituting t = cos θ, we find π π /2 0 ,T0 ) 1 ,T1 ) that c0 = 12 , c1 = 0, which gives p ∗ (t) = 12 T0 (t) + 0 T1 (t) = 12 , so that the linear approximation reduces to a constant in this case.

Example 3.2 Find the polynomial p ∗ in P2 that minimizes ∫

1

−1

[cos t − p(t)]2 dt.

Solution. The choice of the weight w ≡ 1 on [−1, 1] implies that we should work with Legendre polynomials. Since we are looking for a polynomial of degree ≤ 2, it suffices to consider the first three Legendre polynomials, P0 (t) = 1, P1 (t) = t, P2 (t) = 12 (3t 2 − 1). The normalization ∫1 ∫1 ∫1 2 condition  Pn2  2 = 2n+1 (see (3.3)) gives −1 P02 (t) dt = 2, −1 P01 (t) dt = 23 , −1 P22 (t) dt = 25 ,  and so the first three normalized Legendre polynomials are as follows: P 0 (t) = √1 , P 1 (t) = 32 t, 2 1 √ ∫1 √  P 2 (t) = √5 (3t 2 − 1). Since c 0 = ( f , P 0 ) = −1 cos t √1 dt = √1 sin t  = 2 sin 1, we have 2 2 2 2 −1  ∫1 p0∗ (t) = sin 1. Next, c 1 = ( f , P 1 ) = −1 cos t 23 t = 0, which gives p1∗ = p0∗ . Next, c 2 = ( f , P 2 ) =   √ ∫1 2 cos t √5 (3t 2 − 1) = 12 52 (12 cos 1 − 8 sin 1), and so p2∗ (t) = sin 1 + 15 2 cos 1 − 5 sin 1 (3t − 1). −1 2 2

Example 3.3 Find the polynomial p ∗ of degree ≤ 1 that minimizes ∫ ∞ e−x [cos x − p(x)]2 dx. 0

Solution. The family of orthogonal polynomials that correspond to this weight w(x) = e−x on [0, +∞) are Laguerre polynomials. We will L0 (x) = ∫ ∞ need to use the first two Laguerre polynomials: ∫∞ 1, L1 (x) = 1 − x. We have ( f , L0 )w = 0 e−x cos x dx = 12 . Next, ( f , L1 )w = 0 e−x cos x(1 − ∫∞ ∫∞ x) dx = 0 e−x cos x dx − 0 e−x x cos x = 12 − 0 = 12 . Therefore, p1∗ (x) = ( f , L0 )w L0 (x) + ( f , L1 )w L1 (x) = 12 + 12 (1 − x) = 1 − x2 .

3.3

Best Least-Squares Polynomial Approximation . . .

57

Exercise 3.10 Find linear and quadratic polynomials of best L 2 -approximation for the following functions: 1) f (x) = x 3 , [0, 2]; 2) f (x) = e x , [0, 2]; 3) f (x) = x ln x, [1, 3]. Exercise 3.11 Find linear and quadratic polynomials of best L 2 -approximation on [−1, 1] for the following functions: 1) f (x) = x 2 − 2x + 3; 2) f (x) = x 3 ; 3) f (x) = 1/(x + 2); 4) f (x) = e x ; 5) f (x) = ln(x + 2).

Chapter 4

Existence. Compact, Boundedly Compact, Approximatively Compact, and τ-Compact Sets. Continuity of the Metric Projection

The concept of compactness plays an important role in mathematics and applications. A direct generalization of the concept of a compact set is the concept of boundedly compact sets (an intersection of such a set with each closed ball is compact). Further generalization of this concept gives rise to the important concept of approximative compactness (see Definition 4.2 below) introduced by Efimov and Stechkin in the 1950s. An approximatively compact set is always a set of existence. A number of results on proximally of abstract and concrete sets were obtained by employing of the concept of approximative compactness and its variants. Moreover, approximative compactness and its generalizations have proved to be an important tool in proving existence theorems, solarity of Chebyshev sets, and existence of continuous ε-selections of the metric projection operator. In the present chapter, we shall be mostly concerned with problems of existence of best approximation. In Sect. 4.1, we introduce boundedly compact and approximatively compact sets. In Sect. 4.2, we study basic properties of approximatively compact sets, recall the definition of uniformly convex spaces, and prove in particular that a convex closed nonempty subset of a complete uniformly convex space is approximatively compact, and hence a set of existence. The important property of approximative τ-compactness (with respect to a regular τ-convergence) is discussed in Sect. 4.3. Using this concept, which is much more general than that of approximatively compact sets, it proves possible to establish the existence property for many classical and abstract objects such as sets of rational functions, sets of exponential suns, and sets of splines with free knots in the space C[a, b] (see Sect. 4.3.1).

4.1 Boundedly Compact and Approximatively Compact Sets Definition 4.1 A set is called boundedly compact if its intersection with every closed ball is compact. A set is called boundedly weakly compact if its intersection with every closed ball is weakly compact. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_4

59

60

4 Existence and Compactness . . .

Remark 4.1 The subspace Pn of polynomials of degree at most n in the space C[a, b] is boundedly compact. More generally, every finite-dimensional subspace of a normed space is boundedly compact. The arithmetic sum of a nonempty boundedly compact set and a compact set is a boundedly compact set. A finite union of boundedly compact sets is also a boundedly compact set. The set of all C-Lipschitz functions in the space C[0, 1] is another example of a boundedly compact set. Remark 4.2 A boundedly compact set is a set of existence. Indeed, let x be an arbitrary element from X and let y0 ∈ M, where M   is a boundedly compact set. It is clear that the problems inf  x − y,

y ∈M

inf

y ∈M, x−y  ≤ x−y0 

 x − y

are equivalent. The second problem is the problem of minimization of a continuous functional  x − y =: F(y) (x is fixed) on the nonempty compact set M0 = {y ∈ y ∈ M0 M |  x − y ≤  x − y0 }. By Weierstrass’s theorem, there exists an element  such that  x −  y  = inf{ x − y | y ∈ M0 } = ρ(x, M). The proof is complete. Corollary 4.1 A nonempty closed subset of a finite-dimensional Banach space (or of any boundedly compact set in this space) is a set of existence. Definition 4.2 A point x ∈ X is called a point of approximative compactness for a set M (written x ∈ AC(M)) if every sequence (yn )n∈N ⊂ M such that  x − yn  → ρ(x, M) (such a sequence is called minimizing) contains a subsequence converging to some point of M. It is easily checked that each point of approximative compactness x ∈ X of a set M is a point of existence (that is, PM x  ). ˚ 1) in an infinite-dimensional Hilbert space H. We have Example 4.1 Consider the set M = H \ B(0, ˚ 1) in the space X =  ∞ , the set AC(M) consists AC(M) = H \ {0}. For a similar set M = X \ B(0, of only the points of M (AC(M) = M). In this space, the ball B(0, 1) is not an approximatively compact set. In contrast, every ball in a Hilbert space is an approximatively compact set, but in the infinite-dimensional case, it is not compact.

Definition 4.3 A set M ⊂ X is called approximatively compact if each point x ∈ X is a point of approximative compactness for M. Definition 4.4 A set M ⊂ X is called approximatively weakly compact if each minimizing sequence (yn ) ⊂ M has a subsequence weakly converging to an element of M. Remark 4.3 Below (Remark 4.5), we shall see that the set Rm,n of rational functions is not approximatively compact in C[a, b]. However, this set is approximatively compact in every space L p [a, b] (1  p < ∞). The concept of an approximatively compact set was introduced by Efimov and Stechkin [224]; it was later extended by W. Breckner, who considered weak convergence instead of strong convergence.

4.2 Existence of Best Approximation

61

The class of all approximatively compact sets is quite broad. It is clear that every boundedly compact set is approximatively compact. The converse assertion is in general not true — for example, every closed convex set in a uniformly convex Banach space is approximatively compact (see Proposition 4.2 below). Remark 4.4 Briefly touching on the relation between approximatively and boundedly compact sets, we mention the following result: in every infinite-dimensional weakly compactly generated Banach space (that is, in a space X in which there is a weakly compact set whose linear span is dense in X), and in particular in every separable Banach space, there exists a bounded approximatively compact set that is not locally compact, and therefore not boundedly compact (Borodin [114], [118]). Exercise 4.1 Can the convex hull of a compact set in C[a, b] have an interior point? Can the convex hull of a countable set of points in C[a, b] have an interior point? Exercise 4.2 In which normed spaces is the unit ball compact? Exercise 4.3 Check whether {a/(ct + d) ∈ C[0, 1]} is compact. Exercise 4.4 Assume that the linear projection of a certain set on every hyperplane is compact. Is the set itself compact? Exercise 4.5 Let M be an existence set (an approximatively compact set, respectively)   in C[−1, 1]. Is the induced set of symmetrized functions M1 = u(t) = [v(t) + v(−t)]/2 | v ∈ M an existence set? Exercise 4.6 Is the unit ball B(0, 1) of  2 an approximatively compact set? Let L be an infinitedimensional linear manifold. Is the set B(0, 1) ∩ L approximatively compact in L with respect to the norm of  2 ? Is this set an approximatively compact set in  2 ? Exercise 4.7 Let 02 be the space of compactly supported sequences equipped with the  2 -norm. Is every infinite-dimensional linear manifold in 02 an existence set? Construct an example of a proper infinite-dimensional linear manifold 02 that is an approximatively compact set in 02 . Exercise 4.8 Construct an example of nonapproximatively compact subset of c0 (of C[0, 1]). Exercise 4.9 Is it true that a bounded locally compact set is compact?

4.2 Existence of Best Approximation The distance function to a set is continuous (Proposition 1.1), and so it is clear that a compact set is a set of existence. However, the compactness assumption does not generally hold in the majority of interesting cases. Example 4.2 We give examples of some classical abstract and concrete sets of existence: (1) a reflexive subspace (and, in particular, a finite-dimensional subspace) of a normed linear space (Klee [339]); (2) a w ∗ -closed nonempty subset of a dual space (Phelps [468]);

62

4 Existence and Compactness . . .

(3) a closed convex nonempty subset of a reflexive space (Klee [339]); (4) the set R n, m of rational functions on the interval [a, b] with fixed n and m in the space L p [a, b], 1 ≤ p ≤ ∞ (Akhiezer [2], Walsh [594], Efimov and Stechkin [224]); (5) the set of extended exponential sums l     En = u  u = pν (x)e tν x , pν ∈ P n, tν ∈ R,

k :=

ν=1

l 

(1 + deg pν ) ≤ n



ν=1

L p [a,

in the spaces b], 1 ≤ p ≤ ∞, and in C(Q) (Hobby, Rice [292], Kammler [319], Braess [137, Chap. VI]); (6) the splines of variable smoothness in C[a, b] of order n with k free knots (Schumaker [509]; see p. 75 below).

Proposition 4.1 Let X be a normed linear space and M ⊂ X. If (yn ) ⊂ M is a minimizing sequence from M for x ∈ X and if (yn ) weakly converges to y ∈ M, then y is a nearest element in M for x; that is, y ∈ PM x. Proof Indeed, let a functional f ∈ S ∗ be such that f (x − y) =  x − y. Then  x − y = f (x − y) = lim f (x − yn )≤ lim  f  ·  x − yn =ρ(x, M). n→∞

n→∞

Hence y ∈ PM x.



It is easily seen that an approximatively compact set is closed. We mention some more properties of approximatively compact and approximatively weakly compact sets (see, for example, [592]). Proposition 4.2 (a) If M   is approximatively compact, then: (a1) M is a set of existence; (a2) M is closed, (a3) M is P-compact. (b) If M   is approximatively weakly compact, then M is a set of existence and is closed. (c) If M   is approximatively compact (approximatively weakly compact) and if PM x is a singleton for some x ∈ X, then every minimizing sequence for x converges (weakly converges) to PM x. To prove Proposition 4.2, we note that a1), b), and c) are direct consequences of the definitions and Proposition 4.1; assertion a2) follows from a1) and Proposition 1.5; a3) is secured by Proposition 4.1 in view of the fact that each sequence (yn ) ⊂ PM x is a minimizing sequence for x. Remark 4.5 In the space L p [0, 1], 1 < p < ∞, the set Rn,m of rational fractions (see (1.2)) is approximatively compact but is not boundedly compact for m ≥ 2 (see [224]). Indeed, we set 2−1/p

Rk (x) =

bk

x 2 + b2k

,

k = 1, 2, . . . ,

where bk = 4−pk . Let us estimate the function Rk (x) from above and from below on [0, bk ]. We have

4.2 Existence of Best Approximation 2−1/p

Rk (x) ≥

bk

1 −1/p 1 k bk = 4 , 2 2

=

2b2k

63 2−1/p

bl

Rl (x) ≤

b2l

−1/p

= bl

= 4l .

Hence for k > l, ∫

Rk − Rl  L p ≥

bk

|Rk (x) − Rl (x)| p dx

 1/p

0

 1  1 1 1 1/p 4k − 4l bk = 4−k 4k − 4l ≥ − = . 2 2 2 4 4

1

≥ Besides, Rk  L p =

∫ 0

=

∫

0

1

 1/p ∫

2p−1

bk

dx

(xk2 + b2k ) p  1/p ∞ dt (t 2 + 1) p

0

∞

 1/p

2p−1

bk

(xk2 + b2k ) p

dx

< ∞.

Hence the L p -bounded set {Rk (x)} is not boundedly compact. It can be shown that for all n ≥ 0, m ≥ 1, the set Rn,m is sequentially weakly closed in L p , p > 1, and hence is approximatively compact (see also Deutsch [190], Corollary 4.2). However, in the space C[0, 1], the set Rn,m (with m ≥ 1) fails to be approximatively compact. Indeed, on the one hand, it is known (Maehly and Witzgall [414], and also Braess [137]) that the metric projection onto (the Chebyshev set) Rn,m has points of discontinuity. On the other hand, the metric projection onto every approximatively compact Chebyshev set is continuous (see, for example, Proposition 5.1 below or [592, Corollary 2.2]). We give a simple test for approximative compactness of a set (see [224]). Definition 4.5 A set M is called sequentially weakly closed if every point of the space that is the weak limit point of some sequence yn ∈ M lies in M.

Fig. 4.1 Uniformly convex and not uniformly convex spaces; d(l) is the diameter of the set l.

64

4 Existence and Compactness . . .

Definition 4.6 A space X is called uniformly convex (see also Sect. 9.2 below) if for each ε > 0, there exists δ > 0 such that  x − y < ε whenever x, y ∈ X,  x =  y = 1, and  x + y/2 > 1 − δ/2. Every uniformly convex Banach space is reflexive. (For more details on uniformly convex spaces, see Sect. 9.2 below.) Remark 4.6 Consider the unit ball B and draw a hyperplane at a distance h < 1 from 0, thereby cutting a ‘slice of width l’ from the ball B (see Fig. 4.1). Let d(l) be the diameter of the ‘slice’. Consider the limit limh→1 d(l) as h → 1. A space is uniformly convex if the diameter d(l) tends to 0 as h → 1 uniformly over all hyperplanes (over all ‘slices’). Theorem 4.1 Let X be a complete uniformly convex Banach space. If a set M   is sequentially weakly closed, then M is approximatively compact. Theorem 4.1 is a corollary of the more general fact that a complete uniformly convex Banach space is a Efimov–Stechkin space (in such spaces, a sequentially weakly closed set is necessarily approximatively compact; see Chap. 9 below). Theorem 4.2 (B. Szőkefalvi-Nagy [536]) A closed convex subset of a complete uniformly convex Banach space is an approximatively compact Chebyshev set. Proof (of Theorem 4.2) Without loss of generality, we assume that x = 0. We set d = ρ(0, M). The case d = 0 is clear, so we assume that d > 0. Let (un ) be a minimizing sequence from M for 0. We set λn = un  −1 , n = 1, 2, . . . . From a given ε > 0, we choose δ > 0 from the definition of a uniformly convex space (Definition 4.6). For sufficiently large n, m, we have un , um  < d(1 + δ/2). So 1 1 λn un + λm um 2 2 1 = un + um − (1 − dλn )un − (1 − dλm )um  2d 1 1 −1 δ 1 1 (λ−1 − d)λn un  − (λ − d)λm um  > 1 − , ≥ (un + um ) − d 2 2d n 2d m 2 because the distance from 12 (un + um ) ∈ M to the origin is not smaller than d. By uniform convexity, λn un − λm um  < ε

and

un − um  < λn−1 ε + |1 − λm /λn | · um .

It follows that (un ) is a Cauchy sequence, which converges to some point u, because the space is complete. Since M closed, we have u ∈ M, and hence M is approximatively compact.  A convex closed nonempty subset of a complete uniformly convex space is therefore approximatively compact and hence a set of existence (this also follows from James’s reflexivity theorem (Theorem 1.5), because a complete uniformly convex Banach space is reflexive).

4.2 Existence of Best Approximation

65

If a space is not uniformly convex or not reflexive, then a closed convex nonempty set need not be approximatively compact; furthermore, such a set can even be antiproximinal. Consider this situation in the space c0 of null number sequences with standard maximum norm. Example 4.3 In the space c0 (which of course is not uniformly convex), consider the hyperplane M = {x ∈ c0 | f (x) = 0},

where f (x) :=

∞ 

2−k x (k) .

(4.1)

k=1

We claim that PM x =  for all x  M. Above (see Proposition 1.7), it was shown that the kernel Ker ϕ of a linear functional ϕ is a set of existence if and only if ϕ attains its norm. Since clearly, the functional f from (4.1) does not attain its norm, M is not a set of existence. So, since M is linear, no point x  M has a nearest point in M.

Remark 4.7 It is worth pointing out again that if f is a linear continuous functional on c0 that does not attain its norm, then the set M := {x ∈ c0 | | f (x)| ≤ 1} is antiproximinal (of course, M is convex and closed). Considering intersections of sets of this kind in the space X = c0 , Edelstein and Thompson constructed a bounded convex closed body M ⊂ c0 for which E(M) = M (that is, no point outside M has a nearest element in M). Such sets are called antiproximinal. (For more details, see Konyagin [358] and Balaganskii [55].) Remark 4.8 The key point in Example 4.3 is the fact that c0 is not reflexive (in particular, in c0 there is no element on which the functional from (4.1) attains its norm). The following fact is worth noting. If v ∈ S, f ∈ S ∗ are such that  f (v) = v = 1, then y = x − f (x)v is an element of best approximation to x from the hyperplane {z ∈ X | f (z) = 0}. Above (Corollary 4.1), it was noted that for every nonempty closed subset M of a finite-dimensional Banach space X and every point x ∈ X \ M, the set M contains a nearest point to x. This property characterizes the finite-dimensional Banach spaces. Theorem 4.3 In every infinite-dimensional Banach space there exist a nonempty closed set M and a point x ∈ X \ M that has no nearest points in the set M. As a corollary, a Banach space is finite-dimensional if and only if every nonempty closed subset of this space is a set of existence. Proof Since X is infinite-dimensional, by Riesz’s lemma (Lemma 1.1), the unit sphere S of X contains a sequence (xn ) such that  xn − xm  > 1/2 for n  m. We set M := {(1 + 2−n )xn | n ∈ N}. It is clear that the set M is closed and

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4 Existence and Compactness . . .

ρ(0, M) = 1 < 0 − (1 + 2−n xn )

∀n ∈ N.

This shows that for 0, there is no nearest point in M. The second assertion of the theorem is now clear.  Exercise 4.10 Let (X, ρ) be a metric space, M a compact subset of X, x ∈ X. Show that x has a nearest point in X. Exercise 4.11 Check whether the following set is proximinal: a) the unit ball B of a normed space; b) X \ B˚ (the complement of the open ball).  ∫1   Exercise 4.12 Show that the set M = f ∈ C[−1, 1]  0 f (t) dt = 0 is a closed convex set in the space C2 [−1, 1] := (C[−1, 1],  · 2 ) of continuous functions with the mean-square norm that is not an existence set. Hint: Examine PM f for f ≡ 1. Exercise 4.13 Let M be an existence set (an approximatively compact set, respectively) in C[−1, 1].   Check whether the set M1 = u(t) = [v(t) + v(−t)]/2 | v ∈ M is an existence set. Exercise 4.14 Show that the unit ball B(0, 1) in the space c0 is an existence set that is not approximatively compact. Exercise 4.15 Are the union and the intersection of two existence sets also existence sets? Exercise 4.16 Show that M = {x ∈ C[0, 1] | x(0) = 0} is an existence set in C[0, 1] that is not approximatively weakly compact in C[0, 1]. Exercise 4.17 Is every point of the curve γ := {α + α2 t 2 | α ∈ R} a nearest point for some point C[0, 1]? Is the curve γ an existence set in C[0, 1]? Exercise 4.18 Find an example of a smooth curve in  2 that is not proximinal. Exercise 4.19 Check whether M := {1 + a continuous positive function on [0, 1].

f (t ) α 2 +1

| α ∈ R} is an existence set in L 2 [0, 1], where f is

Exercise 4.20 Construct an existence set M in a two-dimensional normed space such that M has no luminosity points for any x  M. Exercise 4.21 Is the affine hull aff M of a convex set of existence (a compact set, respectively) also proximinal? Exercise 4.22 Let M be an existence set and let X be the completion of X. a) Is M also an existence set in X? b) Is the closure of M in X also an existence set in X? Exercise 4.23 Let B be an arbitrary closed ball in a metric space. Discuss its proximinality. Exercise 4.24 Let M be a closed set in X such that M is an existence set in the completion X of X. Check whether M is an existence set in X? Exercise 4.25 Is it true that for every finite-dimensional subset M of C[0, 1], there exists a function f ∈ C[0, 1] for which PM f = f ? Consider the same question for an arbitrary compact set and an arbitrary subset. Does there exist a linear injective mapping of such sets into C[0, 1] whose range consists of nearest points of some function f ∈ C[0, 1]? Exercise 4.26 Check whether M := { f ∈ C[0, 1] | f ≥ 0} is an existence set in C[0, 1]. Is this set a Chebyshev set?

4.3 Approximative τ-Compactness with Respect to Regular τ-Convergence Exercise 4.27 For which C ≥ 0, 1, 2, 4, are the following existence sets? A =    2 2 +1)t +1      α ∈ R , D = αt 2 +(α 2 +c 2 )t +1  α ∈ R α ∈ R , B = αt +(α t +c t +c+1

67  αt 2 +(α 2 +1)t +1   t +c+1

Exercise 4.28 Give an example of an infinite-dimensional metric space in which every nonempty set is proximinal. Does there exist an infinite-dimensional metric space in which every nonempty subset is a Chebyshev set? Exercise 4.29 Is every subspace of an infinite-dimensional Euclidean (inner product) space an existence set? What if the space is Euclidean? Exercise 4.30 a) Show that a closed subset of a finite-dimensional normed space is boundedly compact. b) Show that every finite-dimensional subspace is boundedly compact, but not compact.

4.3 Approximative τ-Compactness with Respect to Regular τ-Convergence Deutsch [190] proposed a very convenient definition of approximative τ-compactness with respect to a regular convergence τ. Using this concept, it proved possible to show proximinality for a wide class of sets. Deutsch put forward a fairly general existence theorem, which implies as a corollary the proximinality of all sets from Example 4.2. (Note that at one time, verification of the existence property for each of the sets from Example 4.2 was quite a challenge.) The concept of approximative τ-compactness generalizes the definition of approximative compactness by N. V. Efimov and S. B. Stechkin, and the definition of approximative weak compactness by W. W. Breckner. Classical examples of τ-convergence include norm convergence, weak convergence, and ∗-weak convergence. It is easily checked that each approximatively τ-compact set is a set of existence, and moreover, the metric projection onto it satisfies certain continuity properties (Theorem 4.4). For example, every set from Example 4.2 is approximatively τcompact with respect to the corresponding τ-convergence. Of special importance here is the fact that in a number of very interesting examples (the set of rational functions, the set of exponential suns, the set of splines with free knots in the space C[a, b]), τ-convergence is not induced by any topology on the underlying space. Approximative τ-compactness was also defined by L. P. Vlasov in 1974, but he required that τ be induced by a topology. Following F. Deutsch, we define a general type of convergence of nets and sequences in normed linear spaces. τ

Definition 4.7 Let X be a normed linear space. By a regular τ-convergence xδ → x of nets (or sequences) in X we mean the convergence τ with the following properties: τ τ (i) τ is translation-invariant; that is, xδ → x implies that xδ + y → x + y for all y ∈ X; τ (ii) τ is norm dominated; that is, xδ → x implies that  x ≤ lim sup  xδ ; τ τ (iii) τ is homogeneous; that is, xδ → x implies that αxδ → αx for every number α.

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4 Existence and Compactness . . .

In this situation, τ is called a regular mode of convergence of nets (respectively, a regular mode of sequential convergence). Clearly, every regular mode of convergence is a regular mode of sequential convergence, but the converse is not true in general. We give some examples of regular convergence [190]. n

Example 4.4 (a) Convergence in norm: xδ → x if and only if  xδ − x → 0. w (b) Weak convergence: xδ → x if and only if x ∗ (xδ ) → x ∗ (x) for all x ∗ ∈ X ∗ . w∗

(c) ∗-weak convergence in the dual space X = Y ∗ : yδ∗ → y ∗ if and only if ∗ yδ (y) → y ∗ (y) for all y ∈ Y . (d) Pointwise convergence in the space C(Q) on a dense subset of a compact  set Q: xδ → x if there exists a dense subset Q0 ⊂ Q on which xδ (t) → x(t) for all t ∈ Q0 . (e) Pointwise convergence in the space C[a, b] at all except possibly finitely many ϕ points of [a, b]: xδ → x if there exists a subset Q0 ⊂ Q such that [a, b] \ Q0 is finite and on which xδ (t) → x(t) for all t ∈ Q0 . a.e. (f) In L p (μ) (1 ≤ p ≤ ∞), convergence of a sequence almost everywhere: xn → x if μ({t | xn (t) → x(t)}) = 0. This is a regular mode of convergence for sequences, but is not, in general, a regular mode of convergence (for nets). Indeed, consider the net (xδ ) of all characteristic functions of finite sets in [0, 1] ordered by containment. Then xδ (t) → 1 for all t ∈ [0, 1]. However,  xδ  = 0 for all δ and 1 p = 1. This shows that condition (ii) from Definition 4.7 of regular convergence is not satisfied. a.w. (g) Almost weak convergence: xn → x if there exists a w ∗ -dense subset Λ of the set of extreme points of the dual unit ball in X ∗ such that x ∗ (xδ ) → x ∗ (x) for all x ∗ ∈ Λ. In C(Q), the almost weak convergence coincides with the Δ-convergence from Example 4.7, (d). Note that various types of regular convergence can also be introduced in the space of bounded linear operators (see [190]). If τ is a regular mode of convergence on X, we write τs for the induced regular ws mode of sequential convergence on X. For example, xn → x means the regular mode of sequential weak convergence. A regular mode of convergence (respectively, a regular mode of sequential convergence) τ is called topological if there is a topology on X such that convergence of a net (respectively, sequence) in this topology is equivalent to τ-convergence. In Example 4.4, (a)–(c), the τ-convergence is topological. In (d)–(g), the regular τ-convergence is not topological (the corresponding counterexamples follow by constructing a sequence that does not τ-converge to zero yet every subsequence has a subsequence that does τ-converge to zero. Such a construction would be impossible if τ were topological.) Definition 4.8 Let τ be a regular mode of convergence (a regular mode of sequential convergence). A subset M of a normed linear space X is called approximatively τ-compact (approximatively sequentially τ-compact) if for all x ∈ X and every

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69

minimizing net (sequence) for x: from M (see Definition 4.2), there exists a subnet1 (a subsequence) that τ-converges to a point of M. For τ = n, we get the classical definition of approximative compactness, and for τ = w, the definition of approximative weak compactness. The main reason for the introduction of approximatively τ-compact sets is that they are proximinal and their metric projections satisfy certain continuity properties. A property somewhat stronger than approximative τ-compactness is bounded τ-compactness. Definition 4.9 Let τ be a regular mode of convergence (a regular mode of sequential convergence). A subset M ⊂ X is called boundedly τ-compact (boundedly sequentially τ-compact) if every bounded net (a regular sequence) from M contains a subnet (a subsequence) that τ-converges to a point of M. (For τ = n, we get the classical definition of bounded compactness.) The following result follows easily from the definition. Lemma 4.1 A boundedly τ-compact set is approximatively τ-compact. We also need definitions of τ-open, τ-closed, and τ-compact sets. Definition 4.10 Let τ be a regular mode of convergence (a regular mode of sequential convergence). A subset M of a space X is called τ-closed if M contains the limit of each of its τ-convergent nets (sequences). A set M is τ-open if X \ M is closed. A set M is countably τ-compact if every sequence from M contains a subnet (a subsequence) that τ-converges to a point of M. A set M is τ-compact if every net (sequence) from M contains a subnet (a subsequence) that τ-converges to a point of M. Definition 4.11 The metric projection onto a set M is called norm-τ-upper semicontinuous (n-τ-upper semicontinuous) at a point x0 if for every net xn ,  xn → x0 , and every τ-open set V ⊃ PM x0 , we have PM xn ⊂ V for all for sufficiently large n. The fundamental properties of approximatively τ-compact sets can now be stated (Deutsch [190]). Theorem 4.4 (F. Deutsch) Let τ be a regular convergence (a regular sequential convergence) and let M be an approximatively τ-compact set. Then (1) M is a set of existence; (2) the metric projection PM onto M is n-τ-upper semicontinuous; (3) the set PM x is countably τ-compact for all x ∈ X. Moreover, if M is boundedly τ-compact, then in addition to (1), (2), the following assertion holds: (4) the set PM x is τ-compact for all x ∈ X. 1 Recall that every mapping of a direct set is called a net. A net g : G → X is called a subnet of a net f : F → X if there exists a mapping h : G → F such that g = f ◦ h, and if moreover, for each s0 ∈ F, there exists t0 ∈ G such that for all t  t0 , h(t)  s0 .

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4 Existence and Compactness . . .

Proof (1) Let x ∈ X and let (yn ) be a minimizing sequence from M for x. By definition, (yn ) has a subnet (a subsequence) (yδ ) that τ-converges to some y0 ∈ M. τ By properties (i) and (ii) of Definition 4.7, we have yδ − x → y0 − x and  y0 − x ≤ lim sup  yδ − x = ρ(x, M). This shows that y0 ∈ PM x and that M is a set of existence. (2) If the result were false, then there would exist a point x0 ∈ X, a sequence (xn ),  xn − x0  → 0, and a τ-open set V ⊃ PM x0 such that PM xn \ V   for each n > 0. We have ρ(x0, M) =  x0 − yn  ≤  x0 − xn  +  xn − yn  =  x0 − xn  + ρ(xn, M) → ρ(x0, M). So (yn ) is a minimizing sequence from M for x0 . Choosing a subnet (a subsequence) τ (yδ ) such that yδ → y0 ∈ M and arguing as in the proof of the first assertion, we see that y0 ∈ PM x0 ⊂ V. However, yδ ∈ X \ V, but X \ V is τ-closed, which gives y0 ∈ X \ V, a contradiction. (3) Let (yn ) be a sequence from PM x. Then (yn ) is a minimizing sequence for x, and hence it has a subnet (a subsequence) that τ-converges to y0 ∈ M. As in the first assertion, we get y0 ∈ PM x. (4) Let M be boundedly τ-compact. By Lemma 4.1, assertions (1)–(3) of the theorem hold. Let x ∈ X and let (yδ ) be a subnet (a subsequence) in PM x. Since PM x is bounded, there exists a subnet (a subsequence) (yγ ) that τ-converges to some  y0 ∈ M. A similar analysis shows that y0 ∈ PM x; that is, PM x is τ-compact. Remark 4.9 The problem of the existence of best approximation in the case that the distance is replaced by functionals and balls are replaced by closed sets is considered in [438], [155]. An important corollary of Theorem 4.4 is the following result (for a proof, see [137]). Lemma 4.2 (F. Deutsch) If each bounded sequence in M ⊂ C(Q), where Q is a metrizable compact set, contains a subsequence that converges pointwise on a dense subset of Q to an element in M, then M is a set of existence. Remark 4.10 In Lemma 4.2 it suffices to consider only minimizing sequences for points outside M. It is worth pointing out that in Lemma 4.2, a set is not assumed (and can fail to be) approximatively compact. We give several more general facts about regular τ-convergence.

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71

Theorem 4.5 (F. Deutsch) Let τ be a regular convergence (a regular sequential convergence) on X and let L be a τ-closed linear subspace of X. Then the following conditions are equivalent: (a) the unit ball BL of the subspace L is τ-compact; (b) each τ-closed subset of L is boundedly τ-compact. Proof (of Theorem 4.5) Let the unit ball BL of L be τ-compact and let M be a τclosed subset of L. Let (xδ ) be a bounded net (a bounded sequence) from M. Since BL is τ-compact, from Definition 4.7,(iii) this property is also satisfied for the ball αBL for every α > 0. As a corollary, every (xδ ) ⊂ M ∩ αBL contains a subnet (a subsequence) that τ-converges to some x ∈ αBL ⊂ L. Since M is τ-closed, it follows that x ∈ M. So M is boundedly τ-compact. For the converse, it suffices to show that BY is τ-closed. Let (yδ ) be a bounded net τ (a bounded sequence) from BL and let yδ → y. Since L is τ-closed, we have y ∈ L. By the norm domination property of Definition 4.7), we have  y ≤ lim sup  yδ ,  whence y ∈ BL . Since the reflexive spaces are characterized by the property that the unit ball is wcompact, Theorem 4.5 has the following corollary. (Here, proximinality of subspaces of reflexive spaces was established earlier by Klee [339].) Corollary 4.2 Let Y be a reflexive subspace of a space X. Then each w-closed subset M ⊂ Y is boundedly w-compact. In particular, M is a set of existence, and the metric projection onto M is n-wupper semicontinuous (See Definition 4.11). Similarly, since the unit ball of a dual space is w ∗ -compact, we have the following. Corollary 4.3 Each w ∗ -closed subset M of a dual space is boundedly w ∗ -compact. In particular, M is a set of existence, and the metric projection onto M is n-w ∗ upper semicontinuous. As another immediate consequence of Theorem 4.5, we can actually characterize the reflexive Banach spaces (Deutsch [190]). Corollary 4.4 In a Banach space X, the following conditions are equivalent: (a) X is reflexive; (b) each w-closed subset of X is boundedly w-compact; (c) each w-sequentially closed subset of X is boundedly ws-compact. In particular, each w-closed (for example, closed convex) subset of a reflexive space is a set of existence with n-w-upper semicontinuous metric projection. In Corollary 4.4, the equivalence (a) ⇔ (b) ( ⇔ c) follows from Theorem 4.5 with regular τ-convergence (sequential regular τ-convergence) generated by the norm topology and in view of the classical Eberlein–Shmul’yan theorem, according to which the reflexive spaces are characterized by weak compactness (sequential weak compactness) of the unit ball. A simple corollary of James’s theorem (Theorem 1.5) is that in the reflexive spaces (and only in them), a (closed) hyperplane is proximinal. Hence from Corollary 4.4, we have the following result.

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4 Existence and Compactness . . .

Corollary 4.5 In a Banach space X, the following conditions are equivalent: (a) X is reflexive; (b) each w-closed subset of X is approximatively w-is compact; (bs ) each w-closed subset of X is approximatively ws-compact; (c) each w-closed subset of X is a set of existence; (cs ) each w-sequentially closed subset X is a set of existence; (d) each closed convex subset X is a set of existence; (e) each (closed) subspace in X is a set of existence. Consider the question as to when an approximatively τ-compact set is approximatively compact. We need the following definition. Definition 4.12 Let τ be a regular mode of convergence (respectively, a regular mode of sequential convergence) in X. Following Deutsch, X is said to have property (Aτ ) τ if  xδ − x → 0 whenever (xδ ) is a net (sequence) with xδ → x and  xδ  →  x|. Spaces satisfying property (Aτ ) with τ = ws are called Kadec–Klee spaces. Definition 4.13 A norm  ·  on a linear space X is called locally uniformly convex if lim  xn − x = 0 whenever x, xn are such that   lim 2 x 2 + 2 xn  2 −  x + xn  2 = 0 (see Fig. 4.2). In other words, if  x =  xn  = 1 and lim  x + xn  = 2, then xn → x. In this case, (X,  · ) is said to be locally uniformly convex (written X ∈ (LUR)).

Fig. 4.2 Definition of a locally uniformly convex space.

Example 4.5 (i) Each locally uniformly convex space has properties (Aws ) and (Aw ). (ii) Each locally uniformly convex dual space satisfies properties (Aw ∗ s ) and (Aw ∗ ). (iii) The spaces L p , 1 ≤ p < ∞, have property (Aa.e. ) (see, for example, [290, p. 209]). The importance of property (Aτ ) stems from the following result (for τ = ws, Theorem 4.6 was established by W. W. Breckner).

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Theorem 4.6 (F. Deutsch) In a space with property (Aτ ), every approximatively τ-compact set is approximatively compact. Proof (of Theorem 4.6) Let τ be a regular mode of convergence (respectively, a regular mode of sequential convergence) on X and let M ⊂ X be approximatively τcompact. For x ∈ X, consider a minimizing sequence (yn ) ⊂ M:  x−yn  → ρ(x, M). By definition, (yn ) contains a subnet (a subsequence) (yδ ) that τ-converges to some τ y ∈ M. Hence y ∈ PM x,  x − y = ρ(x, M) = lim  x − yδ  and x − yδ → x − y. By property (Aτ ), we have  y − yδ  = (x − yδ ) − (x − y) → 0. So, yδ → y in norm, and so M is approximatively compact.  The next result is a consequence of Theorems 4.6 and 4.5. Corollary 4.6 Let τ be a regular mode of convergence (respectively, sequential convergence) on X. If the ball B is τ-compact and X has property (Aτ ), then in X, every τ-closed set is approximatively compact. Corollary 4.7 Let X be a locally uniformly convex dual space. Then in X, every w ∗ -closed set is approximatively compact. In particular, in X, every convex w ∗ -closed subset is a Chebyshev set with continuous metric projection. Corollary 4.7 follows from Corollary 4.6 and the following results: (1) the unit ball of a dual space is w ∗ -compact; (2) a locally uniformly convex dual space has property (Aw ∗ ); (3) in a strictly convex space, every point has at most one element of best approximation from a convex set. Since the weak and w ∗ topologies coincide in reflexive spaces, Corollary 4.7 implies the following well-known result of Efimov and Stechkin (see Theorems 4.1, 4.2, and 5.10). Corollary 4.8 Let X be a reflexive locally uniformly convex space (in particular, a uniformly convex Banach space). Then each convex weakly closed subset of X is approximatively compact. In particular, every convex closed subset of X is a Chebyshev set with continuous metric projection. If the convergence τ is generated by the weak topology, then the converse of Corollary 4.6 is also valid. It is unknown whether Corollary 4.6 is reversible in the general case. (For a partial conversion of Corollary 4.6, see Deutsch [190, Proposition 2.22].) Theorem 4.7 In a Banach space X, the following conditions are equivalent: (a) X is reflexive and has property (Aw ∗ ); (b) each w-closed subset of X is approximatively compact; (c) each closed convex subset of X is approximatively compact; (d) each (closed) hyperplane in X is approximatively compact. Proof The implication (a) ⇒ (b) follows from Corollary 4.6. The implications (b) ⇒ (c) ⇒ (d) are clear. The implication (d) ⇒ (a) is a consequence of James’s reflexivity theorem. 

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4 Existence and Compactness . . .

4.3.1 Applications in C[a, b] Consider now the problem of approximation by one special class of generalized rational fractions (see also Chap. 11). Let V, W be finite-dimensional subspaces of C[a, b] consisting of analytic functions. Consider the following class of rational functions:   RVW := r ∈ C[a, b] | rw = v, w ∈ W, w  0, v ∈ V . Remark 4.11 The function w in the definition of the class RVW may vanish at some points of the compact set (or the interval [a, b]) under consideration. Recall (see Example 4.4) that a net (xδ ) is said to Δ-converge (Deutsch-converge)  in C(Q) to x ∈ C(Q) (xδ → x) if there exists a dense subset Q0 ⊂ Q (which depends on (xδ (t))) on which xδ (t) → x(t) for every t ∈ Q0 . Theorem 4.8 (F. Deutsch) The set RVW is boundedly Δ-compact in C[a, b]. As a corollary, RVW is a set of existence, the metric projection PRW is n-Δ-upper V semicontinuous, and the set PRW x is Δ-compact for all x ∈ C[a, b]. V

Proof (of Theorem 4.8) The last assertion of the theorem follows from the first one in view of Theorem 4.4. To prove the first assertion, consider a bounded net (rδ ) ∈ RVW , rδ  ≤ μ. We have rδ wδ = vδ for some wδ ∈ W, wδ  = 1. For all t ∈ [a, b], we have |vδ (t)| = |rδ (t) wδ (t)| ≤ μ|wδ (t)| ≤ μ,

(4.2)

which gives vδ  ≤ μ for all δ. Since V, W are finite-dimensional and since (wδ ) and (vδ ) are bounded nets, by passing to a subnet, we may assume that vδ − v0  → 0 for some v0 ∈ V and wδ − w0  → 0 for some w0 ∈ W, where clearly, w0  = 1. By analyticity, w0 can have only a finite number of zeros. Passing to the limit in (4.2), we get t ∈ [a, b]. (4.3) |v0 (t)| ≤ μ|w0 (t)|, From (4.3), it is seen that every zero t0 of w0 is a zero of v0 . By analyticity, w0 (t) = (t − t0 )k w 0 (t),



v0 (t) = (t − t0 )k  v0 (t),

where w 0 (t0 )  0,  v0 (t0 )  0, k ≥ k . So the function r0 = v0 /w0 is well defined and continuous on [a, b] \ Z(w0 ), where Z(w0 ) is the set zeros of w0 . Further, no matter how the function r0 is defined on Z(w0 ), we have r0 w0 = v0 . Cancelling the common zero factors (t − t0 )ν of the analytic functions w0 and v0 on Z(w0 ), we see that r0 is well defined everywhere and is continuous. So r0 ∈ C[a, b] and r0 w0 = v0 ; that is, r0 ∈ RVW .

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75

Finally, for t ∈ [a, b] \ Z(w0 ) we have r0 (t) =

vδ (t) v0 (t) = lim = lim rδ (t). w0 (t) wδ (t)



So rδ → r0 , and Theorem 4.8 is proved.



Remark 4.12 In the proof of Theorem 4.8, it was actually shown that the set RVW is boundedly ϕ-compact (in the sense of the definition from Example 4.4). Corollary 4.9 The set Rn,m (see (1.2)) of rational functions Rn,m is boundedly Δ-compact in C[a, b]. In particular, Rn,m is a set of existence, the metric projection PRn, m is n-Δ-upper semicontinuous, and the set PRn, m x is Δ-compact for every x ∈ C[a, b]. Proof We have only to observe that Rn,m = {r ∈ C[a, b] | rw = v for some  w ∈ Pm \ {0}, v ∈ Pn } and apply Theorem 4.8. The set of (extended) exponential sums of order n in C[a, b] is defined as En =

l  i=1

l     pi (t)eλi t  λi ∈ R, (deg pi + 1) ≤ n ,

(4.4)

i=1

where pi is a polynomial and deg pi its degree. The sets of extended exponential sums E n are solutions of linear differential equations with constant coefficients whose characteristic polynomials have only real zeros: an y (n) + · · · + a1 y  + a0 y = c(D − λ1 ) · · · (D − λk )y = 0, d where λ1, . . . , λk ∈ R, k  n, D := dx , a12 + · · · + an2 = 1, y ∈ C n [a, b] (see [137] for more detail). The elements of the set Sn,k of spline functions of order n with free knots form the subset of C[a, b] defined by  Sn,k = x ∈ C[a, b] | there exist knots a = t0 < · · · < tr+1 = b and natural numbers m1, . . . , mr ∈ {1, 2, . . . , n+1}, m1 +. . .+mr = k, such that x ∈ Pn on each interval (ti, ti+1 ), while x has continuous derivatives of order n − mi in a neighbourhood of ti , i = 1, . . . , r .

Theorem 4.9 (F. Deutsch) Let M = E n or Sn,k . Then the set M is boundedly Δscompact in C[a, b]. As a corollary, M is a set of existence, the metric projection PM is n-Δs-upper semicontinuous, and the set PM x is Δs-compact for all x ∈ C[a, b]. Proof (of Theorem 4.9) In the case of approximation by exponential sums, we employ the following result of E. Schmidt (see, for example, [137, p. 173]): if (uk ) is a bounded sequence from En in C[a, b], a < b, then there exists a subsequence (ukν ) that converges uniformly to some u∗ ∈ En on each subinterval [α, β] ⊂ [a, b],

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4 Existence and Compactness . . .

a < α < β < b. So the sequence ukν converges pointwise to u∗ at all points of the interval [a, b], except possibly at the endpoints. Now it remains to use Lemma 4.2. For splines Sn,k with free knots, we also invoke the machinery of Deutsch convergence (Lemma 4.2): every bounded sequence sν ∈ Sn.k contains a subsequence converging to some function u on each compact subset of [a, b] \ {z1, . . . , zl }, where z1, . . . , zl are the limit points of knots of the sequence (sν ). For more details, see de Boor’s lemma in [137, p. 229] and [190, p. 145].  Remark 4.13 There is a subset of C[0, 1] that is boundedly Δs-Compact but not approximatively weak sequentially compact, and hence not approximatively compact (see [190]).

4.3.2 Applications in L p Let 1 ≤ p ≤ ∞ and let V, W be finite-dimensional subspaces of L p [a, b] consisting of analytic functions. Consider the following class of rational functions in L p [a, b]:   R = RVW := r ∈ C[a, b] | rw = v, w ∈ W, w  0, v ∈ V . (4.5) We recall the definition of the pointwise convergence of sequences almost everya.e. where in L p (μ): xn → x if μ({t | xn (t) → x(t)}) = 0. Such a convergence is regular for sequences. Theorem 4.10 (F. Deutsch, R.E. Huff) (1) The set R is approximatively compact for 1 ≤ r < ∞. (2) The set R is boundedly a.e.-compact in the space L ∞ [a, b]. As a corollary, R is a set of existence, the metric projection PR is n-a.e-upper semicontinuous, and the set PR x is a.e-compact for all x ∈ L ∞ [a, b]. Theorem 4.10 was proved by a different method by Blatter [105] and Wolfe [602]. Proof 1) We first show that the set R is boundedly a.e.-compact in L p [a, b], 1 ≤ p < ∞. Let (rn ) be a bounded sequence from R, rn  p ≤ M. We have rn w n = v n

for some wn ∈ W \ {0}, vn ∈ V .

(4.6)

By scaling both sides of this equation, we may assume that wn q = 1, where 1/p + 1/q = 1. Using (4.6) and Hölder’s inequality, we have ∫ vn 1 = |rn wn | ≤ rn  p wn q ≤ M. [a,b]

Since dim V, W < ∞, there exist a sequence (nk ) ⊂ N and points v0 ∈ V, w0 ∈ W such that wnk − w0 1 → 0. vnk − v0 1 → 0,

4.3 Approximative τ-Compactness with Respect to Regular τ-Convergence

77

Since norms on a finite-dimensional space are equivalent, we have vnk −v0 ∞ → 0, wnk − w0 ∞ → 0. It is clear that w0 q = 1, and hence w0 has at most finitely many zeros, since it is analytic. Let Z(w0 ) be the set of zeros of the function w0 . Then on [a, b] \ Z(w0 ), we have rn k =

vnk (t) v0 (t) → . wnk (t) w0 (t)

So on [a, b] \ Z(w0 ), the function v0 /w0 is well defined and continuous. Defining v0 /w0 to be constantly 1 on Z(w0 ), we get a measurable function v0 /w0 on [a, b]. Let x ∈ L p . By Fatou’s lemma, ∫ ∫ p |x − v0 /w0 | ≤ lim inf |x − rnk | p [a,b] [a,b]  p ≤ lim inf  x + rnk  ≤ [ x + M] p, which in turn gives  x − v0 /w0  p ≤  x + M . Hence v0 /w0 ∈ L p [a, b]. a.e. We now show that there exists a function r0 ∈ R such that rnk → r0 . Let t0 be a zero of w0 of multiplicity μ on [a, b], w0 (t) = (t − t0 )μ w(t),

w(t0 )  0.

If v0 does not have a zero at t0 of multiplicity ≥ μ, then in some neighbourhood of t0 , v(t) v0 (t) = , w0 (t) (t − t0 )ν w(t) where v, w are analytic functions, w(t0 )  0  v(t0 ), ν ≥ 1. So for all t and some neighbourhood O(t0 ), we have  v(t)     ≥ δ > 0.  w(t) Since v0 /h0 ∈ L p [a, b], it follows that ∫ ∫  v (t)  p 1  0  p ∞> dt = ∞,  dt ≥ δ  w (t) |t − t0 | pν 0 O (t0 ) O (t0 ) inasmuch as pν ≥ 1. This contradiction shows that every zero of w0 is also a zero of v0 with at least as large multiplicity. Thus the function v0 /w0 can be redefined at the zeros on Z(w0 ) (at the zeros of w0 ) so that the resulting function would be a.e. continuous on [a, b]. It is clear that w0 r0 = v0 , and so r0 ∈ C[a, b] and rnk → r0 . So, the set R is boundedly a.e.-compact in the space L p [a, b], 1 ≤ p < ∞. As a corollary, R is approximatively a.e.-compact in L p [a, b]. Above, it was noted that the space L p , 1 ≤ p < ∞, has property (Aa.e. ) (Example 4.5). By Theorem 4.6, R is approximatively compact. The last assertion of the first part is secured by Theorem 4.4. The proof of the second assertion is similar to that of Theorem 4.8. Theorem 4.10 is proved. 

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4 Existence and Compactness . . .

As in Sect. 4.3.1, setting V = Pn , W = Pm , we arrive at the case of classical rational functions Rn,m on [a, b] (see (1.2)). Theorem 4.10 has the following corollary. Corollary 4.10 1) For 1 ≤ p < ∞, the set Rn,m is approximatively compact in L p . In particular, Rn,m is a set of existence, the metric projection PRn, m is n-n-upper semicontinuous, and the set of best approximants PRn, m x is a.e.-approximatively compact for all x ∈ L p [a, b]. 2) The set Rn,m is boundedly a.e.-compact in L ∞ [a, b]. In particular, Rn,m is a set of existence, the metric projection PRn, m is n-a.e.-upper semicontinuous, and the set of best approximants PRn, m x is a.e.-approximatively compact for all x ∈ L ∞ [a, b]. Efimov and Stechkin proved that in a uniformly convex space, an approximatively compact Chebyshev set is a convex. As a corollary, in L p , 1 < p < ∞, none of the following sets is a Chebyshev set: Rn,m , RV,W (see (4.5)), the set of exponential sums E n (see (4.4)), save, of course, degenerate cases.

Chapter 5

Characterization of Best Approximation and Solar Properties of Sets

In this chapter, we will consider the properties of the best approximants that distinguish it from other best approximants of an approximating set. Much emphasis will be placed on characterization properties of such approximants, from which algorithms for construction of elements of best approximation can be derived. Suns were introduced by Efimov and Stechkin in the 1950s. Such properties were found to be closely related to characterization of elements of best approximation (the Kolmogorov criterion for a nearest element and its generalizations). Later, it turned out that the property of local solarity is connected to important problems of geometrical optics, and in particular, to smoothness properties of the solution to the eikonal equation |∇u| ≡ 1 (see Sect. 5.6). It was found that the study of smooth solutions to the eikonal equation requires consideration of points of continuity of the metric projection and points of differentiability of the distance functions for level surfaces of such solutions. Some general theorems on characterization of an element of best approximation are given in Sect. 5.1. Suns and the Kolmogorov criterion for a nearest element, as well as local and global best approximations and unimodal sets (LG-sets), are discussed in Sect. 5.2. The Kolmogorov criterion of an element of best approximation in the space C(Q) is formulated in Sect. 5.3. Continuity of the metric projection onto Chebyshev sets is discussed in Sect. 5.4. Some facts about differentiability of the distance function are given in Sect. 5.5. The relation of geometric approximation theory to geometric optics is discussed in Sect. 5.6.

5.1 Characterization of an Element of Best Approximation In approximation by convex or linear sets, a characterization of best approximants can be obtained as a corollary to the classical Hahn–Banach theorem and the separation theorem for two sets. Results of such type date back to A. N. Kolmogorov, I. Singer, among others. Algorithmic aspects of best rational approximation are considered, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_5

79

80

5 Characterization of Best Approximation . . .

for example, in [98] and in [455]. The characterization theorem that follows was obtained independently by F. Deutsch and G. Sh. Rubinshtein. Theorem 5.1 (Characterization of an element of best approximation) Let M be a convex subset of a normed linear space X and let x  M. Then y0 ∈ M is element of best approximation to x from M if and only if there exists a functional f = fy0 ∈ X ∗ with the following properties:  f  = 1,

(5.1)

f (x − y0 ) =  x − y0 , f (y − y0 ) ≤ 0 ∀y ∈ M.

(5.2) (5.3)

˚ r), where r :=  x − y0 , are Proof If y0 ∈ PM x, then M and the open ball B(x, disjoint convex sets. By the separation theorem for convex sets, there exist a functional f0 ∈ X ∗ and a constant c ∈ R such that f0 (y) ≤ c, f0 (z) > c,

y ∈ M, ˚ r). z ∈ B(x,

(5.4)

∀z ∈ B(x, r).

(5.6)

(5.5)

Since f0 is continuous, it follows that f0 (z) ≥ c

˚ r), and so We have y0 ∈ M ∩ B(x, r), and hence f0 (y0 ) = c. Next, x ∈ B(x, β := f0 (x) − c = f0 (x − y0 ) > 0. Setting f := r f0 /β, it is clear that f (x − y0 ) =  x − y0 ,

(5.7)

whence  f  ≥ 1. Assume that  f  > 1. Then there exists an element v ∈ X, v < 1, such that f (v) > 1. For w := x − rv, we have f0 (w) = f0 (x) − r f0 (v) = (c + β) − β f (v) < c. However, this contradicts (5.6), because w ∈ B(x, r). So  f  = 1, whence | f (x − y0 ) ≤  x − y0 . This together with (5.7) implies that (5.2). Finally, from (5.4) and (5.6), we get f0 (y − y0 ) ≤ 0 for y ∈ M. Since f differs from f0 by a positive factor, (5.3) is also proved. So condition (5.1)–(5.3) is necessary for an element y0 to be a best approximant to x. The sufficiency is secured by the following lemma, in which the convexity condition of a set is not required.  Lemma 5.1 Let M be a nonempty subset of a normed linear space X and let y0 ∈ M. Assume that there exists a functional f ∈ X ∗ satisfying (5.1)–(5.3). Then y0 ∈ PM x.

5.2

Suns and the Kolmogorov Criterion . . .

81

Proof Since  f (z)| ≤  f  · z, from (5.1)–(5.3) it follows that for all y ∈ M,  x − y ≥  f  −1 f (x − y) = f (x − y0 ) − f (y − y0 ) ≥  x − y0 . This shows that y0 ∈ PM x.



Statements are simpler in the linear approximation setting. Recall the following definition. Let M be a subset of a normed space X. For a set M, the set of functionals M ⊥ := { f ∈ X ∗ | f (x) = 0 for x ∈ M } is called the annihilator of M. It is well known that M ⊥ is a closed subspace of X ∗ . Corollary 5.1 (I. Singer) Let M be a linear subspace of a normed linear space X and let x  M. Then y0 ∈ PM x if and only if there exists a functional f ∈ M ⊥ satisfying (5.1) and (5.2). Theorem 5.2 (Kolmogorov criterion for a nearest element) Let M be a convex subset of a normed linear space X and let x  M. Then y0 ∈ M is an element of best approximation to x from M if and only if for every y ∈ M, there exists a functional f ∈ X ∗ such that  f  = 1, f (x − y0 ) =  x − y0 , f (y − y0 ) ≤ 0. For more details, see, for example, Singer [519], Meinardus [432], Kroó and Pinkus [376], and Petrushev and Popov [466].

5.2 Suns and the Kolmogorov Criterion for a Nearest Element. Local and Global Best Approximation. Unimodal Sets (LG-Sets) Some properties weaker than convexity are frequently found to be useful in nonlinear approximation theory. We start with the following simple example. Proposition 5.1 In a strictly convex normed linear space, a sun is a Chebyshev sun. Indeed, let M be a sun in a strictly convex space and let y0 be a luminosity point from M for a point x  M. If there were a different point y1 ∈ PM x, y1  y0 , then by strict convexity, we would have  x2 − y1  <  x2 − y0  for x2 = 2x − y0 . However, this contradicts the fact that y0 is a luminosity point for x. Definition 5.1 A closed set M   is called a unimodal set (or an LG-set, a global mimimizer; see, for example, [24, Sect. 3.3]) if for all x  M, each local minimum of the function Φx (y) =  y − x, y ∈ M, is global — this explains the origin of the term LG-set (local–global). In other words, for an LG-set, the condition y ∈ PM∩V x, V := B(y, ε) implies that y ∈ PM x.

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5 Characterization of Best Approximation . . .

Proposition 5.2 (B. Brosowski, F. Deutsch) A strict (proto)sun is a unimodal set (an LG-set). Proof Let M be a strict protosun. Assume that y is a point of local minimum for the distance function to M from a point x  M. By definition, this means that ε y ∈ PM∩B(y,ε) x for some ε > 0. Setting xt := y + t(x − y), where t := 3 x−y  , we see that y ∈ PM xt . Finally, y ∈ PM x, since M is a strict (proto)sun. In general, the conclusion of Proposition 5.2 cannot be reversed. Indeed, the complement of the open unit ball in the Euclidean plane is unimodal but is not a sun. Theorem 5.3 (B. Brosowski, W. W. Breckner) Let M be a set of existence in a normed linear space X. Then the following are equivalent: (a) M is a strict sun; (b) for all x ∈ X, y0 ∈ M, we have y0 ∈ PM x if and only if y0 is a nearest point to x from the interval [y0, y] for each y ∈ M; (c) each nearest point from M can be characterized by the Kolmogorov criterion for a nearest element: y0 ∈ PM x if and only if for all y ∈ M, there exists a functional f ∈ X ∗ such that  f  = 1,

f (x − y0 ) =  x − y0 ,

f (y − y0 ) ≤ 0.

Proof (a)⇔(b) M is a strict sun if for all y0 ∈ PM x and y ∈ M,  xt − y ≥  xt − y0 

∀t ≥ 1,

where xt := y0 + t(x − y0 ). Hence  y0 − y + t(x − y0 ) ≥ t  x − y0 

t ≥ 1.

After dividing by t, we get  x − y0 − t −1 (y − y0 ) ≥  x − y0 

0 < t −1 ≤ 1.

This inequality means that y0 is a nearest point for x from the interval [y0, y]. The converse is proved in the same manner. (b)⇒(c). Assume that y0 ∈ PM x. By the hypothesis, for all y ∈ M the point y0 is a nearest point for x from the convex set [y0, y]. By applying the characterization theorem of best approximants (Theorem 5.1), we get a functional required in the Kolmogorov criterion.  Brosowski (see [137], [24]) established a relation between the concept of a strict (proto)sun and the well-known Kolmogorov criterion for a nearest element. A set M is called a Kolmogorov set if x  M, y0 ∈ PM x imply that min f (y − y0 )  0

f ∈E x−y0

∀ y ∈ M,

where Ex−y0 is the set of extreme points of the convex set Px−y0 := { f ∈ S ∗ | f (x − y0 ) =  x − y0 }.

(K)

5.2

Suns and the Kolmogorov Criterion . . .

83

In other words, M is a Kolmogorov set if for all y0 ∈ PM x,   y ∈ M | f (y − y0 ) > 0 for each f ∈ Ex−y0 = . The condition (K) can be written as max

f ∈P y0 −x

f (y − y0 ) ≥ 0 for each y ∈ M.

The concept of a Kolmogorov set was introduced by Brosowski and Wegmann [138], [143]. The condition (K) always implies that y0 ∈ PM x. For Kolmogorov sets (strict protosuns), the condition (K) is also implied by y0 ∈ PM x (just as for the characterization of best approximations for convex sets). According to the geometric form of the Hahn–Banach theorem, the closed convex sets are characterized by the fact that every point outside such a set can be strictly separated from it by a (closed) hyperplane (an open half-space). However, as we already pointed out, some properties weaker than convexity are frequently found to be instrumental in nonlinear approximation theory. It turns out that a similar result also holds for suns (and also for strict (proto)suns) with open half-spaces replaced by open support cones: a point not lying in a sun can be strictly separated from it by an open convex support cone. We formulate the corresponding result, in which the strict protosuns are shown to be the Kolmogorov sets. To formulate this result, we recall that the set    ˚ x) = B˚ −r y + (r + 1)x, (r + 1) x − y , (5.8) K(y, r >0

˚  x − y) with respect to the which consists of homothetic copies of the ball B(x, ˚ homothety centre y, is called the support cone K(y, x) of the ball B(x,  x − y) at its boundary point y. ˚ x). Given We shall also require the following equivalent representation for K(y, a point s on the unit sphere S, we let Ps denote, as before, the set of all functionals from S ∗ that attain their maximum value on S at s (where S ∗ is the dual unit sphere); let Es be the set of all extreme points of the convex w ∗ -compact set Ps ⊂ S ∗ . Lemma 5.2 If x, y ∈ X, x  y, p := (y − x)/ y − x, then   ˚ x) = z | f (z) < f (y) ∀ f ∈ P p K(y,   = z | f (z) < f (y) ∀ f ∈ E p   ˚  x − y)   = z ∈ X | [z, y] ∩ B(x,    B˚ −r y + (r + 1)x, (r + 1) x − y . =

(5.9)

r >0

Proof (of Lemma 5.2) Clearly, it suffices to consider the case x = 0, y = p ∈ S. Let  ˚ 1)   . Choosing q ∈ [p, w] ∩ B, ˚ we have q < 1. If w ∈ z ∈ X | [z, p] ∩ B(0, w−q  r :=  p−q  , then

84

5 Characterization of Best Approximation . . .

 − r p − w =  − r p − [q + r(q − p)] = (r + 1) · q < r + 1, ˚ ˚ 0). that is, w ∈ B(−r p, r + 1)⊂ K(p,     ˚ ˚ Now assume that w ∈ r >0 B −r y + (r + 1)x, (r + 1) x − y = r >0 B −r p, (r +   r 1 1) ; that is, w ∈ B˚ −r p, (r + 1) for some r > 0. We set q := r+1 p + r+1 w. Then p−w  q = −rr+1 < 1. Next, for every f ∈ E p , we have f (q) =

1 r + f (w), r +1 r +1

f (q) ≤  f  q < 1,

whence f (w)< 1.  Let w ∈ z | f (z) < 1 ∀ f ∈ E p . We have sup w(P p ) = max w(P p ) = max w(E p ) < 1; the last equality is secured by the Krein–Milman theorem (the set P p =S ∗ ∩ H, where H := { f ∈ S ∗ | f (p) = 1}, is a convex w ∗ -compact set, and by the Krein–Milman theorem, the set E p of its extreme points is nonempty). As  a corollary, we have w ∈ z | f (z) < 1 ∀ f ∈ P p .     Now let w ∈ z | f (z) < 1 ∀ f ∈ P p . If w  z ∈ X | [z, p] ∩ B˚   , then [p, w] ∩ B˚ = . By the separation theorem of convex sets,there exists f0 ∈ S ∗ such  that f0 (p) = 1 ≤ f0 (w). We have f0 ∈ P p , and hence w  z | f (z) < 1 ∀ f ∈ P p , a contradiction. Lemma 5.2 is proved.  The next result is well known. Lemma 5.3 Let x, y ∈ X, x  y. Then the set ˚ x) X \ K(y, is a sun. ˚ x) is a nearest point for 2x − y, and Moreover, every boundary point z ∈ ∂ K(y, ˚ ˚ x). the ball B(2x − y,  x − y) is contained in the cone K(y, We employ the following separation theorem for strict suns. Theorem 5.4 (B. Brosowski, D. Amir, F. Deutsch) Let M ⊂ X. Then the following conditions are equivalent: (a) M is a Kolmogorov set; ˚ 0, x) =  if v0 ∈ PM x; (b) M ∩ K(v (c) M is a strict protosun. Proof (a)⇔(b) Let v0 ∈ PM x. By the assumption, we have max

f ∈Pv0 −x

f (v − v0 )  0

∀ v ∈ M.

On the other hand, ˚ 0, x) = {v | f (v − v0 ) < 0 ∀ f ∈ Pv0 −x } K(v = {v | max f (v − v0 ) < 0}, f ∈Pv0 −x

˚ 0, x) = . whence M ∩ K(v

5.2

Suns and the Kolmogorov Criterion . . .

85

(b)⇒(c) Let v0 ∈ PM x and let λ > 0. Setting x1 := v0 + λ(x − v0 ), we have ˚ 0, x), whence K(v ˚ 0, x1 ) ∩ M = . From (5.8), we have, in particular, ˚ 0, x1 ) = K(v K(v ˚ 1,  x1 − v0 ) = , M ∩ B(x that is, v0 ∈ PM x1 . (c)⇒(a) Let v0 ∈ PM x and let v ∈ M. If f (v − x0 ) > 0 for all f ∈ Px−v0 , then ˚ 0, x), whence v ∈ K(v ˚ 0 + λ(x − v0 ), λ x − v0 ) v ∈ B(v

for some λ > 0.

Correspondingly, we have v0 + λ(x − v0 ) − v < λ x − v0  = v0 + λ(x − v0 ) − v0 , which contradicts the assumption that M is a sun. Hence min

f ∈P x−v0

f (v − v0 )  0. 

To formulate an analogue of Theorem 5.4 for other classes of sets, we recall one definition. Definition 5.2 A nonempty closed set M is called an α-sun if for every point x  M, there exists a ray  emanating from x such that for all z ∈ , ρ(z, M) = z − x + ρ(x, M). Every sun is an α-sun. Unlike suns, an α-sun can have isolated points — as an example, one may take the two-point set M := {(0, 0)} ∪ {(1, 1)} in the plane with the maximum norm. Theorem 5.5 Let X be a normed liner space. (1) A set M ⊂ X is a sun in X if and only if for each point x  M, there exists ˚ x) ∩ M = . a point y ∈ PM x such that K(y, (2) A set M ⊂ X is a strict sun in X if and only if for each point x  M, the set ˚ x) ∩ M =  holds for every point PM x of its nearest points is nonempty and K(y, y ∈ PM x. (3) A set M ⊂ X is an α-sun in X if and only if for each point x  M, there exists ˚ x) ∩ M = . a point y ∈ S(x, ρ(x, M)) such that K(y, This result also holds in arbitrary asymmetrically normed spaces. Exercise 5.1 Does there exist a unimodal set in a Hilbert space with bounded complement? Exercise 5.2 Is it true than every infinite-dimensional Banach space contains a noncompact strict sun?

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5 Characterization of Best Approximation . . .

5.3 Kolmogorov Criterion in the Space C(Q) Let us consider in more detail the Kolmogorov criterion for a nearest element in C(Q)-spaces. Definition 5.3 A set M ⊂ C(Q) is said to be extremum characterizable (Dunham [212]) if a necessary condition for y0 to be a best approximant (or an element of best approximation) to any continuous x  M is that no y ∈ M exist such that sign(x − y0 )(t) = sign(y − y0 )(t)  0 ∀t ∈ crit(x − y0 ).

(5.10)

Here and in what follows, crit z := {t ∈ Q | |z(t)| = z},

z ∈ C(Q).

The next theorem shows that in C(Q) the definition of extremum characterizability is equivalent to that of a Kolmogorov set. Recall that a topological space is said to be perfectly normal if it is normal and if each closed set in the space is a G δ -set (an intersection of a countable number of open sets). A particular case of a perfectly normal space is a metric space. In the next theorem, the implication b) ⇔ c) is due to Dunham [212]. For the rest, see Sect. 7.9. Theorem 5.6 (characterization of strict protosuns in terms of extrema) Let Q be a perfectly normal space, M ⊂ C(Q). Then the following conditions are equivalent: (a) M is a strict protosun; (b) M is a Kolmogorov set; (c) M is extremum characterizable. The following particular case of Theorem 5.6 is important for applications. This particular case of the sufficiency condition in the Kolmogorov criterion for a nearest element was obtained by G. Meinardus and D. Schwedt (see, for example, [432, Sect. II.8.1]). A necessary and sufficient condition of best approximation in the form of the Kolmogorov criterion is given below in Theorem 5.8. For extensions of Theorem 5.7 to C(Q, Y )-spaces and algorithmic problems of construction of best approximation, see, for example, [452]. Theorem 5.7 (Sufficient condition of best approximation) Let Q be a metrizable compact set, M ⊂ C(Q), x  M, y0 ∈ M. Assume that for all y ∈ M,   min (x − y0 ) (y − y0 )(t) ≤ 0. (5.11) t ∈crit(x−y0 )

Then y0 is an element of best approximation from M to x. For strict protosuns (and in particular, linear and convex sets), Theorem 5.7 also gives a necessary condition of best approximation. In general, condition (5.11) is not necessary (see Example 5.2 of [432, Sect. II.8.1]). We also note two Kolmogorov-type characterizations for a nearest element in the space C(Q) (see [137, p. 10]).

5.3

Kolmogorov Criterion in the Space C(Q)

87

Theorem 5.8 (Kolmogorov criterion for a nearest uniform approximation) Let Q be a compact space, M ⊂ C(Q) a convex set. Then y0 ∈ M is an element of best approximation to x ∈ C(Q) if and only if (x − y0 )(t) [y(t) − y0 (t)] ≤ 0

inf

t ∈crit(x−y0 )

∀y ∈ M.

(5.12)

The necessity of (5.12) is clear. If condition (5.12) is not satisfied for y1 ∈ M, then it is easily checked that for sufficiently small τ > 0, yτ := y0 + τ(y1 − y0 ) gives a better approximation to x than y0 . Theorem 5.9 (Characterization of best uniform approximation by finite-dimensional subspaces) Let Q be a compact space, M an n-dimensional linear subspace of C(Q). Then given x  M, the following conditions are equivalent: (a) y0 is an element of best approximation from M to x ∈ C(Q); (b) there exist m points ti ∈ crit(x − y0 ) and m positive numbers θ i , m ≤ n + 1, such that m  θ i (x − y0 )(ti ) y(ti ) = 0 ∀y ∈ M; i=1

(c) there exist m points ti ∈ crit(x − y0 ), m ≤ n + 1, such that min (x − y0 )(ti ) y(ti ) ≤ 0

1≤i ≤m

∀y ∈ M.

Let us demonstrate how Theorems 5.7 and 5.6 work (see [432]). Example 5.1 Consider approximation of the function x ≡ 1 by the set   M := z a (t) := a − 4a2 (t − 1/2)2 a ∈ R in the space C[0, 1]. We claim that the function z1/2 is an element (even unique) of best approximation to x. For brevity, we write y0 = z1/2 := 12 − (t − 12 )2 . The error function x − y0 = 12 + (t − 12 )2 has two critical points (crit(x − y0 ) = {0, 1}) at which x − y0 is 3/4. For y := z b ∈ M, consider the function y − y0 from (5.11): y − y0 =  2  2 b − 4b 2 t − 1/2 − 1/2 + t − 1/2 . At the points t = 0 and t = 1, this function is nonpositive: 2 2 b − b − 1/4 = −(b − 1/2) ≤ 0 for all b ∈ R. By Theorem 5.7 (or equivalently, by Theorem 5.6), the function y0 = z1/2 = 12 − (t − 12 )2 is an element of best approximation to x. Example 5.2 Consider approximation of the function x ≡ 1 by the set   M := z a (t) := a2 − a(t − 1/2) | a ∈ R

(5.13)

on the interval [0, 1]. We have x ≡ 1  M. Let us check that ρ(x, M) ≥ 1/2. Without loss of generality, we assume that a > 0. Let y := z a ∈ M. Consider the case a ≥ 1. The difference y − x = a2 − a(t − 1/2) − 1 at 0 reads as |a2 + 22 − 1| = a2 − 1 + a2 ≥ 1 − 1 + 1/2 ≥ 1/2; that is, x − y  ≥ 1/2. Now assume that 0 ≤ a ≤ 1. At t = 1, we have |y(t) − x(t)| = |a2 − a2 − 1| = 1 − a2 + a2 . Assume that 1 − a2 + a2 < 1/2. This is equivalent to saying that (a − 1)(a + 1/2) > 0. However, the last equation has no solutions for 0 ≤ a < 1. So in both cases (a ≥ 1 and 0 ≤ a < 1), we have x − y  ≥ 1/2 for all y ∈ M; that is, ρ(x, M) ≥ 1/2. We have x − z±1  = 1/2, and hence z1, z−1 ∈ PM x. Consider y0 = z1 . Then crit(x − y0 ) = {0, 1}; at the points 0 and 1, the error function x − y0 assumes, respectively, the values −1/2 and 1/2. For the function y = z b , consider the difference y − y0 = b 2 − b(t − 1/2) − 1 + (t − 1/2) =

88

5 Characterization of Best Approximation . . .

(b − 1)(b − t + 3/2). For t = 0, the difference y − y0 = (b − 1)(b + 3/2) is negative for −3/2 < t < 1, and for t = 1, the difference y − y0 = (b − 1)(b + 1/2) is positive for t < −1/2 and for t > 1. So for −3/2 < t < 1/2, the difference is negative at 0 and is positive at 1. Hence the minimum in (5.12) is positive. This example shows that condition (5.11) is necessary (it is not implied by the condition y0 ∈ PM x), and also that the set M from (5.13) is not a strict sun in C[0, 1].

5.4 Continuity of the Metric Projection onto Chebyshev Sets We start with the following simple result, which dates back to I. Singer. Theorem 5.10 The metric projection onto an approximatively compact Chebyshev set is continuous. Proof Let M be an approximatively compact Chebyshev set and let x = lim xn , where xn, x ∈ X. We have  x − PM x ≤  x − PM xn  ≤  x − xn  +  xn − PM xn  ≤  x − xn  +  xn − PM x ≤ 2 x − xn  +  x − PM x, whence  x − PM xn  → ||x − PM x. So (PM xn ) is a minimizing sequence from M for x. By compactness, it converges to some element of M, which is a best approximant to x by Proposition 4.1. Since M is a Chebyshev set, such a nearest element is unique.  Remark 5.1 In general, the conclusion of Theorem 5.10 does not hold. For example, it has long been known that there are Chebyshev subspaces with discontinuous metric projection (E. V. Oshman, A. Brown, V. I. Andreev, etc.); in C[0, 1], the metric projection onto the (Chebyshev) set of rational functions Rm,n and the (Chebyshev) set of exponential sums En+ with nonnegative coefficients has points of discontinuity. Each of these sets is a Chebyshev set. The spaces C[a, b], L[a, b] are not strictly convex (and hence neither is uniformly convex). Such spaces contain subspaces onto which the metric projection is not uniformly continuous on a bounded set (for example, on a ball). Consider the corresponding examples. Example 5.3 (S. B. Stechkin) Consider the functions f ,

f ∈ C[−1, 1] defined as follows: the functions f ,

f are linear on [−1, −ε], [−ε, 0], [0, ε], [ε, 1]; at the knots t = 0, ±ε, ±1, 0 < ε < 1/2, they are defined by (see Fig. 5.1) ⎧ ⎪ 0, ⎪ ⎨ ⎪ f (t) := 1, ⎪ ⎪ ⎪ −1, ⎩

t = ±1, t = ±ε, t = 0,

⎧ ⎪ 0, ⎪ ⎨ ⎪

f (t) := 1 ± t, ⎪ ⎪ ⎪ −1, ⎩

t = ±1, t = ±ε, t = 0.

Let L = P1 be the two-dimensional subspace of C[−1, 1] consisting of the functions x(t) = αt + β, α, β ∈ R. Then PL ( f ) is the identically zero function (there are three points of Chebyshev alternant), PL (

f ) = t. So

5.5

Differentiability of the Distance Function

89

1+ε

1 f

−ε

−1

f˜

1−ε

0

ε

1

−1

Fig. 5.1 Illustration to Example 5.3.

 f  ≤ 2,  f˜ ≤ 2,

 fε − f˜ = ε, but PL ( f ) − PL ( f˜) = 1.

Example 5.4 (see [98]) Let L be the one-dimensional subspace of L[−1, 1] consisting of constant functions, and let f := sign(t + ε),

f := sign(t − ε),

0 < ε < 1.

Then PL ( f ) ≡ 1, PL (

f ) ≡ −1. So  f  = 2, 

f  = 2,

f −

f  L 1 = 4ε,

PL ( f ) − PL (

f ) L 1 = 4. Remark 5.2 (see, for example, [98], p. 32) The metric projection onto a subspace in a uniformly convex space is uniformly continuous on every bounded set (more precisely, in every r-neighbourhood of this subspace).

5.5 Differentiability of the Distance Function Asplund [47] seems to have been the first to consider the differentiability of the function ϕ M (x) = 12  x 2 − 12 ρ2 (x, M) (the Asplund function); L. P. Vlasov examined the differentiability of the distance function ρ(x, M). A number of interesting results on differentiability of the distance function were obtained by V. I. Berdyshev both in general and in concrete Banach spaces (see also V. S. Balaganskii, etc.). For example, Berdyshev established the Lipschitz continuity and the one-sided differentiability of

90

5 Characterization of Best Approximation . . .

the (near-best) metric projection to the classes H(ω) and to some other classes of functions. Definition 5.4 Let (X,  · ) and (Y,  ·  ) be normed linear spaces and let U be a nonempty open subset of X. A function f : U → Y is said to be Gâteaux differentiable if there exists a bounded linear operator Tx : X → Y such that lim

t→0

f (x + th) − f (x) = Tx (h) t

for all h ∈ SX . The operator Tx is called the Gâteaux derivative of f at x. If a mapping f is Gâteaux differentiable at every point of U, then f is said to be Gâteaux differentiable on U. Lemma 5.4 Let M be a nonempty closed subset of a normed linear space X. Assume that the distance function ρ( · , M) is Gâteaux differentiable at a point x ∈ X \ M. Let f ∈ X ∗ be the corresponding Gâteaux derivative and let y ∈ PM x. Then  f  = 1 and

f (x − y) =  x − y = ρ(x, M).

Proof Since M is closed and x  M, we have ρ(x, M) =  x − y > 0. Let 0 ≤ t ≤ 1. Since x + t(y − x) ∈ [x, y], it follows that y ∈ PM (x + t(y − x)) by Proposition 1.4, whence ρ(x + t(y − x), M) =  x + t(y − x) − y = (1 − t) x − y. Hence ρ(x + t(y − x), M) − ρ(x, M) (1 − t) x − y −  x − y = lim t→0+ t→0+ t t −t  x − y = − x − y. = lim t→0+ t

f (y − x) = lim

So f (x − y) =  x − y = ρ(x, M). Since the distance function is 1-Lipschitz (Proposition 1.1), it follows that for all z ∈ X, we have ρ(x + tz, M) − ρ(x, M) ρ(x + tz, M) − ρ(x, M) | f (z)| = lim = lim t→0 t→0 t t  x + tz − x ≤ lim = z. t→0 |t| It follows that  f  = 1.



Theorem 5.11 Let M be a nonempty closed subset of a strictly convex normed linear space X. Assume that x is a point of Gâteaux differentiability of the distance function ρ( · , M). Then PM x is at most a singleton (that is, x is a point of uniqueness).

5.5

Differentiability of the Distance Function

91

Proof If x ∈ M, then PM x = {x}. Hence we can assume that x  M. Assume to the contrary that y, z ∈ PM x. Let f ∈ X ∗ be the Gâteaux derivative of the distance function ρ(·, M) at x. By Lemma 5.4,  x−z   x−y  = f = f . f  x − y  x − z Since X is strictly convex, f can attain its norm at at most one point of the unit sphere S. As a corollary, x−y x−z = .  x − y  x − z Since  x − y =  x − z = ρ(x, M), we have y = z, which completes the proof.  Remark 5.3 Let us briefly discuss the development of Theorem 5.11. Balaganskii [56] showed that in a strongly convex1 smooth space X, a closed set M whose distance function is not Fréchet differentiable on a set whose cardinality is smaller than c can be written as M = K \ ∪ B˚ x , where K is a closed convex body, cl(int K = K), ( B˚ x ) is a family of disjoint balls B˚ x ⊂ K with centres at x (of cardinality < c), where the union is taken over points x of approximative compactness of M at which the nearest point is unique. As a corollary, Balaganskii showed that in a strongly convex space with Fréchet differentiable norm, a Chebyshev set is convex if the cardinality of the set of points of discontinuity of the metric projection is smaller than c. This result holds, in particular, for L p -spaces, 1 < p < ∞. This assertion was proved by him earlier in the Hilbert space setting by a different method. Let us give some more results in which the differentiability of the distance function proves useful in classical problems of geometric approximation theory. For a proof of the next two results, see [239]. Theorem 5.12 Let H be a Hilbert space and let M be a Chebyshev set in H. If x → ρ(x, M) is Fréchet differentiable on H \ M, then M is convex. Of course, we are now left wondering when the distance function x → ρ(x, M) is Fréchet differentiable on H \ M. To solve this we consider the following result (see [239]). Theorem 5.13 Let H be a Hilbert space and let M be a Chebyshev set in H. If the metric projection map x → PM x is continuous on H \ M, then the distance function x → ρ(x, M) is Fréchet differentiable on H \ M, and so M is convex. For more results on the convexity of Chebyshev sets and differentiability of the associated distance function, see [271], [58], [56], [603], and [239]. Unfortunately, in this book, we could not pay sufficient attention to the extensive problem of differentiation of metric projection. The differential properties of the operator of best approximation were first studied by Chebyshev. Later, this theory was significantly advanced in the papers of A. Kroó, S, Fitzpatrick, R. R. Phelps, E. Asplund, A. V. Kolushov, J. M. Borwein, J. M. Wolfe, L. Zajíček, M. Bartelt, P. V. Albrekht (this list is by no means complete). 1 A space is strongly convex if (x n ) converges whenever x n ∈ S, f ∈ S ∗ , f (x n ) → 1.

92

5 Characterization of Best Approximation . . .

Exercise 5.3 Does there exist a nonuniqueness set M ⊂ X such that the distance function ρ( · , M) is differentiable on X \ M?

5.6 Relation of Geometric Approximation Theory to Geometric Optics One of the interesting challenges in geometric approximation theory involving the concept of solarity is connected with the Hamilton–Jacobi equations. We will illustrate this by considering the simple example of the eikonal equation, which is the fundamental equation of geometrical optics: |∇ f (x)| = n,

(5.14)

where n = n(x) is the refractive index, | · | is the standard Euclidean norm on Rn . Geometrical optics is a branch of optics in which it is assumed that the wavelength is negligibly small, and the eikonal equation is derived from the wave equation for a complex amplitude (the Helmholtz equation). The function f is called an eikonal.2 The surfaces f = const are called geometrical wave surfaces or geometrical wave fronts. The eikonal equation can also be regarded as the Hamilton–Jacobi equation for the variational problem ∫ δ

n ds = 0,

the optical counterpart of which goes back to Fermat’s principle, also known as the principle of the shortest optical path. As background references for geometrical optics, we mention the books by Born and Wolf [113], Feynman, Leighton, and Sands [234], Bruce and Giblin [151], Kravtsov and Orlov [370], Poston and Stewart [483], and Arnol’d [44]. The purpose of this section is to illustrate the potency of geometric approximation theory in problems connected with the eikonal equation. For an inhomogeneous medium, when n = n(x) is not constant, rays are not straight but rather curved lines, yet even in this case, an analogous machinery of geometric approximation theory can be used. By a solution to the eikonal equation one means either a classical or a generalized solution (see [378], [533]). Here we are concerned with the classical solutions to the eikonal equation. Let Ω be an open subset of Rn . The problem is to find classical solutions to the simplest eikonal equation |∇ f (x)| = 1,

x ∈ Ω,

(5.15)

2 The term ‘eikonal’ (from the Greek εiων ¯ meaning ‘image’) was introduced in 1885 by H. Bruns to denote functions analogous to the characteristic functions of the medium, but later the term came to be used in a wider sense [113].

5.6

Relation of Geometric Approximation Theory to Geometric Optics

93

where f ∈ C 1 (Ω). Equation (5.15) describes the propagation of light rays in a homogeneous isotropic medium, in a geometrical optics approximation. For a statement of this problem (with indication of a number of simple domains for which (5.15) has a solution), the reader may consult Hiriart-Urruty [291, Problem 8]. It will be seen that a study of the geometry of the level surfaces Lθ = {x ∈ Ω | u(x) = θ}

(5.16)

of the solution yields a body of information of great value for this problem. In our setting, Lθ = is a C 1 -smooth surface (if u ∈ C k , then Lθ is C k -smooth). It is well known (see, for example, [233]) that the gradient of the eikonal u(x) is orthogonal to the level surface Lθ at x. Correspondingly, since the potential vector field a = ∇u is continuous in Ω, we have ∫ ∫ (a, dr) = (a, dr) (5.17) γ1

γ2

for all rectifiable curves γ1 , γ2 lying in a simply connected subdomain of Ω, emanating from a given arbitrary point x0  Lθ , and terminating on the same level surface. We have |a| = 1, and hence for every curve γ emanating from x0 , it follows that ∫ θ − θ 0 = (a, dr), γ

and so

∫ ∫ |a| |dr| = |γ|, |θ − θ 0 | = (a, dr)  γ

γ

(5.18)

where u(x0 ) = θ 0 . By a compactness argument, we see that θ − θ 0 is attained on some integral curve γ0 , at each point of which the tangent vector coincides with the gradient vector (or its opposite): ∫ ∫ θ − θ 0 = ± (∇u, dr) = ± ds = ±|γ0 |; (5.19) γ0

γ0

here s is the natural parametrization of γ. Now let us take an arbitrary point x0 in Ω and consider the interval (curve) I from Ω joining the point x0 and some nearest point from the level surface Lθ . The length of this interval equals the distance to the set Lθ . Hence the length of the curve γ that connects Lθ and x0 is not smaller than that of the interval I, and therefore, |I | = |γ0 |. If x0 is the start of the path r = r( · ) of a curve γ for which the tangent vector is different from a(r(s)) with some s, then the inequality in (5.18) is strict, and hence the length of the curve γ is greater than that of the interval I. Summarizing, we see that |θ − θ 0 | = |γ0 | = |I | is equal to the distance of x0 from the level surface. From well-known results in geometric approximation theory (see [61]) it follows that x0 is a unique nearest point, and hence the interval I is also unique and is an integral curve.

94

5 Characterization of Best Approximation . . .

It follows that in Ω ⊂ Rn , a level surface is a ‘local Chebyshev set’ (with locally continuous metric projection) and hence a ‘local sun’. We give the formal definition in terms of a regularity property. Given M ⊂ X, where X is a normed linear space, we set T M := {x | card PM x = 1} (if T M = X, then M is a Chebyshev set). A point x0 ∈ Rn \ M is called a regular point if there exists a neighbourhood O(x0 ) of x0 lying completely in T M (if X = Rn , then this implies the continuity of the single-valued metric projection operator PM onto the set O(x0 ); in the infinite-dimensional setting, this condition is set by definition). For a closed set M, the set of singular points consists, by definition, of the irregular points of Rn \ M and the points of int M. The singular set of a closed set is the set of all its singular points. The singular set is the union of the closure of the nonuniqueness set and the closure of the interior of M; moreover, the set of regular points is always open. Some questions on the structure of the singular set were addressed in, for example, [61], [58], [395]. In passing, we note that if the singular set of a C 1 -smooth hypersurface in Rn is empty, then this hypersurface is a hyperplane; if the singular set is a point, then this hypersurface is a sphere; if the singular set is a subspace (of dimension k, 1  k  n−2), then this set is a ‘spherical cylinder’ whose generators are translations of this subspace. It should be noted further that there does not exist a C 1 -hypersurface in Rn whose singular set is a hyperplane. The last result can be also derived using the machinery of the geometric approximation theory. It should also be noted that there are no C 1 -solutions to the eikonal equation on a smooth compact manifold. In this setting, regular points are points near which the distance to the set is realized on a unique geodesic curve. In this connection we mention that M. I. Karlov gave an example of a nontrivial Chebyshev set on an ellipsoid and described the singular set on which the geodesic curves switch. An account of some results in geometrical optics on manifolds may be found in the survey by Agrachev and Gauthier [1]. The following result was proved by a different method in [291, p. 266]. Theorem 5.14 The C 1 -smooth solutions to the eikonal equation (5.15) on Ω = Rn are precisely the affine functions c ± x ∗ ( · ), where |x ∗ | = 1, x ∗ ( · ) = (x ∗, · ). As a corollary, we have that the affine functions c ± x ∗ ( · ), |x ∗ | = 1, are solutions to equation (5.15) on every open subset of Ω ⊂ Rn . Let us illustrate how this result is related to solarity. For Ω = Rn , the level surfaces of the eikonal equation are Chebyshev suns. Consequently, the level surfaces are smooth as suns in a smooth space (this is proved in Chap. 6), and hence are hyperplanes. It follows easily that only functions of the form c1 ± x ∗ ( · ), |x ∗ | = 1, are C 1 -smooth solutions to the eikonal equation in Ω = Rn . For a description of all C 1 -smooth solutions to the eikonal equation for various domains, see Tsar’kov [569], [565], [566].

Chapter 6

Convexity of Chebyshev Sets and Suns

Convexity of a set is an important structural characteristic of the set. By proving the convexity of Chebyshev sets in some spaces we derive the following fact important for applications: in corresponding spaces, a nonconvex set cannot be a Chebyshev set. As a corollary, at some point either the existence or the uniqueness property is not satisfied. Results of this kind can be useful in solving extremal problems. Convexity of Chebyshev sets is usually studied in smooth spaces. A space is smooth if at each point of the unit sphere, the support hyperplane is unique. This is equivalent to saying that the norm of the space is Gâteaux differentiable (at all points except the origin). In a smooth space, the union of homothetic copies of a ball with homothety centre at its boundary forms an open half-space — this fact underlies the proof of the convexity of Chebyshev suns in such spaces. In Sect. 6.1, we discuss the problem of convexity of suns. Convexity of Chebyshev sets in Rn is considered in Sect. 6.2. In 1966, V. Klee raised the conjecture that if in a Hilbert space, there exists a nonconvex Chebyshev set, then this space also contains a Chebyshev set with convex bounded complement (a Klee cavern). This conjecture was solved in the affirmative by Asplund. Moreover, Asplund showed that the existence of a Klee cavern is equivalent to the existence of a nonconvex Chebyshev set in Hilbert space. The Klee cavern and related questions are discussed in Sect. 6.3. Johnson’s example of a nonconvex Chebyshev set in an incomplete pre-Hilbert space is constructed in Sect. 6.4.

6.1 Convexity of Suns We recall the definition of suns, strict suns, and strict protosuns. Let   M ⊂ X. A point x ∈ X \ M is called a solar point if there exists a point y ∈ PM x   (a luminosity point) such that   y ∈ PM (1 − λ)y + λx for all λ  0 (6.1) © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_6

95

96

6 Convexity of Chebyshev Sets and Suns

(geometrically, this means that there is a ‘solar’ ray emanating from y and passing through x such that y is a nearest point in M for every point of the ray). A point x ∈ X \ M is called a strict solar point if PM x   and condition (6.1) holds for every point y ∈ PM x (see Fig. 1.1 in Sect. 1.1). Further, if for every x ∈ X \ M, condition (6.1) is satisfied for every point y ∈ PM x, then a point x is called a strict protosolar point (in contrast to the case of strict solar points, a nearest point y to x may fail to exist). A closed set M ⊂ X is called a sun (respectively, a strict sun) if every point x ∈ X \ M is a solar point (a strict solar point) of M. A set M ⊂ X is called a strict protosun if every point x ∈ X \ M is a strict protosolar point (see Fig. 1.1). A set M ⊂ X is called a protosun if every x  M such that PM x   has a luminosity point. From the geometric form of the Hahn–Banach theorem, it easily follows that each convex set is a strict protosun and each convex set of existence is a strict sun. We need some more definitions. A nonempty closed set M is called an α-sun if for every point x  M, there exists a ray  emanating from x such that for all z ∈ , we have ρ(z, M) = z − x + ρ(x, M)

(6.2)

(see Definition 5.2 above). Every sun is an α-sun. Unlike suns, an α-sun can have isolated points (as an example, one may take the two-point set M := {(0, 0)}∪{(1, 1)} in the plane with the maximum norm; see Fig. 6.1).

Fig. 6.1 A discrete α-sun M := {(0, 0)} ∪ {(1, 1)} in the plane with the max-norm,

Theorem 6.1 In a smooth space, an α-sun is convex. Proof Assume to the contrary that M is a nonconvex α-sun in a smooth space X. Then there exist a, b ∈ M such that x := (a + b)/2  M. By the definition of an α-sun, there exists a ray  with vertex x such that for all z ∈ , ρ(z, M) = z − x + ρ(x, M).

6.1 Convexity of Suns

97

We continue the ray  beyond the point x to its intersection with the sphere S(x, ρ(x, M)) at the point p. The support hyperplane H to the ball V = B(x, ρ(x, M)) at p is unique, because X is a smooth space. We denote by H˚ the open subspace defined by the hyperplane H and containing ˚ The interval [p, b] contains an the point x. One of the points a, b (say b) lies in H. interior point of the ball B(x, ρ(x, M)) = B(x,  x − p) (in the contrary case, we use the Hahn–Banach theorem to prove that the space is not smooth). But then for a point z sufficiently far from p, the ball B(z, ρ(z, M)) = B(z, z − p), which is similar to the ball B(x,  x − p), with center of similarity at the point p, contains the point b, which is impossible, since b ∈ M. Theorem 6.2 The following conditions on a normed linear space X are equivalent: (a) X is smooth; (b) every α-sun in X is convex; (c) every sun in X is convex; (d) every strict sun in X is convex; (e) every P-convex strict sun in X is convex. Note that for strict protosuns, the answer to the convexity question is different. For example, one can easily construct an example of a nonconvex strict protosun (a closed cavern, which is defined as the complement of an open convex bounded set) in some smooth separable Banach space with Fréchet differentiable norm (see [24]). Proof (of Theorem 6.2) We verify only some of the implications. (a)⇒(c). Let M be a sun in a smooth space X and let u1, u2 ∈ M, 0 < α < 1. Assume to the contrary that x := αu1 + (1 − α)u2  M. Let u0 be a luminosity point of M for x. Arguing as in the proof of the Kolmogorov criterion (see Sect. 5.2), one can show that u0 cannot be characterized in terms of the Kolmogorov criterion; that is, there exist linear functionals f1, f2 ∈ X ∗ such that  fi  = 1,

fi (x − u0 ) =  x − u0 ,

fi (ui − u0 ) ≤ 0, i = 1, 2.

(6.3)

From the first and second equalities of (6.3) we have  12 ( f1 + f2 ) = 1. Next, X is smooth, and hence f1 = f2 . Using the inequality in (6.3) to estimate f1 (x − u0 ), we have f1 (x − u0 ) = f1 (αu1 + (1 − α)u2 − u0 ) = α f1 (u1 − u0 ) + (1 − α) f1 (u2 − u0 ) ≤ 0. However, this contradicts the second equality in (6.3). So x ∈ M and M is convex. (c)⇒(a). Assume that X is not smooth. To construct a nonconvex sun, consider y ∈ S and f1, f2 ∈ X ∗ , f1  f2 , such that f1 (y) = f2 (y) =  y = 1 (such a functional exists, because the space X is nonsmooth). We set Mi := {u ∈ X | fi (u) ≥ 0},

i = 1, 2,

M := M1 ∪ M2 .

Let x ∈ X \ M. Then for i = 1, 2, we have fi (x) < 0. Next, ρ(x, Mi ) = | fi (x)|, and hence by Remark 1.6 and since x − fi (x)y ∈ Mi , we have ρ(x, Mi ) = − fi (x).

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6 Convexity of Chebyshev Sets and Suns

Without loss of generality it can be assumed that c := ρ(x, M1 ) ≤ ρ(x, M2 ) and u0 = x + cy ∈ PM x. Setting xt = u0 + t(x − u0 ), t ≥ 0, we have f1 (xt ) ≥ f2 (xt ). Now ρ(xt , M) = ρ(xt , M1 ) ≤ ρ(xt , M2 ), and hence u0 is also an element of best  approximation to xt from M. So M is a nonconvex sun. Exercise 6.1 Why can a finite set that consists of more than one point never be a sun? Exercise 6.2 Characterize all finite-dimensional spaces in which the class of all suns coincides with the class of all Chebyshev sets. Exercise 6.3 Does there exist a set M that has precisely two luminosity points for only one point x in M? Exercise 6.4 Does there exist in C(Q) a nonconvex compact sun (a nonconvex Chebyshev set)? Exercise 6.5 Let K := {x ∈ X | xi∗ (x) < 1, i = 1, . . . , N }, and let M := X \ K. Is it true that M is a sun for N = 2 and that PM has connected values? Is it true that if N ≥ 3 and PM has connected values, then M is a sun? Exercise 6.6 Let dim X ≥ 2. Is the arithmetic sum of two suns in X a sun? Exercise 6.7 What can be said about a space in which the nonempty intersection of two suns is always a sun? What can be said about suns in such spaces? Exercise 6.8 In which two-dimensional spaces is the union of two overlapping nonnested balls always a sun? Exercise 6.9 In which two-dimensional spaces is the nonempty intersection of two Chebyshev sets always a Chebyshev set? Exercise 6.10 In which two-dimensional spaces is the nonempty intersection of an arbitrary sun with a Chebyshev set always a Chebyshev set? Exercise 6.11 Is the intersection of two suns always a sun? Exercise 6.12 Does there exist a sun with bounded complement? Exercise 6.13 Can a Chebyshev set on a normed plane have two (three, four, five) connected components in its complement?

6.2 Convexity of Chebyshev Sets in R n Recall that a set M is called a Chebyshev set if it is a set of existence and uniqueness; that is, if for each x ∈ X, the set PM x of best approximants to x from M is a singleton. To give a brief history of the problem of convexity of Chebyshev sets, we quote the survey by Balaganskii and Vlasov [61]: (The case R n is often associated with T. Motzkin (1935), and he himself [443] wrote about strictly convex and smooth Minkowski spaces, however his proofs do not carry over even to R3 .) The first was L. N. H. Bunt [154], who in his dissertation in 1934 considered a case slightly more general than R n . Kritikos [374] in 1938, independently of Bunt (but familiar with Motzkin’s paper), proved the convexity of Chebyshev sets in R n . In fact, Efimov and Stechkin [223] were the first (in 1958) to publish a correct, though not sufficiently complete, proof of the convexity of a compact Chebyshev set in a Hilbert space; Klee [340] in 1953 gave a proof with a mistake and F. A. Ficken’s proof of 1951 was published only in Klee’s paper [342] in 1961.

6.2 Convexity of Chebyshev Sets in R n

99

For a more detailed account of the Chebyshev sets convexity problem in abstract and concrete spaces, see [15], [549], [551], [120], [239]. In the infinite-dimensional setting, the most intriguing problem about Chebyshev sets is the following problem of Efimov–Stechkin–Klee. Problem 6.1 Prove or disprove that in an infinite-dimensional Hilbert space, every Chebyshev set is convex. We formulate and give several proofs of the theorem on convexity of Chebyshev sets in finite-dimensional Euclidean spaces. Theorem 6.3 A subset of Rn is a Chebyshev if and only if is closed and convex. That a Chebyshev set is closed is clear (a Chebyshev set is a set of existence, and a set of existence is always closed). The sufficiency in Theorem 6.3 is quite clear and was known long ago. The proof of the necessity is much more challenging. We give several different proofs of the necessity. The proof of the sufficiency and the first proof of the necessity will be given in the plane R2 for clarity. Existence (cf. Proposition 1.4). Assume that a set M ⊂ R2 is closed and convex. Consider x  M. We claim that x has precisely one element of best approximation from the set M.

Fig. 6.2 Existence of best approximation in a compact set M.  Let ξ ∈ M be an arbitrary point and  let r =  x − ξ . Consider the set M = 2 M ∩ B(x, r), where B(x, r) = {y ∈ R   x − y ≤ r } is the closed disc with centre x and of radius r. The set M  is convex and closed as the intersection of two closed convex sets (see Fig. 6.2). Since M  lies in a disc, M  is bounded. So M  is compact. The function  f (y) =  x − y = (α1 − β1 )2 + (α2 − β2 )2

is continuous (here x = (α1, α2 ) and y = (β1, β2 )). By Weierstrass’s theorem, f has a minimum on M  at  y ; that is,  x −  y  ≤  x − η for all η ∈ M . If η ∈ M \ M ,

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6 Convexity of Chebyshev Sets and Suns

then  x − η > r =  x − ξ . These two inequalities show that  y is an element of best approximation to x from M. Uniqueness. Let M be a closed convex set. Assume that some x  M has at least two best approximants y1, y2 ∈ M. Since M is convex, the point z = (y1 + y2 )/2 lies in M. In the (nondegenerate) isosceles triangle with vertices x, y1, y2 , the point z is the foot of the altitude drawn from x to [y1, y2 ]. Correspondingly, its length is less than that of the lateral side, which is  x − y1  =  x − y2 . So there is a point z ∈ M that is closer to x than y1 and y2 , but this contradicts the choice of the points y1 and y2 . Hence M is a Chebyshev set.

6.2.1 Berdyshev–Klee–Vlasov’s Proof of Convexity of Chebyshev Sets in Rn Using the Fixed-Point Theorem We shall employ the classical Brouwer fixed-point theorem.1 For further results, see [323], [599, Theorem 7.5.5], [340]. Theorem 6.4 (Brouwer’s fixed-point theorem) A continuous mapping of a closed ball of a finite-dimensional Banach space into itself has a fixed point. In other words, each continuous mapping f : B(x, r) → B(x, r) has a fixed point z0 ∈ B(x, r), that is, a point z0 such that f (z0 ) = z0 . The infinite-dimensional analogue of Brouwer’s theorem is Schauder’s theorem, which asserts that every continuous mapping of a convex subset into its precompact subset has a fixed point (see Theorem 10.2 below). We shall prove Theorem 6.3 by showing that every Chebyshev set in Rn is a sun and that each Chebyshev sun in Rn is convex. The last assertion is contained in Theorems 6.1 and 6.2 (because clearly, Rn is a smooth space). Lemma 6.1 A Chebyshev set in Rn is a Chebyshev sun. Proof (of Lemma 6.1) Let M be a Chebyshev set. It was noted above that M is nonempty and closed. By  we denote the ray emanating from y = PM x and passing through x. We set 0 < r ≤  x − y and define the mapping g : B(x, r) → B(x, r) by g(z) = gr (z) := x +

r (x − Pz) for z ∈ B(x, r)  x − Pz

(6.4)

(g(z) is the intersection of the sphere S(x, r) with the ray with endpoint at x in the direction from Pz to the point x; see Fig. 6.3). We claim that the mapping g is continuous. To this end, we first show that the metric projection PM onto M is continuous.2 Assume to the contrary that PM is 1 Berdyshev’s proof depends on a result weaker than Brouwer’s theorem. In 1960, he was awarded the gold medal for his work in a student competition. 2 In the infinite-dimensional setting, the metric projection onto a Chebyshev set (and even onto a Chebyshev subspace) can be discontinuous.

6.2 Convexity of Chebyshev Sets in R n

101

Fig. 6.3 Proof of the solarity of a Chebyshev set using the fixed-point theorem.

discontinuous at a point u. Then there exist a number ε > 0 and a sequence (un )n∈N , un → u, such that vn − v ≥ ε for all n ∈ N, where PM un = vn , PM u = v (we can always assume that u  v). Since the distance function is uniformly continuous v be its limit point. It is (Proposition 1.1), the sequence (vn )n∈N is bounded. Let  clear that  v  v. Again using the uniform continuity of the distance function, we v ∈ M; have u −  v  = limn→∞ un − vn  = ρ(u, M). Since M is closed, we have  that is, v and  v are both best approximants to u, which contradicts the fact that M is a Chebyshev set. So the mapping g is continuous as a composition of the continuous metric projection PM and the reflection about the ball’s origin. By Brouwer’s theorem, there exists a fixed point z0 ∈ B, g(z0 ) = z0 . From the definition of g it follows that x lies in the interval joining the point z0 with its nearest point PM z0 . From Proposition 1.4 and since M is a Chebyshev set, it follows that the point PM z0 is a nearest point to every point from the interval [z0, PM z0 ]. But x ∈ [z0, PM z0 ], and its nearest point is y = PM x. So PM z0 = y. Applying the above argument (which was carried out above for the point x) to the point z0 , we can move farther along the ray . As a result, we show that for every point w ∈ , the point y is a unique best approximant from M. This completes the proof of Lemma 6.1. It is easily seen that the proof of the solarity of Chebyshev sets with the help of the fixed-point theorem can be carried over to any infinite-dimensional normed linear space for boundedly compact sets.

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6 Convexity of Chebyshev Sets and Suns

6.2.2 Asplund’s Proof of the Convexity of Chebyshev Sets in Rn Using the Inversion of the Unit Sphere We give the proof of Theorem 6.3 in Rn by the inversion method. The inversion method can be also carried over to an infinite-dimensional Hilbert space for sets with continuous metric projection. Unlike the method based on the fixed point theorem, which is capable of delivering the convexity of Chebyshev sets with continuous metric projection in arbitrary smooth spaces, the inversion method works only in inner product spaces. We first recall the definitions of a farthest point and a uniquely remotal set. Let x ∈ Rn ,   M ⊂ X. A point y0 ∈ M is called a farthest point from the set M to a point x (see, for example, [599]) if  x − y0  = sup{ x − y | y ∈ M }; that is,  x − y0  ≥  x − y for every point y ∈ M. A set M is uniquely remotal (or a set with unique farthest-point property, or a max-Chebyshev set) if for each point x from X, there exists a unique farthest point from M for x. For a uniquely remotal subset M of X, we denote by F M (x) = F(x) the unique farthest point from M to x. The corresponding mapping F M : Rn → A is called the max-projection operator (or the farthest-point mapping, the metric antiprojection) onto M. It is clear that a singleton is a uniquely remotal set. The following lemma shows that in Rn there are no other sets with this property. A number of questions about existence and uniqueness of farthest points for subsets of a Banach space were considered by E. S. Polovinkin, M. V. Balashov, and G. E. Ivanov, among others [68], [303], [62], [66], [71]. Lemma 6.2 (B. Jessen) The only uniquely remotal sets in Rn are the singleton sets. Proof (of Lemma 6.2) We prove the lemma only for closed sets. Let M be a closed uniquely remotal set in X. It is clear that M is bounded. We first show that F M is continuous and then apply Brouwer’s fixed-point theorem to every ball that contains M. Assume that F M is not continuous. Then there exists a sequence (xk ) ⊂ X converging to some point x such that the sequence F M (x1 ), . . . , F M (xk ), . . . is not convergent to F M (x). Since the set M is clearly compact, there exists a subsequence (xik ) of x1, . . . , xk , . . . such that the sequence F M (xi1 ), . . . , F M (xik ), . . . converges to some point y, y  F M (x). Moreover, we have y ∈ M, since M is closed. The points F M (x) and y are farthest points from M to x. This, however, contradicts the definition of M. So the mapping F M is continuous. The restriction of F M to every (compact convex) ball B(x0, r0 ) that contains M maps the ball B(x0, r0 ) into itself. By Brouwer’s fixed-point theorem, there exists a point y0 ∈ B(x0, r0 ) such that F M (y0 ) = y0 . So y0 ∈ M. But y0 is a farthest point from M to y0 . This shows that set M is a singleton; that is, M = {y0 }.

6.2 Convexity of Chebyshev Sets in R n

103

The following result, which we give without proof, is an extension of Lemma 6.2. Lemma 6.3 (J. Blatter [106]) Let X be a reflexive space, M ⊂ X a uniquely remotal set. If the max-projection F M : x → F M (x) is continuous, then M is a singleton. Definition 6.1 The mapping σ : Rn \ {0} → Rn defined by σ(x) = x/ x 2 is called the inversion in the unit sphere. The mapping σ is clearly continuous at every point x, x  0. The next lemma describes the images of Euclidean balls and spheres under the inversion σ. Lemma 6.4 Let X = (X, ·, ·) be a Euclidean  space. Then   r x if  x  r, , (1) σ S(x, r) = S 2 2 2 2    x − r | x − r | (2) σ S(x,  x) = {y | y, x = 1/2},    x r if  x > r, (3) σ B(x, r) = B ,  x 2 − r 2  x 2 − r 2    x r (4) σ B(x, r) \ {0} = X \ B˚ if  x < r; in particular, ,  x 2 − r 2 r 2 −  x 2   x r if y  B(x, r), then σ(x) ∈ B˚ ; , 2 2 2 2  x − r r −  x   (5) σ B(x,  x) \ {0} = {y ∈ X | y, x ≥ 1/2}.   Proof (of Lemma 6.4) Let us prove (5). Let w ∈ σ B(x,  x) \ {0} . Then w = σ(z) for some z ∈ B(x,  x) \ {0}. We have z ∈ B(x,  x) \ {0} ⇔ z  0 and

z − x ≤  x

⇔ z  0 and

z − x 2 ≤  x 2

⇔ z  0 and

z 2 − 2z, x +  x 2 ≤  x 2

⇔ z  0 and z 2 ≤ 2z, x

1 z ⇔ ≤ , x = w, x 2 z 2 1   =⇒ w ∈ y ∈ X  ≤ y, x . 2   x r Let us prove (4). Let w ∈ X \ B˚ . Then w = σ(y) for some ,  x 2 − r 2 r 2 −  x 2 y ∈ X \ {0}. We have   x r w ∈ X\ B˚ ,  x 2 − r 2 r 2 −  x 2

y r −x

− ⇔ y ∈ X \ {0} and

≥ 2 2 2 2  y r −  x r −  x 2

2 ⇔ y ∈ X \ {0} and y(r −  x 2 ) + x y 2 ≥ r  y 2

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6 Convexity of Chebyshev Sets and Suns

⇔ y ∈ X \ {0}

and

2

y(r −  x 2 ) + x y 2 2 ≥ r 2  y 4

⇔ y ∈ X \ {0}

and

 y 2 (r 2 −  x 2 )2

+ 2(r 2 −  x 2 ) y 2 x, y +  x 2  y 4 ≥ r 2  y 4 ⇔ y ∈ X \ {0}

and

(r 2 −  x 2 )2 + 2(r 2 −  x 2 )x, y +  x 2  y 2 ≥ r 2  y 2 ⇔ y ∈ X \ {0}

and

(r 2 −  x 2 ) + 2x, y ≥  y 2

⇔ y ∈ X \ {0}

and

 y 2 − 2y, x +  x 2 ≤ r 2

⇔ y ∈ X \ {0}

and

 y − x 2 ≤ r 2

y ∈ B(x, r) \ {0}, whence w ∈ σ(B(x, r) \ {0}), proving (4).



Let us now prove the theorem on convexity of Chebyshev sets in Rn using inversion in spheres. Assume that there exists a nonconvex Chebyshev set M in Rn . It can be assumed without loss of generality that 0  M, 0 ∈ conv M (where conv M is the convex hull of M). The inversion σ : x → x/ x 2 (x  0) transforms the set M into the bounded set  G = σ(M) := {x/ x 2  x ∈ M }. (6.5) ˚ d) has The set G is bounded, because 0  M, and hence, for some d > 0, the ball B(0, ˚ no common points with M, i.e., M ⊂ X \ B(0, d). The mapping (( · )) acts identically on X \ {0}, and hence by assertion (4) of Lemma 6.4 we get G ⊂ B(0, d), the result required. Next, if a ball B(x, r) contains G, then it also contains the origin in its interior. Indeed, the case 0 ∈ S(x,  x) is impossible by assertion (5) of the lemma, because by the hypothesis 0 ∈ conv M. Now, if 0  B(x, r) ⊃ G, then  x > r, and hence by assertion (3) of the lemma, we have   r x ⊃ M, , 0B  x 2 − r 2  x 2 − r 2 which is impossible, since 0 ∈ conv M by the assumption. For each point x ∈ Rn , there exists a smallest ball B(x, t(x)) that contains G and ˚ t(x)), and such that G  B(x, r) for all r < t(x). Since by the above, we have 0 ∈ B(x, so t(x) >  x, it follows by assertion (4) of the lemma that the inversion transforms the ball B(x, t(x)) into Rn \ V, where   x t(x) ˚ , . V=B  x 2 − t 2 (x) t 2 (x) −  x 2 Since G is clearly closed, the open ball V is the largest ball with centre at x( x 2 − t 2 (x))−1 such that its complement contains M. Since M is a Chebyshev set, such a ball intersects M in a unique point PM (x/( x 2 − t 2 (x))). So the function

6.2 Convexity of Chebyshev Sets in R n

x → q(x) =

105

P(x/( x 2 − t 2 (x))) , P(x/( x 2 − t 2 (x))) 2

q : Rn → G,

(6.6)

is such that  x − y <  x − q(x),

if y ∈ G and y  q(x).

Thus, G is a uniquely remotal set. By Jessen’s lemma (Lemma 6.2), G, and therefore M, is a singleton. In this case, our hypothesis 0  M and 0 ∈ conv M cannot hold. This shows that M is convex. The theorem on convexity of Chebyshev sets in Rn is proved. Asplund’s proof of convexity of Chebyshev sets in Rn using the inversion in the unit sphere can be carried over to an arbitrary Hilbert space for sets with continuous metric projection.

6.2.3 Konyagin’s Proof of Convexity of Chebyshev Sets in Rn Using the Decomposition Lemma The origin of the decomposition lemma can be traced back to the works of P. L. Chebyshev; more explicit constructions date back to E. Ya. Remez (namely, algorithms for construction of polynomials of near-best uniform approximation). Decomposition-type results (‘removal of unnecessary matter’) are based on the remarkable Helly’s theorem from finite-dimensional convex geometry, which implies the finite-dimensional character of such theorems. Various variants of the decomposition lemma can be found, for example, in the books by Tikhomirov and MagarilIl’yaev [544], Polovinkin and Balashov [481, Sect. 1.17]. Lemma 6.5 (Decomposition lemma) Let T be a compact set in Rn and let g : T × Rn → R be a function that is convex with respect to x and continuous with respect to (t, x). We set ϕ(x) = max g(t, x). t ∈T

Assume that  x is a point of minimum of the function ϕ. Then there exist positive x ), numbers α1, . . . , αr , α1 +· · ·+αr = 1, r ≤ n+1 and points y1, . . . , yr , yi ∈ ∂g(τi, ·)( where τi ∈ T0 ( x ) = {t ∈ T | g(t,  x ) = ϕ( x )},

∂g(τi, ·)( x ) := {y ∈ Rn | g(τi, x) − g(τi,  x ) ≥ x −  x, y, n i = 1, . . . , r, ∀x ∈ R }, such that 0=

r  i=1

In Lemma 6.5, the set

αi yi .

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6 Convexity of Chebyshev Sets and Suns

 ∂ f ( x ) = z ∈ Rn | f (x) − f ( x ) ≥ z, x −  x

 ∀ x ∈ Rn ,

(6.7)

as defined from a convex function f : Rn → R x ∈ Rn , is called nand a point  zi (xi −  xi ), z = (z1, . . . , zn ), the subdifferential of f at  x . Here z, x −  x  = i=1 x = ( x1, . . . ,  xn ). x = (x1, . . . , xn ), and  The decomposition lemma will be proved in Sect. B.2. Konyagin’s proof of the convexity of Chebyshev sets in Rn using the decomposition lemma is as follows. Assume that there exists a nonconvex Chebyshev  set M in Rn . Let G = {x/ x 2  x ∈ M } be the image of M under the inversion x → x/ x 2 . The set G ⊂ Rn is bounded and closed, and hence is compact. Above, in the proof of the convexity of Chebyshev sets in Rn by the inversion methods, we showed that G is a uniquely remotal set. We apply the decomposition lemma to the set T = G and the function g(t, x) = t − x, t ∈ G. Let  x be a point of minimum of the function ϕ and let numbers αi , points τi , yi (i = 1, . . . , n), and the set T0 be as in the conclusion of the lemma. x , which Let r > 1. By definition of T0 , τ1, . . . , τr are farthest points from G to  contradicts the fact that G is a uniquely remotal set. Assume now that r = 1. Then T0 = {τ1 }, and by the decomposition lemma, x ). 0 = α1 y1 . Since α1 = 1, we have y1 = 0. On the other hand, y1 ∈ ∂g(τ1, ·)( x  ≥ 0 for all x ∈ Rn , which gives This by definition means that τ1 − x − τ1 −  x  = 0; that is, τ1 =  x . But τ1 is a farthest point from G to  x . By Jessen’s τ1 −  lemma (Lemma 6.2), the set G (and therefore, the set M) is a singleton. However, such a set cannot be nonconvex. This contradiction completes the proof.

6.2.4 Vlasov’s Proof of Convexity of Chebyshev Sets in Rn Vlasov’s proof of the convexity of Chebyshev sets in Rn is based on the following result (see [590]) involving the δ-suns machinery. Theorem 6.5 (L. P. Vlasov) Let M be a Chebyshev set with continuous metric projection in a normed linear space X. Then M is a δ-sun; that is, ρ(xn, M) − ρ(x, M) −→ 1  xn − x

(6.8)

for every point x  M and some sequence (xn )n∈N such that n ∈ N, xn → x. Definition 6.2 A nonempty closed set M is called a δ-sun if for every x  M, there exists a sequence (xn ), xn  x, xn → x, satisfying condition (6.8). Proof (of Theorem 6.5) Let x ∈ Rn \ M, y = PM x, and let a sequence (xn )n∈N be such that x ∈ (xn, y] and xn → x. We set yn = PM xn . If y = yn for some n ∈ N, then for each point ξ ∈ [yn, x], we have PM ξ = y, and hence the fraction in (6.8) is equal to 1.

6.2 Convexity of Chebyshev Sets in R n

107

Assume now that y  yn for all n. In the two-dimensional plane spanned by the points x, xn, y, yn , consider the point z that is the intersection of the line xn z parallel to [x, yn ] with the line yyn . Since the triangles xyyn and xn yz are similar, we have  yn − y  xn − y  x − y z − yn  = and = , (6.9)  xn − x  x − y  xn − z  x − yn  and since PM x = {y} and PM xn = {yn }, we find that  x − y ≤  x − yn ,

 xn − yn  ≤  xn − y.

(6.10)

Next, using (6.9) and (6.10), we get  xn − y ≤  xn − z.

(6.11)

Applying the equality  xn − y =  xn − x +  x − y, inequality (6.10), (6.11), the inequality  xn − z ≤  xn − yn  +  yn − z, and inequality (6.9), yields ρ(xn, M) − ρ(x, M)  xn − x  xn − y −  xn − yn   xn − yn  −  x − y = =1−  xn − x  xn − x (6.11)  xn − z −  xn − yn  ≤  xn − x  yn − y  yn − z (6.9)  yn − y = −→ 0, ≤ =  xn − x  x − y ρ(x, M) (6.10)

0 ≤ 1−

on letting yn → y. Theorem 6.5 holds in an arbitrary normed linear space for approximatively compact Chebyshev sets (by Theorem 5.10, the metric projection on an approximatively compact Chebyshev set is continuous). Proof of the convexity of Chebyshev sets in Rn by Vlasov’s lemma. Let M be a Chebyshev set. By Theorem 6.5, (6.8) is satisfied for M. Let x0 ∈ Rn \ M, δ > 0, R = ρ(x0, M). Given ξ ∈ Rn , we define f (ξ) = (1 − 2δ)ξ − x0  − ρ(ξ, M). The continuous function f attains its minimum value on the compact set B(x0, R) at some point x. We claim that x ∈ S(x0, R). ˚ R) Assume to the contrary that  x− x0  < R. By (6.8), there exists a point y ∈ B(x, such that ρ(y, M) − ρ(x, M) > (1 − δ) x − y. Hence f (y) = (1 − 2δ) y − x0  − ρ(y, M) < (1 − 2δ)( y − x +  x − x0 ) − ρ(x, M) − (1 − δ) x − y = (1 − 2δ) x0 − x − ρ(x, M) − δ x − y = f (x) − δ x − y. But this inequality contradicts the fact that f (x) ≤ f (y). So the assumption that  x − x0  < R was false, and hence  x0 − x = R.

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6 Convexity of Chebyshev Sets and Suns

Since f (x) ≤ f (x0 ), we have ρ(x, M) ≥ ρ(x0, M) + (1 − 2δ)R. Assume that δn → 0. The function fn (ξ) = (1 − 2δn )ξ − x0  − ρ(ξ, M) attains its minimum value on the ball B(x0, R) at some point xn . By the above,  x0 − xn  = R (n = 1, 2, . . . ), and hence ρ(xn, M) ≥ ρ(x0, M) + (1 − 2δn )R.

(6.12)

We set y0 = PM x0,

s=

x0 − y0 x0 − y0 = ,  x0 − y0  R

sn =

xn − x0 xn − x0 = .  xn − x0  R

Since  xn − y0  ≥  xn − yn , from (6.12) we have  xn − y0  ≥  x0 − y0  + R − 2Rδn , which gives s + sn  ≥ 2 − 2δn . The parallelogram identity for s, sn reads as s + sn  2 + s − sn  2 = 2(s 2 + sn  2 ) = 4.

(6.13)

We have δn → 0, and hence by (6.13) and the previous inequality, s − sn  → 0 as x = 2x0 − y0 as n → ∞ (the point  x is opposite n → ∞. This implies that xn →  to y0 on the sphere S(x0, R)). Letting n → ∞ in (6.12), we get ρ( x, M) ≥ ρ(x0, M) + R.

(6.14)

On the other hand, ρ( x, M) ≤  x − y0  ≤  x − x0  + ρ(x0, M) = R + ρ(x0, M).

(6.15)

Using (6.14) and (6.15), we obtain ρ( x, M) =  x − y0  = 2R, that is, ρ( x, M) = 2R, whence PM  x = y0 . x from the ray  (that So, for the point x0  M, we have shown that the point  emanates from y0 = PM x0 and passes through x0 ) has y0 as a best approximant x ]. As a corollary, every other point in the in M. Recall that x0 is a midpoint of [y0,  x ] has y0 as a nearest element in M. Repeating for  x the above argument interval [y0,  (which was given above for x0 ), we proceed farther along the ray . Repeating this process further, we prove that y0 is a nearest point from M to every point w of the ray . Vlasov’s proof of convexity of Chebyshev sets in Rn can be carried over to Chebyshev sets with continuous metric projection in arbitrary Banach spaces with strictly convex dual.

6.2 Convexity of Chebyshev Sets in R n

109

6.2.5 Brosowski’s Proof of Convexity of Chebyshev Sets in Rn Let   M ⊂ X. For 0 < r ≤  x − y, consider the (set-valued) mapping g(z) = gr (z) := x +

r (x − PM z) for z ∈ B(x, r)  x − PM z

(this is the mapping (6.4) from Sect. 6.2.1). It turns out that if the (set-valued) mapping PM is locally Lipschitz, then there exists a number r > 0 such that the mapping gr is a contraction. To formulate the appropriate fixed-point theorem, we require the definition of the Hausdorff distance between sets A and B: h(A, B) := max{d(A, B), d(B, A)}, where d(A, B) := sup{ρ(a, B) | a ∈ A}. The following result is well known (see, for example, [279, Sect. 21]). Its proof follows the same approach as that of the Banach contraction theorem for (singlevalued) contractions. Theorem 6.6 (Contraction principle for set-valued contractions) Let ϕ : K → 2K be a contraction on a complete metric space (K, d); that is, there exists 0 ≤ c < 1 such that h(ϕ(x), ϕ(y)) ≤ c · d(x, y). Then the mapping ϕ has a fixed point. Definition 6.3 A set M ⊂ X is called a metasun [592] if for every x  M, there exist z ∈ X and y ∈ PM z such that x ∈ [y, z). Theorem 6.7 (B. Brosowski) Let X be Banach space, M ⊂ X a set of existence with locally Lipschitz metric projection; that is, for all x ∈ X, h(PM y, PM z) ≤ Cx  y − z for all y, z ∈ B(x, rx ) with some constants Cx, rx > 0.

(6.16)

Then M is a metasun. In addition, if M is a Chebyshev set or X is strictly convex, then M is a Chebyshev sun. From Theorem 6.7 it follows that in a smooth space X, an arbitrary Chebyshev set with locally Lipschitz metric projection is convex. As a corollary, we get the following extension of Phelps’s criterion (Theorem 3.2) for convexity of Chebyshev sets. Theorem 6.8 Let M be a Chebyshev set in a Hilbert space. Then the following conditions are equivalent: (a) M is convex; (b) the metric projection is 1-Lipschitz; (c) the metric projection is locally Lipschitz.

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6 Convexity of Chebyshev Sets and Suns

Proof (of Theorem 6.7) Let x ∈ M. Without loss of generality we assume that x = 0, ρ(x, M) = 1. Let r be such that 0 < r < rx ,

r(2r + 1)−1 Cx ≤ 1/2,

where the constants Cx, rx are defined in (6.16). Consider the auxiliary mapping ϕ(z) := −

r PM z. 2r + 1

Let z ∈ B(0, r), y ∈ PM z. Then  y ≤ z + z − y = z + ρ(z, M) ≤ z + z + ρ(0, M) ≤ 2r + 1. Hence ϕ : B(0, r) → B(0, r). Next, for all y, z ∈ B(0, r), we have r h(PM y, PM z) 2r + 1 r 1 Cx ·  y − z ≤  y − z. ≤ 2r + 1 2

h(ϕ(y), ϕ(z)) =

By the contraction principle for set-valued contractions (Theorem 6.6), the mapping ϕ has a fixed point z0 ; that is, z0 ∈ ϕ(z0 ). Then z0 = −r(2r + 1)−1 y0 for some y0 ∈ PM z0 . It follows that 0 ∈ [y0, z0 ], z0  0; that is, M is a metasun. To complete the proof it remains to observe that a Chebyshev metasun is a sun. Exercise 6.14 Prove the convexity of a Chebyshev set in R2 without recourse to a fixed-point theorem.

6.3 The Klee Cavern In 1966, Klee [343] raised the conjecture that if in a Hilbert space there exists a nonconvex Chebyshev set, then this space also contains a Chebyshev set with convex bounded complement. This conjecture was solved in the affirmative by Asplund [47], who proposed to call such a set a Klee cavern. Moreover, Asplund showed that the existence of a Klee cavern is equivalent to the existence of a nonconvex Chebyshev set in Hilbert space. Later, C. Franchetti and I. Singer put forward duality theorems for approximation by caverns and characterized best approximants by such sets. In the space  2 , Klee [342] constructed a cavern that is a set of uniqueness (he called such a cavern a semi-Chebyshev cavern) but that is not an existence set. Vlasov [590] proved, in particular, that in a Banach space, a Chebyshev set with continuous metric projection cannot be the complement of a bounded set. Recall Asplund’s construction of a Klee cavern in an infinite-dimensional Euclidean space X. Let M be a nonconvex Chebyshev set in X, 0 ∈ conv M.

6.3 The Klee Cavern

111

As in Sect. 6.2.2, let G ⊂ X be the image of M under the inversion σ( · ) in the unit sphere. Since M is closed and 0 ∈ conv M, the set G is bounded. For each point x, there is a smallest ball B(x, t(x)) ⊃ G such that G  B(x, r) for all r < t(x). Every ball B(x, r) ⊃ G must contain the origin in its interior, whence  x < r (see Sect. 6.2.2). ˚ t(x)), we have t(x) >  x, and hence by assertion 4) Since by the above 0 ∈ B(x, of Lemma 6.4, the inversion maps the ball B(x, t(x)) onto the complement X \ V of the open ball   x t(x) ˚ , , (6.17) V=B  x 2 − t 2 (x) t 2 (x) −  x 2 which is the largest ball with centre in x/( x 2 − t 2 (x)) such that the closure of its complement contains M. Since M is a Chebyshev set, such a ball intersects M in a unique point q(x) := PM (x/( x 2 − t 2 (x))). So the function x → q(x) =

P(x/( x 2 − t 2 (x))) , P(x/( x 2 − t 2 (x)))

q : X → G,

(6.18)

is such that  x − y <  x − q(x),

if y ∈ G and y  q(x).

Let y be a point of minimum of the function t( · ) on X. That this point exists and is unique for y follows from simple two-dimensional geometric considerations. In other words, y is a Chebyshev centre of the bounded set G.) Consider the set   C := x ∈ X | t(x) ≥ t(y) + 1 . We claim that such a set C is the required Klee cavern. We first show that t( · ) is a convex function. Indeed, given x, z ∈ X, we have ˚ t(x)), q((x + z)/2) ∈ B(x,

˚ t(z)). q((x + z)/2) ∈ B(z,

(6.19)

Applying the parallelogram identity to the points x, z, q((x + z)/2), (x + z)/2 − q((x + z)/2), we find that  2  x − z 2 + 2t((x + z)/2)) = 2 x − q((x + z)/2) 2 + 2z − q((x + z)/2) 2, which by (6.19) gives  4t

x+z 2



2 ≤4

2 t(x) + t(z) , 2

proving the convexity of the function t( · ). ˚ t(x)), and hence  x ≤ t(x). Next, let d be such that G ⊂ B(0, d). We have 0 ∈ B(x, Further, t(x) =  x − q(x) ≤  x + q(x) ≤  x + d, and so t(x) ≤  x + d, which gives  x ≤ t(x) ≤  x + d ∀x ∈ H.

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6 Convexity of Chebyshev Sets and Suns

So X \ C is a nonempty bounded convex open set. Let x ∈ X \ C. Consider the function   (t(y) + 1)x− t(y) + 1 − t(x) q(x) x → b(x) := t(x) Since b(x) − q(x) = t(y) + 1 and since  x − q(x) = t(x) by the construction, we have     B b(x), t(y) + 1 ⊃ B x, t(x) ⊃ G, whence t(b(x)) = t(y) + 1; that is, b(x) ∈ C. Next, we have b(x) − x =

t(y)(x − q(x)) + (x − q(x)) + t(x)(q(x) − x) t(x)

(6.20)

and  x − q(x) = t(x), and therefore  x − b(x) = t(y) + 1 − t(x). Assume that in C there is a point z such that  x − z ≤ t(y) + 1 − t(x). Then t(z) = z − q(z) ≤ z − x +  x − q(z) ≤ t(y) + 1,

(6.21)

which together with the definition of C shows that all inequalities in (6.21) become equalities. This is possible only when q(z) = q(x) and the vector z − x is equal to x − q(x) multiplied by a positive constant (see (6.20)); that is, z = b(x). So  x − z >  x − b(x) if x ∈ X \ C, z ∈ C, z  b(x). Finally, we have b(x) = x for x ∈ C, thereby showing that C is a Chebyshev set (a Chebyshev Klee cavern). Exercise 6.15 Is it possible to construct a Chebyshev cavern in a finite-dimensional space? (A cavern is the complement of an open convex bounded set.)

6.4 Johnson’s Example of a Nonconvex Chebyshev Set in an Incomplete Pre-Hilbert Space In 1987, Johnson published an example of a Chebyshev set with bounded convex complement in the incomplete pre-Hilbert space 02 of  2 -sequences with finite support. In [61], Balaganskii and Vlasov gave a very similar example, but it was based on geometrical considerations rather than on calculations. A detailed exposition of the construction of a nonconvex Chebyshev set in 02 can also be found in Fletcher and Moors [239].

6.4

Johnson’s Example of a Nonconvex Chebyshev Set . . .

113

Johnson’s example depends substantially on the incompleteness and structural properties of the space 02 , so it cannot be carried over to the Hilbert space setting. Recall Johnson’s construction (see, for example, [312], [239]). We set En = {(x1, x2, . . . , xn, 0, 0, . . . )} | xi ∈ R, 1 ≤ i ≤ n}, ∞  E= En . n=1

Given x = (x1, x2, . . . , xn, 0, 0, . . . ) ∈ En , as the norm of x we consider the standard  2 -norm. So E = 02 is the linear manifold in  2 consisting of finitely supported sequences. Next, we set a0 = 2, A1 = 1, F0 = 1, L0 = 1, d1 = {(x1, 0, 0, . . . ) | −F0 ≤ x1 ≤ a0 F0 }, L1 (x1 ) = a0 F02 + (a0 − 1)F0 x1 − x12,

(x1, 0, 0, . . . ) ∈ d1,

= 2L1 (x1 )/[a0 + 1], S1 = {(x1, −F(x1 ), 0, 0, . . . ) | (x1, 0, 0, . . . ) ∈ d1 }. F12 (x1 )

Define by induction an = 1 + An Ln,

for some An > 0,

dn+1 = {(x1, x2, . . . , xn+1, 0, 0, . . . ) | −Fn ≤ xn+1 ≤ an Fn }, 2 Ln+1 = an Fn2 + (an − 1)Fn xn+1 − xn+1 ,

(x1, x2, . . . , xn+1 0, 0, . . . ) ∈ dn+1,

Sn+1

2 Fn+1 (x1 ) = 2Ln+1 /[an + 1], = {(x1, x2, . . . , xn+1, −F(xn+1 ), 0, 0, . . . ) |

(x1, x2, . . . , xn+1 0, 0, . . . ) ∈ dn+1 }. Hence M=

∞ 

Sn

(6.22)

n=1

is a nonconvex Chebyshev set in 02 . Remark 6.1 Johnson [313] showed that the  2 -closure of his nonconvex Chebyshev set M from 02 (see (6.22)) is not a Chebyshev set in the space  2 . We also note that there is still no answer to the following less famous problem on Chebyshev sets in Hilbert spaces. In the 1990s, D. Kölzow posed the following problem. Namely, Johnson’s example of a nonconvex Chebyshev set was constructed in a concrete incomplete pre-Hilbert space 02 . According to the well-known Kadec’s theorem (see, for example, [229, Theorem 10.35]), all separable infinite-dimensional Banach spaces are homeomorphic (and by the Banach–Mazur theorem, the separable

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6 Convexity of Chebyshev Sets and Suns

infinite-dimensional Hilbert spaces are isometrically isomorphic). This is not so for incomplete spaces. For example, Shkarin [511] constructed an example of an infinitedimensional separable incomplete pre-Hilbert space that is not homeomorphic to its closed hyperplane. Kölzow’s problem is stated as follows. Problem 6.2 Does there exist a nonconvex Chebyshev set in every infinite-dimensional incomplete pre-Hilbert space? (Johnson constructed his example of a nonconvex Chebyshev set in the particular space 02 .)

Chapter 7

Connectedness and Approximative Properties of Sets. Stability of the Metric Projection and Its Relation to Other Approximative Properties

In this chapter, we study various structural connectedness-type properties of approximating sets (among which we consider Chebyshev sets, suns, moons, uniqueness sets, and so on). By structural characteristics of sets one usually understands properties of linearity, finite-dimensionality, convexity, connectedness of various kinds, and smoothness of sets. From results of such kind one may derive necessary and sufficient conditions for a set to have certain important approximative properties such solarity and lunarity. We shall give direct theorems of geometric approximation theory in which approximative properties of sets are derived from their structural characteristics and put forward converse theorems in which from approximative characteristics of sets one derives their structural properties. As approximative properties, we shall consider the properties of uniqueness and existence of an element of best approximation, the Chebyshev property, approximative compactness, solarity, and the stability of the operators of best and near-best approximation. In terms of applications, converse theorems play the following role. Once it is found that an object under study does not have ‘good’ structural properties, converse theorems might be employed to infer that it also fails to have ‘good’ approximative properties. In this way, one usually may succeed in showing that a certain object also fails to have the approximative properties under consideration (see, for example, [558], [559], [560], [562]). Classes of connectedness of sets are introduced and studied in Sect. 7.1. The problem of connectedness of suns is considered in Sect. 7.2. The main result here is that a sun in every finite-dimensional normed linear space X is path-connected and locally path-connected. In Sect. 7.4, we present Klee’s example of a discrete Chebyshev set. Koshcheev’s example of a disconnected sun (in infinite-dimensional space) is constructed in Sect. 7.5. Several important properties of radial continuity of the metric projection are introduced and studied in Sect. 7.6. It is shown, in particular, that an approximatively compact Chebyshev sun is B-connected. Important definitions of spans, segments, Menger-connected sets, and monotone path-connected

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_7

115

116

7

Connectedness and Approximative Properties of Sets . . .

sets are given in Sect. 7.7. Problems of continuous and semicontinuous selections of the metric projection and their relations to solarity and proximinality of sets are examined in Sect. 7.8. In particular, we prove here that the metric projection onto a strict protosun is ORL-continuous. In the so-called MS-spaces, it is shown that a set M is a strict protosun if and only if it has ORL-continuous metric projection (or equivalently, M is a unimodal set or a moon). In this section, we also introduce and study the important property of Brosowski–Wegmann-connectedness. Solarity of sets of generalized rational fractions is studied in Sect. 7.10. Approximative properties of sets lying in a subspace are examined Sect. 7.11. We finish this chapter by formulating the available results on approximation by products (see Sect. 7.12).

7.1 Classes of Connectedness of Sets We consider the following classes of sets M ⊂ X: ˚ is the class of B-connected ˚ ˚ r) is ( B) sets (that is, the sets M such that M ∩ B(x, connected for all x ∈ X, r > 0); (B) is the class all of B-connected sets (that is, the sets M such that M ∩ B(x, r) is connected for all x ∈ X, r > 0); (P) is the class of all P-connected sets of existence (that is, for any M ∈ (P) the set PM x is nonempty and connected); (P0 ) is the class of P0 -connected sets (that is, for any M ∈ (P0 ) the set PM x is connected). (AC) is the class of approximatively compact sets (see Definition 4.3). (E ) is the class of proximinal sets (sets of existence). ˚ B-connected sets were first introduced by Wulbert [604] (under the name ‘boundedly connected sets’). It is easily shown that (see [592, Proposition 0.5]) ˚ (B) ⊂ ( B).

(7.1)

Indeed, for any r > 0 ˚ r) ∩ M = B(x,

n  r ∩ M, n+1 n=1 ∞  ˚ n) ∩ M. M= B(0,

∞   B x,

(7.2)

n=1

˚ The first equality in (7.2) shows that M is B-connected, because any union of connected sets with a common point is a connected set (see, for example, [380, p. 141]). The last of (7.2) implies that the set M is connected, because if each pair of points of a set lies in some connected set, then the set is also connected (see also [380, p. 141]). A partial conversion of inclusion (7.1) is given in the next theorem (Koshcheev [362, Theorem 8]).

7.1 Classes of Connectedness of Sets

117

˚ Theorem 7.1 In a normed linear space, an approximatively compact B-connected set is B-connected; that is, ˚ ⊂ (B). (AC) ∩ ( B) Proof (of Theorem 7.1) Assume the contrary; that is, M ∩B(x, r) = A∪C, where x ∈ X, r > 0, A, C are nonempty closed disjoint sets. The case ρ(x, A) < r, ρ(x, C) < r is ˚ impossible, since M is B-connected. Let ρ(x, A) = ρ(x, C) = r. By compactness of A and C, there exists b > 0 such that the nonempty sets Ab := {x ∈ X | ρ(x, A) ≤ b}, Cb := {x ∈ X | ρ(x, C) ≤ b} are disjoint. Besides, for some δ > 0, ˚ ρ(x, M) + δ) ∩ M ⊂ (Ab ∪ Cb ), B(x, ˚ which contradicts the B-connectedness of the set M. The case ρ(x, A) < ρ(x, C) = r is easily reduced to the one just considered. Indeed if ρ(x, C) > ρ(x, A), then since the distance function ρ( · , M) is continuous (Proposition 1.1), there exists a point x0 ∈ (x, xb ), where xb ∈ C, such that ρ(x0, A) =  ρ(x0, C). Theorem 7.1 is proved. We recall the following definitions. Definition 7.1 A mapping F : X → 2Y is called lower semicontinuous at a point x0 if for any neighbourhood O(y) of any point y ∈ F(x0 ), there exists a neighbourhood O(x0 ) such that F(x) ∩ O(y)  ∅ for each x ∈ O(x0 ). As usual, a mapping F is lower semicontinuous if it is lower semicontinuous at any point x0 ∈ X. Definition 7.1 can be equivalently stated as follows: (1) a mapping F is lower semicontinuous if, for any open set U ⊂ Y , its inverse image F −1 (U) := {x ∈ X | F(x) ∩ U  ∅} is open; (2) a mapping F is lower semicontinuous if, for any x0 ∈ X, xn → x0 and any y ∈ F(x0 ), there exist yn ∈ F(xn ), n ∈ N, such that yn → y as n → ∞. Definition 7.2 A mapping F : X → 2Y is called upper semicontinuous at a point x0 if, for any open set V ⊂ Y such that F(x0 ) ⊂ V, there exists a neighbourhood O(x0 ) such that F(x) ⊂ V. A mapping F is upper semicontinuous if it is upper semicontinuous at any point x0 ∈ X. Definition 7.2 can be equivalently stated as follows: (1) a mapping F is upper semicontinuous if, for any closed set U ⊂ Y , its inverse image F −1 (U) := {x ∈ X | F(x) ∩ U  ∅} is closed; (2) a mapping F is upper semicontinuous if, for any x0 ∈ X, xn → x0 and any open set V ⊂ Y such that F(x0 ) ⊂ V, there exists a number n0 , depending on V, such that F(xn ) ⊂ V for all n ≥ n0 . Remark 7.1 The metric projection onto a boundedly compact set is upper semicontinuous (in particular, the metric projection onto a closed subset of a finite-dimensional space is upper semicontinuous). Theorem 7.2 (L. P. Vlasov) In a normed linear space, a P-connected set with upper ˚ semicontinuous metric projection is B-connected.

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Connectedness and Approximative Properties of Sets . . .

˚ Proof (of Theorem 7.2) Assume the contraty: M is not B-connected; that is, there ˚ r) such that B˚ ∩ M = A ∪ C, where A, C are nonempty exists an open ball B˚ = B(x, ˚ and for definiteness we open (in M) disjoint sets. Hence ρ(x, M) < r, PM x ⊂ B, assume that PM x ⊂ A. Let v ∈ C, z ∈ [x, v], z  ∈ PM z. Then x − z  ≤ x − z + z − z  ≤ x − z + z − v = x − v < r ˚ PM z ⊂ A ∪ C for any z ∈ [x, v]. The restriction of PM to the that is, PM z ⊂ B, interval [x, v] is upper semicontinuous, and hence the nonempty disjoint sets A := {z ∈ [x, v] | PM z ⊂ A},

B  := {z ∈ [x, v] | PM z ⊂ C}

are open in [x, v]. However, [x, v] = A ∪ C , since M is P-connected, contradicting the connectedness of the interval [x, v].  The conditions on the set in Theorem 7.2 can be weakened at the expense of a strong restriction on the space. Theorem 7.3 (L. P. Vlasov [592]) In a uniformly convex Banach space, a Pconnected set is B-connected. In particular, in a Hilbert space any Chebyshev set is B-connected (and a fortiori connected). Below we shall show that an approximatively compact Chebyshev sun is Bconnected (Theorem 7.15). The following result is due to Nevesenko [445] and Koshcheev [364]. Theorem 7.4 In a normed linear space X, a P-connected set is B-connected if at least one of the following conditions is satisfied: (a) PM is upper semicontinuous; (b) PM is lower semicontinuous. Nevesenko [446] also obtained different conditions for a P-connected set to be B-connected. We also note the following result (Tsar’kov [550]), in which the condition in Theorem 7.4 on a set is relaxed at the expense of a restriction on the space (for the definition of Efimov–Stechkin spaces, see Sect. 9.1 below). ˚ Theorem 7.5 In a Efimov–Stechkin space, a closed P0 -connected set M is B-con˚ nected; that is, (P0 ) ⊂ ( B). For the metrizable asymmetrically normed uniformly convex spaces, Theorem 7.5 was partially extended by Borodin [117]: in a metrizable asymmetrically normed uniformly convex space a P-connected set is B-connected. Sosov [527] extended Theorem 7.4 to uniformly convex geodesic spaces. Theorem 7.5 does not hold in the general case: in an arbitrary Banach space a Chebyshev set need not be connected — Dunham’s well-known example of a disconnected Chebyshev set in C[0, 1] (see Sect. 7.3) shows that (P)  (B) ∩ (E ).

7.2 Connectedness of Suns

119

˚ 1) B-connected, ˚ Exercise 7.1 Is the set X \ B(0, B-connected? ˚ Exercise 7.2 Is the closure of a B-connected set B-connected? What about the closure of a Bconnected set? Exercise 7.3 Let M ⊂ C[0, 1] be the set consisting of all functions that have no zeros. Is this set ˚ B-connected (B-connected)? Exercise 7.4 Let M ⊂ C[0, 2] be the set consisting of all functions with finite number of zeros on ˚ [1, 2]. Is this set B-connected (B-connected)? ˚ Exercise 7.5 Is it true that the arithmetic sum of a B-connected (B-connected) set and a convex set ˚ is B-connected (B-connected)? Exercise 7.6 Does there exist a nonconvex compact B-connected set in  2 ? Exercise 7.7 Does there exist a B-connected nonconvex polygonal line in  2 ? Consider the same question in C[0, 1]. Exercise 7.8 Show that the sphere S is a B-connected set. Exercise 7.9 In which cases the union of two balls in  2 is a B-connected set? Exercise 7.10 Let M ⊂  2 be the union of the unit ball B(0, 1) and and arbitrary compact set ˚ K  B(0, 1). Show that M is not B-connected (not B-connected).

7.2 Connectedness of Suns On a normed plane (in particular, on an asymmetrically normed plane) any sun is B-contractible (that is, its intersection with any ball is contractible), and hence, is B-connected (H. Berens, L. Hetzelt, A. R. Alimov). In the general finite-dimensional case, the first result on the connectedness of suns was obtained by Koshcheev [362] in 1975. Theorem 7.6 In a finite-dimensional normed linear space, any sun is connected In an attempt to solve the problem of B-connectedness of suns in normed spaces of finite dimension ≥ 3, Berens and Hetzelt gave the first nontrivial example of a nonsmooth space of arbitrary finite dimension ≥ 3 (namely, X = n∞ ) in which any sun is B-connected (and even B-contractible). The best general result on the connectedness of suns in arbitrary finite-dimensional spaces is due to Brown [146]. Theorem 7.7 If M is a sun in finite-dimensional normed linear space X, then it is path-connected and locally path-connected. Moreover, there exist positive constants L and α, depending only on X, such that, for any distinct points x, y ∈ M, there exists a path s : [0, 1] → M joining x and y such that s(ξ) − s(η)  L x − y · |ξ − η| α for all ξ, η ∈ [0, 1].

120

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Connectedness and Approximative Properties of Sets . . .

In view of Theorem 7.34 below, Theorem 7.7 has the following corollary for suns contained in a finite-dimensional subspace of a normed linear space. Corollary 7.1 Let X be a normed linear space and let M be a sun in X contained in a finite-dimensional subspace. Then M is path-connected and locally path-connected. In the problem of connectedness of strict suns, we mention the following results (see, respectively, [24], [571], and [19]). Theorem 7.8 A boundedly compact strict sun in a normed space is B-connected (B-path-connected, if X is a Banach space). Theorem 7.9 In a finite-dimensional asymmetrically normed space, a strict sun is B-connected. Theorem 7.10 Let M be a strict sun in a finite-dimensional Banach space X, ˚ dim X ≤ 3. Then M is P-contractible, P-solar, B-acyclic, B-infinitely connected, ˚ ˚ B-contractible, is a B-retract, and admits a continuous additive (multiplicative) ε-selection for any ε > 0. Much stronger results on the connectedness of suns can be obtained in the socalled (BM)-spaces introduced by Brown [145]. It turned out that many results on suns in n∞ are also true in such spaces (see Sect. 10.4.1 below). Problem 7.1 Recently I. G. Tsar’kov constructed an example of a non B-connected sun in a four-dimensional polyhedral space. But it is still unknown at present whether each sun in a three-dimensional normed space is B-connected (or, equivalently, Pconnected). The answer to this question of B-connectedness of suns is positive in the two-dimensional setting, in the so-called (BM)-spaces (in particularly, in n∞ ), in 3-dimensional (cylindrical) spaces of the form Y ⊕∞ R (where dim Y = 2), and in smooth spaces (in which any sun is convex).

7.3 Dunham’s Example of a Disconnected Chebyshev Set with Isolated Point In a finite-dimensional space, any Chebyshev set is a strict sun, and hence by Theorem 7.7 a Chebyshev set in a finite-dimensional Banach space is connected (and moreover, B-connected, B-path-connected, locally connected, and is a B-retract). The infinite-dimensional situation is different. Ch. Dunham (see [214]) constructed an example of a disconnected locally compact Chebyshev set in C[0, 1]. Similar examples were also constructed by Braess [134], and Brosowski, Deutsch, Lambert, and Morris [142]. Note that the Chebyshev sets constructed in these examples are not suns. In this section, we recall the famous Ch. Dunham’s example of a Chebyshev set with isolated point. That the set in his example is proximinal can be proved using the above definition of τ-compactness (Lemma 4.2).

7.4 Klee’s Example of a Discrete Chebyshev Set

121

Example 7.1 (a Chebyshev set with an isolated point) Let ϕ : R+ → R be a strictly monotone function such that ϕ(0) = 1, limt→∞ ϕ(t) = 0. (For example, as ϕ one can take ϕ(t) = (1 + t)−1 .) We set  (2 + a)ϕ(t/a), a > 0, va (t) = 0, a=0 and define M = {va |[0,1] | a ≥ 0} ⊂ C[0, 1].

(7.3)

It is easily seen that the zero function is an isolated point of this set. We claim that M is a set of existence. Let (un ) be a bounded sequence and let un = van . We have va = |va (0)| ≥ a, and hence the sequence (an ) is bounded and has a limit point a∗ . If a∗ > 0, then the subsequence (un ) converges to va∗ . Next, if a∗ = 0, then this subsequence converges pointwisely to the function v0 ≡ 0 on the open interval (0, 1). By Lemma 4.2, the set M is a set of existence. We now claim that M is a set of uniqueness. Assume that va, vb ∈ PM f for some f . It can be supposed that a < b. The pointwise inequality va (t) < v(a+b)/2 (t) < vb (t) for t ∈ [0, 1] implies that f − v(a+b)/2 < max{ f − va , f − vb }. Hence v(a+b)/2 gives a better approximation for f than va and vb , a contradiction. Let us show that the above set M is not a sun. We need following simple result. Remark 7.2 A proto sun cannot have proper isolated points. Indeed, if y were a proper isolated point of some proto sun N, when we could choose w ∈ N \ {y} and consider x ∈ (w, y) such that x − y ≤ 12 ρ(y, (N \ {y})). This gives PN x = y, PN w = w; that is, N cannot be a sun. Remark 7.3 The Chebyshev set M constructed by Dunham is not a sun, because it has an isolated point { f0 } and by Remark 7.2 a sun cannot have isolated points. Remark 7.4 We also note that the Chebyshev set M in (7.3) is not approximatively weakly compact. Indeed, consider the functional x ∗ ( f ) = f (0). Then the bounded sequence (v1/n ) has no limit point u for which x ∗ (v1/n ) → x ∗ (u). It is also worth pointing out that for such M the set of near-best points PM f := M ∩ B( f , ρ( f , M) + δ) need not be connected (cf. [137], Exercise 3.17).

7.4 Klee’s Example of a Discrete Chebyshev Set At one time many researches were surprised by Dunham’s example [214] of a Chebyshev set with an isolated point in C[0, 1]. Even against this background Klee’s example looks fantastic. We give a simplified exposition of Klee’s example of a discrete Chebyshev set following Balaganskii and Vlasov [61].

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Suppose that a cardinal number τ is such that τ ℵ0 = τ (for example, τ = 2ℵ0 ). In  this section,  = (Γ) is the Banach space of all real functions x on Γ such that s ∈Γ |x(s)| = x  < ∞; Γx = {s ∈ Γ : x(s)  0}; Bc = B(c, 1), c ∈ ; δs (t) = 1 for t = s and 0 for t  s. Note that |Γx | ≤ ℵ0 and that x is an extreme point of the ball Bc (x ∈ ext Bc ) if and only if x = c ± δs ; if |Γ| = τ, then || = |(Γ)| = τ ℵ0 = τ = |Γ|. Lemma 7.1 Let {B j } j ∈J be a family of balls B j = Bc j , |J | < |Γ| = τ. Then U =  j ∈J B j  , and for each x ∈  \ U, there exists a ball Bc such that x ∈ ext Bc, Bc ∩ B j =   Proof Setting J  = j ∈J Γc j , we have

∀ j ∈ J.

|J  | ≤ ℵ0 |J | = max{ℵ0, |J |} < τ. Next, |Γ| = τ, and hence there exists s0 ∈ Γ \ J , c j (s0 ) = 0 for any j ∈ J . This means that all c j lie in the hyperplane {z ∈  | z(s0 ) = 0} and then, clearly, U  . We now take t  J  ∪ Γx (|J  ∪ Γx | < |Γ|). Then c j (t) = x(t) = 0 for any j ∈ J. We set c = x − δt . It is clear that x ∈ ext Bc . Next, for any j ∈ J, c j − c = c j − x + δt = c j − x + δt > 2, the lemma is proved.

Bc ∩ B j =  

The Chebyshev set is constructed by transfinite induction. Let ωτ be the smallest of the ordinal numbers b such that |{a : a < b}| = τ, I = {a : a < ωτ }. It is known that, for any b ∈ I, |{a ∈ I : a < b}| < τ and |I | = τ. We denote by ψ some bijection from  onto I. Step 0 of transfinite induction. We put c0 = 0, x0 = δ1 . Induction hypothesis. Let k ∈ I, k  0. Suppose that, for all i ∈ I, i < k, we have  already constructed xi , ci , Bi = Bci such that Bi ∩ B j =  ∀ i, j < k (i  j), xi  j 1 ≥ ci − x for i  j. Exercise 7.11 Show that a sun may not have isolated points. Can an α-sun have isolated points?

7.5 Koshcheev’s Example of a Disconnected Sun

123

7.5 Koshcheev’s Example of a Disconnected Sun As distinct from α-suns (see (6.2)), which can be disconnected even on a normed plane (see Fig. 6.1 in Sect. 6.1), the question of connectedness of suns proved to be quite involved. The only known example of a disconnected sun in an infinite-dimensional1 subspace of C[0, 1] was constructed by Koshcheev [365] in a renormed infinite-dimensional subspace of C[0, 1]. We recall Koshcheev’s construction. In C[0, 1] consider the subspace 

1   (x) < ∞ Y = x ∈ C[0, 1]  x(0) = 0,

0

with the norm



−1

x := max x C[0,1], 8

1 (x) , 0

1

where 0 (x) is the variation of a function x, x C[0,1] is the uniform norm. It is clear that Y is a Banach space. Let  1   1   , n = 1, 2, . . . , A1 := x ∈ Y  x 2n 2n   1    1  C1 := x ∈ Y  x − , n = 1, 2, . . . , 2n + 1 2n + 1 A := A1 ∩ Y, C := C1 ∩ Y, M := A ∪ C. The sets A, C are closed in Y and convex. 1We claim that the set M is a disconnected sun in Y . Let x ∈ A1 ∩ C1 . Then 0 (x) = ∞ and x  Y . Hence A ∩ C = . Since A, C are closed, M is disconnected. Let qn := (2n + 1)−1, n = 1, 2, . . . . pn := (2n)−1, Given an arbitrary fixed element g of the space Y , g  M, let us prove that the best approximation ρ(g, A) to this element is given by ρ(g, A) = max {pn − g(pn )} = pn0 − g(pn0 ). n=1,2,...

(7.4)

Note that pn0 − g(pn0 ) > 0, for otherwise we would have g(pn ) ≥ pn for any natural n, and hence g ∈ A ⊂ M. Consider a nondecreasing continuous function g(t) ¯ such that ¯ C[0,1] = pn0 − g(pn0 ). Since ¯ = 0, g(p g(0) ¯ n ) ≥ pn − g(pn ), n = 1, 2, . . . and g 1 ¯ = pn0 − g(pn0 ), we have g¯ ∈ Y , g ¯ = pn0 − g(pn0 ). We set v1 (t) = g(t) + g(t). ¯ 0 (g) Next, ¯ n ) ≥ g(pn ) + pn − g(pn ) = pn, n ∈ N, (7.5) v1 (pn ) = g(pn ) + g(p and hence the function v1 lies in the set A. Therefore, 1 Recall that in a finite-dimensional space any sun is path-connected (see Sect. 7.2).

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ρ(g, A) ≤ v1 − g = g ¯ = pn0 − g(pn0 ). On the other hand for any y ∈ A we have y(pn0 ) ≥ pn0 , which shows that y − g ≥ y − g C[0,1] ≥ y(pn0 ) − g(pn0 ) ≥ pn0 − g(pn0 ). So, ρ(g, A) = pn0 − g(pn0 ), proving (7.4). A similar argument shows that ρ(g, C) = max {qm + g(qm )} = pm0 + g(pm0 ), m=1,2,...

(7.6)

To complete the proof it suffices to find a point v0 ∈ PM g such that the intersection ˚ 0, g) with the set M is empty (see Theorem 5.5). of the cone K(v There are two cases to consider: ρ(g, A) ≤ ρ(g, C) and ρ(g, A) > ρ(g, C). We set d := ρ(g, M) and consider, for example, the first case. We have d = ρ(g, A) ≤ ρ(g, C). Consider the continuous function  g(t), ¯ x(t) = −d,

(7.7)

t ∈ [0, pm0 +1 ] ∪ [pm0 , 1], t = qm0 ;

on the intervals [pm0 +1, qm0 ], [qm0 , pm0 ] the function x(t) is defined as a linear function. It is clear that ¯ C[0,1] = d, x C[0,1] = g

1 0

(x) ≤

1 (g) ¯ + 4d = 5d, 0

and hence x = d. Let v0 (t) = g(t) + x(t). The values of the functions x and g¯ are equal at the points pn , n = 1, 2, . . . ,. Hence v0 and v1 are also equal at these points. As a result, v0 ∈ A by (7.5). Consider a functional fn0 ∈ Y ∗ , fn0 (z) = z(pn0 ). It is easily shown that fn0 Y ∗ = 1. By the construction, the function g¯ is such that g(p ¯ n ) ≥ pn − g(pn ), n = 1, 2, . . . . Hence, using (7.4) and (7.7), d = pn0 − g(pn0 ) = g ¯ C[0,1] ≥ g(p ¯ n0 ) ≥ pn0 − g(pn0 ); that is, g(p ¯ n0 ) = d. It follows that fn0 (v0 ) = g(pn0 ) + x(pn0 ) = g(pn0 ) + d = pn0 , fn0 (v0 − g) = x(pn0 ) = g(p ¯ n0 ) = d = x = v0 − g . Next, for any v ∈ A we have fn0 (v) = v(pn0 ) ≥ pn0 = fn0 (v0 ), and therefore, ˚ 0, g) ∩ A = ∅. K(v Let the functional fm0 ∈ Y ∗ be defined by fm0 (z) = −z(qm0 ). Hence fm0 Y ∗ = 1, and so, since x(qm0 ) = −d and using (7.6) and (7.7), we find that

7.6

Radial Continuity of the Metric Projection . . .

125

fm0 (v0 ) = −v0 (qm0 ) = −g(qm0 ) + d ≤ −g(qm0 ) + ρ(g, C) = qm0 , fm0 (v0 − g) = g(qm0 ) − v0 (qm0 ) = d = x = v0 − g . ˚ 0, g) ∩ C = ∅, inasmuch fm0 (w) = −w(gm0 ) ≥ gm0 ≥ fm0 (v0 ) It follows that K(v for any w ∈ B. So, if g is an arbitrary element of the space Y , g  M, then there exists an element ˚ 0, g) ∩ M = ∅. Hence M is a sun by the of best approximation v0 such that K(v Kolmogorov criterion (see Theorem 5.5). Note that for x  M the sets of nearest points P A x and PC x are nonempty. Choosing x  M such that ρ(x, A) > ρ(x, C), we find a local minimum of the distance function ρ( · , M) which is not a global one at points from P A x. So, M is not a unimodal set (is not an LG-set), and hence by Proposition 5.2 (see also Theorem 7.26 below), M is not a strict sun. We also note that the above set M is not approximatively compact.

7.6 Radial Continuity of the Metric Projection. B-Connectedness of Approximatively Compact Chebyshev Suns In parallel with the upper or lower semicontinuity (see Definitions 7.1 and 7.2), one often considers the Hausdorff upper or lower semicontinuity. Definition 7.3 For metric spaces X and Y , a set-valued mapping ϕ : X → 2Y is called Hausdorff upper (lower) semicontinuous if xn → x implies that d(ϕxn, ϕx) → 0 (d(ϕx, ϕxn ) → 0). We also need the concept of the one-sided Hausdorff distance (or the deviation) of a set A from a set C: d(A, C) := sup{ρ(a, C) | a ∈ A}. The Hausdorff distance between sets A and C is defined as follows: h(A, C) := max{d(A, C), d(C, A)}. We note at once that the upper semicontinuity of a mapping F at a point x0 implies its Hausdorff upper semicontinuity at x0 . It is well known (see, for example, [193]) that if F(x0 ) is compact, where F : X → 2Y \ {∅}, then F is lower semicontinuous at a point x0 if and only if F is Hausdorff lower semicontinuous at x0 ; that is, for any ε > 0, there exists a neighbourhood U of x0 such that d(F(x0 ), F(x)) < ε for all x ∈ U. Correspondingly if M ⊂ X is boundedly compact, then PM is always upper semicontinuous, and the Hausdorff continuity of the operator PM is equivalent to its lower semicontinuity [108]. Definition 7.4 Let M ⊂ X and x0 ∈ X. The metric projection PM is called ORUcontinuous (outer radially upper continuous) at a point x0 (see, for example, [24]) if,

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for any y0 ∈ PM x0 and any open W ⊃ PM x0 , there exists a neighbourhood U of x0 such that PM x ⊂ W for any x ∈ U ∩ {y0 + λ(x0 − y0 ) | λ  1}. Lemma 7.2 (see [141]) Let M ⊂ X and x0 ∈ X. Consider following properties: (1) the operator PM is ORU-continuous at x0 ; (2) for each v0 ∈ PM x0 and ε > 0, there exists δ > 0 such that d(PM x, PM x0 ) < ε for any x ∈ {v0 + λ(x0 − v0 ) | 1 ≤ λ < 1 + δ}; (3) for each point v0 ∈ PM x0 and any sequence (xn ) from the set {v0 + λ(x0 − v0 ) | λ ≥ 1}, xn → x0 , d(PM xn, PM x0 ) → 0; (4) for each point v0 ∈ PM x0 , any sequence (xn ) from the set {v0 + λ(x0 − v0 ) | λ ≥ 1}, xn → x0 , and any sequence (vn ), vn ∈ PM xn , ρ(vn, PM x0 ) → 0; (5) for each point v0 ∈ PM x0 , any sequence (xn ) from the set {v0 + λ(x0 − v0 ) | λ ≥ 1}, xn → x0 , and any sequence (vn ), vn ∈ PM xn and vn → v, v ∈ PM x0 . Then (2)⇔(3)⇔(4)⇒(5). If PM x0 is compact, then (4) ⇒(1) and the first four conditions are equivalent. If M is compact, then (5)⇒(1) and the first five conditions are equivalent. Remark 7.5 In general, the implications (4)⇒(1) and (5)⇒(1) do not hold. Let M be a subset of the Euclidean plane defined as follows: M = {(ξ, η) | ξ ≥ 1} ∪ {(ξ, η) | ξ ≤ −1}  ∪ {(ξ, η) | |ξ | < 1, η ≥ 1 − ξ 2 } \ {(1, 0), (−1, 0)}. We set x0 = (0, 0). Then PM x0 = {(ξ, η) | ξ 2 + η2 = 1, η > 0}. This shows that property (4) holds, and hence so holds (5). We set v0 = (0, 1) ∈ PM x0 , and consider the open set W = {(ξ, η) | η > 0}, W ⊃ PM x0 . Then, for each x = (ξ, η) from the set {v0 + λ(x0 − v0 ) | λ > 1}, we have η < 0 and moreover PM x = {(1, η), (−1, η)}, so that PM x ∩ W = ∅ and the operator PM is not ORU-continuous at x0 . Lemma 7.3 (see [141]) If M is closed, then property (5) of Lemma 7.2 holds. Proof Let v0 ∈ PM x0 , xn ∈ {v0 + λ(x0 − v0 ) | λ ≥ 1}, xn → x0 , vn ∈ PM xn , vn → v. Then v ∈ M and

7.6

Radial Continuity of the Metric Projection . . .

127

x0 − v ≤ x0 − xn + xn − vn + vn − v = x0 − xn + ρ(xn, M) + vn − v → ρ(x0, M), that is, x0 − v ≤ ρ(x0, M), which gives v ∈ PM x0 .



Theorem 7.11 (B. Brosowski, F. Deutsch) Let M be a P-compact strict protosun. Then the metric projection PM is ORU-continuous. Proof Let x0 ∈ X and v0 ∈ PM x0 . Given εn > 0, εn → 0, we put xn := v0 + (1 + εn )(x0 − v0 ). Then xn → x0 . We first show that PM x0 =

∞ 

PM xn .

(7.8)

n=1

Indeed, let v ∈ PM x0 . Then, for each n, xn − v ≤ xn − x0 + x0 − v = xn − x0 + x0 − v0 = xn − v0 = ρ(xn, M).  This shows that v ∈ PM xn and PM (x0 ) ⊂ ∞ n=1 PM xn . ∞ Conversely, let v ∈ n=1 PM xn . Then xn − v = ρ(xn, M) for each n, and so x0 − v = ρ(x0, M), which implies that v ∈ PM x0 . This proves (7.8). ∞Now let W be an open set containing PM x0 . By (7.8) we have PM x0 = n=1 PM xn , where PM xn is a decreasing sequence of compact sets, and hence there exists a number N such that PM xn ⊂ W for all n ≥ N. So, for some δ > 0 we have PM x ⊂ W for all x ∈ {v0 + λ(x0 − v0 ) | 1 ≤ λ < 1 + δ}. As a corollary, there exists a neighbourhood O(x0 ) of x0 such that if x = v0 +λ(x0 −v0 ) for λ ≥ 1 and x ∈ O(x0 ), then 1 ≤ λ < 1 + δ. This shows that the metric projection  PM is ORU-continuous. Theorem 7.11 is proved. Corollary 7.2 The metric projection onto a Chebyshev sun is ORU-continuous. Definition 7.5 The metric projection PM is called IRL-continuous (inner radially lower continuous) at a point x0 if, for any y0 ∈ PM x0 and any open W such that W ∩ PM x0  , there exists a neighbourhood U of x0 such that PM x ∩ W   for any x ∈ U ∩ {y0 + λ(x0 − y0 ) | 0  λ  1}. It is clear that the lower semicontinuity of the metric projection implies its IRLcontinuity. We note the following properties equivalent to the IRL-continuity of the operator PM at a point x0 (see [141]). Lemma 7.4 Let M ⊂ X and x0 ∈ X. Then the following properties are equivalent: (1) the operator PM is IRL-continuous at x0 ;

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(2) for each v0, v1 ∈ PM x0 and each ε > 0, there exists δ > 0 such that PM x ∩ B(v1, ε)  ∅ for all x ∈ {v0 + λ(x0 − v0 ) | 1 − δ < λ ≤ 1}; (3) for each v0, v1 ∈ PM x0 and any sequence (xn ) from the set {v0 + λ(x0 − v0 ) | 0 ≤ λ ≤ 1}, xn → x0 , there exists a point vn ∈ PM xn , such that vn → v1 . Proof (of Lemma 7.4) Let us show that (2)⇒3). If (3) were not true, then there would exist a number ε > 0, points v0, v1 ∈ PM x0 , and a sequence (xn ) from the interval {v0 + λ(x0 − v0 ) | 0 ≤ λ ≤ 1} such that xn → x0 , but ρ(v1, PM xn ) > ε for ˚ 1, ε)  ∅ for any x from the half-open some n. We choose δ > 0 such that PM x ∩ B(v interval {v0 + λ(x0 − v0 ) | 1 − δ < λ ≤ 1} =: Rδ . Then, for all sufficiently large n, we have xn ∈ Rδ and ρ(v1, PM xn )  ε, which contradicts the previous inequality ρ(v1, PM xn ) > ε. The implication (1)⇒(2) is clear. (3)⇒(1) Assume that (3) holds, but (1) is not satisfied. Then there would exist a point v0 ∈ PM x0 and an open set W, PM x0 ∩ W  ∅, such that, for any neighbourhood U of x0 , there would exist a point x from the intersection U ∩ {v0 + λ(x0 − v0 ) | 0 < λ < 1} such that PM x ∩ W = ∅. We choose v1 from PM x0 ∩W. Then for any n there exists xn := v0 + λn (x0 − v0 ) c 1 − 1/n < λn < 1 such ˚ 1, ε) ⊂ W. Then PM xi ∩ B(v ˚ 1, ε) = ∅ that PM xn ∩ W = ∅. Let ε > 0 be such that B(v for i = 1, 2, 3, . . . . We have xn → x0 , but ρ(v1, PM (xn )) ≥ ε for any n. This contradiction proves the lemma.  Theorem 7.12 If, for some x, the set of nearest points PM x is convex, then the metric projection is IRL-continuous at x. Proof If PM x = ∅, then there is nothing to prove. Let v0, v1 ∈ PM x and let xn ∈ [x, v0 ], xn → x. Then it is clear that xn = v0 + (1 − εn )(x − v0 ) for 0 ≤ εn ≤ 1 and εn → 0. We set vn = (1 − εn )v1 + εn v0 . Then vn ∈ PM x ⊂ M and vn → v1 . Next, we have xn − vn = (1 − εn ) x − v1 = (1 − εn ) x − v0 = xn − v0 = ρ(xn, M), which shows that vn ∈ PM xn ; that is, the metric projection is IRL-continuous by Lemma 7.4.  Remark 7.6 In general, the converse of Theorem 7.12 does not hold. For example, on the plane with maximum norm, consider the set M consisting of two points {(1, 0)} and {(1, 12 )}. It is clear that the set PM 0 = M is not convex, but the operator PM is IRL-continuous at 0. Corollary 7.3 If a set M is convex or is a Chebyshev set, then the metric projection PM is IRL-continuous. Theorem 7.13 (B. Brosowski, F. Deutsch) Let M be an existence set with IRL- and ORU-continuous metric projection. Then M is P-connected.

7.7

Spans, Segments. Menger Connectedness, and Monotone Path-Connectedness

129

Remark 7.7 Theorem 5.1 of [140] asserts that under the hypotheses of Theorem 7.13 ˚ the set is also B-connected. However, the proof from [140] of this result raises objections. At present, the validity of this assertion is unknown. The following two results are consequences of Theorem 7.13. Theorem 7.14 An approximatively compact set with IRL-continuous metric projection is B-connected. Proof (of Theorem 7.14) Let a set M satisfy the hypotheses of the theorem. By Theorem 4.4, the metric projection onto an approximatively compact set is upper semicontinuous, and hence, is ORU-continuous. By Theorem 7.13, the set M is ˚ P-connected. By Theorem 7.2, the set M is B-connected, and by Theorem 7.1, is B-connected.  Theorem 7.15 An approximatively compact Chebyshev sun is B-connected. Theorem 7.15 is a consequence of Theorem 7.14 and Corollary 7.3. ˚ Remark 7.8 The question whether an arbitrary Chebyshev sun is B-connected is at present open (the proof of this result in [140] has a gap). Proof (of Theorem 7.13) Assume that the set M is not P-connected. Then there exist x0 ∈ X and r > 0 such that PM x0 is disconnected. We assume that x0 = 0. Let PM x0 ∩ M = A ∪ C, where A, C are nonempty disjoint sets which are open in PM x0 . Let y ∈ C. Then there exists λ0 ∈ (0, 1) such that PM (λy) ⊂ C for all λ ∈ [λ0, 1]. We set β := inf{λ ∈ [0, 1] | PM (λy) ⊂ C}. Note that PM (βy) ⊂ C. Indeed, if this were not so, then we would have ∅  PM (βy) ⊂ A, since M is P-connected. Let v0 ∈ PM (βy) ∩ A. Then the IRL-continuity of the metric projection implies that, for any sequence xn ∈ (βy, y) such that xn → βy, there exists a sequence vn ∈ PM xn ⊂ C such that vn → v0 ∈ A. But this is impossible, because A is open in PM 0 and vn ∈ C \ A for any n. So, PM (βy) ⊂ C. On the other hand, the metric projection PM is ORU-continuous, and hence by definition there exists ε > 0 such that PM (λy) ⊂ C for all λ ∈ (β − ε, β). However, this contradicts the definition of β. So, the set M is P-connected. Theorem 7.13 is proved. 

7.7 Spans, Segments. Menger Connectedness, and Monotone Path-Connectedness We first consider the space C(Q) (Q is a metrizable compact set) with Chebyshev norm. By a segment in the space C(Q) we mean the set [[x, y]] := {z | z(t) ∈ [x(t), y(t)]}

(7.9)

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(here the interval [x(t), y(t)] is not directed, that is, we do not assume that x(t) ≤ y(t)). The relative interior of a segment [[x, y]] is defined as follows:   [[x, y]]0 := z ∈ [[x, y]] | z(t) ∈ (x(t), y(t)) ∀t ∈ Q : x(t)  y(t) . It is clear that [[x, y]] = [[y, x]], α[[x, y]] = [[αx, αy]], α ≥ 0, −[[x, y]] = [[−x, −y]]. Definition 7.6 A span is a set M with the property: [[x, y]] ⊂ M, whenever x, y ∈ M. As examples of spans in C(Q) we mention: a closed ball, an open ball, and [[x, y]]0 . A closed unit ball B(0, 1) in C(Q) is the segment [[−1, 1]], the open unit ball ˚ x) (see (5.8), Lemma 5.2) is ˚ 1) is the segment [[−1, 1]]0 . The support cone K(y, B(0, an example of an open span. According to Vasil’eva [581], [582], a subset Π ⊂ C(Q), where Q is a Hausdorff compact set, is a closed span if and only if Π can be written as a generalized segment   [[ f1, f2 ]]gen := f ∈ C(Q) | f (t) ∈ [ f1 (t), f2 (t)] ∀t ∈ Q , where f1, f2 : Q → R, f1 ≤ f2 , f1 is upper semicontinuous on Q, f2 is lower semicontinuous (in the definition of [[ f1, f2 ]]gen , the functions f1, f2 are not required to lie in C(Q)). Let A ⊂ Q, A  ∅. A relative segment and its relative interior are defined as   [[x, y]] A : = z | z(t) ∈ [x(t), y(t)], ∀ t ∈ A , (7.10)   [[x, y]]0A : = z | z(t) ∈ (x(t), y(t)), ∀ t ∈ A : x(t)  y(t) . It is easily checked that [[x, y]] A := [[x, y]] A. It is also clear that [[x, y]]0 ⊂ [[x, y]] ⊂ [[x, y]] A .

7.7.1 The Banach–Mazur Hull Definition 7.7 Following Brown [145], the Banach–Mazur hull m(M) (also called the spindle or the ball hull) of a bounded set ∅  M ⊂ X) is defined as the intersection of all closed balls that contain M. Definition 7.8 A set M ⊂ X is said to be m-connected (Menger connected) [145] if m({x, y}) ∩ M  {x, y} for every pair of distinct points x, y ∈ M. For brevity, we set

7.7

Spans, Segments. Menger Connectedness, and Monotone Path-Connectedness

131

m({x, y}) = m(x, y). An important property of the Banach–Mazur hull m(·, ·) (at least in the separable spaces X) is that z ∈ m(x, y) if and only if z lies metrically between x and y with respect to the so-called (Brown) associated norm | · | (see Lemma 7.5 below), which, in turn, is equivalent to the inclusion z ∈ [[x, y]] (see Sect. 7.7.2). It is clear that  DR (x, y), (7.11) m(x, y) = 

R ≥ x−y /2

where DR (x, y) := x,y ∈B(z,R) B(z, R) is the R-strongly convex interval [481], [301], [302], [70] with endpoints x, y. In the space C(Q), the structure of the Banach–Mazur hull m(M) is fairly transparent (see, for example, Brown [145, Theorem 3.1]):    m(x, y) = z  z(q) ∈ [x(q), y(q)], q ∈ Q =: [[x, y]]. (7.12) A similar representation also holds in the space C0 (Q) (Q is a locally compact Hausdorff space) — this follows from the characterization of extreme elements (‘evaluation at a point’) of the unit sphere of the dual space to C0 (Q), as obtained by Brosowski, Deutsch, and Morris in [142].

7.7.2 Segments and Spans in Normed Linear Spaces In analogy with (7.9), we define a segment in an arbitrary normed linear space X as follows:   [[x, y]] := z ∈ X | min{ϕ(x), ϕ(y)} ≤ ϕ(z) ≤ max{ϕ(x), ϕ(y)} ∀ϕ ∈ ext S ∗   (7.13) = z | ϕ(z) ∈ [ϕ(x), ϕ(y)] ∀ϕ ∈ ext S ∗ ; recall that ext S ∗ is the set of extreme points of the dual unit sphere S ∗ of X ∗ . (The reason for the analogy with (7.9) is that each extreme functional f ∈ C(Q)∗ (or f ∈ C0 (Q)∗ ) can be written as f (x) = ±x(t), where x ∈ C(Q) or x ∈ C0 (Q), t ∈ Q.) Indeed, if one takes an arbitrary subset A ⊂ ext S ∗ such that A ∪ (−A ) = ext S ∗ , then   [[x, y]] = z ∈ X | f (z) ∈ [ f (x), f (y)] ∀ f ∈ A . The relative interior of a segment [[x, y]] is defined as follows:  [[x, y]]0 := z ∈ [[x, y]] | ϕ(z) ∈ (ϕ(x), ϕ(y)) ∀ϕ ∈ ext S ∗ : ϕ(x)  ϕ(y)}.

(7.14)

Having at our disposal the concept of a segment, we may, in analogy with the above, define a span in a normed linear space X as follows: a set   Π ⊂ X is called a span if [[x, y]] ⊂ Π for all x, y ∈ Π.

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Every closed ball is a closed span. Indeed, let u, v ∈ B. If for some w ∈ B we have f (w) ∈ [ f (u), f (v)] for all f ∈ ext S ∗ , then by the Krein–Milman theorem, this inclusion can be extended to all f ∈ S ∗ , which implies that w  1, and so w ∈ B. Spans of functions (7.13) arise naturally in convex and nonlinear analysis, in problems of approximation theory and optimal control, in estimating the distance to a Chebyshev subspace, in related problems of strong uniqueness of extreme elements [334], and also in problems of estimating the widths of the intersections of function classes with spans [583]. Following Franchetti and Roversi [242], we consider the class (MeI) of spaces X such that m(x, y) = [[x, y]] for all x, y ∈ X (MeI) (the abbreviation (MeI) comes from the phrase ‘The hull m(x, y) equals the interval (segment) [[x, y]] for all x, y’). The inclusion m(x, y) ⊃ [[x, y]] (7.15) holds in every normed linear space X (see, for example, [242, Theorem 3.1]). Indeed [281, p. 55] (see also [469]), a closed convex set M is an intersection of closed balls if and if and only if for every point outside M, there exists a closed ball containing M but not that point. Now it remains to invoke the clear result (following from the fact that x = sup f ∈ext S ∗ f (x)) that a point not lying in a closed ball can always be strictly separated from it by an extreme hyperplane. Remark 7.9 The equality m(x, y) = [[x, y]]

(7.16)

holds in a broad class of Banach spaces. Indeed, Franchetti and Roversi ([242], Theorem 3.2) showed that (7.16) holds in the spaces for which smooth points are norm-dense in the unit sphere. This class of spaces includes the weakly Asplund spaces (in particular, the weakly compactly generated spaces, and hence the separable spaces and reflexive spaces). We also note that if a space X is such that ext S ∗ lies in the closure of the set of w ∗ -semidenting points of the dual ball B∗ (Moreno’s condition), then [[x, y]] = m(x, y) for all x, y ∈ X; this condition, for example, is satisfied for finite-dimensional spaces and spaces with the Mazur intersection property [14]. Recall that a point f ∈ S ∗ is called a w ∗ -semidenting point of the dual ball B∗ (see, for example, [270]) if for every ε > 0, there exists a w ∗ -slice S of the ball B∗ such that diam({ f } ∪ S) < ε. Here, S(B∗, x, δ) := {g ∈ S ∗ | g(x) > 1 − δ}, 0 < δ < 1, x ∈ X. Note that Moreno’s condition is not satisfied for the space  1 (there are no w ∗ semidenting points on the unit ball of the space  ∞ ; see [439]). However, (7.16) holds for the space  1 by the above Franchetti and Roversi theorem. In the finitedimensional setting, the equality [[x, y]] = m(x, y) was established by Brown [145]. Remark 7.10 Phelps [469] showed that ext S ∗ = S ∗ for a given finite-dimensional space X if and only if each convex bounded closed subset of X is an intersection of closed balls; as a corollary, in such a space, we have

7.7

Spans, Segments. Menger Connectedness, and Monotone Path-Connectedness

[[x, y]] = [x, y]

133

∀x, y ∈ X.

We shall also need the following class of normed linear spaces introduced by Franchetti and Roversi [242]: ext S ∗ is w ∗ -separable.

(Ex-w ∗ s)

In the definition of the class (Ex-w ∗ s), it will be always assumed that F = ( fi )i ∈I ⊂ ext S ∗ is w ∗ -dense in ext S ∗, card I  ℵ0, F = −F (the label (Ex-w ∗ s) is taken from the German ‘Die Extrempunktmenge der konjugierten Einheitskugel ist w ∗ -separabel’). Every space from the class (Ex-w ∗ s) has w ∗ -separable unit ball. Indeed, since the dual unit ball B∗ is the w ∗ -closure of the convex hull of ext S ∗ , the w ∗ -separability of the set ext B∗ (the James boundary) implies that of the ball B∗ , and as a corollary ([177], p. 253), we have the w ∗ -separability of X ∗ . In [177], assertion (2), it was shown that the w ∗ -separability of the ball B∗ is equivalent to the space X being isometrically isomorphic to a subspace of  ∞ (the latter condition is known to hold for all separable Banach spaces). We also note that [177] puts forward an example of a Banach space such that X ∗ is w ∗ -separable, but the ball B∗ is not w ∗ -separable. Further, it is well known (see, for example, [4, Lemma 1.4.1.]) that if X is a separable normed linear space, then the w ∗ -topology on the dual unit ball B∗ is metrizable. Hence every separable space lies in the class (Ex-w ∗ s). The class (Ex-w ∗ s) contains the nonseparable space  ∞ (as the space of continuous functions on the Stone–Čech compactification of N). We also note that C(Q) on an nonseparable Q and c0 (Γ) on an uncountable Γ fail to lie in the class (Ex-w ∗ s). Summarizing the above with regard to spaces in (MeI) and (Ex-w ∗ s), we note that: the class (MeI) ∩ (Ex-w ∗ s) contains all separable Banach spaces (in particular, all spaces C(Q) on a metrizable compact space Q) and the nonseparable space  ∞ . Definition 7.9 Let a space X lie in the class (Ex-w ∗ s) and let F = ( fi )i ∈I be a family of functionals from the definition of the class (Ex-w ∗ s) (we  always assume that F = −F); also, let (αi ) ⊂ R, αi > 0, i ∈ I, card I  ℵ0 , and αi < ∞. Given x ∈ X, we put  αi | fi (x)|. (7.17) |x| = i ∈I

Then | · | is a norm on X (see [145]), and in honour  of A. L. Brown, we call it the (Brown) associated norm. It is clear that |x|  x αi . The importance of the associated norm is illustrated by the following result [14], which is a natural generalization of Corollary 3.2 in [145] to the infinite-dimensional setting.

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Lemma 7.5 Let X be a Banach space from the class (MeI)∩(Ex-w ∗ s) (in particular, let X be a separable Banach space) and let x, y ∈ X. Then the following conditions are equivalent: (a) z ∈ m(x, y); (b) for all i ∈ I, | fi (x) − fi (y)| = | fi (x) − fi (z)| + | fi (z) − fi (y)|, where F = ( fi )i ∈I is the family in the definition of the class (Ex-w ∗ s); (c) |x − y| = |x − z| + |z − y| (i.e., z is | · |-between x and y). Proof (of Lemma 7.5) Let z ∈ m(x, y). Since X ∈ (MeI), it follows from the definition that f (z) ∈ [ f (x), f (y)] for all f ∈ ext S ∗ . In particular, condition b) holds, and therefore, condition c) holds by (7.17). Conversely, assume that z is | · |-between x and y. It is clear that | fi (x − y)| ≤ | fi (x − z)| + | fi (z − y)|. Hence, using (7.17), we have | fi (x) − fi (y)| = | fi (x) − fi (z)| + | fi (z) − fi (y)| for all i. As a corollary, | f (x) − f (y)| = | f (x) − f (z)| + | f (z) − f (y)| for all f ∈ ext S ∗ , inasmuch as the  family ( fi ) is w ∗ -dense in ext S ∗ . Finally, z ∈ m(x, y), because X ∈ (MeI).

7.7.3 Monotone Path-Connectedness Let k(τ), 0  τ  1, be a continuous curve in a normed linear space X. Following [142], we say that the curve k( · ) is monotone if f (k(τ)) is a monotone function with respect to τ for all f ∈ ext S ∗ . Definition 7.10 A closed set M ⊂ X is said to be monotone path-connected [11] if every two points from M can be joined by a continuous monotone curve (arc) k( · ) ⊂ M. Note that a monotone path-connected set is always B-connected (i.e., its intersection with every closed ball (and hence with every open ball [592]) is connected; cf. [142, Proposition 1.3]). Definition 7.11 A set M ⊂ X is said to be strictly monotone path-connected if every two points x, y from M can be joined by a continuous curve k( · ) ⊂ M that lies (except for the points x and y) in [[x, y]]0 ([[x, y]]0 is defined in (7.14)). Note that there are examples of finite-dimensional spaces containing Chebyshev sets and suns that are not m-connected (and a fortiori not monotone path-connected) (see Example 10.3 in Sect. 10.4.1 below). Remark 7.11 In a two-dimensional space, every sun is monotone path-connected. A. R. Alimov and B. B. Bednov [23] characterized the three-dimensional spaces in which every Chebyshev set is monotone path-connected. A (different) characterization of the three-dimensional spaces in which every closed set with lower semicontinuous (continuous) metric projection is monotone path-connected is due to Alimov [21].

7.8

Continuous and Semicontinuous Selections . . .

135

Remark 7.12 The finite-dimensional spaces with the property ext S ∗ = S ∗ are the only spaces in which the monotone path-connectedness of a closed set is equivalent to its convexity (see Example 7.10). Remark 7.13 Using (7.15), one can easily show that a monotone path-connected set is necessarily m-connected. The converse assertion may fail to hold even for closed sets, as seen from the example in C[0, 1] given by Franchetti and Roversi [242]: let M = M1 ∪ M−1, where Mσ = {x ∈ C[0, 1] | x(0) = σ}, σ = ±1. Then M consists of two disjoint closed convex components, but at the same time, M is easily seen to be m-connected. However, in c0 and in an arbitrary finite-dimensional space Xn , these properties are equivalent for closed sets (see [24, Sect. 9.1]). A sufficient condition than an m-connected subset of a normed linear space be monotone path-connected is, for example, its bounded compactness (see [24, Theorem 9.1]). Note that in finite-dimensional spaces with the property ext S ∗ = S ∗ , the class of monotone path-connected (m-connected) closed sets coincides with the class of closed convex sets (in such an X, one always has m(x, y) = [[x, y]] for all x, y ∈ X by Phelps’s theorem [469]). We also note that the set of generalized rational fractions RV,W is monotone path-connected even in every weighted space Cϕ (Q, X). On solarity of monotone path-connected Chebyshev sets, see Sect. 10.4.1 below. For monotone path-connectedness of suns in C(Q), see [11] and [571].

7.8 Continuous and Semicontinuous Selections of Metric Projection. Relation to Solarity and Proximinality of Sets Let M be a convex nonempty closed subset of a Banach space X. E. W. Cheney put forward a conjecture that the property of existence of a continuous selection of the metric projection on M and the condition that the set C([a, b], M) of all continuous mappings in M be a proximinal set in the space of all continuous functions C([a, b], X) are related. S. V. Konyagin and I. G. Tsar’kov gave an affirmative answer to this question in weakly compactly generated spaces (in particular in reflexive spaces and in separable spaces). Recall that a Banach space X is said to be weakly compact generated if there exists a weakly compact set K ⊂ X whose linear span is dense in X. Theorem 7.16 (S. V. Konyagin, I. G. Tsar’kov) Let M be a nonempty closed convex subset of a weakly compactly generated Banach space X. Then the following conditions are equivalent: a) there exists a single-valued continuous selection ϕ : X → M of the metric projection; i.e., ϕ(x) ∈ PM (x) for all x ∈ X; b) the set C([a, b], M) is an existence set in the space C([a, b], X).

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Theorem 7.17 (I. G. Tsar’kov) Let X be a Banach space and let M ⊂ X be an existence set admitting a set-valued lower semicontinuous selection Ψ : X → 2 M of the metric projection; that is, ∅  Ψ(x) ⊂ PM (x) for all x ∈ X. Then the set M is B-connected, and hence path-connected. We first prove an auxiliary result. Lemma 7.6 Let M be a closed subset of a normed linear space X such that for some x ∈ X, the set PM x is disconnected. We denote by D the closure of the convex hull of the union of the point x and the set PM x. Assume that D is a complete metric space. Then the metric projection PM does not admit a lower semicontinuous selection on D. Proof Assume to the contrary that PM has a lower semicontinuous set-valued selection Ψ on D. By the hypothesis, PM x is a union of two nonempty disjoint closed sets A1 and B1 . Setting r1 = ρ(x, M) and x1 = x, we argue by induction on i. (1) Assume that there exists a point xi ∈ X such that PM xi = Ai  Bi , where Ai and Bi are nonempty disjoint closed sets, ri = ρ(xi, M). We fix an arbitrary point ai ∈ Ai and consider the number ki = inf{k ∈ [0, 1] | Hkai (B(xi, ri )) ∩ Bi  ∅}. Here Hka is the homothety with centre of similarity a and coefficient of similarity k. Let xi+1 and B(xi+1, ri+1 ) be obtained from xi and B(xi, ri ) by the homothety with centre ai and coefficient di = min{ki + 2−i ki, 1}. We set Bi+1 = Ai ∩ B(xi+1, ri+1 ) and Ai+1 = Bi ∩ B(xi+1, ri+1 ). (2) We have thus constructed a nested system of balls {B(xi, ri )}, which is such that A1 ∩ B(xi, ri )  ∅, B1 ∩ B(xi, ri )  ∅. Since the balls are nested, we have xm − xn  rm − rn for all m, n ∈ N. Hence the sequences (xi ) and (ri ) are Cauchy sequences. Since D and R are both complete, the limits x0 ∈ D and r0  0 of these sequences exist. We have B(x0, r0 ) ⊂ B(xi, ri ), and so PM x0 ⊂ PM xi ⊂ A1 ∪ B1 . Consider the sequence yi := ai + (1 − 2−i ) ki (xi − ai ). Since Ai and Bi are nonempty disjoint sets, we have ki > 0, and further, by the definition of ki , we have ρ(yi, Ai ) < ρ(yi, Bi ), i ∈ N. Hence Ai ⊃ PM yi  ∅, i ∈ N. Moreover, yi − xi+1  2−(i−1) ki xi − ai → 0, i → ∞, and therefore yi → x0 , i → ∞. By the construction, we have A2i+1 ⊂ A1 and A2i ⊂ B1 , and hence Ψy2i+1 ⊂ A1 and Ψy2i ⊂ B1 (i ∈ N). As a result, x0 ∈ {y ∈ D | A1 ⊃ Ψy  ∅} ∩ {y ∈ D | B1 ⊃ Ψy  ∅}. However, this leads to a contradiction, because A1 ∩ B1 = ∅. The proof is complete.  Corollary 7.4 Let M be an existence set in a Banach space and let M admit a lower semicontinuous selection of the metric projection. Then M is P-connected.

7.9

Suns, Unimodal Sets, Moons, and ORL-Continuity . . .

137

Let us now proceed with the proof of the theorem. Proof Assume to the contrary that there exists a ball B(x, r) such that the intersection M ∩ B(x, r) consists of nonintersecting closed nonempty sets A and C. Let a ∈ A and b ∈ C be some points, and let Ψ be a set-valued lower semicontinuous selection of the metric projection. Then Ψ(a) = PM a = {a} and Ψ(b) = PM b = {b}, and furthermore, since M is P-connected, the set PM y (and hence Ψy) is disjoint from both A and C. Hence the connected set K = [a, x] ∪ [x, b] can be represented as two nonempty disjoint closed sets {y ∈ K | Ψ−1 (y) ⊂ A}

and

{y ∈ K | Ψ−1 (y) ⊂ C}. 

This contradiction proves the theorem. We note three more results (see, for example, [24], [16], [570]).

Theorem 7.18 (I. G. Tsar’kov) Let M be an existence set in a finite-dimensional space X with lower semicontinuous metric projection. Then M is a sun. Theorem 7.19 (A. R. Alimov) Let M be a closed set with lower semicontinuous metric projection in a Banach space of dimension at most 3. Then M is a sun, is B-contractible, is a B-retract, and there is a continuous selection of the metric projection onto M. Theorem 7.20 (I. G. Tsar’kov) Let X be an asymmetric finite-dimensional space and let M ⊂ X. If the metric projection onto M is lower semicontinuous, then it has a continuous selection (that is, there exists a continuous mapping ϕ : X → M such that ϕ(x) ∈ PM x for all x ∈ X). For more results on the existence of continuous selections of the near-best metric projections and applications thereof, see also [552], [564], [563], [561], [568], [14], [16], [17], [18], [19].

7.9 Suns, Unimodal Sets, Moons, and ORL-Continuity. Brosowski–Wegmann-connectedness The Moone Hee hath for seasons set, the Sun his setting knows. Psalm CIV, 19

In this section, we consider natural restrictions on approximating sets that guarantee that such sets are suns, moons, or unimodal sets (LG-sets) in a broad class of spaces. As such natural properties we consider the Brosowski–Wegmannconnectedness (regularity) and ORL-continuity of the metric projection. In a broad class of spaces (which contains the space of continuous functions C(Q)), in terms of

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such properties one can characterize, for example, strict suns, and prove that some specific sets are suns or fail to be suns. Results of this kind will be used to prove the solarity of certain Chebyshev sets in some concrete and abstract infinite-dimensional spaces. The problem of solarity of Chebyshev sets was studied by N. V. Efimov, S. B. Stechkin, V. Klee, L. P. Vlasov, B. Brosowski, I. G. Tsar’kov, S. V. Konyagin, among others. We also note the papers by Vlasov [592], Balaganskii and Vlasov [61], Karlov and Tsar’kov [323], Brown [145], Brosowski, Deutsch, Lambert, and Morris [142], Alimov [12]. In every finite-dimensional space, a Chebyshev set is a sun. Dunham [214] (see also [142] and Sect. 7.3) constructed an example of a locally compact Chebyshev set in C[0, 1] that is not a sun. In general, the solarity of a set is usually proved under compactness constraints or certain continuity conditions of the metric projection operator onto this set (see Vlasov [592, Chap. 3]), Balaganskii and Vlasov [61, Chap. I, Sect. 2]). Among classical results here, we mention a theorem of L. P. Vlasov to the effect that a boundedly compact P-acyclic set (in particular, a Chebyshev set) in a normed space is a sun (see Theorem 10.1 below). It is known that in a normed space, a locally compact Chebyshev set with continuous metric projection PM is a sun (see, for example, [323]). The concept of a Brosowski–Wegmann-connected (regular) subset of a normed linear space was introduced by B. Brosowski [138, Sect. 1] under the name ‘regular set’ (see also [132]) and independently by Ch. Dunham [212] under the name ‘set with closed-sign property’ for C(Q)-spaces (for general spaces, the definition was given in [143]; see also [520] and [480]). We give the definition of Brosowski– Wegmann-connectedness (regularity) in the particular case of C(Q)-spaces. Definition 7.12 A closed set M ⊂ C(Q) is said to be Brosowski–Wegmannconnected (regular) if for every x, y ∈ M and nonempty compact set A ⊂ Q with (7.18) inf |x(t) − y(t)| > 0, t ∈A

there exists a sequence (vn ) of points from M such that vn → y and sign(vn (t) − y(t)) = sign(x(t) − y(t)) for all t ∈ A

(7.19)

(in general, the sign can be different for different t). This definition is equivalent to saying that for every closed set A ⊂ Q satisfying condition (7.18), there exists a sequence (vn ) of points from M such that vn → y and vn ∈ [[x, y]]0A (here, we recall, [[x, y]]0A is the relative interior of a segment [[x, y]] A; see (7.10)). Example 7.2 The sets depicted in Fig. 7.1 are monotone path-connected, but not Brosowski– Wegmann-connected in the space 3∞ . Example 7.3 For each 0 ≤ t ≤ 1 in the (x, y)-plane, define the curve Mt as follows:

7.9

Suns, Unimodal Sets, Moons, and ORL-Continuity . . .

139

Fig. 7.1 Sets that are not Brosowski–Wegmann-connected in 3∞ .

Mt = {(x, y) | x, y ≥ 0, y = (1 − x t )1/t },

0 < t < 1,

M0 = {(x, y) | x, y ≥ 0, x + y = 1}, M1 = {0 ≤ x ≤ 1, y = 0} ∪ {0 ≤ y ≤ 1, x = 0}. It is easily checked that in the space 3∞ , the set   (Mt + (0, 0, t)) | t ∈ R M= is Brosowski–Wegmann-connected (moreover, such a set is a strict sun in 3∞ ; see Theorem 7.22 below). It is worth noting that the section of this set by the coordinate plane z = 1 is not Brosowski– Wegmann-connected.

For example, if M is a Brosowski–Wegmann-connected subset of the space n∞ and x, y ∈ M, then x and y can be jointed by a monotone curve from [[x, y]] (see [24, Sect. 9], [14]). In addition, if all the coordinates of x and y are distinct, then they can be jointed by a strictly monotone curve k(τ) lying in M. Below, we shall consider sets with ORL-continuous metric projection. This property is weaker than the lower semicontinuity of the metric projection, but it turns out that it is shared by unimodal sets and by moons, and moreover, in many spaces this property characterizes the strict (proto)suns (see Theorem 7.26 below). We shall show that in the space C(Q) (and C0 (Q)), a set with ORL-continuous metric projection (in particular, an approximatively compact Chebyshev set, and more generally, a Chebyshev set with continuous metric projection) is Brosowski– Wegmann-connected, and hence is a strict protosun. Definition 7.13 The metric projection PM is said to be ORL-continuous (outer radially lower continuous) [141] at a point x if y ∈ PM x,

(xn ) ⊂ {y + λ(x − y) | λ  1},

xn → x,

implies that ρ(y, PM xn ) → 0. Remark 7.14 The following properties are equivalent to the ORL-continuity of the operator PM at a point x0 (see [141]): (1) for each v0, v1 ∈ PM x0 and each ε > 0, there exists δ > 0 such that PM x ∩ B(v1, ε)  ∅ for all x ∈ {v0 + λ(x0 − v0 ) | 1 ≤ λ < 1 + δ};

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(2) for each v0, v1 ∈ PM x0 and sequence xn from the set {v0 + λ(x0 − v0 ) | λ ≥ 1}, xn → x0 , there exists a sequence of points vn ∈ PM xn such that vn → v1 . It is clear that a lower semicontinuous metric projection is ORL-continuous. However, the converse implication need not hold. Recall that a space X satisfies the (P)-Brown property if the condition x + z ≤ x for every fixed x, z ∈ X implies that there exist positive constants c, b such that y + cz ≤ y , provided that x − y ≤ b. It is known that in every finite-dimensional space without the (P)-Brown property (see, for example, [108], [150]), there exists a finite-dimensional subspace L (a strict sun) for which the metric projection onto it is not lower semicontinuous. Nevertheless (see, for example, [24, Sect. 9]), the mapping PL is ORL-continuous. Below, we shall see that the metric projection onto a strict protosun is always ORL-continuous (Theorem 7.23). We also note that in a three-dimensional space, a strict sun (and even a subspace) may fail to have a continuous selection of the metric projection (Brown; see, for example, [317]). It is worth pointing out that in Theorem 7.21, the strict solarity of a set is proved only from a continuity-type property without any compactness constraints. Theorem 7.21 Let M be a closed set with ORL-continuous metric projection in the space C(Q) (where Q is a metrizable compact set). Then M is Brosowski–Wegmannconnected. We shall give two different proofs of Theorem 7.21. We first note one result important for applications ([141]; see also Theorem 7.26 below). Theorem 7.22 (Brosowski–Wegmann) In the space C(Q), Q a metrizable compact set, the following properties are equivalent for a set M ⊂ C(Q): (a) M is a strict protosun; (b) M is Brosowski–Wegmann-connected (regular); (c) M is a Kolmogorov set; (d) M is extremum characterizable. In this theorem, the implication ‘strict protosolarity’ ⇒ ‘Brosowski–Wegmannconnectedness’ follows from Theorems 7.21 and 7.23, which will be proved below. For the implication (b) ⇒ (a), see, for example, [138], [143], [600]; for the implication (d) ⇔ (a), see [212], [143], and Sect. 5.2. Proof (of Theorem 7.21) Assume to the contrary that the set M is not Brosowski– Wegmann-connected. Then by definition, there exist a pair of points v, v0 ∈ M and a nonempty compact set A ⊂ Q such that mint ∈ A |v0 (t) − v(t)| > 0. Next, there exists a number λ > 0 such that    (7.20) Bλ := v˜ ∈ M  v˜ − v0 < λ, v˜ ∈ [[v0, v]]0A = ∅. By Urysohn’s lemma (see, for example, [5]), there exists a continuous function ψ ∈ C(Q) such that

7.9

Suns, Unimodal Sets, Moons, and ORL-Continuity . . .

⎧ ⎪ 1, ⎪ ⎨ ⎪ ψ(t) = 0, ⎪ ⎪ ⎪ 0 < ψ < 1, ⎩

141

t ∈ A, t ∈ {τ | v(τ) = v0 (τ)}, otherwise.

Next, we set ϕ(t) := ψ(t) · sign[v(t) − v0 (t)] (here by definition we assume that sign 0 = 0). It is clear that ϕ is continuous on Q. By continuity, for some δ ∈ (0, v − v0 /2) we have ψ(t) < 1/2 for all t ∈ {τ ∈ Q | |v(τ) − v0 (τ)| ≤ δ}. Next, given α ≥ 0, we set fα := v0 + α · ϕ.

(7.21)

Since A is nonempty and |ϕ(t)| = 1 for all t ∈ A and |ϕ(t)| < 1 for all t ∈ Q \ A, it follows from (7.21) that fα − v0 = α

∀α ≥ 0,

(7.22)

the function fα − v0 attaining its norm at every point of A. For every point  v ∈ M from (7.21), we have fα − v˜ = v0 − v˜ + αϕ.

(7.23)

Consider two cases. (a) Assume that there exists a point t0 ∈ A at which   sign v0 (t0 ) −  v (t0 ) = sign v(t0 ) − v0 (t0 ) . By definition, sign ϕ(t) = sign(v(t) − v0 (t)), and hence   sign v0 (t0 ) −  v (t0 ) = sign v(t0 ) − v0 (t0 ) = sign ϕ(t0 ). Using (7.23) and since |ϕ(t)| = 1, t ∈ A, we find that   | fα (t0 ) −  v (t0 )| =  v0 (t0 ) −  v (t0 ) + αϕ(t0 ) ≥ α |ϕ(t0 )| ≥ α = fα − v0 ,

(7.24)

because the numbers (v0 (t0 ) −  v (t0 )) and ϕ(t0 ) have the same sign. (b) Now let   sign  v (t) − v0 (t) = sign v(t) − v0 (t) ∀t ∈ A.

(7.25)

Since  v ∈ M, it follows from (7.20) and (7.25) that  v − v0 ≥ λ. As a corollary, for all α ≤ λ/2, the point v0 is a nearest point from M for fα . On the other hand, setting β = v − v0 , we have by (7.25) that     | fβ (t) − v(t)| = v0 (t) − v(t) + βϕ(t) =  |v0 (t) − v(t)| − βψ(t). (7.26) If t is such that |v(t) − v0 (t)| ≤ δ, then the right-hand side of (7.26) is smaller than β/2, inasmuch as 0 < δ < β/2 and 0 ≤ ψ(t) < 1/2.

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Next, if t is such that |v(t)− v0 (t)| > δ, then the right-hand side of (7.26) is smaller than β. Indeed,      |v0 (t) − v(t)| − βψ(t) ≤ max |v(t) − v0 (t)| − βψ(t), βψ(t) − |v(t) − v0 (t)|   ≤ max β − δ0, β − δ < β,    where δ0 := min ψ(t)  t ∈ {τ | |v0 (τ) − v(τ)| > δ > 0. So (7.27) fβ − v < β = fβ − v0 , and hence v0  PM fβ . We set γ := sup{α ∈ R | v0 ∈ PM fα }.

(7.28)

From (7.27), we have γ < ∞. It is clear that v0 ∈ PM fγ by the continuity of the metric function (see Proposition 1.1). Since the metric projection is ORL-continuous, from an arbitrary sequence γn → γ, γn > γ, we can find a sequence vn ∈ PM ( fγn ), n = 1, 2, 3, . . . , such that vn → v0 . We have fγn − vn = v0 − vn + γn ϕ. We claim that the case in which v0 − vn and ϕ have the same sign at some point t ∈ A is impossible. Assume to the contrary that such a point t ∈ A exists. Then, using (7.24), we obtain | fγn (t) − vn (t)| ≥ γn = fγn − v0 . However, vn ∈ PM ( fγn ), and hence from the previous inequality we get v0 ∈ PM ( fγn ), contradicting the definition of γ. So we can assume that sign(vn − v0 ) = sign ϕ on A. Since Bλ = ∅ (see (7.20)), we have vn − v0 ≥ λ. In this case, we again arrive at a contradiction: the inequality vn − v0 ≥ λ contradicts the convergence vn → v0 . This contradiction shows that the assumption that the set M was not Brosowski–Wegmann-connected was false. Theorem 7.21 is proved.  For the second proof of Theorem 7.21 we require some results and definitions. Note that the metric projection onto a Chebyshev subspace is always IRL- and ORU-continuous (see Sect. 7.6, Corollary 7.3, and Theorem 7.11), but still it can be discontinuous. Let us show that the metric projection operator onto a strict protosun is ORL-continuous. Theorems 7.23–7.25, 7.29 can be found in the papers of Brosowski, Wegmann, Amir, and Deutsch [139], [141], [33]. Theorem 7.23 The metric projection onto a strict protosun is ORL-continuous. Proof (of Theorem 7.23) Let x0 ∈ X, ε > 0, v0, v1 ∈ PM x0 . It suffices to show that ˚ 1, ε)  ∅. if x = v0 + λ(x0 − v0 ), λ > 1, then PM (x) ∩ B(v

7.9

Suns, Unimodal Sets, Moons, and ORL-Continuity . . .

143

Since M is a strict protosun, we have v0 ∈ PM x, and moreover, x − v1 ≤ x − x0 + x0 − v1 = (1 − λ)(v0 − x) + x0 − v1 = (λ − 1) v0 − x0 + x0 − v0 = λ x0 − v0 = x − v0 ≤ x − v1 . This shows that x − v1 = x − v0 ; that is, v1 ∈ PM x.



Recall that a closed set M   is a unimodal set (LG-set) if for all x  M, each local minimum of the function Φx (y) = y − x , y ∈ M, is a global minimum (see Sect. 5.2). Theorem 7.24 A closed set with ORL-continuous metric projection is unimodal (is an LG-set). Proof (of Theorem 7.24) If M were not a unimodal set, then there would exist ˚ 0, ε) ∩ M, x0 ∈ X, v0 ∈ M, and ε > 0 such that x0 − v0 ≤ x0 − v for all v ∈ B(v but v0  PM x0 . Let x1 be the last point on the half-open interval [v0, x0 ) that has v0 as a best (global) approximation in M (that such a point exists follows from the continuity of the metric function ρ( · , M)). Hence ρ(x, M) < x − v0 for all x ∈ (x1, x0 ]. Using property 1) of ORL-continuous metric projection (see ˚ 0, ε)  ∅ for all x ∈ Vδ := Remark 7.14), we can choose δ > 0 such that PM x ∩ B(v ˚ 0, ε), then {v0 + λ(x1 − v0 ) | 1 < λ < 1 + δ}. If xλ ∈ Vδ and vλ ∈ PM xλ ∩ B(v xλ − vλ < xλ − v0 and x0 − v0 = x0 − xλ + xλ − v0 > x0 − xλ + xλ − vλ ≥ x0 − vλ , which contradicts the fact that v0 is a local best approximation to x0 from M.



In some spaces, the strict (proto)solarity is equivalent to the property of being a moon. We give the corresponding definitions. Definition 7.14 Let   M ⊂ X. A point y0 ∈ M is called a lunar point if

whenever

x∈

−1 PM (y0 )

and

˚ 0, x) y0 ∈ M ∩ K(y ˚ 0, x)  . M ∩ K(y

Definition 7.15 A set M is called a moon2 if all its points y0 are lunar. It is clear that every protosun is a moon. We shall show (Theorem 7.26) that a strict protosun is always a unimodal set (an LG-set, a global minimizer), and in turn, a unimodal set (an LG-set) is a moon. Theorem 7.25 In a normed linear space, a unimodal set (an LG-set) is a moon. 2 To quote from [33]: “Such sets were originally called sign regular (by B. Brosowski and F. Deutsch). The present name ‘moon’ was given on Monday 21.09.1969 (‘Moonday’), for obvious reasons.”

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Proof (of Theorem 7.25) Let M be a unimodal set. If M is not a moon, then there −1 (v ) such that exist v0 ∈ M and x ∈ PM 0 ˚ 0, x)  , M ∩ K(v

but

˚ 0, x). v0  M ∩ K(v

In other words, there exists ε > 0 such that ˚ 0, x) ⊂ X \ M. ˚ 0, ε) ∩ K(v B(v ˚ 0, x) ∩ M. Then by definition of the support cone K(v ˚ 0, x) (see (5.8) and Let u ∈ K(v Lemma 5.2), for some λ > 0 we have ˚ 0 + λ(x − v0 ), λ x − v0 ). u ∈ B(v ˚ 1, x1 − v0 ), K(v ˚ 0, x1 ) = K(v ˚ 0, x), and We set x1 := v0 + λ(x − v0 ). Then u ∈ B(x ˚ 1, x1 − v0 ) ⊂ X \ M, ˚ 0, ε) ∩ B(x B(v i.e., v0 is a locally nearest point from M for x1 . However, x1 − u < x1 − v0 , and  hence v0 is not a globally nearest point. Definition 7.16 A space X is called an (MS)-space [33], [141] if in X, every moon is a strict protosun (or, what is the same, a Kolmogorov set [24, Sect. 3.2]). In the next theorem, which was proved in papers by B. Brosowski, R. Wegmann, F. Deutsch, and D. Amir (see [33], [139], [600]), we summarize the above results. Theorem 7.26 Let M be a nonempty subset of a normed linear space X. Then we have the implications “i) ⇒ i + 1)”, i = 1, 2, 3: (1) M is a strict protosun; (2) the metric projection PM is ORL-continuous at all points; (3) M is a unimodal set (an LG-set, a global minimizer); (4) M is a moon. If X is an (MS)-space, then the conditions (1)–(4) are equivalent. Remark 7.15 The class of (MS)-spaces contains, in particular, the spaces C(Q) (Q is a compact Hausdorff space), C0 (Q) (Q is a locally compact Hausdorff space),  1 (Γ), and the finite-dimensional polyhedral spaces. So in such spaces, conditions (1)–(4) are equivalent. Remark 7.16 A sun (unlike a strict sun; see Theorem 7.26) can fail to be a unimodal set (an LG-set) even in the two-dimensional setting. As an example, consider the complement of the open quadrant in the plane with the maximum norm. Then the points lying near one of the sides of the quadrant have local nearest points from the opposite side, but they are not global nearest points. Remark 7.17 The implications (2)⇒(3), (3)⇒(4) in Theorem 7.26 are not in general reversible. To see that (4)(3), it suffices to consider the set M := {(ξ, η) | ξ 2 /4 + η2 ≥ 1} in the Euclidean plane. It is easily seen that M is a moon but not a unimodal

7.9

Suns, Unimodal Sets, Moons, and ORL-Continuity . . .

145

set: for the point (0, −1/2), the point (0, 1) is a local but not global point of best approximation. Next, let M be the complement of the open unit ball in the Euclidean plane. It is easily seen that the function Φx (y) = y − x , y ∈ M, has only global minima; that is, M is a unimodal set (and hence a moon). On the other hand, it is clear that M is not a sun, because M has bounded complement. Correspondingly, (3)(2). The implication (3)⇒(1) in Theorem 7.26 can be reversed under the assumption that the set under consideration is a B-sun (that is, its intersection with every closed ball is a sun or empty). The following results hold [17], [18], [16]. Theorem 7.27 (A. R. Alimov) A B-solar unimodal set is a strict sun. Theorem 7.28 (A. R. Alimov) Let M be a B-sun. Then the following conditions are equivalent: (1) the metric projection PM onto M is ORL-continuous; (2) M is a strict sun. Definition 7.17 A point v0 of the unit sphere S is said to be strongly nonlunar if for ˚ 0, x) and ˚ 0, 0), there exists a point x ∈ B˚ such that u ∈ K(v every point u ∈ K(v ˚ 0, x) ∩ S. v0  K(v A space X is said to be strongly nonlunar [33], [141] if every point v0 of the unit sphere S is strongly nonlunar. In testing the strongly nonlunar property, the following equivalence is frequently ˚ helpful [33]: (here v0 ∈ S, x ∈ B) ˚ (a) v0  K(v0, x) ∩ S; ˚ 0, x) ⊂ B. ˚ ˚ 0, ε) ∩ K(v (b) there exists ε > 0 for which B(v Remark 7.18 Brosowski and Wegmann [141] showed that in the space P[0, 1] of algebraic polynomials on the interval [0, 1] (and also in the space of analytic functions) with uniform norm, there is a moon that is not a strict protosun. Problem 7.2 Suppose that in a space X, every moon is a strict protosun. Is it true that X is strongly nonlunar? The next theorems, Theorems 7.29 and 7.30, are due to Amir and Deutsch [33]. Theorem 7.29 In a strongly nonlunar normed space, a moon is a strict protosun. In other words, a strongly nonlunar space lies in the class (MS). Proof (of Theorem 7.29) By Theorem 7.26, every strict protosun is a moon. Assume that M is a moon that is not a strict protosun. Then there exists a point v0 ∈ M that ˚ 0, x) ∩ M  ∅. Without loss is a best approximation to some x ∈ X but for which K(v ˚ 0, 0) ∩ M  ∅. Since of generality, we assume that x = 0, ρ(x, M) = 1. Let v ∈ K(v

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the sphere S is strongly nonlunar at the point v0 , there exists a point x1 ∈ B˚ such that ˚ 0, x) ∩ S. This means that there exists ε > 0 such that ˚ 0, x) and v0  K(v u ∈ K(v ˚ 0, x) ⊂ B˚ ⊂ X \ M. ˚ 0, ε) ∩ K(v B(v ˚ 0, x1 ) ∩ M. But this contradicts the fact that M is a moon. So v0  K(v



In the next theorem we show that the space C0 (Q) (and in particular, C(Q)) is strongly nonlunar. Here Q is a locally compact Hausdorff space, C0 (Q) is the space of all continuous functions x on Q such that the set {t ∈ Q | |x(t)| ≥ ε} is compact for all ε > 0; C0 (Q) is equipped with the Chebyshev norm. It is known that each extreme functional f on the space C(Q) (Q is a Hausdorff compact set) or C0 (Q)∗ reads as f (x) = ±x(t), where x ∈ C(Q) or x ∈ C0 (Q), t ∈ Q. Given a function x ∈ C0 (Q), we define crit x ± := {t ∈ Q | x(t) = ± x }, crit x = crit x + ∪ crit x − . So for v0, x ∈ X we have by Lemma 5.2 that  ˚ 0, x) = v ∈ X | v(t) < v0 (t) if t ∈ crit(v0 − x)+, K(v  v(t) > v0 (t) if t ∈ crit(v0 − x)− . Theorem 7.30 The space C0 (Q) is strongly nonlunar. ˚ 0, 0). Choose 0 < δ < 1 such that Proof Let v0 ∈ S, v1 ∈ K(v δ < min{|v0 (t) − v1 (t)| | t ∈ crit v0 } and define   2δ δ K + = t | v0 (t) ≥ 1 − > 1 − ≥ v1 (t) , 3 3   2δ δ − ≤ v1 (t) . K = t | v0 (t) ≤ −1 + < −1 + 3 3 Let V + and V − be, respectively, disjoint neighbourhoods of K + and K − . Note that K + and K − are compact G δ -sets, and moreover, K + ⊃ crit v0+ , K − ⊃ crit v0− . By Urysohn’s lemma (see, for example, [5]), there exists a function f ∈ C0 (Q) such that ⎧ ⎪ 1/2 ⎪ ⎨ ⎪ f = −1/2 ⎪ ⎪ ⎪ 0 ⎩

on K +, on K −, off V − ∪ V +,

| f | < 1/2 off K + ∪ K − . Setting x = v0 − f , we have x − v0 = 1/2 and crit(v0 − x)+ = K +,

crit(v0 − x)− = K − .

˚ 0, x). Since v1 < v0 on K + , v1 > v0 on K − , we get v1 ∈ K(v

7.9

Suns, Unimodal Sets, Moons, and ORL-Continuity . . .

147

We set J = {t | |v0 (t)| ≥ 1/2}. Since crit v0 ⊂ int(K + ∪ K − ) and J \ int(K + ∪ K − ) is compact, it follows that   sup |v0 (t)| | t ∈ J \ int(K + ∪ K − = 1 − δ1 ˚ 0, ε) ∩ K(v ˚ 0, x); that is, for some δ1 > 0. We set ε = min{δ/6, δ1 /2}. Let v ∈ B(v + − v0 − v < ε, v < v0 on K , v > v0 on K . In particular, |v| < 1 on K + ∪ K − . We have that if t ∈ J \ (K + ∪ K − ), then |v(t)| < |v0 (t)| + ε ≤ 1 − δ1 + ε < 1. If t  J, then |v(t)| < |v0 (t)| + ε < 1/2 + ε ≤ 1. ˚ 0, x) ⊂ B; ˚ that is, v0 is strongly nonlunar.  ˚ 0, ε) ∩ K(v So v < 1, and hence B(v The following result is a consequence of Theorems 7.29 and 7.30. Theorem 7.31 In C0 (Q), a set is a sun if and only if it is a moon. Let us now proceed with the second proof of Theorem 7.21. By Theorem 7.26, the set M is a unimodal set, and hence a moon. Now it remains to employ Theorem 7.31, according to which M is a strict protosun (in our setting, M is a Chebyshev sun). From Theorems 7.31 and 7.26, we have the following result. Theorem 7.32 Let M be a set with ORL-continuous metric projection in an (MS)space (in particular, in the space C0 (Q), Q is a locally compact Hausdorff space, or in the space  1 (Γ)). Then M is a strict protosun. Now Theorem 7.21 follows from Theorem 7.32 as a corollary. Definition 7.18 A point y ∈ S is called a quasipolyhedral point (in the sense of Amir ˚ 0) ∩ S. A space is quasipolyhedral if all points of its and Deutsch [33]) if y  K(y, unit sphere are quasipolyhedral. Note that (see [33, p. 184]) a point y ∈ S is quasipolyhedral if there exists ε > 0 such that ˚ 0) = B(y, ˚ ε) ∩ B(0, 1); ˚ ε) ∩ K(y, B(y, in turn, the last condition is satisfied if there exists ε > 0 such that ˚ ε) ∩ S ˚ ε) ∩ bd K(v ˚ 0, 0) = B(y, B(y, (here and below, bd A is the boundary of a set A). It is easily checked that the property of a space being quasipolyhedral is hereditary: if X is quasipolyhedral, then so is every subspace of X. The definition of a quasipolyhedral space can be equivalently stated as follows. A space is quasipolyhedral [33], [240, p. 1751] if for every x ∈ S, there exists a neighbourhood O(x) of x such that [x, y] ⊂ S for all y ∈ O(x) ∩ S. In the

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Connectedness and Approximative Properties of Sets . . .

finite-dimensional setting, the quasipolyhedrality of a space is equivalent to its polyhedrality; this ceases to be true if dim X = ∞: a polyhedral space (all sections of the unit ball of such a space by finite-dimensional subspaces are polytopes) need not be quasipolyhedral [240]. Amir and Deutsch [33], see also Wegmann [600], showed that every quasipolyhedral space is an (MS)-space, and hence is strongly nonlunar [600]. In particular, in such a space, a Chebyshev set with continuous metric projection is a sun. Exercise 7.12 Show that the unit sphere of a strictly convex space is a moon. Exercise 7.13 Show that the set M = {(x 1, x 2 ) ∈ R2 | (x 1 )2 + 4(x 2 )2 ≥ 1} in R2 is a nonunimodal moon.

7.10 Solarity of the Set of Generalized Rational Fractions It is well known that the set Rn,m of rational fractions,



p  p ∈ Pn, q ∈ Pm, q  0 , Rn,m := q  on the interval [a, b] is a Chebyshev sun in the space C[a, b] with Chebyshev norm. Akhiezer and Walsh independently showed that the best Rn,m -approximation is unique, and they established the existence of a best approximant (see, for example, [162]). We consider a more general case of approximation by generalized rational functions:

 p  p ∈ V, q ∈ W, q(t) > 0, t ∈ Q , RV,W = q  where V, W ⊂ C(Q). As distinct from the case of classical rational fractions Rm,n , a best approximant by the set of generalized rational fractions RV,W may fail to exist or may be nonunique (see, for example, Sect. 11.4 below). Theorem 7.33 The set RV,W of generalized rational functions (V, W are convex nonempty subsets of C(Q)) is a strict protosun in the space C(Q). Proof (of Theorem 7.33) Let us check that RV,W is Brosowski–Wegmann-connected. Consider g0 = p0 /q0 , g1 = p1 /q1 ∈ RV,W . Given n = 1, 2, . . . , define gn =

1 n p1 + (1 − 1 n q1 + (1 −

1 n )p0 1 n )q0

.

Since V, W are convex, we have gn ∈ RV,W . Simple algebra gives gn − g0 =

p1 + (n − 1)p0 p0 p1 q0 − p0 q1 − = , q1 + (n − 1)q0 q0 (q1 + (n − 1)q0 )q0

7.11

Approximative Properties of Sets Lying in a Subspace

which implies that

 gn − g0 = (g1 − g0 )

 q1 . q1 + (n − 1)q0

149

(7.29)

We have q1 > 0, q2 > 0 on Q, and hence, using (7.29), sign(gn − g0 )(t) = sign(g1 − g0 )(t)

(7.30)

for all t ∈ A, where A ⊂ Q is such that mint ∈ A |g1 (t) − g2 (t)| > 0 (A is a nonempty closed subset of Q). It is clear that gn → g0 for n → ∞, and hence RV,W is Brosowski–Wegmann-connected by (7.30). Now the strict protosolarity of RV,W is secured by Theorem 7.22.  Remark 7.19 There is a simpler proof of Theorem 7.33. Given two unequal numeric fractions a1 /b1 and a2 /b2 , the fraction a1 t + (1 − t)a2 , b1 t + (1 − t)b2

t ∈ (0, 1),

lies in the interval (a1 /b1, a2 /b2 ). This shows that the set RV,W of generalized rational functions is strictly monotone path-connected (see Definition 7.11), and hence is Brosowski–Wegmann-connected and is a strict protosun by Theorem 7.22. Example 7.4 Recall Dunham’s example of a Chebyshev set with isolated point (see Sect. 7.3). Let ϕ : R+ → R be a strictly monotone function such that ϕ(0) = 1, limx→∞ ϕ(x) = 0. (For example, as ϕ one may put ϕ(x) = (1 + x)−1 .) We set  (2 + a)ϕ(x/a), a > 0, va (x) = 0, a = 0, and define

M = {va |[0,1] | a ≥ 0} ⊂ C[0, 1].

The zero function is an isolated point of this set. The trace of the curve va (x), a > 0, is a strictly monotone path-connected set (see Definition 7.11), and hence the set M \ {0} is strictly monotone path-connected, and thus is a strict protosun. If the isolated point {0} is added to M, the strict protosun M \ {0} becomes the Chebyshev set M, but this set is not a sun (because it has an isolated point — see Remark 7.2).

7.11 Approximative Properties of Sets Lying in a Subspace In this section, we consider the problem of approximatively geometric properties of sets M in normed linear or asymmetrically normed spaces (X, · ) under the additional condition that M ⊂ H, where H is a subspace of X. As before, let B be the unit ball of X. We claim that if | · | H,θ is the asymmetric norm on H defined by the Minkowski functional of the set (B − θ) ∩ H with respect to 0, where θ < 1 is

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an arbitrary point, then M is a Chebyshev set in (H, | · | H,θ ) for all θ. Similar results also hold for suns and strict suns. ˚ We define the following two norms: Let H be a subspace in X and let θ ∈ B. | · |θ is the asymmetric norm on X defined by the Minkowski functional of the body B˚ − θ with respect to the point 0; | · | H,θ is the asymmetric norm on H induced by the asymmetric norm | · |θ . Here we recall that by an asymmetric norm on X we mean a nonnegative sublinear functional · such that for all x, y ∈ X, (1) x = 0 ⇔ x = 0; (2) αx = α x for all α ≥ 0; (3) x + y ≤ x + y . In general, x  −x . Note that the function x sym = max{ x , −x }, x ∈ X, is a norm. A survey of modern results on the general theory of asymmetrically normed spaces with applications to approximation theory can be found in Cobzaş [174]. We denote the ball, sphere, etc., of radius r centred at x with respect to the asymmetric norm | · |θ by Bθ (x, r), Sθ (x, r), etc.; the subscripts indicate the corresponding (asymmetric) norms. Proposition 7.1 The asymmetric norms · and | · |θ are equivalent for every choice ˚ and θ ∈ B, Bθ (0, r) ⊂ B(0, 1) ⇐⇒ 0 < r ≤ (1 + − θ )−1, (7.31) B(0, R) ⊂ Bθ (0, r) ⇐⇒ 0 < R ≤ r(1 − θ ). Moreover, for a fixed H, the asymmetric norms | · | H,θ are equivalent on H for all ˚ θ ∈ B. We shall prove this assertion later. Let Hkx (·) denote the homothety with ratio k ≥ 0 and centre x ∈ X. The following result is clear. Proposition 7.2 Let ω ∈ B(y, r), k ≥ 0. Then k Hω (B(y, r)) = B(ω + k(y − ω), kr).

(7.32)

Proof (of Proposition 7.1) By definition, Bθ (0, 1) = B(−θ, 1). Next, Bθ (0, r) = Hr0 (Bθ (0, 1)) = Hr0 (B(−θ, 1)) = B(−rθ, r), where the last equality follows from (7.32). By Proposition 1.2, Bθ (0, r) = B(−rθ, r) ⊂ B(0, 1) if and only if r ≤ 1/(1 + − θ ), which proves the first inclusion in (7.31). The second inclusion in (7.31) is proved similarly.  The next result (see Alimov [10]) looks surprising at first sight. However, in the context of asymmetric approximations, it is fairly natural. The question we deal with here, that of approximation by sets lying in a subspace, is one such problem: a section

7.11

Approximative Properties of Sets Lying in a Subspace

151

of a symmetric body (the ball B) by an affine subspace H need not be symmetric! Such a section defines an (asymmetric) norm on H, and moreover, if a set M ⊂ H was a Chebyshev set in X, it is also a Chebyshev set in H by Theorem 7.34. Theorem 7.34 Let M be, respectively, a Chebyshev set, a sun, or a strict sun in a space (X, · ). Then: (1) the set M is a Chebyshev set, a sun, or a strict sun, respectively, in the space ˚ (X, | · |θ ) for all θ ∈ B. (2) If (H, | · | H,θ ) is an affine subspace of X and M ⊂ H, then M is a Chebyshev set, ˚ a sun, or a strict sun, respectively, in the space (H, | · | H,θ ) for all θ ∈ B. A similar result also holds for α-suns and semisuns (see Definition on p. 190). In assertion (2) of Theorem 7.34, the condition M ⊂ H cannot be relaxed. One can easily construct an example showing that the intersection M ∩ H, where M is a Chebyshev set in X, M  H, need not be a Chebyshev set in H. Proof (of Theorem 7.34) I. Let M be a Chebyshev set (X, · ). We claim that the set M preserves its property in the space (X, | · |θ ). Let x  M. By Proposition 7.1, ρθ (x, M) := inf z ∈M |z − x|θ > 0. Without loss of generality, we assume that x = 0, ρθ (x, M) = 1. Let us show that ρ(−θ, M) = 1. By definition, Bθ (0, 1) = B(−θ, 1). Hence ˚ B(−θ, 1) ∩ M = ∅. Assume to the contrary that ˚ B(−θ, 1 + δ) ∩ M = ∅

for some

δ > 0.

From (7.32), we see that B(−θ, 1 + δ) = Bθ (δθ, 1 + δ). Next, | − θ|θ < 1 − ε0 for some ˚ 1 + δ) for all 0 < ε < ε0 by Proposition 1.2. ε0 > 0, and hence Bθ (0, 1 + ε) ⊂ B(δθ, This is a contradiction, because on the one hand, Bθ (0, 1 + ε) ∩ M = ∅, while on the other hand, 1 = ρθ (0, M) = inf z ∈M |z|θ . By the assumption, the point −θ has a unique nearest point yˆ from M in (X, · ). By the above, yˆ + θ = 1. Now we have Bθ (0, 1) = B(−θ, 1), and hence yˆ is a unique | · |θ -nearest point for 0; i.e., M is a Chebyshev set in the space (X, | · |θ ). Now assume that M is a sun in (X, · ). Consider a point x  M. According to the above, ρθ (x, M) > 0. We assume without loss of generality that x = 0 and ρθ (0, M) = 1. As before, we have Bθ (0, 1) = B(−θ, 1),

˚ B(−θ, 1) ∩ M = ∅,

S(−θ, 1) ∩ M  ∅.

Let yˆ be a luminosity point in (X, · ) for −θ. By Theorem 5.5, the support cone   ˚ yˆ, −θ) = K( B˚ −αθ + (1 − α) yˆ, α (7.33) α>1

has no common points with M. By Proposition 7.2,   B −αθ + (1 − α) yˆ, α = Hαyˆ (B(−θ, 1)) = Hαyˆ (Bθ (0, 1)) = Bθ yˆ (1 − α), α .

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Hence the support cone K˚ θ ( yˆ, 0) =

  B˚ yˆ (1 − α), α α>1

is disjoint from M, which in view of Theorem 5.5 implies that M is a sun in the space (X, | · |θ ). The assertion for strict suns is proved similarly. II. Let H be an affine subspace in X. We assume for simplicity that 0 ∈ H. Let M ⊂ H be a Chebyshev set in (X, · ). We claim that M preserves this property in the space (H, | · | H,θ ). ˚ Since M is a Chebyshev set in (X, · ), we Let x ∈ H \ M. We fix a point θ ∈ B. have ρ H,θ (0, M) > 0. We assume without loss of generality that ρ H,θ (0, M) = 1. We have B˚ H,θ (0, 1) ∩ ˚ ˚ 1)∩H. Next, M ⊂ H, and so B(−θ, 1)∩M = M = ∅. By definition, B˚ H,θ (0, 1) = B(−θ, ∅. Further, ρ H,θ (0, M) = 1, and therefore, S(−θ, 1) ∩ M  ∅. Moreover, since M is a Chebyshev set in (X, · ), this intersection is a singleton. Hence M is a Chebyshev set in (H, | · | H,θ ). For strict suns and α-suns the arguments are similar. Theorem 7.34 is proved.  Using Theorem 7.34 and the well-known fact that a sun in a two-dimensional space X is P-contractible3 (see, for example, Berens and Hetzelt [101], Alimov [8], Brown [148]), one can get without much effort the following result (Alimov [10]), which extends a theorem of Brown’s ([148, Sect. 4]) that he obtained in the particular case of finite-dimensional (symmetric) normed spaces. Theorem 7.35 Let X be an asymmetrically normed linear space, H a subspace in X, dim H = 2, and let M ⊂ H be a sun in X. Then M is P-contractible, and for all ε > 0, there exists a continuous ε-selection of the metric projection onto M.

7.12 Approximation by Products The simplest nontrivial family of products is the set M1,1 of functions of the form   M1,1 := (at + b)(ct + d), a, b, c, d ∈ R . We shall prove the following result. Theorem 7.36 The set M1,1 is not a sun in the space C[−1, 1]. The weaker result that M1,1 is not a strict sun in C[−1, 1] can be easily derived from the Kolmogorov criterion for a nearest element. Indeed, following Dunham [215], consider approximation of the function x(t) = 2t 2 + 1 on the interval [−1, 1]. 3 A topological space X is contractible (to a point) if the identity mapping from X into X is homotopic to a constant mapping. A set M is P-contractible if PM x is contractible for every point x ∈ X.

7.12

Approximation by Products

153

We claim that the function y0 = 2 is an approximant for x in M1,1 on [−1, 1]. On this interval, the difference x − 2 has extrema only at the points {−1, 0, 1}, besides x − 2 = 1, and hence if there were an element y ∈ M1,1 that approximated x better than 2, then it would satisfy the inequalities y(−1) > 2,

y(0) < 2,

y(1) > 2.

(7.34)

If a function y had a zero on [−1, 1], then we would have x − y ≥ 1, because x(t) ≥ 1 on [−1, 1], which shows that y cannot approximate x better than 2. If y has no zeros on [−1, 1], then y cannot lie in the class M1,1 , because the product of two monomials must have two zeros (possibly equal) on R. In view of (7.34), this gives at least three zeros of the derivative y  (one zero on the interval (−1, 1) and two zeros on the set |t| > 1), which is impossible. So the function y0 = 2 is an element of best approximation to x = 2t 2 + 1 from M1,1 on [−1, 1]. We have x − 3t 2 = 1 on [−1, 1], and so the best approximation to x is not unique (moreover, αt 2 ∈ PM1,1 x for all α ≥ 2). We claim that the set M1,1 is not extremum characterizable in the sense of Definition 5.10. Indeed, the function x − y0 , where y0 := 3t 2 , x = t 2 + 1, has a unique extremum on [−1, 1] (the point 0), and at this point, (x − y0 )(2 − y0 ) > 0. Hence the family x, y0 , y := 2 satisfies condition (5.10). Finally, by Theorem 7.22, the set M1,1 is not a strict protosun, but since this set is proximinal (see Theorem 7.37 below), M1,1 is not a strict sun. Theorem 7.36 will be proved below. We first show that the set M1,1 is not P-connected.

(7.35)

Consider the more general case of approximation by generalized products   MV,W := v · w | v ∈ V, w ∈ W , where V, W are finite-dimensional subspaces in C(Q). It is clear that MV,W = V · W is a subset of the finite-dimensional subspace L := span MV,W

(7.36)

(of dimension at most dim V + dim W) generated by products of the form {gi h j | i = 1, . . . , n, j = 1, . . . , m}, where g1, . . . , gn is a basis for the subspace V, h1, . . . , hm is a basis for the subspace W, n = dim V, m = dim W. Hence MV,W is an existence set if and only if it is closed. We first give a simple example from [272] showing that in general, the set MV,W need not be closed.

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Connectedness and Approximative Properties of Sets . . .

Example 7.5 Let V = {ag1 + bg2 | a, b ∈ R}, W = {ch1 + dh2 | c, d ∈ R} be the subspaces in C[0, 5] generated, respectively, by linear combinations of the functions g1, g2 and h1, h2 depicted in Fig. 7.2.

Fig. 7.2 Basis functions for the subspaces V and W .

It is clear that g1 h1 = h1, g2 h1 = 0, g2 h2 = g2, the functions h1, g1 h2, g2 are linearly independent.

(7.37) (7.38)

Consider the sequence vn wn ∈ MV,W defined by vn = g1 + ng2,

wn = h + 1 + n−1 h2,

n ∈ N.

From (7.37), we have vn wn = (g1 + ng2 )(h1 + n−1 h2 ) = g1 h1 + n−1 g1 h2 + ng2 h1 + g2 h2

(7.39)

−1

= h1 + n g1 h2 + g2 . So vn wn → h1 + q2, MV,W . Then

n → ∞, that is, h1 + g2 ∈ cl MV,W . Assume that h1 + g2 ∈ h1 + g2 = (ag1 + bg2 )(ch1 + dh2 )

for some a, b, c, d ∈ R. From (7.37), it follows that h1 + g2 = ach1 + adg1 h2 + bdg2 ; hence, using (7.38), we get the system ac = 1,

ad = 0,

bd = 1,

which clearly has no solution. This shows that the set MV,W = V · W is not closed.

7.12

Approximation by Products

155

Note that in Example 7.5, we have g2 h1 = 0, whereas g2  0, h1  0. In the next theorem, we show that the existence of best approximation by MV,W can be guaranteed in the absence of such a degeneracy. Theorem 7.37 (G.A. Gislason [272]) Let V, W be finite-dimensional subspaces of C(Q). If v ∈ V, w ∈ W, and vw = 0 imply that v = 0 or w = 0, then the set MV,W is closed in C(Q) (and, as a corollary, is an existence set). Proof Let g ∈ cl MV,W and let g − vn wn → 0, n → ∞, for some vn, wn ∈ MV,W . If g = 0, then clearly, g ∈ MV,W , because 0 ∈ V, 0 ∈ W. So we can assume without loss of generality that vn wn  0

and

vn = 1

∀n.

By compactness, it suffices to show that the sequence (wn ) is bounded. Assume the contrary. Then by passing to subsequences if necessary and letting n → ∞, it follows that vn wn → 0, (7.40) wn because vn wn → g and since the sequence ( wn ) is not bounded. By compactness, there exist functions v ∈ V and w ∈ W such that vn → v, wn / wn → w, and as a corollary, vn wn → vw (7.41) wn (by passing to further subsequences if necessary). In (7.41), the functions v, w are nonzero, since by the assumption, we have g  0. By the hypothesis of the theorem, vw  0. However, vw = 0 by (7.40), a contradiction.  To prove Theorem 7.36, we need two results by Ch. Dunham. We first give the necessary definitions. A point vw ∈ MV,W is called an interior point if there exists in the subspace L (see (7.36)) a neighbourhood of vw contained in MV,W . By the quadratic formula and continuity, it is clear that all polynomials of degree 2 with two distinct zeros are interior points for MV,W . A monomial p with one real simple root is an interior point, since polynomials near p have a real root nearby, and hence no imaginary root exists. On the other hand, constants and polynomials with a double zero are not interior points, since they can be perturbed into a polynomial that is not a product of monomials. Remark 7.20 Let vw ∈ MV,W be an interior point. Then vw ∈ PMV ,W x ⇐⇒ vw ∈ PL x,

x ∈ C[a, b].

Theorem 7.38 Let y(t) = (at + b)(ct + d) be an interior point for M1,1 , x  M1,1 . Then: (1) y ∈ PM1,1 x if and only if the difference x − y has an alternant of length 3; (2) if y ∈ PM1,1 x, then y is a unique element of best approximation to x; that is, PM1,1 x = {y}.

156

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Connectedness and Approximative Properties of Sets . . .

Proof (of Remark 7.20) Since MV,W ⊂ L, the sufficiency is clear. To prove the necessity, note that it follows from linearity that if an element vw is not an approximant for x, then for x there exists an element that is closer to it than vw in every neighbourhood of vw.  Theorem 7.38 follows from Remark 7.20 and the Chebyshev alternation (equioscillation) theorem (see Sect. 2.1). In the particular case of M1,1 , the enveloping subspace consists of polynomials of degree ≤ 2, which gives three alternant points in Chebyshev’s theorem. We note the following auxiliary result of independent interest. Lemma 7.7 Let M be a sun and y0 a one-point connected component of the set of nearest points PM x for some x ∈ X. Then y0 is a luminosity point for x. Proof (of Lemma 7.7) For small λ > 0, the point xλ := λx + (1 − λ)y0 clearly ˚ 0, xλ ) ∩ M = ∅. It is clear that has a unique nearest point y0 . By Theorem 5.5, K(y ˚ x). This shows that y0 is a luminosity point for x. ˚ 0, xλ ) = K(y,  K(y Proof (of Theorem 7.36) As before, consider the functions x = 2t 2 + 1 and y0 = 2, 2 ∈ PM1,1 x. By Theorem 7.38, an interior point cannot be a nearest point for x. It is also clear that no constant c  2 lies in PM1,1 x. Consider the remaining functions from M1,1 : these are binomials p with a root of multiplicity 2. Since 2 ∈ PM1,1 x and since p ∈ PM1,1 x, it follows that p(−1) ≥ 2, p(0) ≤ 2, p(1) ≥ 2. Next, p  const, and hence on the interval [−1, 1], the derivative p has a zero. Further, since p reads as p = α(a − t)2 , the polynomial p also has a zero on [−1, 1]. Now it is clear that p − 2 ≥ 2. So the function y0 = 2 is a one-point connected component of the set of nearest points PM1,1 x. By Lemma 7.7, y0 is a luminosity point for x. Note that x = (2+(4t 2 ))/2 and that the functions y0 := 2 and y := 4t 2 are elements ˚ 0, x). On the other hand, of best approximation from M1,1 for x. As a result, y ∈ K(y ˚ since y0 is a luminosity point for x, we have K(y0, x) ∩ M = ∅ by Theorem 5.5. This, however, contradicts the previous inclusion, because y ∈ M. This contradiction proves Theorem 7.36.  Remark 7.21 In C[a, b], there exists a nonconvex Chebyshev set contained in a twodimensional subspace (see, for example, [26]). Such a set consists of the union of two rays emanating from a common point. Of course, such a Chebyshev set is a Chebyshev sun.

Chapter 8

Existence of Chebyshev Subspaces

The subspace of polynomials of degree at most n serves as a classical example of a Chebyshev subspace in the space C[0, 1]. Interest in the study of approximative properties of more general subspaces stems from consideration of Chebyshev (Haar) systems of functions that extend the classical Chebyshev system composed of polynomials of degree at most n (see Chap. 2). Of course, every space X contains trivial Chebyshev subspaces: X and {0}. In this chapter, we outline the known results on the existence of nontrivial Chebyshev subspaces in finite- and infinite-dimensional spaces. For the problem of Chebyshev subspaces in finite-dimensional spaces, which we consider in Sect. 8.1, we formulate the classical Zalgaller’s theorem, which asserts that every finite-dimensional normed space of dimension n contains a Chebyshev subspace of every dimension k ≤ n. In Sect. 8.2, we study Chebyshev subspaces in infinite-dimensional spaces. We give some examples of infinite-dimensional Chebyshev subspaces of some infinitedimensional spaces. Next, we formulate and prove Garkavi’s theorem, which states that there exists a separable Banach space that contains no nontrivial Chebyshev subspaces. Further, we recall the well-known (presently unsolved) problem of whether there exists a Banach space that contains no nontrivial Chebyshev subspaces. For L 1 -spaces, we give the classical M. G. Krein’s theorem (Theorem 8.3) to the effect that in L 1 [0, 1] (with Lebesgue measure), there are no finite-dimensional Chebyshev subspaces. Next, we formulate several extensions of Krein’s theorem, and in particular, provide Garkavi’s theorem (Theorem 8.4), which gives a necessary and sufficient condition that L 1 (Q, Σ, μ) contain a Chebyshev subspace of dimension n < ∞. This result is followed by the result of P. Ørno, which says that there exists a separable reflexive Banach space without finite-dimensional Chebyshev subspaces. We conclude this section with two related results: the first one is the theorem of P. Ørno– Yu. A. Brudnyi–E. A. Gorin, which asserts that every Chebyshev set in L 1 [0, 1] is either a singleton or is infinite-dimensional, and Theorem 8.8, which supplements and extends this result, says that every finite-dimensional sun in L 1 [0, 1] is convex.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_8

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8 Existence of Chebyshev Subspaces

8.1 Chebyshev Subspaces in Finite-Dimensional Spaces We first formulate without proof the well-known Zalgaller’s theorem (1972), which answers in the affirmative the problem of existence of Chebyshev subspaces of every dimension in an arbitrary finite-dimensional Banach space. A simple proof of this result can be found in Singer [520, p. 31]. In the three-dimensional setting, a similar result was obtained earlier by T. J. McMinn (1960). Theorem 8.1 (Zalgaller) If dim X = n < ∞, then for every k = 1, . . . , n, the space X contains a Chebyshev subspace of dimension k.

8.2 Chebyshev Subspaces in Infinite-Dimensional Spaces The study of approximative properties of infinite-dimensional subspaces is complicated first of all by the fact that in such a space, an element of best approximation may fail to exist. Nontrivial examples of Chebyshev subspaces can be easily constructed in some infinite-dimensional spaces. Example 8.1 The subspace  L := ϕ ∈ L 1 [0, 1] | ϕ(t) = 0

∀ t ∈ [0, 1/2]



∫1 is a nontrivial Chebyshev subspace in the space L 1 [0, 1] with the standard norm  f  = 0 | f (t)| dt. In Fig. 8.1, the function ϕ ∗ is a unique element of best approximation to f ∈ L 1 [0, 1] from the subspace L.

f ϕ∗

1/2

1

Fig. 8.1 Nontrivial Chebyshev subspace in L 1 [0, 1].

Note that if a space X is reflexive, then according to one well-known result of J. Lindenstrauss, in such a space there is an exposed point on the unit ball B —this is

8.2 Chebyshev Subspaces in Infinite-Dimensional Spaces

159

a point x ∈ S at which there exists a support hyperplane H to the ball that intersects the ball B exactly in the point x. It is clear that in this case, the hyperplane H − {x} is a nontrivial Chebyshev subspace in the space X. We say that a subspace L of X is nontrivial if |L  {0} or L  X. Theorem 8.2 (A. L. Garkavi) There exists a Banach space that contains no nontrivial Chebyshev subspaces. Proof Let Γ be a set whose cardinality is greater than that of the continuum: card Γ > c. Consider the space 0∞ (Γ) of essentially bounded functions f : Γ → R with at most countable supp f := {γ ∈ Γ | f (γ)  }. This space 0∞ (Γ) is equipped with the norm sup{| f (γ)| | γ ∈ Γ}. Let L ⊂ 0∞ (Γ) be a closed subspace. There are two cases to consider: (1) card L ≤ c; (2) card L > c. Since the union of at most a continuum of countable sets supp f has cardinality ≤ c, in the first case, supp f does not cover Γ. Hence, for some γ0 ∈ Γ, we have f (γ0 ) = 0 for all f ∈ L. Then, for the indicator function χγ0 (which is 1 at γ0 and 0 at the remaining points), we have 1 =  χγ0  = | χγ0 (γ0 ) − f (γ0 )| ≤  χγ0 − f  = 1 for all f ∈ BL := B ∩ L. Hence ρ( χγ0 , L) = 1, and all the functions f from the unit ball BL lie in PL ( χγ0 ). So in the first case, L is not a Chebyshev subspace. Now assume that card L > c and let L be a Chebyshev subspace. Then for ϕ  L, there exists a unique element y ∈ L such that ϕ − y = ρ(ϕ, L). The function ϕ − y has countable support {γ1, γ2, . . . } ⊂ Γ. However, the cardinality of all real functions defined on the countable set supp(ϕ − y) is c. Hence there exist functions y1  y2 ∈ L that agree on supp(ϕ − y). As a result, for sufficiently small ε, the element y − ε(y2 − y1 ) is also an element of best approximation to the function ϕ, a contradiction to the fact that L is a Chebyshev subspace.  The space considered in Theorem 8.2 is not separable. It is worth noting that the following question is still unanswered. Problem 8.1 Does there exist a separable Banach space without nontrivial Chebyshev subspaces? That the separability assumption is essential in Problem 8.1 is shown in Theorem 8.2. The completeness condition is also essential. Namely, V. Klee and independently Singer [520, p. 31] constructed an example of an incomplete separable space without nontrivial Chebyshev subspaces. As such a space one may consider the linear manifold in c0 consisting of all finitely supported sequences (that is, sequences in which only finitely many terms are nonzero). The following well-known question is also open at present: Does every separable space C(Q) contain a Chebyshev subspace of infinite dimension and codimension. In 2016, G. M. Ustinov showed that there is no such subspace in any space C(Q),

160

8 Existence of Chebyshev Subspaces

where Q is a countable metrizable compact set with finite number of limit points. An example of such a space is the space of convergent sequences c = (x (0), x (1), x (2), . . . ), where x (0) = lim x (n) ,  x = max{|x (k) |} = C({0, 1; 12 ; 13 , . . . }). It is also worth pointing out that earlier, Borodin [115] showed that in such spaces, a proper subspace is approximatively compact if and only if it is finite-dimensional. We also give an example of a subspace of infinite dimension and codimension in C[0, 1] onto which there is a 1-Lipschitz selection of the metric projection that is a linear projection onto this subspace. Example 8.2 Consider in the space C[0, 1], the subspace L = {ϕ ∈ C[0, 1] | ϕ|[1/2,1] ≡ const}. The operator P f associating with a function f its extension f|[1,1/2] to the interval [1/2, 1] by the value f (1/2) is a norm-1 linear projection; it is also a 1-Lipschitz selection of the metric projection onto L. Exercise 8.1 Does the space X = { f ∈ C[0, 1] | f (0) = 0} with the Chebyshev norm contain a nontrivial Chebyshev subspace? ∫1 Exercise 8.2 Let C[0, 1] be equipped with the norm  f  = ( 0 f (t)2 dt)1/2 . Does this space contain a nontrivial Chebyshev subspace? Exercise 8.3 Is the arithmetic sum of Chebyshev subspaces in C[0, 1] a Chebyshev subspace? Exercise 8.4 Does C[0, 1] contain two Chebyshev subspaces whose intersection is not a Chebyshev subspace? Consider the same problem in L 1 [0, 1].

8.3 Finite-Dimensional Chebyshev Subspaces in L 1 (µ) The problem of best L 1 -approximation dates back to P. L. Chebyshev, A. A. Markov, E. I. Zolotarev, and A. N. Korkin. These studies were continued by Jackson, S. M. Nikol’skii [448], Krein [372], Khavinson [287], [288], V. N. Nikol’skii [450], Garkavi [260], [265], Phelps [468], [468], Moroney [440], Motornyi [442], Strauss [532], Sommer [526], Kroó [377], Pinkus [476], Pinkus and Wajnryb [477], Brown [147], among others. In the majority of problems considered by these authors, the polynomials and approximants were assumed to be continuous. For a survey of results in this field, see the books [519], [520], [476]. S. Ya. Khavinson obtained a number of interesting results that extend and further develop the well-known Jackson’s theorem, which states that in L 1 [a, b], a Chebyshev system of functions is a set of uniqueness of best approximation with respect to the linear manifold of continuous functions on [a, b]. Khavinson showed that Jackson’s theorem also remains valid for the real space L 1 ([a, b], μ) with an arbitrary finite measure μ supported in [a, b]. At the same time, there are some Chebyshev systems that do not have this property for a fixed measure μ. Khavinson’s studies were continued by Kripke and Rivlin [373], who refined some of Khavinson’s results

8.3 Finite-Dimensional Chebyshev Subspaces in L 1 (μ)

161

and extended them to the case of complex-valued functions. They found that if the subspace spanned by a Chebyshev system of complex-valued functions contains the real parts of all its elements, then such a system possesses properties analogous to those of a real Chebyshev system. The next theorem, which was obtained by M. G. Krein (pronounced “crane”) in 1938 (see, for example, [152], [288]), shows that in L 1 [0, 1] (with Lebesgue measure), there are no finite-dimensional Chebyshev subspaces. Theorem 8.3 (Krein) In L 1 [0, 1], there is no finite-dimensional Chebyshev subspace of nonzero dimension. In more general spaces L 1 (Q, Σ, μ), a similar result was obtained by Phelps [468] and Moroney [440]. A stronger result is due to Garkavi (1964, [260, Lemma II]): in an atomless (Q, Σ, μ), for every n-dimensional subspace Ln ⊂ L 1 (Q, Σ, μ), there exists x ∈ L 1 (Q, Σ, μ) for which the set of nearest elements in the subspace Ln has dimension n. Moreover, if a measure μ is atomless and Ln is an n-dimensional subspace in L 1 (Q, Σ, μ), then {x | dim PLn x = n} is everywhere dense in the space L 1 (Q, Σ, μ). Theorem 8.4 (Garkavi [260]; see also [476, Theorem 2.9]) A necessary and sufficient condition that L 1 (Q, Σ, μ) contain a Chebyshev subspace of dimension n < ∞ is that the space (Q, Σ, μ) have at least n atoms. Remark 8.1 In the infinite-dimensional space L 1 (Q, Σ, μ), there exist infinitedimensional Chebyshev subspaces of infinite dimension (R. Phelps [468], [519, p. 332]) and finite defect (A. L. Garkavi [262]). The property from Krein’s and Garkavi’s theorems can be considerably strengthened. In fact, the unit sphere of the sphere L 1 [0, 1] contains ‘pads’ of every finite dimension in ever ‘direction’. n be a system of linearly independent elements Theorem 8.5 (P. Ørno) Let {gi }i=1 1 from L [0, 1]. Then there exist a function f ,  f  L 1 = 1, and a number ε > 0 such that n      μk gk  1 = 1 for all μk ∈ [0, ε], 1 ≤ k ≤ n. f + i=1

L

Proof Let M ≥ 0 and I, J be two nonintersecting subsets of the set {1, . . . , n} (possibly empty). Let the set Γ(I, J) ⊂ [0, 1] be defined by   Γ(I, J) = t | gi (t) > M, i ∈ I, g j (t) < −M, j ∈ J, |gk | ≤ M, k  I ∪ J . (8.1) It is clear that the family of measurable sets Γ(I, J) forms a partition of the interval [0, 1]. Given a fixed set Γ := Γ(I, J) of nonzero measure, consider the set VΓ of vectors from Rn defined by

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8 Existence of Chebyshev Subspaces

 ∫

∫ g1, . . . ,

VΓ := σ

 gn σ ⊂ Γ ,

σ

where σ runs over measurable subsets of Γ. According to the well-known Lyapunov’s theorem (see, for example, [406, p. 159]), the set VΓ is convex and compact. Consider the mapping from VΓ to VΓ defined by ∫ ∫   ∫   ∫ (8.2) T g1, . . . , gn := g1, . . . , gn . σ

σ

Γ\σ

Γ\σ

It is clear that this mapping is continuous. Applying the (infinite-dimensional) Brouwer fixed-point theorem to the mapping T, we have, for some σ0 ⊂ Γ, ∫ ∫ gk = gk , 1 ≤ k ≤ n. σ0

Γ\σ0

So each set Γ(I, J) can be split into two disjoint subsets Γ1 (I, J) and Γ2 (I, J) such that ∫ ∫ gk = gk , 1 ≤ k ≤ n. (8.3) Γ1 (I,J)

Γ2 (I,J)

Given c > 0, we set f := c(ϕ, − ψ, ) +

   (I,J)



|gk | (ϕI,J − ψI,J ) .

(8.4)

k ∈I∪J

Here ϕI,J is the indicator function of the set Γ1 (I, J), ψI,J is the indicator function of the set Γ2 (I, J), the sum being taken over (I, J)  (, ). It is clear that one can choose c and M so as to have  f  L 1 = 1. We set  c 1 ε := min , nM n and choose 0 ≤ μk ≤ ε, 1 ≤ k ≤ n. Hence, using (8.4), we have on Γ(, ) that  n n  h, := c + k=1 μk gk f+ μk gk = n g, := −c + k=1 μk gk k=1 see also (8.1). Further, since n  μk gk ≤ nMε ≤ c k=1

on Γ(, ), it follows that

on Γ1 (, ), on Γ2 (, );

8.3 Finite-Dimensional Chebyshev Subspaces in L 1 (μ)

∫ ∫ n  μk gk = f + k=1

Γ(,)



c+

n 

163



μk gk +

k=1

Γ1 (,)

∫



c−

n 

 μk gk ;

k=1

Γ2 (,)

in view of (8.3), the terms μk gk cancel out after integration. Similarly, for (I, J)  (, ) on Γ(I, J), we have    ⎧ ⎪ ⎨ hI,J := k ∈I (1 + μk )gk − k ∈J (1 − μk )gk + kI∪J μk gk , ⎪ f+ μk gk =    ⎪ (1 + μk )gk + μk gk , ⎪ gI,J := − (1 − μk )gk + k=1 ⎩ k ∈I k ∈J k ∈I∪J n 

where hI,J ≥ (1 − nε)M ≥ 0 gI,J ≤ (−1 + nε)M ≤ 0

on Γ1 (I, J), on Γ2 (I, J),

by the choice of ε and in view of (8.1). Therefore, ∫ ∫ ∫ n  μk gk = hI,J − f + k=1

Γ(I,J)

Γ1 (I,J)

(8.5) (8.6)

gI,J .

Γ2 (I,J)

After termwise integration on the right, the terms involving μk gk cancel out in view of (8.3). Finally, taking into account (8.3) and (8.4), we find that ∫ 1 n n  ∫   f + = f + μ g μk gk k k 0

∫ =

Γ(,)

k=1

c+

So, for all μk ∈ [0, ε]

 (I,J)(,)

(I,J)Γ(I,J)

∫



Γ(I,J) k ∈I∪J

k=1



|gk | =

∫

1

| f | = 1.

0

n    f + μk gk  L 1 = 1, i=1

which proves the theorem.



Krein’s theorem (Theorem 8.3) follows from Ørno’s theorem as a corollary. The next result [453] can be obtained by a similar argument (for a proof, see, for example, [152]). Theorem 8.6 (P. Ørno) There exists a separable reflexive Banach space without finite-dimensional Chebyshev subspaces. Another classical extension of Krein’s theorem 8.3 is as follows (see [453], [152]). Theorem 8.7 (P. Ørno–Yu. A. Brudnyi–E. A. Gorin) Every Chebyshev set in L 1 [0, 1] is either a singleton or is infinite-dimensional.

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8 Existence of Chebyshev Subspaces

The next theorem [26] supplements and extends the above result for the case of finite-dimensional suns in the space L 1 [0, 1]. Theorem 8.8 Every finite-dimensional sun in L 1 [0, 1] is convex. To prove Theorem 8.8, we need one result of independent interest (Theorem 8.9), which shows that if  is an arbitrary finite-dimensional subspace in L 1 [0, 1], dim   1, then there exists a set of translations of , dense in the unit ball B ⊂ L 1 [0, 1], that intersect the unit ball of L 1 [0, 1] in smooth sets. Theorem 8.9 Let 0 ⊂ L 1 [0, 1] be an arbitrary finite-dimensional subspace of dimension  1. Then there exists a set of translations of 0 by points from an everywhere dense set in the unit ball B of L 1 [0, 1] such that each translation intersects the unit sphere S of L 1 [0, 1] only at smooth points of S. The next remark shows that the condition that a subspace is finite-dimensional is essential in Theorem 8.9. Remark 8.2 There exists an infinite-dimensional subspace in L 1 [0, 1] such that every ˚ intersects the translation of this subspace (by a vector from the open unit ball B) unit ball B of L 1 [0, 1] in a nonsmooth set. For a proof, it suffices to consider some subspace containing the subspace L0 of functions that vanish a.e. on one-half of the interval [0, 1] (say on [0, 1/2]). Next, every point  s of the unit sphere supporting every translation of L0 is a function that vanishes a.e. on the other half of the interval [0, 1]. Indeed, if there were a point s ∈ S not vanishing a.e. on [1/2, 0], then for the function  0, t ∈ [0, 1/2], e(t) = s(t), t ∈ (1/2, 1], e ∈ L0, we would have s − te < 1 for small t > 0, and hence s could not be a point s is a nonsmooth point of the of support of a translation of L0 . Correspondingly,  sphere S due to the well-known characterization of smooth points of the L 1 -sphere (see, for example, [345, Sect. 26.5]): a point f0 of the unit sphere S of L 1 is a smooth point of S if and only if f0 (t)  0 a.e. Theorem 8.9 is a consequence of the following auxiliary lemma and the corollary thereof. Lemma 8.1 Let  be a finite-dimensional subspace in L 1 [0, 1] of dimension ≥ 1. Then for every ε ∈ (0, 1), there exists a function ϕ ∈ L 1 [0, 1], ϕ  ε, such that A = {t | (y0 + ϕ)(t) = 0} is a null set for all y0 ∈ . Proof (of Lemma 8.1) We argue by induction. (1◦ ) Let B(0, 2) be the closed ball with centre at 0 and radius 2. For the subspace 1 := , there exists a finite partition of the interval [0, 1] into measurable subsets y0 that is {B1j } such that for every function y0 ∈ 1 ∩ B(0, 2), there exists a function  constant on every B1j and for which  y0 −  y0  < δ1 :=

ε2 164



ε2 . 162

Let B1j = Fj  Fk,

8.3 Finite-Dimensional Chebyshev Subspaces in L 1 (μ)

165

μ(F ) = μ(Fj), and define x1 ∈ L ∞ [0, 1] to be 1 on Fj and −1 on Fj. It is clear that ∫ j x (t) y0 (t) dt = 0 for every set C that is a union of sets from {B1j }. C 1  = {t | | y0 (t) + ϕ1 (t)| < ε4 }, and consider an arbitrary We set ϕ1 (t) := ε3 x1 (t), A function y(t) = y0 (t) + ϕ1 (t) from ( + ϕ1 ) ∩ B(0, 2). We have ∫ ∫ ε  ε  μ( A) + δ1  − δ1 . x1 (t)y(t) dt  x1 (t)( y0 (t) + ϕ1 (t)) dt − δ1 = μ( A) 4 3   A A  ≤ 2δ1  ε/2. By Chebyshev’s inequality, we have This implies that μ( A) ε/12 μ{t |  y0 −  y0  ≥ ε8 }  (8/ε) y0 −  y0   ε4 . Hence the measure of the set ε ε A1 := {t | |y0 (t) + ϕ1 (t)| < 8 } is smaller than 3ε 4 . We put Δ1 = 16 . ◦ (2 ) Assume that we have already constructed a subspace k (k  2) containing  and the functions {ϕ1, . . . , ϕk−1 }. There exists a finite partition of the interval [0, 1] into measurable sets {Bkj } such that for every function y0 ∈ k ∩ B(0, 2), there exists a function  y0 that is constant on each Bkj and for which  y0 −  y0  < δk := ε 2 Δ2k−1 162(k+1)

ε 2 Δ2k−1

16(k+1)

2



. There exists a function xk (t) ∈ L ∞ [0, 1] assuming the value 1 or −1 at each point and such that ∫ xk (t) y0 (t) dt = 0 C

for every set C from {B1j }. xk (t), D := {t | | y0 (t) + ϕk (t)| < εΔ4k−1 }, and We set ϕk (t) := εΔ3k−1 k k arbitrary function y(t) = y0 (t) + ϕk (t) from (k + ϕk ) ∩ B(0, 2). Hence

consider an

∫ ∫ εΔk−1  μ(D) + δ  x (t)y(t) dt x (t)( y (t) + ϕ (t)) dt − δk k k k 0 k 4k D D εΔk−1 = μ(D)−δk . 3k

As a result, μ(D) ≤

2δk Δ k−1 (ε(3−k −4−k ))

 ε/2k . By Chebyshev’s inequality, we have

 εΔk−1 εΔk−1  (8k /(εΔk−1 ) y0 −  μ t |  y0 −  y0  ≥ y0   . k 8 8k } is smaller than 3ε . Hence the measure of the set Ak := {t | |y0 (t) + ϕ1 (t)| < εΔ8k−1 k 4k εΔ k−1 We put Δk = 16k . N (3◦ ) So for every y0 ∈ , the value of the function y0 + k=1 ϕk is smaller than εΔ N −1 on a set of measure  43εN . Since 8N ∞ ∞   εΔ N −1 εΔk−1 εΔ N 16  ϕk (t)   , N k 16 15 16 N 3 k=N +1 k=N +1

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8 Existence of Chebyshev Subspaces

 εΔ N −1 we see that the values of the function y0 + ∞ k=1 ϕk are smaller than 16 N on a set  of measure  43εN . Setting ϕ := ∞ k=1 ϕk , we finally get that for all y0 ∈ , the set of zeros of the function y0 + ϕ has measure zero. Lemma 8.1 is proved.  Corollary 8.1 Let 0 be a finite-dimensional subspace in L 1 [0, 1], dim 0  1. Then for every ε ∈ (0, 1) and a finite set of functions { f1, . . . , fn } ⊂ L 1 [0, 1], there exists a function ϕ ∈ L 1 [0, 1], ϕ  ε, such that for every i = 1, n, the section of the unit sphere S of L 1 [0, 1] by the plane i := 0 + fi + ϕ (i = 1, n) intersects the unit sphere of L 1 [0, 1] in a smooth set, provided that i intersects the interior B˚ of the unit ball B ⊂ L 1 [0, 1]. Proof (of Corollary 8.1) Here we use the fact that every point y(t) of the sphere S for which the set of zeros is a null set is a smooth point of S (see [345, Sect. 26.5]). We set  := lin0, f1, . . . , fn  and apply Lemma 8.1 to . All the points of the section ( + ϕ) ∩ S are smooth points of S. Hence the sections of S by the planes i also consist only of smooth points of S, provided that i intersects the interior of the unit ball B.  Remark 8.3 The results of this section extend verbatim to spaces L 1 ([0, 1], μ) with atomless σ-finite measure. On the other hand, if the measure has atoms and if dim L 1 > 2, then all translations of a two-dimensional subspace will give nonsmooth sections of the unit L 1 -ball (as in the C(Q)-case in [26]). Now Theorem 8.8 follows from Theorems 8.9, 7.34 and the well-known straightforward result that every sun in a smooth space is convex (see, for example, Theorem 6.1 or [24, Sect. 4]). In the opposite direction, we mention that in every normed linear space, every convex proximinal set (in particular, a convex closed finite-dimensional set) is a strict sun. The following result is a corollary to Theorem 8.8. Corollary 8.2 No nonconvex subset of the set of rational fractions is a sun in L 1 [0, 1].

Chapter 9

Efimov–Stechkin Spaces. Uniform Convexity and Uniform Smoothness. Uniqueness and Strong Uniqueness of Best Approximation in Uniformly Convex Spaces

In this chapter, we consider the class of Efimov–Stechkin spaces (reflexive spaces with the Kadec–Klee property). This class is a natural class of spaces in which sets with ‘good structure’ have ‘many’ points of approximative compactness (points of stability of the metric projection operator). In particular, in such spaces every nonempty closed convex set is approximatively compact (and hence an existence set); moreover, if the space is in addition rotund (strictly convex), then every nonempty closed convex set is a Chebyshev set. The Efimov–Stechkin spaces are characterized by the following important property: every hyperplane in such a space is approximatively compact (see Theorem 9.1 below). The class of complete uniformly convex spaces constitutes the well-known subclass of Efimov–Stechkin spaces. We start with several approximative–geometric properties that characterize Efimov–Stechkin spaces (Theorem 9.1). Further, in Sect. 9.2, we describe some classical properties of uniformly convex spaces and their dual spaces (uniformly smooth spaces). In Sect. 9.3 and Sect. 9.4, we consider problems of uniqueness, strong uniqueness, and stability in such spaces.

9.1 Efimov–Stechkin Spaces In their study of Chebyshev sets, Efimov and Stechkin [224] proved that in a uniformly convex Banach space, a sequentially weakly closed set is approximatively compact (see Theorem 4.1 above). Efimov–Stechkin spaces were introduced by I. Singer to characterize the spaces in which every sequentially weakly closed set is approximatively compact. As before, let X be a real Banach space, X ∗ a dual space, S = {x |  x = 1}, ∗ S = {x ∗ ∈ X ∗ |  x ∗  = 1}. Definition 9.1 A Banach space X is called a Efimov–Stechkin space if (ES)

for all xn ∈ S and f ∈ S ∗ such that f (xn ) → 1, the sequence (xn ) has a convergent subsequence.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_9

167

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9 Efimov–Stechkin Spaces. Uniform Convexity . . .

The (ES) property is shared by all finite-dimensional spaces, Hilbert spaces, L p - and  p -spaces, 1 < p < ∞, and also, more generally, by complete uniformly convex spaces. Sometimes Efimov–Stechkin spaces are called reflexive spaces with the Kadec–Klee property (or with the Radon–Riesz property). The closed unit ball in an infinite-dimensional Efimov–Stechkin space gives an example of a bounded convex approximatively compact set that not boundedly compact. By James’s theorem (Theorem 1.5), every Efimov–Stechkin space is reflexive. An example of a reflexive space that is not a Efimov–Stechkin space was constructed in [359] — this is the space W21 of absolutely continuous real functions x on [0, 1] with ∫ 1  1/2 (x (t))2 dt 0 for all n. Setting 1 1 xn, x, yn = n = 1, 2, . . . , y= f (xn ) f (x) we have f (yn ) = 1, n = 1, 2, . . . , f (y) = 1,  yn  →

 x = 1; | f (x)|

(9.1)

that is, (yn ) weakly converges to y. Since for the closed hyperplane H = {z ∈ X | f (z) = 1} we have ρ(0, H) = 1/ f  = 1 (see Proposition 1.8)), it follows from (9.1) that (yn ) is a minimizing sequence for 0 from H. By the assumption of e), there exists a subsequence (ynk ) ⊂ (yn ) that converges to an element z ∈ H. Since (ynk ) is weakly convergent to y, we have z = y. As a corollary, xnk = f (xnk )ynk → f (x)y = x.

(9.2)

Singer [517] showed that in every Banach space Y , the condition (KK) is equivalent to the condition (KK1 )

if (xn ) weakly converges to x and  xn  →  x, then  xnk − x → 0 for some subsequence (xnk ).

Let us prove this result. The implication (KK)⇒(KK1 ) is clear. To prove the converse implication, we assume that there exist a sequence (xn ) ⊂ Y and a point x ∈ Y such w that xn → x,  xn  → x, but  xn −x → 0. Then there exist ε0 > 0 and a subsequence (xnk ) ⊂ (xn ) such that  xnk − x ≥ ε0,

k = 1, 2, . . . .

(9.3)

w

However, because  xnk  →  x and xnk → x, it follows by the condition (KK1 ) that there exists a subsequence (xnk m ) ⊂ (xnk ) such that  xnk m − x → 0, which contradicts (9.3). Now assertion b) follows from (9.2) and the equivalence (KK1 )⇔(KK). The implication a)⇒(KK1 ) is evident. Let us show that b)⇒c). Let M be a sequentially weakly closed set in X. Let x ∈ X and let yn be a minimizing sequence from M for x. Then  x − yn  → ρ(x, M); that is, the sequence (yn ) is bounded. Since X is reflexive, by James’s theorem (yn ) contains a subsequence (ynk ) that is weakly convergent to some y ∈ M. Now choose f ∈ S ∗ , f (x − y) =  x − y. We have  x − y = f (x − y) = lim f (x − ynk ) ≤ lim  x − ynk  k→∞

= ρ(x, M) ≤  x − y,

k→∞

9.2 Uniformly Convex Spaces

171

which shows that  x − ynk  →  x − y. Since the sequence (x − ynk ) also weakly converges to (x − y), the condition of assertion a) guarantees that there exists a subsequence nkm for which x − ynk m → x − y; that is, ynk m → y. This shows that M is approximatively compact. The implication b)⇒c1 ). Since X is reflexive and the sphere S is bounded, the sequential weak closeness of a set E ⊂ S implies that E is weakly compact. In turn, by (KK), this is equivalent to saying that E is norm-compact. The implication c1 )⇔d1 ) follows from the following well-known result (see, for example, Deutsch [190]), which states, in particular, that a Banach space is reflexive if and only if each of its weakly closed subsets is boundedly weakly compact (each sequentially weakly closed subset is boundedly sequentially weakly compact).  We also note that every subspace and every quotient space of a Efimov–Stechkin space are Efimov–Stechkin spaces. Let X be a given normed linear space. Denote by (P AC ) the class of all sets M ⊂ X for which the set PM x of all nearest points is connected for all points x ∈ AC(M) (for points of approximative compactness of the set M). The following inclusions hold: (P) ⊂ (P0 ) ⊂ (P AC ), ˚ ⊂ (P AC ). (B) ⊂ ( B) We note another characterization of Efimov–Stechkin spaces (in Theorem 9.2, (F ) is the class of nonempty closed sets). Theorem 9.2 (I. G. Tsar’kov) For a Banach space X, the following properties are equivalent: (a) X is a Efimov–Stechkin space; ˚ (b) (P AC ) ∩ (F ) ⊂ ( B); ˚ (c) (P AC ) ∩ (F ) = ( B) ∩ (F ); (d) for every nonempty closed set M, the set AC(M) of points of its approximative compactness is nonempty and connected. Exercise 9.1 Let X = X1 × X2 , where (X1,  · 1 ), (X2,  · 1 ) are Efimov–Stechkin spaces. We equip X with the norm (x1, x2 )  := x1  X1 + x2  X2 . Check whether X is a Efimov–Stechkin space. Consider the same question for the norm 2 + x  2 )1/2 . (x1, x2 )  := ( x1  X 2 X2 1

9.2 Uniformly Convex Spaces The unit sphere of a normed space completely determines all the properties of that space; however, the unit sphere is hard to visualize. Hence many researches have

172

9 Efimov–Stechkin Spaces. Uniform Convexity . . .

aimed at finding simple computable and pictorial geometric–numeric characteristics of the ball, which, of course, are incapable of completely describing the space, but still are related to some of its properties. As such characteristics, we first mention the modulus of convexity (introduced by J. Clarkson) and the modulus of smoothness (introduced by M. Day). Definition 9.2 Let X be a normed linear space. For every ε ∈ (0, 2], define the (Clarkson) modulus of convexity of the space X as follows: x + y       δX (ε) = inf 1 −    x, y ∈ BX ,  x − y ≥ ε . 2 A norm  ·  is said to be uniformly convex if δX (ε) > 0 for all ε ∈ (0, 2]. In this case, the space X = (X,  · ) is said to be uniformly convex. (Recall that BX = B := B(0, 1) is the unit ball of X.) Note that δX (ε) = inf{δY (ε) | Y , where Y is a two-dimensional subspace of X }. It is clear that the function δX is nondecreasing and X is uniformly convex if and only if δX (ε) > 0 for all ε > 0. Lemma 9.1 Let X be a normed linear space and let δX (ε) be the modulus of convexity of X. Then x + y     δX (ε) = inf 1 −   | x, y ∈ SX ,  x − y = ε . 2 In other words, Lemma 9.1 shows that in a uniformly convex space X, if x, y ∈ S,  x − y = ε, then y − x x + y  y − x  ε      (9.4)   = x +  ≥  x −   ≥ 1− , 2 2 2 2 which gives

ε ∀ε ∈ [0, 2]. 2 From (9.4), it follows that the above definition of a uniformly convex space coincides with Definition 4.6. δ(ε) ≤

Remark 9.1 From Definition 9.2, it follows that if  x ≤ r,  y ≤ r, r > 0, then x + y   x − y 

  . (9.5)  ≤ r 1 − δX  2 r Indeed, to prove (9.5), it suffices to put ξ = x/r, η = y/r,  ε := ξ − η ≤ 2, and  take  apply Definition 9.2, which gives δX (ξ − η) ≤ 1 −  ξ+η 2 , and hence (9.5). Remark 9.2 Let ε = εX (δ) be the infimum of the diameters of sets cut from the unit sphere (ball) by hyperplanes defined by functionals of unit norm and that lie at the distance 1 − δ from 0. Then the function inverse to εX (δ) coincides with the modulus of convexity δX (ε) of the space X. This result was proved by M. I. Kadec (1955).

9.2 Uniformly Convex Spaces

173

Proof (of Lemma 9.1) To begin with, we note that if x, y ∈ X,  x − y ≥ ε, then on the interval [x, y] there exist points x , y  such that x + y x+y = 2 2

and

 x  − y   = ε.

So it suffices to consider the case x, y ∈ B,  x − y = ε. We need to show that sup{ x + y | x, y ∈ B,  x − y = ε}

(9.6)

= sup{ x + y | x, y ∈ S,  x − y = ε}. Without loss of generality, we assume that X is two-dimensional, so that the suprema in the previous formula are attained. Assume that the supremum on the left is attained at points u0, v0 ∈ B. We claim that u0, v0 ∈ S. Assume to the contrary that v0  < 1. We set A := {w ∈ B | w − u0  = ε}. Let x ∗ be a functional from the dual unit sphere for which x ∗ (u0 + v0 ) = u0 + v0  (such a functional exists by the corollary to the Hahn–Banach theorem). Hence for w ∈ A, we have x ∗ (u0 + w) ≤ u0 + w ≤ u0 + v0  = x ∗ (u0 + v0 ), because the maximum in (9.6) is attained on u0, v0 . This inequality shows that the functional x ∗ has a local maximum on A at v0 . We have v0  < 1, and hence x ∗ attains its norm on v0 − u0 , which gives x ∗ (v0 − u0 ) = v0 − u0  = ε. Hence 1 u0 + v0  + v0 − u0  2 1 = x ∗ (u0 + v0 ) + x ∗ (v0 − u0 ) = x ∗ (v0 ) < 1. 2

u0  ≤

We claim that this is impossible. Setting δ :=

 1 min 1 − u0 , 1 − v0  > 0, 2

we have u  := u0 + δ(u0 + v0 ) ∈ B, v  := v0 + δ(u0 + v0 ) ∈ B, u  − v   = ε, and u  + v   = (1 + 2δ)u0 + v0  > u0 + v0 , contradicting the maximality of u0 + v0 . Lemma 9.1 is proved. Note that if  x =  y = 1 and  x − y = ε, then y − x x + y  y − x  ε        = x +  ≥  x −  =1− , 2 2 2 2 and hence δX (ε) ≤ ε/2 for all ε ∈ (0, 2]. Remark 9.3 It is easily seen that the spaces n1 , n∞ are not uniformly convex. As a corollary, the spaces c0 ,  1 ,  ∞ are also not uniformly convex.

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9 Efimov–Stechkin Spaces. Uniform Convexity . . .

Remark 9.4 Note also that each Hilbert space H is uniformly convex. Indeed, from the parallelogram identity, which characterizes the inner product spaces, we have, for all ε ∈ (0, 2], x + y       x, y ∈ S,  x − y = ε δH (ε) = inf 1 −   2   x − y 2    1 1    = inf 1 −  x 2 +  y 2 −    x, y ∈ S,  x − y = ε 2 2 2  2 ε =1 − 1 − > 0, (9.7) 4 inasmuch as  x + y 2 = 4 −  x − y 2 for  x =  y = 1 by the parallelogram identity. Moreover, the classical G. Nördlander’s theorem (see, for example, [198], [229]) states that a Hilbert space is the ‘most uniformly convex’ among all (uniformly  convex) Banach spaces; that is, δX (ε) ≤ 1 − 1 − ε 2 /4 for every uniformly convex space X (cf. (9.7)). Remark 9.5 A uniformly convex space is strictly convex, because for a uniformly convex space, the difference 1 −  x+2 2  is uniformly estimated from below for all x, y,  x =  y = 1,  x − y ≥ ε. However, there exist strictly convex spaces that are not uniformly convex. A compactness argument shows that in finite-dimensional spaces, the strict convexity (rotundity) of a space is equivalent to its uniform convexity. Theorem 9.3 (J. Clarkson) Let (Ω, μ) be a measurable space, p ∈ (1, ∞). Then the space L p (Ω, μ) is uniformly convex. We need the following auxiliary lemma. ˜ > 0 such that Lemma 9.2 Let p ∈ (1, ∞) and let ε > 0. Then there exists δ˜ = δ(ε)  u + v p  p p   ˜ |u| + |v|  < (1 − δ)  2 2 whenever u, v ∈ R are such that |u − v| ≥ ε max{|u|, |v|} > 0. Proof (of Lemma 9.2) It can be assumed that |u| = max{|u|, |v|}. By homogeneity of the inequality from the hypothesis of the lemma, it can be assumed that x = 1 and 1 − ε ≥ y ≥ 0. We have 1 + vp   1 + v  p  p  1 + v  p−1 p = > v p−1 = , 2 2 2 2 2 p

1+v p y ∈ (0, 1). As a result, f (v) = 1+v 2 − ( 2 ) is a decreasing function on [0, 1]. It follows that f (v) ≥ f (1 − ε) > f (1) = 0 for all v ∈ (0, 1 − ε), which gives the ˜ existence of the required δ. 

9.2 Uniformly Convex Spaces

175

˜ where δ˜ from Lemma 9.2 is Proof (of Theorem 9.3) For ε ∈ (0, 2], we set δ = δ, constructed from ε · 4−1/p . Let x, y ∈ L p (Ω, μ),  x − y ≥ ε and  x =  y = 1, where  ·  is the standard norm on L p (Ω, μ). Setting

  M := ω | ε p |x(ω)| p + |y(ω)| p ≤ 4|x(ω) − y(ω)| p , ∫

we claim that max

∫ |x| p dμ,

M

 |y| p dμ ≥

M

εp 21/p+1

.

(9.8)

Putting off the proof of inequality (9.8), we now show how the assertion of the theorem follows from it. Using the convexity of the function | · | p , Lemma 9.2, and inequality (9.8) gives ∫  |x(ω)| p + |y(ω)| p  x(ω) + y(ω)  p  −  dμ 2 2 ∫  |x(ω)| p + |y(ω)| p  x(ω) + y(ω)  p  − ≥  dμ 2 2 ∫M  δε p |x(ω)| p + |y(ω)| p  dμ ≥ 1/p+1 . ≥ δ 2 2 M It follows that ∫ ∫   εp |x(ω)| p + |y(ω)| p  x(ω) + y(ω)  p dμ − δ 1/p+1  dμ ≤  2 2 2 εp ≤ 1 − δ 1/p+1 . 2 This implies that

 ε p  1/p 1  x + y ≤ 1 − δ 1/p+2 , 2 2 p

which gives δL p (ε) ≥ 1 − (1 − δ 21/εp+2 )1/p > 0. To prove (9.8), consider the complement M c of the set M. We have ∫ ∫  εp p |x(ω)| p + |y(ω)| p dμ |x(ω) − y(ω)| dμ ≤ 4 Mc Mc ∫  εp εp |x(ω)| p + |y(ω)| p dμ ≤ ≤ . 4 Mc 2 ∫ Hence  x − y p ≥ ε p , and so M |x(ω) − y(ω)| p dμ ≥ ε p /2. As a result,  x − y M ≥ ε , where  ·  M is the norm of the space L p (M, μ). Finally, we have 21/ p max{ x M ,  y M } ≥ Theorem 9.3 is proved.

ε 1 · 1/p . 2 2 

Theorem 9.4 Let X be a Banach space. Then the following conditions are equivalent: (i) X is uniformly convex;

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9 Efimov–Stechkin Spaces. Uniform Convexity . . .

(ii) if xn, yn ∈ X, n ∈ N, a sequence (xn ) is bounded, and lim (2 xn  2 + 2 yn  2 −  xn + yn  2 ) = 0,

n→∞

(9.9)

then lim  xn − yn  = 0;

n→∞

(iii) if xn, yn ∈ B, n ∈ N, and if  xn + yn  → 2, then  xn − yn  → 0. Remark 9.6 The condition that the sequence (xn ) in Theorem 9.4 is bounded cannot be relaxed. Proof (of Theorem 9.4) The implication (ii)⇒(i) is clear. Let us prove the implication (iii)⇒(ii). Let (xn ), (yn ) ⊂ X, let a sequence (xn ) be bounded, and let condition (9.9) hold. The estimate ≥ 2 xn

2

2 xn  2 + 2 yn  2 −  xn + yn  2 + 2 yn  2 − ( xn  +  yn )2 = ( xn  −  yn )2 ≥ 0

shows that  xn  −  yn  → 0 and that the sequence (yn ) is bounded. Passing to subsequences, we can assume that  xn  → a,  yn  → a. If a = 0, then the assertion is satisfied.   Let a > 0. Then  xn + yn  → 2a, and hence  xxnn  + yynn   → 2. By (iii), we have  x yn   n −  → 0, and hence  xn − yn  → 0.   xn   yn  Let us prove that (i)⇒(iii). Assume to the contrary that (iii) is not satisfied. Then there exist sequences (xn ) and (yn ) such that  12 (xn + yn ) → 1, but  xn − yn  → 0. This means that there exists ε > 0 such that lim  xn − yn  ≥ ε, but  12 (xn + yn ) → 1 (here we again pass to subsequences if required). This shows that X cannot be uniformly convex. We claim that (iii)⇒(i). Assume that X is not uniformly convex. Then we can find sequences (xn ) and (yn ) from the unit sphere S and a number ε > 0, for which there is no δ > 0 for which from the inequality  xn − yn  ≥ ε it follows that  12 (xn + yn ) < 1 − δ for each n ∈ N. Hence  12 (xn + yn ) → 1, but at the same  time,  xn − yn  → 0, which contradicts assertion (iii). We note without proof the following fact (see, for example, [83, Chap. II, Sect. 1]). Theorem 9.5 Let 1 < p < ∞. A space X is uniformly convex if and only if for every ε > 0, there exists a number δ > 0 such that for every x, y ∈ X,  x + y p  1    x − y   ·  x p +  y p .  ≤ 1−δ  2 max{ x,  y} 2 Theorem 9.6 Every subspace and quotient space of a uniformly convex space is uniformly convex.

9.3 Uniqueness of Best Approximation by Convex Closed Sets . . .

177

A proof of Theorem 9.6 can be found in the book [83, Chap. II, Sect. 1]. Example 9.1 (see [229]) On the space C[0, 1], consider the norm | · | defined as follows: 2 +  f 22, | f | 2 =  f ∞ where  · ∞ is the standard uniform norm on [0, 1] and  · 2 is the standard meansquare norm of the space L 2 [0, 1]. It is easily checked that the norm | · | is strictly convex. Consider the function fn = 1 and the function gn defined, for each n, as a broken line connecting in succession the points (0, 0), (1/n, 1), and (1, 1). Consequently, the sequence of pairs of functions fn, gn does not satisfy the condition of uniform convexity. So the space (C[0, 1], | · |) is not uniformly convex.

9.3 Uniqueness of Best Approximation by Convex Closed Sets in Complete Uniformly Convex Spaces In this section, we prove a well-known result that extends the classical Theorem 3.1 from the Hilbert space setting to the case of uniformly convex spaces. Theorem 9.7 In a complete uniformly convex space, every closed convex set is a Chebyshev set. Remark 9.7 Note that the (single-valued) metric projection operator onto a convex closed set in a uniformly convex space need not be linear (even in the case of approximation by subspaces). Remark 9.8 For strictly convex spaces, the conclusion of Theorem 9.7 may fail to hold. Indeed, V. S. Balaganskii [55] constructed an example of a strictly convex Banach space X (a renormed subspace of c0 of finite codimension) with Fréchetdifferentiable norm that contains an antiproximinal closed convex bounded body. Proof (of Theorem 9.7) Let C be a closed convex set in a uniformly convex space X. We need to show that for every x ∈ X, the set of nearest points from C for x is a singleton. Above (Theorem 3.1), it was shown that this is so in Hilbert spaces. Let x ∈ X, x  C. Given n ≥ 1, we set   1 An := z ∈ C   x − z ≤ ρ(x, C) + . n Setting p = 2 in Theorem 9.5, we see that if z1, z2 ∈ A, then   1   z1 + z2 2 z1 − z2    x − z1  2 +  x − z2  2 .  ≤ 1−δ x − 2 max( x − z1 ,  x − z2 ) 2 Since C is convex, we have d := ρ(x, C). As a result,

(z1 +z2 ) 2

∈ C, and hence  x −

z1 +z2 2 2 

≥ d 2 , where

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9 Efimov–Stechkin Spaces. Uniform Convexity . . .

 d2 ≤ 1 − δ

  1 2 z1 − z2  · d+ . max( x − z1 ,  x − z2 ) n

Hence sup

z1,z2 ∈ A n

z1 − z2  → 0,

n → ∞,

that is, the diameter of An tends to zero. Since the space is complete, there exists a point y lying in the intersection of all the An . It is clear that y ∈ PC x. That the best approximation is unique is secured by strict convexity of the space X (see Proposition 1.4). Theorem 9.7 is proved.  Remark 9.9 A more transparent proof can be derived from the Hahn–Banach theorem and Remark 9.2. Indeed, there exists a functional x ∗ ∈ S ∗ that separates the ball B(x, ρ(x, C)) and the set C. Hence An lies in the set z ∈ B(x, ρ(x, C) + 1/n) |  x ∗ (z)  1 ; in view of Remark 9.2, the diameter of such a set tends to zero as n → ∞.

9.4 Strong Uniqueness in Uniformly Convex Spaces Let X be a uniformly convex space of dimension ≥ 2 and let L be a subspace of X. According to Sect. 9.3, the subspace L is a set of uniqueness (if L is proximinal, then L is a Chebyshev sun). Recall (see Definition 2.2) that the metric projection PM onto the set   M ⊂ X is said to be strongly unique at x ∈ X if PM x = {y} is a singleton and there exists a number γ > 0 such that for every y  ∈ M,  x − y   −  x − y ≥ γ y − y  .

(9.10)

The smallest constant γ in (9.10) is called the strong uniqueness constant. The strong uniqueness property in its classical form (9.10) occurs very rarely. For example, it was noted above that in L p -spaces (1 < p < ∞), the metric projection onto a proper subspace V ⊂ L p fails to possess the strong uniqueness property at any point. Consider the strong uniqueness problem in uniformly convex spaces in its nonclassical form. Here we speak about the existence of a nonnegative strictly increasing function ϕ on R+ (which can depend on x and M, and hence on y ) such that  x − y   −  x − y ≥ ϕ( y − y  ) for all y ∈ M. The following result can be found, for example, in [376]. Theorem 9.8 Let L be a linear subspace of a uniformly convex space X and for x ∈ X, let a point y ∗ be an element of best approximation from L. Then for all y ∈ M, we have

9.4 Strong Uniqueness in Uniformly Convex Spaces

179

 x − y −  x − y ∗  ≥  x − y δ

  y − y∗    x − y

.

Remark 9.10 We have  y − y ∗  ≤  x − y +  x − y ∗ , and hence  x − y +  x − y ∗   x − y∗   y − y∗  ≤ =1+ < 2,  x − y  x − y  x − y  ∗ because  x − y ∗  ≤  x − y. So the function δ y−y x−y  is well defined. Remark 9.11 It is known that the modulus of convexity δ( · ) is a nondecreasing function. If  x − y − ||x − y ∗  ≤ σ, then  x − y −  x − y ∗  ≥  x − y ∗  δ



 y − y∗   ,  x − y∗  + σ

because  x − y ≥  x − y ∗ . Proof (of Theorem 9.8) Since L is a linear space, it suffices to show that   y  z − y − z ≥ z − y δ z − y for every z for which 0 is an element of best approximation from L. According to (9.5), if u, v ≤ r, r > 0, then u + v   u − v 

  .   ≤ r 1−δ 2 r Setting u = z, v = z − y, we have r = z − y ≥ z and  2z − y    y 

  .   ≤ z − y 1 − δ 2 z − y Next 0 ∈ PL z, and so

As a result,

 2z − y       = z −  2

y   ≥ z. 2

  y 

z ≤ z − y 1 − δ , z − y

which gives the required inequality.



A similar argument proves the following corollary. Corollary 9.1 Let L be a linear subspace of a uniformly convex space X. Assume that δ(ε) ≥ Cε q for all ε ∈ (0, 2] and q ≥ 1. Next, let y ∗ be the element of best approximation to x from L. Then

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9 Efimov–Stechkin Spaces. Uniform Convexity . . .

 x − y q −  x − y ∗  q ≥ C y − y ∗  q for all y ∈ L, y  y ∗ . Moreover, if  x − y −  x − y ∗  ≤ σ, then

 x − y −  x − y ∗  ≥ Ck  y − y ∗  q,

where k :=  x − y ∗ /( x − y ∗  + σ)q . Proof The first inequality is a direct consequence of Theorem 9.8. For q = 1, there is nothing to prove. Let q > 1. We have   y − y∗    x − y −  x − y ∗  ≥  x − y  x − y   y − y∗   q ≥  x − yC .  x − y It remains to multiply this inequality by  y − x q−1 and use the fact that  x − y ∗  ≤  x − y. The second inequality is secured by Remark 9.11.  Remark 9.12 O. Hanner showed that the modulus of convexity δp ( · ) of the spaces L p and  p is estimated as follows: ⎧ (p − 1)ε 2 ⎪ ⎪ + o(ε 2 ), ⎨ ⎪ δ p (ε) = ε p 8 ⎪ ⎪ ⎪ p + o(ε p ), ⎩ p2 In particular, for L p ,

1 < p < 2, 2 ≤ p < ∞.

δ p (ε) ≥ cp εr

for all ε ∈ [0, 2], where r := max{2, p}, ⎧ p−1 ⎪ ⎪ ⎪ ⎨ 8 , ⎪ cp = 1 ⎪ ⎪ ⎪ ⎪ p2 p , ⎩

1 < p < 2, 2 ≤ p < ∞.

9.5 Uniformly Smooth Spaces Definition 9.3 Let X be a Banach space. Given τ > 0, define the modulus of smoothness of the space X as follows:    x + τh +  x − τh − 2   ρX (τ) := sup   x=h=1 . 2

9.5 Uniformly Smooth Spaces

181

A space X is said to be uniformly smooth if lim τ↓0

ρX (τ) = 0. τ

Since 2 x = (x + τh) + (x − τh) ≤  x + τh +  x − τh, the function ρX ( · ) is nonnegative. A space X is uniformly smooth if for every ε > 0, there exists δ > 0 such that for every x ∈ S and y ∈ δB,  x + y +  x − y ≤ 2 + ε y. It is clear that a subspace of a uniformly smooth space is uniformly smooth. It is also easily checked that a uniformly smooth space is smooth (that is, at each point of its unit sphere, the tangent hyperplane is unique). Definition 9.4 Let E ⊂ X, and let x0 be an interior point of E, f : E → Y . The function f is said to be Fréchet differentiable at x0 if Δ f := f (x) − f (x0 ) can be written as A(x − x0 ) + α(x, x0 ) x − x0 , where α(x, x0 ) = o(1) for x → x0 , and A is a continuous linear operator (the derivative f (x0 ) of f at x0 ). A norm f (·) =  ·  is said to be differentiable if it is differentiable at every point x ∈ X \ {0}, or equivalently, if it is differentiable at each point x ∈ S. Uniform differentiability of a norm on the sphere S means that the mapping α(x, x0 ) tends to zero uniformly as  x − x0  → 0 on the set X × S. p

Example 9.2 Let L p = L p (Ω, μ), 1 < p < ∞, F( f ) := p1  f  L p , f ∈ L p , f  0. Let us show that ∫ | f (x)| p−1 sign f (x)h(x) dμ(x). F ( f )[h(t)] = Ω

We need the following well-known inequalities:  α   |a| sign a − |b| α sign b 2|a − b| α ∀ α ∈ (0, 1], a, b ∈ R and

 β   |a| sign a − |b| β sign b  β(|a| + |b|)β−1 |a − b|

for all β  1, a, b ∈ R. Indeed, by the Lagrange mean-value theorem, we have for ϕ(t) = F( f + th), F( f + th) − F( f ) = ϕ(t) − ϕ(0) = ϕ (ξ) ∫ = | f (x) + ξ h(x)| p−1 sign ( f (x) + ξ h(x))h(x) dμ(x), Ω

and hence

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9 Efimov–Stechkin Spaces. Uniform Convexity . . .

∫     | f (x)| p−1 sign f (x)h(x) dμ(x) F( f + th) − F( f ) − Ω ∫  ≤ | f (x) + ξ h(x)| p−1 sign ( f (x) + ξ h(x))h(x) dμ(x) Ω ∫   | f (x)| p−1 sign f (x)h(x) dμ(x) − Ω ∫   ≤  f (x) + ξ h(x)| p−1 sign ( f (x) + ξ h(x)) Ω   − | f (x)| p−1 sign f (x)|h(x)| dμ(x) ∫ ≤ 2p (| f (x) + ξ h(x)| + | f (x)|)σ |ξ h(x)|γ |h(x)| dμ(x) Ω ∫ σ+1 p (| f (x)| + |h(x)|)σ |h(x)|γ+1 dμ(x), ≤2 Ω

where γ = min{p − 1, 1} and σ = 0 for p  2 and σ = p − 2 for p > 2. We use Hölder’s inequality to estimate the last integral. First, consider the case p  2. We have ∫ (| f (x)| + |h(x)|)σ |h(x)|γ+1 dμ(x) Ω ∫ |h(x)| p−1 |h(x)| dμ(x) = Ω ∫  p1  ∫  q1 ≤ |h(x)| p dμ(x) |h(x)| (p−1)q dμ(x) =

Ω 1+ p h L p q

Ω

= o(h L p ).

Now let p  2. Applying Hölder’s inequality twice (for the pair (p, q) and the pair ( p−1 p−2 , p − 1)), we get ∫ (| f (x)| + |h(x)|)σ |h(x)|γ+1 dμ(x) Ω ∫ = (| f (x)| + |h(x)|) p−2 |h(x)||h(x)|dμ(x) Ω ∫  p1  ∫  q1 p  |h(x)| dμ(x) × (| f (x)| + |h(x)|)(p−2)q |h(x)| q dμ(x) Ω

 h L p

∫

Ω

p−1

(| f (x)| + |h(x)|)(p−2)q p−2 dμ(x) Ω 1 ∫  q1 p−1 × |h(x)| q(p−1) dμ(x) Ω

=

h L2 p 

p−2

f + h L p = o(h L p ).

 q1

p−2 p−1

9.5 Uniformly Smooth Spaces

183

Summarizing, this gives ∫ F( f + h) − F( f )= | f (x)| p−1 sign f (x)h(x) dμ(x) + o(h L p ) Ω

and hence 

F ( f )[h(t)] =

∫ Ω

| f (x)| p−1 sign f (x)h(x) dμ(x).

Theorem 9.9 Let X be a Banach space. Then the following assertions are equivalent. (1) X is a uniformly smooth space; (2) the norm of X is uniformly Fréchet differentiable on the sphere S; (3) the norm of X is Fréchet differentiable on the sphere S, and the mapping x → x ∗ from the sphere S to the dual sphere S ∗ , where x ∗ ∈ S ∗ : x ∗ (x) = 1, is uniformly continuous. For proofs of Theorems 9.10 and 9.11, see, for example, the book [229, Sect. 9.1]. Theorem 9.10 (J. Lindenstrauss) Let X be a Banach space, δX (ε) the modulus of convexity of X, and ρX ∗ (τ) the modulus of convexity of the dual space X ∗ . Then for each τ > 0, we have  ε  ρX ∗ (τ) = sup τ − δX (ε) | 0 < ε ≤ 2 . 2 Similarly, if ρX (τ) is the modulus of smoothness of X and if δX ∗ (ε) is the modulus of convexity of the dual space X ∗ , then  ε  ρX (τ) = sup τ − δX ∗ (ε) | 0 < ε ≤ 2 . 2 Proof (of Theorem 9.10) We claim that δX (ε) + ρX ∗ (τ) ≥ τε/2 for ε ∈ (0, 2] and τ > 0. Indeed, let x, y ∈ S,  x−y ≥ ε. We choose f , g ∈ S ∗ for which f (x+y) =  x+y and g(x − y) =  x − y. By definition of the modulus of smoothness ρX ∗ (τ), we have 2ρX ∗ (τ) ≥  f + τgX ∗ +  f − τgX ∗ − 2 ≥ ( f + τg)(x) + ( f − τg)(y) − 2 = f (x + y) + τg(x − y) − 2 =  x + y + τ x − y − 2 ≥  x + y + τε − 2. As a corollary, 2 −  x + y ≥ τε − 2ρX ∗ (τ). Now by definition of the modulus of smoothness δX (ε), we have δX (ε) + ρX ∗ (τ) ≥ τε/2. As a result, ρX ∗ (τ) ≥ sup{τε/2 − δX (ε) | 0 < ε ≤ 2}. To prove the converse inequality, consider arbitrary τ > 0 and f , g ∈ S ∗ . For every η > 0, there exist x, y ∈ S such that ( f + τg)(x) ≥  f + τgX ∗ − η

and

( f − τg)(y) ≥  f − τgX ∗ − η.

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9 Efimov–Stechkin Spaces. Uniform Convexity . . .

Consequently, 1 τ 1  f +τgX ∗ +  f − τgX ∗ − 2 ≤ f (x + y) − 2 + g(x − y) + η 2 2 2  τ  x + y    ≤   − 1 +  x − y + η 2 2 τ ≤ −δX ( x − y) +  x − y + η 2  ε  ≤ sup τ − δX (ε) | 0 < ε ≤ 2 + η. 2 Hence, since η > 0 is arbitrary, we get the first assertion of Theorem 9.10. The proof of the second assertion is similar. Remark 9.13 In the case of a Hilbert space H, some simple algebra shows that  ρ H (τ) = 1 + τ 2 − 1. In particular, it follows that each Hilbert space H is uniformly smooth. Indeed, Hilbert spaces are the ‘most uniformly convex’ and ‘most uniformly smooth’ spaces, because for any Banach space X (see [198]), we have   δX (ε) ≤ 1 − 1 − ε 2 /4 and ρX (τ) ≥ 1 + τ 2 − 1. Remark 9.14 For L p -spaces, 1 < p < ∞, there is the following asymptotic formula for the modulus of smoothness (the formula for the modulus of convexity δp (ε) is given above in Remark 9.12):  τ p /p + o(τ p ), 1 < p < 2, ρ p (τ) = 2 2 (p − 1)τ /2 + o(τ ), 2 ≤ p < ∞. Theorem 9.11 (Yu. L. Shmul’yan) Let X be a Banach space with dual X ∗ . (1) A space X is uniformly convex if and only if the norm  · X ∗ of the dual space is uniformly Fréchet differentiable. (2) The norm of X is uniformly Fréchet differentiable if and only if the dual space X ∗ is uniformly convex. Proof (of Theorem 9.11) Let X be a uniformly convex space and let δX (ε) be the modulus of convexity of X and let ρX ∗ (τ) be the modulus of smoothness of X ∗ . Next, let ε0 > 0. From the definition of the modulus of smoothness, we have δX (ε) ≥ δX (ε0 ) > 0 for all ε ∈ [ε0, 2]. Consider τ ∈ (0, δX (ε0 )). For ε ∈ [ε0, 2] we have

9.5 Uniformly Smooth Spaces

185

ε ε δX (ε) ε δX (ε0 ) − ≤ − ≤ − 1 ≤ 0, 2 τ 2 τ 2 and hence, using Theorem 9.10,  ε δ (ε)  ε ε ρX ∗ (τ) X 0 = sup − ≤ sup = . τ 2 τ 2 2 0 0,  ετ  ε τ 0 − δX (ε) ≥ . ρX ∗ (τ) = sup 2 0 0, there exists δ > 0 such that x + y  + x − y  ≤ 2 + ε y  wherever x  = 1, y  < δ.

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9 Efimov–Stechkin Spaces. Uniform Convexity . . .

Exercise 9.7 Let M be a closed convex (Chebyshev) subspace of a uniformly smooth and uniformly convex Banach space. Show that there exists k > 0 such that the metric projection is k-Lipschitz in the following sense:  PM x − y  ≤ k x − y  for all x ∈ X, y ∈ M.

Chapter 10

Solarity of Chebyshev Sets

There is another glory of the sun, and another glory of the moon 1 Corinthians, 15:41

Besides its own importance, the solarity of Chebyshev sets turned out to be an important tool in the study of Chebyshev sets (very frequently, in order to show that a Chebyshev set is convex, it suffices to prove its ‘solarity’ (in some sense or other) and then employ Theorem 6.1 on the convexity of suns in smooth spaces (of course if the corresponding space is smooth). To begin with, we note that in general, a Chebyshev set may fail to be a sun — as a corresponding example one may consider Dunham’s construction of a disconnected Chebyshev set from Sect. 7.3 (such a set is not a sun). Similar examples were also constructed by Braess [134], Brosowski, Deutsch, Lambert, and Morris [142]. A common point in these constructions of nonsolar Chebyshev sets is that each of them contains an isolated point. But according to a simple result of V. Klee (see Remark 7.2), a sun may not have proper isolated points. Correspondingly, the examples of [142], [134], [214] of Chebyshev sets with isolated point are not suns. In Sect. 10.1, we start with classical results on solarity of Chebyshev sets that are either boundedly compact or locally compact and have continuous metric projection. Several extensions of these results will be discussed in Sect. 10.4, in which, in particular, the solarity of boundedly compact P-acyclic sets is studied. In Sect. 10.2, we examine relations between various classes of suns (suns, strict suns, α-suns, β-suns, γ-suns, δ-suns). In Sect. 10.2, we also consider approximative properties of b-extensions of sets. The problem of solarity of Chebyshev sets is discussed in Sect. 10.3. At present, no complete description has been given of the spaces in which every Chebyshev set with continuous metric projection is a sun. In particular, it is unknown in which spaces every approximatively compact Chebyshev set is a sun. We discuss these problems in Sect. 10.3. In Sect. 10.4, we give some other results

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_10

187

188

10 Solarity of Chebyshev Sets

in which the solarity of a set is proved under certain constraints on its geometric properties and continuity of the metric projection. We give a general theorem on solarity of monotone path-connected Chebyshev sets (see Sect. 10.4.1) and prove a theorem on solarity of boundedly compact P-acyclic sets; this result is a direct extension of Theorem 10.1 (see Sect. 10.4.3). We also prove some other results on the relation between solarity and the geometry of a set and stability of the metric projection operator.

10.1 Solarity of Boundedly Compact Chebyshev Sets The following result, which asserts the solarity of boundedly compact Chebyshev sets, is classical in the problem of solarity of Chebyshev sets. This fact depends on the fixed point method, which in the study of Chebyshev sets dates back to V. I. Berdyshev, F. A. Ficken, V. Klee, B. Brosowski, and L. P. Vlasov. Theorem 10.1 In a normed linear space, a boundedly compact Chebyshev set is a sun. Remark 10.1 Theorem 10.1 can be directly extended to the case of boundedly compact P-acyclic sets (see Theorem 10.13) below). Remark 10.2 Theorem 10.1 is usually stated for Banach spaces. However, its proof is based (in addition to clear geometric considerations) in essence only on Schauder’s1 fixed point theorem, which in the actual fact holds in arbitrary normed linear spaces (not necessarily complete). Let us formulate the corresponding result (see, for example, [280, p. 119]). Theorem 10.2 (Schauder fixed-point theorem for normed spaces) Let C be a convex (not necessarily closed) subset of a normed linear space. Then any continuous mapping F : C → C such that F(C) is precompact in C has a fixed point. The following result [323] is an extension of Theorem 10.1. Theorem 10.3 In a normed linear space, a locally compact Chebyshev set with continuous metric projection is a sun. Theorem 10.1 follows from Theorem 10.3, because the metric projection onto an approximatively compact (in particular, onto a boundedly compact) Chebyshev set is continuous (see Theorem 5.10). From Theorem 10.1 it follows as a corollary that in a finite-dimensional normed linear space a Chebyshev set is a sun. 1 More precisely, on one of its extensions: J. Schauder himself formulated his theorem for closed convex subsets of a complete space.

10.2 Relations Between Classes of Suns

189

Proof (of Theorem 10.3) Assume on the contrary that M satisfies the hypotheses of the theorem, but M is not a sun. Then there is a point x  M such that on the ray  := {(1 − λ)y + λx | λ ≥ 0} (where PM x = {y}) the point x is a farthest point from y among all points z of the ray  for which PM z = {y}. We assume without loss of generality that x = 0. Since M is locally compact and PM is continuous, we can find a sufficiently small ball V = B(0, r) such that the closure of PM V is compact. We construct a mapping f of the ball V into itself as follows: u f (v) = − r for all v ∈ V, PM v = {u}. u It is clear that f is continuous and maps the ball V into its compact part. By Schauder’s fixed-point theorem (Theorem 10.2), the mapping f has a fixed point v0 , v0 = −

u0 r, u0 

where PM v0 = {u0 }.

Since 0 ∈ [v0, u0 ], we have u0 = y (indeed, 0 − u0  ≤ 0 − y = ρ(0, M), for otherwise v0 − y ≤ v0 − 0 + 0 − y < v0 − 0 + 0 − u0  = ρ(v0, M), and so, u0 ∈ PM 0, which gives u0 = y, because M is a Chebyshev set). But this contradicts the choice of the point 0 (the point v0 lies on the ray  further from y than 0). Theorem 10.3 is proved.  Remark 10.3 The continuity condition of the metric projection in Theorem 10.3 is essential (the disconnected Chebyshev set in Dunham’s example is locally compact, but is not a sun).

10.2 Relations Between Classes of Suns Recall that a closed set M ⊂ X is called a sun (respectively, a strict sun) if each point x ∈ X \ M is a point of solarity (respectively, a point of strict solarity) for M (see (1.1)). A set M ⊂ X is called a strict protosun if each point x ∈ X \ M is a point of strict protosolarity. Clearly, a strict sun is a sun. A nonempty closed set M is called an α-sun, if, for any point x  M, there exists a ray  emanating from x such that ρ(z, M) = z − x + ρ(x, M) for any z ∈  (see p. 96). Any sun is an α-sun. A nonempty closed set M is called a β-sun if, for any x  M, r > 0, there exists a point z such that ρ(z, M) − ρ(x, M) = r. Any α-sun is a β-sun. Recall that a nonempty closed set M is called a δ-sun (see also p. 106) if, for any x  M, there exists a sequence (xn ), xn  x, xn → x, satisfying condition (10.1): ρ(xn, M) − ρ(x, M) −→ 1.  xn − x

(10.1)

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10 Solarity of Chebyshev Sets

A closed set   M ⊂ X is called a γ-sun if, for any x  M and r > 0, there exists a sequence (zn ) ⊂ X such that ρ(zn, M) − ρ(x, M) → r,

 x − zn  = r.

(10.2)

A set M ⊂ X is called: a) a semi-sun if, for any x  M and r > 0, there exist z ∈ X and y ∈ PM z such that z − x = r, x ∈ [y, z]; b) a meta-sun if, for any x  M, there exist z ∈ X and y ∈ PM z such that x ∈ [y, z). Note (see [592, Theorem 3.2]) that in a Banach space a meta-sun is a δ-sun (= γ-sun); b) a P-convex meta-sun is a semi-sun. We consider examples of non-identical classes of ‘suns’. In the space 2∞ (on the plane x, y with  ∞ -norm), the set {(x, y) | x ≥ −1, y ≥ −1} is a sun, but is not a strict sun; the set {(x, y) | y = 0} is a strict sun, but is not a Chebyshev sun; the set consisting of two points (1, 1) and (−1, −1) is an α-sun, but is not a sun (see Fig. 6.1). Example 10.1 This is a more involved example of an α-sun which is not a sun (the example is due to L. P. Vlasov). In the space  2 , consider the set ∞     M= x i=2

xi2

(1 + 1/i)2

 ≥ 1, x1 = 0 ,

and define a = (1, 0, . . . ). As a new unit ball, consider the convex hull of the set {a }

∞      xi2 ≤ 1, x1 = 0 {−a }. x i=2

For x = 0 there is no nearest point in M. Geometrically, this means that the set M is an α-sun with ‘rays’ parallel to a.

Definition 10.1 A nonempty closed set M is called almost convex if, for any closed ball B(x, r) lying at a positive distance from M, there exists a closed ball B(y, R) ⊃ B(x, r) of arbitrary large radius R which is also at a positive distance from M. Example 10.2 Let us construct an example of a δ-sun which is not a γ-sun (L. P. Vlasov [591]). Let X be an infinite-dimensional Hilbert space and let X0 ⊂ H be a nonclosed dense linear manifold, c ∈ X \ X0 , M1 = {z ∈ X | x − c  ≥ 1}, M = M1 ∩ X0 . The set M is closed in X0 , but, clearly, M is not almost convex; hence, M is not a γ-sun (see Theorem 10.4 below). However M is a δ-sun in X0 . Indeed, for any x ∈ X0 \ M and any ε, 0 < ε < x − c  (x  c), we have ρ(z, M) − ρ(x, M) z − y  − x − y  = = 1, z − x  z − x  where z := x + ε(c − x), y := c + (x − c)/ x − c . It remains to consider z n ∈ X0 , z n → z. We M )−ρ(x, M ) have ρ(z n ,z → 1. It should be pointed out that the set M is also not a Chebyshev set. n −x 

Theorem 10.4 (L. P. Vlasov) Consider the following conditions on a subset M of a normed linear space X: (a) M is δ-sun;

10.2 Relations Between Classes of Suns

191

(b) M is γ-sun; (c) M is almost convex. Then (b) ⇒ (a), (b) ⇔ (c). In a Banach space all three conditions are equivalent. An example of an incomplete normed linear space containing a δ-sun which is not a γ-sun is given above in Example 10.2. For the asymmetric setting, see [20]. Proof (of Theorem 10.4) (b) ⇒ (a). Let x  M, M ⊂ X be a γ-sun. By definition of a γ-sun, for any n ∈ N, there exists an element zn ∈ X such that  x − zn  = n1 and ρ(zn, M) − ρ(x, M)  Hence

1 1 − . n n2

ρ(zn, M) − ρ(x, M) 1  1− ,  x − zn  n

because  x − zn  = n1 . Next, by (6.1) we have ρ(zn, M) − ρ(x, M) → 1,  x − zn 

n → ∞,

that is, M is a δ-sun. (a) ⇒ (b) Let M be a δ-sun in a Banach space X. Consider arbitrary x  M, r > 0, σ > 1. Following R. Phelps and L. P. Vlasov, we define the partial order in the ball V := B(x, r) as follows: we set z ≺ z  if   z − z    σ ρ(z , M) − ρ(z, M) . (10.3) That this relation is antisymmetric and transitive is clear. We claim that ≺ satisfies the hypotheses of Zorn’s lemma. Let (zα ) be a net in V. For α  α , we have   zα − zα   σ ρ(zα, M) − ρ(zα, M) . By (10.3), the net ρ(zα, M) ⊂ R is monotone in α. Next ρ(z, M)   x − z + ρ(x, M) and since z ∈ B(x, r), this net is bounded. Correspondingly, ρ(zα, M) is convergent. This implies that the net (zα ) is a Cauchy sequence. Since X is complete, the net (zα ) converges to some z ∈ V; that is, v − zα  → 0. It is clear that z − zα   σ(ρ(zα, M) − ρ(z, M)), and hence, zα ≺ z for any α; that is, the net (zα ) has an upper bound z. So we can apply Zorn’s lemma, from which there exists a maximal element v ∈ V such that v  x. ˚ r). Since ρ(v, M)  ρ(x, M) > 0, we have v  M. By Assume that v ∈ B(x, definition of a δ-sun, there exists a sequence zn  v, v − zn  → 0, for which ρ(zn, M) − ρ(v, M) → 1, v − zn 

n → ∞.

˚ r), and hence, for some n0 and for all n  n0 Hence we have σ > 1 and v ∈ B(x, we have zn ∈ V and

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10 Solarity of Chebyshev Sets

  v − zn   σ ρ(zn, M) − ρ(v, M) , which contradicts the maximality of v. So, v lies on the boundary of  V; that is, v ∈ S(x,  r). The condition x ≺ v means by definition that  x − v  σ ρ(v, M) − ρ(x, M) . We set vσ = v. By (6.1) we have 1 ρ(vσ, M) − ρ(x, M)   1, σ  x − vσ  which gives

ρ(vσ, M) − ρ(x, M) → 1,  x − vσ 

σ → 1 + 0.

Now vσ − x = r, and so M is a γ-sun by definition, the result required. (b) ⇒ (c) Let ρ(B(x, r), M) > 0; that is, ρ(x, M) > r (we recall that ρ(M, N) := inf{ρ(x, N) | x ∈ M }). For any r  > r, by the definition of a γ-sun, there exists a sequence (zn ) such that zn − x = r , ρ(zn, M) → r  + r. We have ρ(x, M) > r, and hence there is z := zn such that z − x = r  and ρ(z, M) > r  + r. Let x  := z + Then

r (x − z). r

r

 x − x   = 1 −   x − z = r  − r, r

 x  − z =

r  x − z = r, r

and now Proposition 1.2 gives B(x, r) ⊂ B(x , r ) ⊂ B(z, r  + r). Since ρ(z, M) > r  + r; that is, ρ(B(z, r  + r), M) > 0, we have ρ(B(x , r ) > 0, the result required. (c) ⇒ (b) Let x  M and let r ∈ (0, ρ(x, M)), R > 0. By definition of almost convexity, there exist x  ∈ X and r  > R + ρ(x, M) such that B(x, r) ⊂ B(x , r ), ρ(B(x , r ), M) > 0, whence  x − x   ≤ r  − r, ρ(x , M) > r  by Proposition 1.2. R   Setting z := x + x−x   (x − x) (in the case x = x we take any z with z − x = R), we claim that z ∈ B(x , r ). In the case x  = x this result follows from the relations z − x   = z − x = R < r , and in the general case, from the relations 0 < r −ρ(x, M) − R ≤ ρ(x , M) − ρ(x, M) − R ≤  x−x   − R ≤ r  − r − R ≤ r  . As a result,

  R   z−x   = 1 −   x − x  x − x   =   x−x   − R =  x − x   − R ≤ r  .

So, z ∈ B(x , r ), and besides, M ⊂ X \ B(x , r ). This gives

(10.4)

10.2 Relations Between Classes of Suns

193 (10.4)

ρ(z, M) ≥ ρ(z, X \ B(x , r )) = r  − z − x   = r  −  x  − x + R ≥ R + r. Since z = zn depends only on r = rn ∈ (0, ρ(x, M)), it follows by making rn → ρ(x, M) that M is a γ-sun. Theorem 10.4 is proved.  The following result (L. P. Vlasov [592]) will be required in the proof of Theorem 10.13. Proposition 10.1 In a normed linear space X a semi-sun M is a sun if at least one of the following conditions is satisfied: (a) M is P-compact; (b) M is P-w-compact, the graph of PM is w-sequentially closed; (c) X is a dual space, M is P-w ∗ -compact, and the graph of PM is w ∗ -sequentially closed. Proof Let 0  M. It can be assumed without loss of generality that x = 0, ρ(0, M) = 1. By definition of a semi-sun, for any n there exist zn and yn ∈ PM zn such that 0 ∈ [yn, zn ], zn  = n. Since (yn ) ⊂ PM 0 and since PM 0 is compact, there exists a subsequence ynk → y ∈ M. Let r ≥ −1 be arbitrary, z := −r y. To prove the proposition, it suffices to show that y ∈ PM z for all r ≥ −1. Setting zn := −r yn , we have zn ∈ [yn, zn ] for all n ≥ r, inasmuch as  yn − 0 = ρ(0, M) = 1, zn = −nyn . It is clear that zn k → z. It is also clear that there exists k0 such that nk ≥ r for all k ≥ k0 . Hence zn k ∈ [ynk , znk ] for all k ≥ k0 . Next, zn k − ynk  = ρ(zn k , M), which gives in the limit z − y = ρ(z, M). As a result, y ∈ PM z, because y ∈ M. In cases (b) and (c) the argument involves subnets and is similar (see [592, Proposition 3.1]).  In the study of approximative properties of sets one frequently has to know approximative properties of b-neighbourhoods (b-extensions) of sets. Properties of b-extensions are also useful in problems of geometric optics and in problems of separation (almost convexity) of sets. Definition 10.2 Let b > 0, M ⊂ X; the b-extension of a set M is defined as Mb := {x ∈ X | ρ(x, M) ≤ b} (N. V. Efimov and S. B. Stechkin [222]). The following properties are straightforward (see [222]): – Mb is a closed set; – if M is connected, then so is Mb ; – if M is bounded, then so is Mb ; – bd Mb is the set of all x ∈ X for which ρ(x, M) = b; – if x  Mb , then ρ(x, Mb ) = ρ(x, bd Mb ); – if x  Mb , then ρ(x, M) = ρ(x, Mb ) + b; – Ma+b = (Ma )b ; – if M is bounded, then there exists a number a0 such that bd Ma for any a ≥ a0 has the strong star-like property; that is, any ray emanating from some point y ∈ X intersects bd Ma exactly once;

194

10 Solarity of Chebyshev Sets

– if M is a set of existence, then so is Mb . We also note some properties of b-extensions Mb of a set M: (a) If ρ(x, M) ≥ b, then ρ(x, M) = ρ(x, Mb ) + b. This fact was proved by Efimov and Stechkin [222]. To prove it, consider an arbitrary point z ∈ Mb . Then ρ(x, M) ≤  x − z + ρ(z, M) ≤  x − z + b, whence ρ(x, M) ≤ inf z ∈Mb  x − z + b = ρ(x, Mb ) + b. Conversely, let y ∈ M and let a point z ∈ [x, y] be such that ρ(z, M) = b (such a point z exists, since the function z → ρ(z, M) is continuous). We have  x − y =  x − z + z − y ≥ ρ(x, Mb ) + ρ(z, M) = ρ(x, Mb ) + b, whence ρ(x, M) = inf y ∈M  x − y ≥ ρ(x, Mb ) + b. (b) If M is a sun, then so is Mb . Let x  Mb , x  ∈ PM x, y ∈ [x, x ], ρ(y, M) = b. Then, by (a) and Proposition 1.4,  x − y =  x − x   −  y − x   = ρ(x, M) − ρ(y, M) = ρ(x, M) − b = ρ(x, Mb ); that is, y ∈ PMb x. So far, we have only proved the following fact. (c) If M is a set of existence, then so is Mb . Now assume that M is a sun. Hence if x1 lies on the ray emanating from x  and passing through x, then as before we get  x1 − y = ρ(x1, Mb ). This shows that Mb is a sun. In general, the converse of (b) is not true: it suffices to consider the ‘two-point set’ on the plane (see Fig. 6.2). A similar analysis shows that (d) If M is a strict sun, then so is Mb . (e) If M is an α-sun, then so is Mb . (d) If M is a δ-sun, then so is Mb . (f) If M is a γ-sun, then so is Mb . To prove (e), one should employ the separation theorem for α-suns (Theorem 5.5). We set Qb (M) = {x ∈ X | ρ(x, M) ≥ b}. (g) Let M ⊂ X, b > 0. Then (see [56]) if

ρ(z, M) = b, x ∈ PM z, then z ∈ PQb (M) x;

(h) Let M ⊂ X, b > 0. Then (see [56]) if

x  Qb (M), z ∈ PQb (M) x, then ρ(z, M) = b.

(i) Let X be a normed linear space, M ⊂ int M, Q = X \ int M is a convex set, b > 0. Then Qb (M) is a convex set (possibly empty; see [57]). The following result was proved by N. V. Efimov and S. B. Stechkin and independently by L. P. Vlasov. (j) Let b > 0. In order that the b-extension of any Chebyshev set in a normed linear space be a Chebyshev set, it is necessary and sufficient that the space be strictly convex.

10.2 Relations Between Classes of Suns

195

Further, we set  T(M) = x ∈ X | lim diam(M ∩ B(x, ρ(x, M) + δ) = 0 . δ→0

(k) (see [57]) Let X be a locally uniformly convex space, M ⊂ X be a closed subset of X, b > 0. Then T(M) ⊂ T(Mb ). We set δS(M) = {x ∈ X | x is a point of δ-solarity of M } (that is, for x (6.8) is satisfied), that is, ρ(xn, M) − ρ(x, M) −→ 1  xn − x for any point x  M and some sequence (xn )n∈N such that xn → x, n ∈ N. (l) (see [61]) If b > 0 and M is a closed set, then δS(Mb ) = δS(M) \ Mb (the proof of (l) follows from (a)). (m) (see [60, Lemma 13]) Let M ⊂ X be closed and convex, b > 0. Then, for any x ∈ X \ M \ Qb (M), ρ(x, M) + ρ(x, Qb (M)) = b. In assertion (m), the convexity condition of a set cannot be relaxed. As an example, consider as M the complement of a square in X = R2 with the max-norm. For strictly convex spaces, assertion (m) is due to Klee [343]. Definition 10.3 A set M ⊂ X is called a-convex (a > 0) if every point x  M is contained in some open ball of radius a which is disjoint from M (Efimov and Stechkin [222]). The following result partially strengthens Theorem 10.4 (see [592], Theorem 3.3). Theorem 10.5 (L. P. Vlasov) The following conditions on a closed subset M of a Banach space X are equivalent: (a) M is δ-sun; (b) M is γ-sun; (c) M is almost convex; (d) for any a, b > 0 the set Mb is a-convex. Proof (d) ⇒ (c) Let x  M, b := ρ(x, M), a > 0. By property (a) on p. 194, the conditions ρ(y, M) ≤ a + b and ρ(y, Mb ) ≤ a are equivalent; that is, Ma+b = (Mb )a . Consider b ∈ (0, b). Since x  Mb and since Mb is a-convex, there exists a ball

196

10 Solarity of Chebyshev Sets

˚ a), B(z, ˚ a) ∩ Mb = . The last equality is equivalent to B(z, a) such that x ∈ B(z, ˚ a + b − b) ∩ (Mb )b−b = , that is, saying that B(z, ˚ a + b − b) ∩ Mb = , B(z,

˚ a + b) ∩ M = . B(z,

˚ a), we have  x − z < a = a + b − b; that is, by Proposition 1.1 Next, since x ∈ B(z,  we have B(x, b ) ⊂ B(z, a + b). If b ∈ (0, b), then B(x, b) ⊂ B(z, a + b) and ρ(B(z, a + b), M) = ρ(z, M) − a − b > ρ(z, M) − a − b ≥ 0. This means that M is almost convex. The implication c) ⇒ d) is clear. The remaining implications are contained in Theorem 10.4.  Other results on relations between classes of suns can be found in the works of Vlasov [592], [591], Koshcheev [363] and in the survey paper by Balaganskii and Vlasov [61].

10.3 Solarity of Chebyshev Sets 10.3.1 Solarity of Chebyshev Sets with Continuous Metric Projection Above we have shown (Theorem 10.1) that a boundedly compact Chebyshev set is a sun. More intriguing is the problem of solarity of approximatively compact Chebyshev sets and a more general problem of solarity of Chebyshev sets with continuous metric projection. Here the answer is known only in some particular cases. In C(Q) and  1 (Γ) (and more generally, in an arbitrary strongly nonlunar space, see Theorems 7.29, 7.24 and 7.25) the answer to this question is positive for Chebyshev sets with continuous metric projection (Theorem 7.32). The authors do not know an example of a space in which there exists a nonsolar Chebyshev set with continuous metric projection. However, for Chebyshev sets with continuous metric projection (in particular, for approximatively compact Chebyshev sets) the following weaker assertion holds in the general setting: a Chebyshev set with continuous metric projection in a normed linear space is a δ-sun; that is, ρ(xn, M) − ρ(x, M) −→ 1  xn − x

(10.5)

for any point x  M and some sequence of points (xn )n∈N such that x ∈ (xn, PM x), n ∈ N, xn → x (L. P. Vlasov, Theorem 6.5). We also note the following well-known result (Asplund [47], Vlasov [592]).

10.4 Solarity and Structural Properties of Sets

197

Theorem 10.6 A Chebyshev set in a Hilbert space is a sun (is convex) if and only if the metric projection onto it is continuous. Theorem 10.6 is partially strengthened in the following Theorems 10.7, 10.8, and 10.9 due to Vlasov (see [592], [591]). Theorem 10.7 Let X be a uniformly convex normed linear space, and let M ⊂ X be an approximatively compact set. Then M is a Chebyshev set if and only if M is a sun. Here and in what follows, the upper w-semicontinuity of a mapping ϕ : X → Y means that this mappings is upper semicontinuous from X to Y , where X and Y are equipped with weak topologies. A similar convention will be applied for the w ∗ -topology. Theorem 10.8 In a reflexive space, a P-convex set with upper w-semicontinuous metric projection is a sun. Theorem 10.8 will be proved in Sect. 10.4.3. Theorem 10.9 In a locally uniformly convex Banach space, a Chebyshev set with continuous metric projection is a sun (is convex in a smooth space). Theorem 10.10 In a uniformly convex Banach space, a locally compact Chebyshev set is a sun (is convex in a smooth space). Problem 10.1 Characterize spaces in which any Chebyshev set with continuous metric projection is a sun. In particular, in which spaces any approximatively compact Chebyshev set is a sun? In connection with Problem 10.1 note that in a normed linear space every Chebyshev set with continuous metric projection is a δ-sun (Theorem 6.5). Recall another well-known problem. Problem 10.2 Must a weakly compact Chebyshev set in a Banach space be a sun?

10.4 Solarity and Structural Properties of Sets Above, we proved a classical result (Theorem 10.1) on solarity of boundedly compact Chebyshev sets. Here we give some other results, in which the solarity of a set is proved under certain constraints on its geometric properties and continuity of the metric projection. We first give a general theorem on solarity of monotone pathconnected Chebyshev sets (see Sect. 10.4.1). Next, we prove a theorem on solarity of boundedly compact P-acyclic sets; this result is a direct extension of Theorem 10.1 (see Sect. 10.4.3). We also prove some other results on the relation between solarity and the geometry of a set and stability of the metric projection operator.

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10 Solarity of Chebyshev Sets

10.4.1 Solarity of Monotone Path-Connected Chebyshev Sets No progress has been made in a long time in strengthening Koshcheev’s results (see [24, Sects. 8.1 and 8.2]) on connectedness of compact suns in arbitrary normed linear spaces (and even in specific finite-dimensional spaces). Considerable achievements in this direction are due to H. Berens and L. Hetzelt, who established metric convexity of suns in arbitrary two-dimensional spaces and in the space n∞ (see also [24]). As a corollary, every sun in n∞ is monotone path-connected. We note at once that there are examples of finite-dimensional spaces that contain non-m-connected (not monotone path-connected) Chebyshev suns (see [145, Theorem 4.3], [242, p. 19]). We give one more example. Example 10.3 Let X be a finite-dimensional normed linear space with the property ext S ∗ = S ∗ . Phelps [469] showed that the property ext S ∗ = S ∗ is satisfied for a space Xn if and only if each convex bounded closed subset of Xn can be represented as the intersection of closed balls (in other words, such an Xn lies in the class (MIP); that is, it satisfies the Mazur intersection property); as a corollary, in such a space the monotone path-connectedness of a closed set is equivalent to its convexity. Next, for every n  3, I. G. Tsar’kov [549] constructed an example of a space X of dimension n with the property ext S ∗ = S ∗ that contains an unbounded nonconvex Chebyshev set M  ; moreover, in such X, every bounded Chebyshev set in X is convex. So M  serves as an example of a B-acyclic (P-acyclic) set (Chebyshev sun) that is not monotone path-connected.

Remark 10.4 In connection with Example 10.3, P. A. Borodin raised the question whether a bounded Chebyshev set is monotone path-connected. With regard to this question, B. B. Bednov constructed an example of a bounded Chebyshev set in 31 that is not monotone path-connected. Moreover, A. R. Alimov and B. B. Bednov [23] showed that in a three-dimensional Banach space X, every Chebyshev set is monotone path-connected if and only if either X has a cylindrical norm or each exposed point of the unit sphere of X is a smooth point (in the latter case, every Chebyshev set in X is convex by Berdyshev–Brøndsted’s theorem). A similar characterization for sets with continuous (lower semicontinuous) metric projection was given by A. R. Alimov [21]. We note the following result [12], in which the solarity of an arbitrary Chebyshev set in a normed linear space is established under connectedness-type structural constraints. Theorem 10.11 Let M be a monotone path-connected subset of a normed linear space. Assume that PM x = {y} for some x  M. Then x is a point of solarity (y is a luminosity point). As a corollary, a monotone path-connected Chebyshev set is a sun. Theorem 10.11 can be looked at as the first result in which the solarity of a Chebyshev set is established under connectedness-type structural restrictions. Proof (of Theorem 10.11) I. Singer and A. L. Garkavi put forward the fortunate idea of using extreme functionals (elements of S ∗ ) in general theorems. Under this approach, use was made of the lemma on continuation of extreme functionals, which

10.4 Solarity and Structural Properties of Sets

199

was formulated by Singer in [516] and further developed by G. Choquet [166] and A. L. Garkavi [254], [257]. In particular, I. Singer obtained the following result [518]. Let G ⊂ X be a linear subspace, x ∈ X \G. Assume that for x, there exists an element of best approximation g0 from G. Then there exists an extreme hyperplane H0 passing through the point g0 separating 0 from the ball B(x,  x − g0 ) that is a hyperplane of support to this ball. As a corollary, for an arbitrary x ∈ X,  x = sup f (x) = sup f (x) f ∈S ∗

f ∈ext S ∗

(10.6)

(that is, ext S ∗ is the James boundary for X). Next, let y, x ∈ X, y  x, p := (y − x)/ y − x. We denote by Σ p the set of all functionals from S ∗ that attain their maxima on the unit ball B at the point p. By Lemma 5.2,  ˚ x) = z | f (z) < f (y) ∀ f ∈ Σ p K(y,  (10.7) = z | f (z) < f (y) ∀ f ∈ Σ p ∩ ext S ∗ . ˚ x) is the open support cone to the ball B(x,  x − y) at its boundary point y Here K(y, (see Sect. 5.2). From the above result of Singer, it follows that every point not lying in a closed ball can be separated from it by an extreme (affine) hyperplane (this result is a particular corollary of the characterization of bars in terms of strict separation by extreme functionals [13]). Resuming the proof of Theorem 10.11, we suppose that M is a monotone pathconnected Chebyshev set in X that is not a sun. By Theorem 5.4, there exists a point ˚ x) ∩ M  , where PM x = {y}. We assume without loss of x  M such that K(y, ˚ x) and let k(τ), 0 ≤ τ ≤ 1, generality that x = 0 and ρ(x, M) = 1. Let u ∈ M ∩ K(y, be a continuous monotone curve joining y and u, k(0) = y, k(1) = u. By Lemma 5.2, there exists r > 1 such that ˚ r , αr ) ⊂ K(y, ˚ x), u ∈ B(z

y ∈ S(zr , αr ),

where zr = −r y + (r + 1)x, αr = (r + 1). We set ˚ := k(·) \ {u, y}. k(·) Then by (10.6) and since PM x = {y}, we have ˚ ⊂ B(z ˚ x). ˚ r , αr ) ⊂ K(y, k(·) ˚ ∩ B˚ = . We choose a sequence of points vn ∈ k(·), ˚ Assume that k(·) vn → y. By the above result of Singer’s on separation by extreme hyperplanes, for each n there exist a functional fn ∈ ext S ∗ and a number dn ≥ 1 such that the extreme hyperplane {z | fn (z) = dn } separates the point vn  B from the closed ball B. Since

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˚ ∩ B =  because M is a Chebyshev set, we have the curve k(·) is monotone and k(·) fn (vn ) ∈ [ fn (y), fn (u)]. Next, since the ball B∗ is w ∗ -compact, the sequence ( fn ) has a ∗-weak limit point f ∈ S ∗ . Further, dn → 1, and hence on passing to subnets, we get f (y) = 1 and f (u) ≥ 1. ˚ x). However, by (10.7), this contradicts the original assumption that u ∈ K(y, ˚ ˚ ˚ ˚ So the case k(·) ∩ B =  is impossible. But the case k(·) ∩ B   is also impossible, since by the above B˚ ∩ M =  and k(·) ⊂ M. This contradiction proves Theorem 10.11.  Remark 10.5 In general, a Chebyshev set can fail to be monotone path-connected even in the finite-dimensional case (see Example 10.3 on p. 198). Here, the monotone path-connectedness can be guaranteed for the finite-dimensional (BM)-spaces, in which every sun (and hence every Chebyshev set) is monotone path-connected (see [24]). Note that a boundedly compact monotone path-connected set is a sun ([24, Theorem 9.3]). Moreover, from Theorems 7.15 and 7.9 of [24], we have the following result. Theorem 10.12 An approximatively compact monotone path-connected subset of a Banach space is a δ-sun (in a locally uniformly convex space, a sun).

10.4.2 Acyclicity and Cell-Likeness of Sets Acyclicity and cell-likeness were found to be important in the study of approximative properties of sets and in particular, their solarity. In this auxiliary section, we recall some concepts from geometric topology and discuss in particular the definitions of acyclicity and cell-likeness. In our setting, acyclicity arises in connection with the following direct theorem of the geometric theory of approximations (Theorem 10.13): in a normed linear space, a P-acyclic boundedly compact set is a sun. The answer to the inverse question is unknown in general (see, for example, [19]). The conjecture here is that acyclicity (P-acyclicity) is the ‘right term’ for this problem; in other words, acyclicity is a property that is believed to be shared by all suns in finite-dimensional spaces (this is so in the two-dimensional setting, in the space n∞ , and, of course, in smooth spaces). A homology (cohomology) theory associates with a topological space X a sequence of abelian groups Hk (X), k = 0, 1, 2, . . . (homology groups), and H k (X), k = 0, 1, 2, . . . (cohomology groups), which are homotopy invariants of a space: if two spaces are homotopy invariant, then the corresponding (co)homology groups are isomorphic. There are several ways to construct (co)homology groups, of which we mention the following: the construction based on nerves of covers proposed by

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P. S. Alexandroff and further extended by E. Čech; the Vietoris construction based on the concept of true cycles, which applies to metric spaces; the construction based on the concept of singular chains. Let A be a nontrivial abelian group. A metrizable space is said to be acyclic if its Čech cohomology group with coefficients from A is trivial. Thus, the definition of acyclicity depends on the chosen group of coefficients. It is worth noting that the (Alexandroff–)Čech homology is not a homology theory, since it fails to satisfy the exactness axiom, whereas Čech cohomology forms a cohomology theory of topological spaces. For a comprehensive account of (co)homologies on compact sets, topological spaces, and uniform spaces, the reader is referred to the survey by Melikhov [433]. If a (co)homology has compact support (satisfying the compact supports axiom) and if the coefficients of the (co)homology group lie in a field, then the notions of homological and cohomological acyclicity coincide [429]. However, this is not the case for an arbitrary abelian coefficient group (see, for example, [24, Sect. 6]). Below, unless otherwise stated, acyclicity will be understood in the sense of Čech cohomology with coefficients in an arbitrary abelian group. A compact nonempty space is called an Rδ -set (see, for example, [279, formula (2.11)]) if it is homeomorphic to the intersection of a countable decreasing sequence of compact absolute retracts (or contractible compact sets [279], Theorem 2.13). We note that Rδ -sets arise naturally as spaces of solutions to the Cauchy problem for autonomous and nonautonomous differential inclusions (see [24, Sect. 6]). Results of this kind date back to N. Aronszajn, who established the Rδ property for the set of local solutions of the Cauchy problem in the finite-dimensional space Rn . A compact space Y is said to be cell-like (or having the shape of a point) if there exist an absolute neighbouring retract Z and an embedding i : Y → Z such that the image i(Y ) is contractible in any of its neighbourhoods U ⊂ Z (see [279], formula (82.4)). A cell-like set need not be contractible. A space X having the homotopy type of a point is said to be contractible. From the well-known Hyman’s characterization of Rδ -sets it readily follows that an Rδ -set is necessarily cell-like (see [24, Sect. 6], [379]). Since every mapping of a compact set of trivial shape into an absolute neighbourhood retract is homotopically trivial, it follows that a compact set of the shape of a point (a cell-like set) is contractible in each of its neighbourhoods in every ambient absolute neighbourhood retract. As a corollary, the classes of Rδ -sets and cell-like (of trivial shape) compact sets coincide. Note that cell-likeness implies acyclicity (with respect to any continuous (co)homology theory ([379], p. 854), there being acyclic sets that are not cell-like, as well as cell-like sets that are not path-connected (e.g., topologist’s sine curve).

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10.4.3 Solarity of Boundedly Compact P -Acyclic Sets The main result of this section is the following theorem, which was proved by L. P. Vlasov in the case of Banach spaces. Theorem 10.13 In a normed linear space, a P-acyclic boundedly compact set is a sun. We need some auxiliary results. Let M be a set of existence with metric projection PM , x  M, r > 0. Consider the set-valued mapping ϕ(z) = ϕx,r = x +

r (z − PM z), r + ρ(x, M)

z ∈ X.

Proposition 10.2 The following conditions are equivalent: (a) z is a fixed point of ϕx,r ; (b) z − x = r and there exists y ∈ PM z such that x ∈ (z, y). r Indeed, if z ∈ ϕx,r (z), then by definition, z = x + r+ρ(x, M) (z − y), where y ∈ PM z. It follows that

r r z+ y, x = 1− r + ρ(x, M) r + ρ(x, M)

that is, z ∈ (z, y), which gives z − x = z − x +

r z − y, r + ρ(x, M)

ρ(x, M) z − x = z − y = z − x +  x − y = z − x + ρ(x, M) r

and z − x = r. Let us prove the implication b) ⇒ a). If x ∈ (z, y), then z = x + λ(z − y), where λ > 0. From the equality z − x = r we obtain λ = r/z − y. Further, y ∈ PM z, and hence z − y = z − x +  x − y = r + ρ(x, M) and r z = x + r+ρ(x, M) (z − y) ∈ ϕ x,r (z). The next result follows from Proposition 10.2 and definitions. Proposition 10.3 Let M be a set of existence. Then: (a) M is a semisun if and only if for every x  M and r > 0, the mapping ϕx,r has a fixed point; (b) M is a metasun if and only if for every x  M, there exists r > 0 such that the mapping ϕx,r has a fixed point. Lemma 10.1 Let M be a set of existence and let ρ(0, M) = 1, r > 0, Vr := B(0, r), Kr := −r · conv PM Vr , Wr := Vr ∩ Kr , ϕ := ϕ0,r . Then Wr  0 and ϕVr ⊂ Vr ,

ϕWr ⊂ Wr .

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Proof Let v ∈ ϕVr . Then v = α(z − y), where α = r(r + 1)−1 , z ∈ Vr , y ∈ PM z. We have z−y = ρ(z, M) ≤ z−0+ ρ(0, M) ≤ r +1, v ∈ Vr . Since = r ·PM 0 ⊂ Vr ∩Kr , we have Wr  . Let v ∈ ϕWr . As a result, v = α(z − y), where z ∈ Wr , y ∈ PM z. It follows that −r y ∈ Kr , v = αz + (1 − α)(−r y), and since z ∈ Kr and Kr is convex,  we get v ∈ Kr . By the above, v ∈ Vr and v ∈ Vr ∩ Kr = Wr . Proof (of Theorem 10.13) Let x  M, r > 0. We can assume without loss of generality that x = 0, ρ(x, M) = 1. We use the notation of Lemma 10.1. Let z ∈ Vr , y ∈ PM z. Then 0−y ≤ 0−z+z−y = 0−z+ρ(z, M) ≤ 2·0−z+ρ(0, M) ≤ 2r + 1; that is, (10.8) PM Vr ⊂ V2r+1 ∩ M =: P2r 0. Since M is boundedly compact, the set P2r 0 (and hence the set PVr ) is compact. Further, since in a normed linear space the closed convex hull of a compact set is compact, the set Kr := −r · convPM Vr = −r · conv PM Vr (and therefore, the set Wr ) is compact. Next, the set Wr is convex, and for every z ∈ Wr , the set ϕz is acyclic as the homothetic image of an acyclic set. By Lemma 10.1, ϕWr ⊂ Wr . Next, an appeal to the classical fixed-point theorem for acyclic upper semicontinuous mappings on convex compact spaces (see, for example, [459], [460, Corollary 7.4], [461, Theorem 3.2]) shows that the mapping ϕ has a fixed point x0 ∈ Wr for all r > 0. By Proposition 10.3, M is a semisun. Finally, M is a sun by Proposition 10.1.  Theorem 10.14 (L. P. Vlasov) Let X be a dual space and let M ⊂ X be a P-convex P-w ∗ -closed set of existence with upper w ∗ -semicontinuous metric projection. Then M is a sun. Proof (of Theorem 10.14) Let x  M (we assume without loss of generality that x = 0, ρ(0, M) = 1), r > 0. In the notation of Lemma 10.1, ϕ := ϕ0,r is a set-valued mapping of a w ∗ -compact convex set Vr into itself, and moreover, for all z ∈ Vr , the set ϕz is nonempty, convex, and w ∗ -compact, because so is the set PM z (the w ∗ -compactness of PM z is a consequence of the fact that it is w ∗ -closed and lies in the w ∗ -compact ball B(z, ρ(z, M))). That ϕ is w ∗ -upper semicontinuous follows from the easily verifiable identity {z | ϕz ⊂ G} = {z | PM z ⊂ z − α−1 G}, the w ∗ -upper semicontinuity of the metric projection, and the fact that the set z − α−1 G is w ∗ -open together with G. It is known that if K is a convex compact set in a locally convex space X and ψ : K → 2K is a set-valued upper semicontinuous mapping such that for each x ∈ K, the set ψ(x) is nonempty, convex, and compact, then ψ has a fixed point x0 ∈ K; that is, x0 ∈ ψ(x0 ). From this result and Proposition 10.3 it follows that M is a semisun. Next, it is well known (see, for example, [380, p. 184]) that if E, F are subsets of a normed linear space X [X ∗ ] equipped with any of the topologies n, w [w ∗ ] induced from X [X ∗ ], if f : E → 2F is upper semicontinuous and for each x ∈ X, f x closed, then the graph of f is closed (in the topological product E × F). Now M is a sun by Proposition 10.1, c).  The next result (see [22]) extends Theorem 10.14.

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Theorem 10.15 If the metric projection onto a set M ⊂ X ∗ is w ∗ -upper semicontinuous and has nonempty w ∗ -closed acyclic values, then M is a sun. Now we can prove Theorem 10.8. Indeed, a reflexive space is a dual space, w ∗ = w. Further, PM x is always closed, and being convex, is w ∗ -closed. It remains to invoke Theorem 10.14. Definition 10.4 A set M is said to be ε-compact (see [592]) if for all x  M, there exists ε > 0 such that Pε, M x is compact; here   Pε, M x := M ∩ B x, ρ(x, M) + ε (is the additive ε-projection or the set of near-best approximations). Every ε-compact set is approximatively compact; a boundedly compact set is always ε-compact. Definition 10.5 Let X, Y be metric spaces. A set-valued mapping ϕ : X → 2Y is said to be upper (lower) Hausdorff semicontinuous if xn → x implies that d(ϕxn, ϕx) → 0 (d(ϕx, ϕxn ) → 0), where d(M, N) := supx ∈M ρ(x, N). We point out the following auxiliary fact ([286, p. 150]). Proposition 10.4 Let X, Y be metric spaces, ϕ : X → 2Y Hausdorff upper semicontinuous, and ϕx compact for each x ∈ X. Then ϕ is upper semicontinuous. Theorem 10.16 (L. P. Vlasov) Let M be a P-acyclic subset of a Banach space X. Then M is a metasun and a β-sun if one of the following conditions is satisfied: (a) M is locally compact and the metric projection is Hausdorff upper semicontinuous; ˚ (b) M is locally compact and B-connected; (c) M is ε-compact. Proof (of Theorem 10.16) We claim that (a) ⇒ (b) ⇒ (c). The implication (a) ⇒ (b) follows from Theorem 7.2, Proposition 10.4, and the fact that an acyclic set is connected and compact. The implication (b) ⇒ (c) holds because in a normed space, ˚ a locally compact, P-compact, and B-connected set is ε-compact, approximatively compact, and its metric projection is upper semicontinuous (see [592, Theorem 2.2]). So it suffices to prove the theorem under condition c). Let x  M (we can assume that x = 0, ρ(0, M) = 1). As in the proof of Theorem 10.13, we get PV Vr ⊂ P2r 0 for all r > 0 (see (10.8)). Since M is ε-compact, Pε, M 0 is compact for some ε > 0. If 0 < r < ε/2, then PM Vr ⊂ Pε 0, and moreover, PM Vr , convPM Vr are compact. In the notation of Lemma 10.1, the set Wr is compact. The metric projection PM is upper semicontinuous, because an ε-compact set is approximatively compact and since the metric projection onto an approximatively compact set is upper semicontinuous (see [592, Proposition 2.9]). The fact that ϕ = ϕ0,r is upper semicontinuous is proved in the same way as upper w ∗ -semicontinuity was proved in Theorem 10.14. Finally, ϕz and PM z are both acyclic. By the classical fixed-point theorem for acyclic upper

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205

semicontinuous mappings on convex compact spaces (see, for example, [459], [460, Corollary 7.4]), the mapping ϕ has a fixed point x0 ∈ Wr for all r > 0. Hence M is a meta-sun by Proposition 10.3. By Theorem 3.2, a) of [592], M is a β-sun. Theorem 10.16 is proved.  Theorem 10.17 (L. P. Vlasov) Let M be a P-convex set in a Banach space X, and suppose that one of the conditions of Theorem 10.16 holds. Then M is a sun. Proof (of Theorem 10.17) If M is locally compact, then PM x is locally compact, and since PM x is convex bounded and closed, it is compact. If M is ε-compact, then PM x and some PM,ε x ⊃ PM x are compact. Since a convex compact set is acyclic, on applying Theorem 10.16, we see that M is a metasun. Further, by Theorem 3.2 b) of [592], M is a semisun, and Proposition 10.1 implies that M is a sun.  The following theorem [592, Theorem 4.13] is a summary of the preceding theorems, especially for the case of Chebyshev sets. Theorem 10.18 In a Banach space X, a Chebyshev set M with metric projection PM x is a sun if one of the following conditions is satisfied: (a) M is boundedly compact; (b) M is ε-compact; (c) M is locally compact and the metric projection is continuous; ˚ (d) M is locally compact and B-connected; (e) X is uniformly convex and M is locally compact; (f) X is reflexive and the metric projection is w-continuous; (g) X is a dual space and the metric projection is w ∗ -continuous; (h) X is strongly nonlunar (in particular, X = C(Q) or  1 (Γ)) and the metric projection is continuous; (i) X is locally uniformly convex and the metric projection is continuous; (j) for all x ∈ X, there exist r > 0 and C > 0 such that d(PM y, PM z) ≤ C d(y, z) for all

y, z ∈ B(x, r).

Exercise 10.1 Is it true that a monotone path is necessarily a sun? Exercise 10.2 Does there exist a monotone path-connected set that is not a Chebyshev set (a sun)?

Chapter 11

Rational Approximation

Two important objects often used in applications are algebraic rational fractions and rational generalized fractions. It is well known that rational approximations are often superior to approximation by subspaces. In this chapter, we discuss approximative characteristics of rational approximations, that is, problems of existence, uniqueness, and stability of the metric projection operator onto such objects. Characterizations of the elements of the best approximations are given, and the solarity and structure of such nonlinear approximation objects are studied. The approximation of functions by rational expressions is important in different disciplines of analysis and numerical mathematics. Existence of best rational approximation is addressed in Sect. 11.1. We prove the classical result of Walsh–Akhiezer that the set of rational functions Rn,m , m, n ≥ 0, is an existence set in C[a, b]. The proof is greatly facilitated by Deutsch’s machinery of approximative τ-compactness (with respect to regular τ-convergence), which was studied in Sect. 4.3. In Sect. 11.2, we give de la Vallée Poussin’s estimate and Chebyshev’s characterization theorem of an element of best rational approximation. These results extend the classical Chebyshev and de la Vallée Poussin theorems from Sect. 11.2. In Sect. 11.3, we show that the set Rn,m of rational functions in L p , 1 ≤ p < ∞, is a set of existence. However, by the classical Efimov–Stechkin theorem. the set Rn,m , m ≥ 1, is not a Chebyshev set in L p , 1 < p < ∞. Several extensions of this theorem are discussed. In Sect. 11.4, we obtain theorems on characterization, existence, and uniqueness of best approximation by generalized rational fractions when the numerator is taken from one linear subspace and the denominator from a different one. Results on characterization of best generalized rational approximation are given in Sect. 11.5. Uniqueness of general rational approximation is discussed in Sect. 11.6. Continuity properties of the best rational approximation operator are addressed in 11.7. In this chapter, we consider the following approximation problem. Given a function f ∈ C[a, b] and a pair of integers m, n ≥ 0, the function f is approximated by functions of the form u = p/q, where © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_11

207

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p(t) = a0 + a1 t + . . . + an t n ∈ Pn, q(t) = b0 + b1 t + . . . + bm t m ∈ Pm . We can always assume that the fraction p/q is irreducible; that is, p and q have no common factors other than constants (the irreducibility of zero is 0/1). In order that a rational function be bounded on [a, b], it is necessary and sufficient that the denominator q have no zeros on [a, b]. So in the problem of approximation in the uniform norm, we can assume without loss of generality that q(t) > 0 on [a, b]. The family thus obtained is denoted by Rm,n (or Rm,n [a, b]):   p  Rn,m = u =  p ∈ Pn, q ∈ Pm, q > 0 on [a, b] . q Uniform approximation by rational fractions from the class Rn,m on the interval [a, b] was considered first by Chebyshev. Elaborating on the alternance idea of Chebyshev, Akhiezer and Walsh justified the uniqueness of best approximations by rational fractions and proved the existence theorem. So Rn,m is a Chebyshev set in the space C[a, b]. There is one essential difference from uniform approximation in linear Chebyshev sets: the possibility of degeneracy. Rational fractions p1 /q1 and p2 /q2 are said to be equivalent if p1 q2 = p2 q1 . A rational function p/q of degree (m, n), where n ≥ 1, is said to be degenerate if p/q ≡ 0 or p/q ∈ Rm−1,n−1 . Though degeneracy does not affect uniqueness of best approximation, it does spoil the continuity of the metric projection operator (see Sect. 11.7 below).

11.1 Existence of a Best Rational Approximation To begin with, we note that the family of rational functions Rn,m is not convex and is not boundedly compact (or even approximatively compact) in the space C[a, b]. For example, in the space C[0, 1], the absence of bounded compactness is clear if we consider the sequence uk (t) =

1 , 1 + kt

k = 1, 2, . . . . 

Since lim uk (t) =

k→∞

1, 0,

(11.1)

t = 0, t > 0,

the sequence (uk ) has no subsequence converging to a continuous function. The situation for L p -approximation, 1 < p < ∞, is similar. The sequence (u¯k ), u¯k := uk /uk , where uk is defined in (11.1), k = 1, 2, . . . , is bounded in L p but does not converge in L p to a rational function (cf. Theorem 11.4 below). The existence of best approximation is based on a generalized Deutsch’s existence theorem (Lemma 4.2) and the following result [137, Sect. V.1].

11.1 Existence of a Best Rational Approximation

209

Lemma 11.1 (D. Braess) Assume that a sequence (uk ) ⊂ Rn,m [a, b] is bounded in the uniform norm of C[a, b] or in the L r -norm (1 ≤ r < ∞). Then there exist l ≤ m/2 + 1 points t1, . . . , tl and a subsequence (ukl ) that converges to some u∗ ∈ Rn,m uniformly on each compact interval [α, β] ⊂ [a, b] \ {t1, . . . , tl }. Moreover, if u∗ is nondegenerate, then the subsequence (ukl ) converges uniformly on the entire interval [a, b]. Proof Assume that uk := pk /qk ∈ Rn,n , uk C[a,b] ≤ c1 , k = 1, 2, . . . . One may normalize the denominator qk of the fraction uk = pk /qk so that qk ∞ = max |qk (t)| = 1 and t ∈[a,b]

qk (t) > 0, t ∈ [a, b].

Then pk ∞ ≤ uk C[a,b] ≤ c1 . Using a compactness argument and passing if required to subsequences, we can assume that pk → p∗ ∈ Pm and qk → q∗ ∈ Pn uniformly on [a, b]. As a corollary, q∗ ∞ = lim qk j ∞ = 1 and q∗ (t) ≥ 0 for t ∈ [a, b]. This implies that q∗ has at most l ≤ m/2 + 1 zeros (of even multiplicity) t1, . . . , tl on [a, b]. Let [α, β] be a subinterval of [a, b] without a zero of q∗ . Then (pk j /qk j ) converges uniformly to p∗ /q∗ on [α, β]. So if (uk ) is  · ∞ -uniformly bounded, then   max |p∗ (t)/q∗ (t)| | t ∈ [α, β] ≤ c1 . If (uk ) is L r -uniformly bounded, then ∫ β ∫ β |p∗ /q∗ | r dμ ≤ lim |pk /qk | r dμ ≤ lim uk  r ≤ c1r . α

k→∞

α

k→∞

The above bounds are independent of α and β. Since α and β can be chosen arbitrarily close to the zeros of q∗ , it follows that p∗ /q∗ has no poles on [a, b] but only removable singularities. As a corollary, we have p∗ (t) p1 (t) = , q∗ (t) q1 (t)

if q∗ (t)  0,

where q1 (t) > 0 on [a, b] and the fraction p1 /q1 is irreducible; that is, p1 and q1 have no common factors other than constants. Putting u∗ = p1 /q1 ∈ Rm,n , we get the first assertion of the lemma. If the fraction u∗ is not degenerate, then no division has been performed, and  hence p1 = p∗ , q1 = q∗ , proving the second assertion. From the preceding lemma and Lemma 4.2 we get the following classical result. Theorem 11.1 (J. Walsh, N. I. Akhiezer) The set of rational functions Rn,m , m, n ≥ 0, is an existence set in C[a, b]. We note (see Remark 4.5) that the set Rn,m is not approximatively compact for any m ≥ 1 in C[a, b]. For example, in C[0, 1], the best approximation from the set R0,1

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for the function f = 1 − 2t is the function u0 = 0 (there are two points of alternance), but  f − u0  = 1. The sequence un = (nt + 1)−1 is a minimizing sequence for f that contains no convergent subsequences. This shows that R0,1 is not approximatively compact. However, in view of Lemmas 11.1 and 4.2, the set Rn,m is approximatively τ-compact in the sense of pointwise convergence in the space C(Q) on a dense subset Δ

of the compact set Q (Deutsch convergence): by definition, xδ → x if there exists a dense subset Q0 ⊂ Q on which xδ (t) → x(t) for all t ∈ Q0 (see Sect. 4.3.1). The next result pertains to discrete sets. Remark 11.1 For m ≥ 1, the set Rn,m may fail to be closed, and in particular, may fail to be an existence set in the space C(K) on a discrete compact set K (whose cardinality exceeds the sum of degrees of the denominator and numerator in the class Rn,m ). Indeed, consider N > n, m ≥ 1, and define K = { j/N | j = 0, 1, . . . , N }. We set f (t) = 1 for t = 0 and f (t) = 0 at all other points of K. It is clear that f  Rn,m |K . However, if we consider the minimizing sequence (uk |K ), where uk (t) = (1 + kt)−1 ∈ Rn,m , k = 1, 2, . . . , from (11.1), then we get ρ( f , Rn,m |K ) = 0; that is, for f , there exists no nearest point. In fact, this example also works in the case that a compact set has an isolated point and the cardinality of the remaining part of it exceeds n. Remark 11.2 In the case of approximation by rational fractions from the class Rn,m , we have only to consider functions without zeros common to the numerator and denominator. Meinardus [432, p. 155] noted that this is no longer possible in generalized rational approximation. Indeed, let V = {t1, t2 ∈ R2 | t12 + t22 ≤ 1}. On the ball V, the function ⎧ t 4 + t 4 + (t12 + t22 )Tn (t1 ) ⎪ ⎪ ⎨ 1 2 ⎪ , f (t1, t2 ) = t12 + t22 ⎪ ⎪ ⎪ 0, ⎩

(t1, t2 )  (0, 0), (t1, t2 ) = (0, 0),

where Tn is the Chebyshev polynomial of the first kind of degree n, is approximated by rational fractions of the form p/q, where

aik t1i t2k , q(t1, t2 ) = bik t1i t2k . p(t1, t2 ) = i,k ≥0 i+k ≤4

i,k ≥0 i+k ≤2

Meinardus (see [432], [137, p. 109]) showed that for n ≥ 7, an element of best approximation to f from the class Rn,m is unique and reads as u∗ =

t14 + t24 p∗ := . q∗ t12 + t22

11.2 Characterization of Best Rational Approximation in the Space C[a, b]

211

Exercise 11.1 Verify that the best rational approximation of a continuous function on the interval [0, 1] may fail to exist with respect to the seminorm  f  := max | f (ti )|, where ti ∈ [0, 1], i runs over a finite set of values. Hint: consider points t0 = 0, t1 = 1, functions with values f (t0 ) = 1, f (t1 ) = 0, and the function a/(bt + c). Exercise 11.2 Verify that if Π is a span in C[a, b], then the coefficient vectors of the rational functions from the set Π ∩ R n, m form a convex set. Prove that R n, m , m ≥ 1, is not convex. Does this assertion hold for an arbitrary convex set Π?

11.2 Characterization of Best Rational Approximation in the Space C[a, b] In the case of approximation by rational fractions, there are analogues of the classical theorems on upper estimates for best approximation and characterization of an element of best approximation. Best real rational approximants are characterized by equioscillation (equal ripple) properties. Theorem 11.2 (Ch.-J. de la Vallée Poussin) Let f ∈ C[a, b], r = p/q ∈ Rn,m , deg p = n − ν, deg q = m − μ (0 ≤ μ ≤ m, 0 ≤ ν ≤ n). We set N := n + m + 2 − min{ν, μ}, Δ := f − r. If there exists a family of points t1 < . . . < t N from the interval [a, b] such that Δ(ti ) Δ(ti+1 ) < 0,

i = 1, . . . , N − 1,

then ρ( f , Rn,m ) ≥ min |Δ(ti )|. i=1,..., N

Let r = p/q ∈ Rn,m , deg p = n − ν, deg q = m − μ (0 ≤ μ ≤ m, 0 ≤ ν ≤ n). Here and below, the quantity d := min{n − ν, m − μ} is called the defect of a rational fraction r = p/q. For example, the defect of the fraction r = (1 − x 2 )/(1 + x 2 ) in R2,2 is zero, but in R2,3 , the defect is 1. Proof Assume that  f −  r  < mini=1,..., N |Δ(ti )| for some  r = p/ q ∈ Rn,m . Then for the difference  r − r = ( f − r) − ( f −  r ), we have sign( r − r)(ti ) = sign Δ(ti ), i = 1, . . . , N, and hence it has at least N − 1 zeros. Next,  r − r = ( pq − p q)/q q, and hence the polynomial pq − p q also has at least N − 1 zeros, but deg( pq − p q) ≤ max{n + m − μ, n + m − ν} = N − 2, a contradiction. Theorem 11.2 is proved.  Theorem 11.3 (P. L. Chebyshev) Let f ∈ C[a, b], r = p/q ∈ Rn,m , deg p = n − ν, deg q = m − μ (0 ≤ μ ≤ m, 0 ≤ ν ≤ n). We set N := n + m + 2 − min{ν, μ}, Δ := f − r. A necessary and sufficient condition for r to be a best rational approximant for f is that there exist a family of points t1 < . . . < t N from the interval [a, b] such that

212

11 Rational Approximation

Δ(ti ) = α(−1)i Δ,

i = 1, . . . , N,

(11.2)

where the constant α = ±1 is independent of i. Proof Necessity. Assume that there does not exist a family (ti ) of N points satisfying condition (11.2). Proceeding as in the proof of the necessity in Chebyshev’s theorem for polynomials (see Sect. 2.1), we construct closed intervals [a, τ2 ], [τ2, τ3 ], . . . , [τm, b], m < N, such that Δ(τi ) = 0, i = 2, . . . , m, each interval having nonempty intersection only with Q+ or only with Q− , while the neigbouring intervals intersect both Q+ and Q− . Here Q± := {t ∈ Q | Δ(t) = ±Δ}. We write the polynomial  := (t − τ2 )(t − τ3 ) · · · (t − τm ) in the form  = pϕ − qψ, where ϕ ∈ Pm , ψ ∈ Pn . This is possible, because m < N. Then for small λ, we have q = λϕ > 0 on [a, b] and p − λψ ∈ Rn,m, q − λϕ λ pϕ − qψ =Δ− . f − rλ = Δ + r − rλ = Δ − λ q(q − λϕ) q(q − λϕ) rλ =

Arguing as in the proof of Kolmogorov’s theorem (see Sect. 5.1), we can choose λ so as to have  f − rλ  <  f − r . Sufficiency is secured by de la Vallée Poussin’s theorem (Theorem 2.2).  From Theorem 11.3, one can prove the uniqueness of best rational approximation for f ∈ C[a, b]. Assuming that there exist two fractions r   r ∈ Rn,m of best approximation and applying (11.2), one can prove that the numerator of the difference r − r=

pq − q p = (f − r ) − ( f − r) q q

has at least N − 1 zeros, whereas deg(p p − q − p q) < N − 1. Remark 11.3 It was noted above (see Remark 2.1 in Sect. 2.1) that if f is an odd function, then the polynomial of best approximation on the interval [−1, 1] of any degree is odd, and if f is an even function on [−1, 1], then the polynomial of best approximation is even. A similar fact also holds for best rational approximation in C[−1, 1]. Indeed, if f is an even function on [−1, 1] and r = p/q is an irreducible form of its best approximation from C[−1, 1], then p, q are even. Indeed, the fraction r ∗ (t) = p(−t)/q(−t) is also an irreducible best approximation to f . Since irreducible representations are unique up to a constant factor, we have p(−t) = cp(t), q(−t) = cq(t). Comparing the leading coefficients yields c = ±1. The case c = −1 is impossible, since then both p and q would be odd and hence have a common factor t. A similar analysis shows that if f is an odd function, then p is an even polynomial and q is odd.

11.3 Rational L p -Approximation

213

11.3 Rational L p -Approximation In this section we shall show that the set Rn,m of rational functions in L p , 1 ≤ p < ∞, is a set of existence. In Remark 4.5, we mentioned that general considerations yield that Rn,m is approximatively compact for 1 < p < ∞ (N. V. Efimov and S. B. Stechkin) and for p = 1. J. Blatter (1968) proved this using the fact that for a rational L p -approximation, the degree of both the numerator and denominator of the best rational approximant is maximal; that is, if u∗ is a locally nearest element of best L p -approximation from Rn,m , 1 < p < ∞, for f  Rn,m , n ≥ 1, then u∗  Rn−1,m−1

(11.3)

(see Cheney and Goldstein [164], Braess [137, p. 59], Vyacheslavov and Ramazanov [593]). Let us prove (11.3). We show first that linear combinations of the functions (t − c)−1 , c > b, are dense in C[a, b], hence in all spaces L p , 1 ≤ p < ∞. Indeed, by Weierstrass’s theorem, polynomials P(t) are dense in C[a, b], and every polynomial n+1 (t − ck )−1 , P(t) ∈ Pn can be approximated with any accuracy by R(t) = P(t) k=1 where ck are sufficiently large. It remains to expand R(t) into a sum of its partial fractions. Next, let r = u/v ∈ Rn−1,m−1 [a, b], v > 0, be an element of best approximation to f ∈ L p [a, b] \ Rn,m from the class Rn,m . The function rλ := r + λ(t − c)−1 lies in Rn,m for all c > b. Since it does not approximate f better than r, the integral ∫ b ϕ(λ) := | f − rλ | p dt a

attains its maximum at λ = 0; that is, ϕ (0) = 0. Since ∫ b drλ dt, ϕ (λ) = −p | f − rλ | p−1 sign( f − rλ ) dλ a we have

∫ a

b

| f − r | p−1 sign( f − r)

1 dt = 0. t−c

This implies that f − r = 0 a.e. This contradiction proves (11.3). Dunham [213] pointed out that the nondegeneracy property (11.3) ceases to hold in L 1 . Indeed, let f0 ∈ L 1 [−1, 1], f0 (t) > 0 on (−1/2, 1/2), f (t) = 0 elsewhere. Then that |r | ∈ PR0,1 =: r0 = 0. Indeed, it is clear that if r ∈ R0,1 [−1, 1], this implies ∫ R0,1 [−1, 1], and further, since the function |r | is convex, we have |t | ≥1/2 |r | dt ≥ ∫ |r | dt. As a corollary, if r ∈ R0,1 [−1, 1], then |t | ≤1/2

214

11 Rational Approximation

∫ ∫ ∫    f − r 1 ≥  f − |r | 1 ≥ | f | dt − |r | dt + |r | dt |t | ≤1/2 |t | ≤1/2 |t | ≥1/2 ∫ ≥ | f | dt =  f − r0 1 . |t | ≤1/2

A similar example can be constructed in the space L 1 [−1, 1] for the set R0,2 (see [409, p. 247]). Theorem 11.4 (Existence theorem for rational L p -approximation) Let m, n ≥ 0. Then the set Rn,m is approximatively compact in L p [a, b], 1 ≤ p < ∞. As a corollary, Rn,m is a set of existence in L p [a, b]. Proof Let (uk ) be a minimizing sequence for f ∈ L p . By Lemma 11.1, there exist l ≤ m/2+1 points t1, . . . , tl such that the sequence (uk ) (or a subsequence) converges uniformly to u∗ ∈ Rn,m on the set Iδ := {x ∈ [a, b] | |x − ti | ≥ δ, i = 1, 2, . . . , l} for each δ > 0. As a corollary, ∫ ∫ | f − u∗ | p dμ ≤ lim | f − uk | p dμ k→∞

Iδ

Iδ

≤ lim  f − uk  p = ρ( f , Rn,m ) p . k→∞

Since this inequality is satisfied for all δ > 0 and since the function f − u∗ is integrable, we have ∫ | f − u∗ | p dμ ≤ ρ( f , Rn,m ) p . [a,b]

So

u∗

is a nearest element for f . Finally, let 1 < p < ∞. By (11.3), an element of best approximation u∗ for f  Rn,m cannot be a degenerate rational fraction. By Lemma 11.1, the minimizing sequence contains a uniformly convergent subsequence, which is also convergent  in L p . So the set Rn,m is approximatively compact in L p . Theorem 11.5 (N. V. Efimov and S. B. Stechkin) The set Rn,m , m ≥ 1, is not a Chebyshev set in L p , 1 < p < ∞. This result was obtained by N. V. Efimov and S. B. Stechkin from their more general result that asserts that an approximatively compact subset of a uniformly convex and smooth Banach space is a Chebyshev set if and only if it is convex. Theorem 11.5 was extended by I. G. Tsar’kov, who showed in a more general setting, for an approximatively compact nonconvex set in L p , that not only can one guarantee the nonuniqueness of best approximation, but for some point, the set of nearest points is not acyclic. In particular, this is true for the set of nonconvex approximatively compact generalized fractions in L p (Ω, T, Σ), 1 < p < ∞: there is a point for which the set of its nearest points is not acyclic. This extension of Theorem 11.5 is also true if one considers generalized rational fractions, in which the

11.3 Rational L p -Approximation

215

numerator and denominator are taken from some fixed convex subsets of L p (Ω, T, Σ), 1 < p < ∞. The next theorem (see [137, p. 111], [136], [409, p. 247]), which also strengthens Theorem 11.5, shows that for n ≥ 1, the space L p , 1 < p < ∞, contains sufficiently many functions that have more than one nearest point in Rn,m . Theorem 11.6 (D. Braess) Let 1 < r < ∞, n ≥ 0, m ≥ 1. Then every (n + 2)dimensional subspace E of L r [α, β] such that E ∩ Rn,m = {0} contains a function that has at least two best approximations from Rn,m . Proof By Theorem 11.4, Rn,m is a set of existence in L r . Let S n+1 := { f ∈ E |  f  = 1}. Arguing by contradiction, assume that for each f ∈ E in Rn,m there is precisely one element of best approximation. Since by Theorem 11.4 the set Rn,m is approximatively compact in L r , the metric projection PRn, m |E on Rn,m is continuous on E (see the proof of Theorem 5.10). Moreover, by (11.3), the range of PRn, m |E contains no degenerate rational functions. Let the denominators be normalized by maxx ∈[α,β] {q(t} = 1. Then the mapping ϕ : Rn,m \ Rn−1,m−1 → Rn+1 , which sends n ak t k to the vector of coefficients (a1, . . . , an ), is continuous. u = p/q with p = k=0 It is clear that the mapping ϕ is odd; that is, ϕ ◦ P(− f ) = −ϕ ◦ P( f ). By the Borsuk–Ulam antipodality theorem (see, for example, [484, Theorem 8.13]), if Q is a symmetric bounded open neighbourhood of the origin in Rn+1 and if T : ∂Q → Rn is a continuous odd mapping, then there exists a point x ∈ ∂Q such that T(x) = 0. Applying this result, we see that there exists f0 ∈ S n+1 such that ϕ ◦ P( f0 ) = 0. But then the fraction P f0 is degenerate, a contradiction. This shows that the mapping P|E cannot be continuous, and hence there is an element f ∈ E with at least two  nearest points from Rn,m . By combining the arguments of Theorem 11.6, a function with (at least) four best rational L p -approximations may be constructed (see [137, Chap. 5, Sect. 1]). The following result holds [136]. Theorem 11.7 (D. Braess) Let 1 < r < ∞, n ≥ 0. Then every (n + 2)-dimensional subspace E of the space L r [−1, 1] such that E ∩ Rn,1 = {0} and such that f (−t) = (−1)n+1 f (t)

∀f ∈E

contains a function that has at least four approximations in Rn,1 . Remark 11.4 Braess [136] notes that an analysis similar to that carried out in the proof of Theorem 11.6 shows that each (n + 2)-dimensional subspace in C[α, β] contains an element for which the best approximant from Rn,m is degenerate (that is, for which the degrees of the numerator and denominator are not maximal).

216

11 Rational Approximation

11.4 Existence of Best Approximation by Generalized Rational Fractions Our further aim is to obtain theorems on characterization, existence, and uniqueness of best approximation by generalized rational fractions when the numerator is taken from one linear subspace and the denominator from a different one. The first studies of problems of approximation by generalized rational fractions were performed by E. W. Cheney, H. L. Loeb, G. Sh. Rubinshtein, among others, in the 1960s. As distinct from the above particular case Rn,m [a, b], in the generalized approximation setting one cannot guarantee the existence and uniqueness of best approximation. So let V and W be two finite-dimensional subspaces of C(Q), where Q means a compact metric space (or in some problems, a finite real interval). We assume that W contains at least one function that is positive on Q. As an approximation class, we consider the class  p   R = RV,W :=  p ∈ V, q ∈ W, q(t) > 0 on W . q A particular case here is the set of classical rational functions Rn,m [0, 1]. In general, RV,W is not a set of existence (see Example 11.1 below). The existence of best approximation by RV,W can be guaranteed in the case of classical rational approximations (Theorem 11.1) and in the case that dim W = 1 and the subspace V is arbitrary (in the latter case, the problem is linear). Example 11.1 Consider the problem of approximation of the function x(t) = t by functions of the form at 2 /(b0 + b1 t) on the interval [0, 1]. Taking a = 1, b1 = 1 and letting b0 ↓ 0, the function x(t) = t can be approximated arbitrarily closely by functions of this form, x  R2,1 [0, 1] (that is, the set R2,1 [0, 1] under consideration is not closed). Example 11.2 Consider the function f (t1, t2 ) =

2 2 ⎧ ⎪ ⎨ (t1 + 1) + (t2 + 1) , ⎪ t1 + t2 + 2 ⎪ ⎪ t1 + 1, ⎩

−1 ≤ t1 ≤ 1, −1 < t2 ≤ 1, −1 ≤ t1 ≤ 1,

t2 = −1,

defined on the square [−1, 1]2 . We consider the class R = R [−1, 1] of approximants of the form 2  2  i=0 k=0 1  1  j=0 l=0

For ε > 0, we set rε (t1, t2 ) :=

ai k t1i t2k j

.

b j l t1 t2l

(t1 + 1)2 + (t2 + 1)2 . t1 + t2 + 2 + ε

The set R is not closed, because rε (t1, t2 ) → f (t1, t2 ) uniformly on [−1, 1]2 , but f  R . Exercise 11.3 Show that the set of functions at/(b |t | + c), where a, b, c ∈ R, is not a set of existence in C[−1, 1]. Hint: consider a piecewise linear function f such that f (−1) = f (1) = 0, f (1/2) = − f (−1/2) = 1.

11.5 Characterization of Best Generalized Rational Approximation

217

11.5 Characterization of Best Generalized Rational Approximation As was noted above, in the general case, RV,W (Q) is not a set of existence in the space C(Q) (or even in the space C[a, b]). However, even in the general setting in which the set RV,W is not proximinal, it becomes possible to characterize elements of best approximation. First results in this direction were obtained by E. W. Cheney and H. L. Loeb. Given r ∈ R, we set V + rW := {v + rw | v ∈ V, w ∈ W }. So V + rW is a (closed1) linear subspace in C(Q). If {g1, . . . , gn } is a basis for V and {h1, . . . , hm } is a basis for W, then V + rW is the linear hull of the vectors {g1, . . . , gn, r h1, . . . , r hm }. However,   even in the  case r 0, this family is not a basis. Indeed, if r = ai gi / bi hi , then ai gi − bi r hi = 0. So dim(V + rW) ≤ n + m − 1. In what follows, we shall require the following well-known fact: if L, G are linear subspaces, then dim(L + G) + dim(L ∩ G) = dim L + dim G.

(11.4)

The following result (characterization theorem) is due to Cheney and Loeb (see [162]). Theorem 11.8 An element r ∈ RV,W (Q) is an element of best approximation to f  RV,W (Q) if and only if no element ϕ ∈ V + rW has the same signs as f − r on the set   Y := t | | f (t) − r(t)| =  f − r  . of critical points of f − r. Proof If r is not an element of best approximation to f , then we can choose r ∗ = v ∗ /w ∗ ∈ R that approximates f better than r. We set ϕ = w ∗ (r ∗ − r). It is clear that ϕ ∈ V + rW. Next, if τ  Y and σ(τ) := sign( f − r)(τ), then σ(τ)( f − r ∗ )(τ) ≤  f − r ∗  <  f − r  = σ(τ)( f − r)(τ), whence σ(τ)(r ∗ − r)(τ) > 0 and σ(τ)ϕ(τ) > 0. For the converse, let ϕ agree in sign with f − r on Y . We write ϕ = v0 + rw0 , r = v/w and define 1 The sum of finite-dimensional subspaces is always closed.

218

11 Rational Approximation

rλ =

v + λv0 . w − λw0

The rest of the proof is concerned with the choice of λ to satisfy the inequality  f − rλ  <  f − r . We set δ = inf σ(t)ϕ(t). t ∈Y

By continuity and compactness, δ > 0. Let e = f − r. We define   K1 = t ∈ Q | σ(t)ϕ(t) > δ/2, |e(t)| > e/2 , K2 = Q \ K1 . It is clear that K1 ⊃ Y and K2 is a compact set that has no common points with Y . Hence there exists a number μ such that |e(t)| ≤ μ < e,

t ∈ K2 .

(11.5)

To find the required constraints on λ we need some estimates. For t ∈ K2 , we have | f (t) − rλ (t)| ≤ | f (t) − r(t)| + |r(t) − rλ (t)| ≤ μ + r − rλ . Since r − rλ  → 0 for λ → 0, it follows by (11.5) that for small λ, the last term is majorized by e. Now if we choose λ to be sufficiently small that the difference f (t) − rλ (t) has the same sign on K1 as f (t) − r(t), then for t ∈ K1 , we have | f (t) − rλ (t)| = σ(t)( f − r)(t) + σ(t)(r − rλ )(t) λσ(t)ϕ(t) ≤ e − (w − λw0 )(t) λδ ≤ e − < e. 2w − λw0  Now it suffices to take λ > 0 sufficiently small that the difference w − λw0 is positive on Q.  In the next theorem, r ∈ RV,W and ϕ1, . . . , ϕ N is a basis for the space V + rW. Theorem 11.9 (E. W. Cheney, H. L. Loeb) An element r ∈ RV,W is an element of best approximation to f if and only if the origin of R N lies in the convex hull of the set   Y := σ(t) tˆ | |e(t)| = e , where e = f − r, σ(t) = sign[ f (t) − r(t)], tˆ = (ϕ1 (t), . . . , ϕ N (t)). We also note the following result (see [475]). Theorem 11.10 (E. W. Cheney, H. L. Loeb) An element r ∈ RV,W is an element of best approximation from RV,W to f in C(Q), where Q is a finite union of nonsingleton connected compact sets in Rd , if and only if the origin is an element of best approximation to f − r from the subspace V + rW.

11.5 Characterization of Best Generalized Rational Approximation

219

To formulate the Chebyshev alternation theorem, we require some definitions. Recall (see Sect. 2.3) that a finite-dimensional subspace M of C[a, b] is said to be a Haar subspace (or a Chebyshev subspace) if it has a basis satisfying the Haar condition (see Definition 2.1). It is clear that the property that a subspace is a Haar subspace is independent of the choice of a basis in this space. Let a = (a1, . . . , al ) be a real vector of length l. We denote by S + (a) the maximal number of sign changes in the sequence a1, . . . , al , where the zero values of ai are replaced arbitrarily by 1 or −1. For example, S + (−1, 0, 1, −1, 0, −1) = 4. Given a function f ∈ C[a, b], the number of sign changes on [0, 1] is defined as S + ( f ) := sup S + ( f (t1 ), . . . , f (tm )), where the supremum is taken over all tuples of points a ≤ t1 < . . . < tm ≤ 1 and m ∈ N. Given a subspace M (not necessarily a Haar subspace), we denote by ν(M) − 1 the supremum of the number of sign changes of elements in this subspace (in general, the number of sign changes can be infinite). As usual, dim M denotes the dimension of a subspace M. It is clear that a subspace M is a Haar (Chebyshev) subspace if and only if dim M = ν(M). Next, by η(M) we denote the dimension of a maximal Haar subspace from M. So a subspace M is a Haar subspace if and only if dim M = η(M). In summary, ν(M) = 1 + maximum number of variations in sign by members of the subspace M, η(M) = max{dim H | H is a Haar subspace of M }. Note that if M is a finite-dimensional space in C[a, b], then ν(M) ≥ dim M ≥ η(M). Given an element r ∈ R, we again form the subspace V +rW. Note that the indices ν(V + rW) and η(V + rW) depend only on r. Recall (see Sect. 2) that a function e is said to have an alternant of length k if there exist points t1 < . . . < tk such that e(ti ) = (−1)λ,

where

|λ| = e.

Theorem 11.11 (Alternation theorem [162]) If the error function e := f − r has alternant of length at least 1 + ν(V + rW), then r is an element of best approximation to f from R. Conversely, if r is an element of best approximation to f from R, then e := f − r has at least 1 + η(V + rW) alternations. We also give the following analogue of de la Vallée Poussin’s theorem from Chapter 2 (for a proof, see [408]). Theorem 11.12 Let r ∈ RV,W and f ∈ C[a, b]. Assume that at some points a ≤ t1 < . . . < tk ≤ b, where k > ν(V + rW), the difference e := f − r alternates sign (sign e(ti ) · sign e(ti+1 ) < 0, i = 1, . . . , k − 1). Then ρ( f , RV,W ) ≥ min | f (ti ) − r(ti )|. 1≤i ≤k

220

11 Rational Approximation

Proof (of Theorem 11.11) If r is not an element of best approximation to f , then by the characterization theorem (Theorem 11.8), there exists a function ϕ ∈ V + rW such that |e(t)| = e ⇒ ϕ(t)e(t) > 0 for all t at which |e(t)| = e. This shows that if the function e has k points of alternant, then ϕ has k − 1 sign changes. Hence the function e has at most 1 + ν(V + rW) alternations. To prove the converse result we need the following lemma (see [162, Chap. 3, Sect. 4]). Lemma 11.2 (A. A. Goldstein) Let {g1, . . . , gn } be a Haar system in C[a, b], let a ≤ t0 < t1 < . . . < tn ≤ b, and let λ0, . . . , λn be nonzero constants. Then in order that 0 ∈ conv{λ0 tˆ0, . . . , λn tˆn }, where tˆi = (g1 (ti ), . . . , gn (ti )), it is necessary and sufficient that λi alternate in sign; that is, i = 1, . . . , n. λi λi−1 < 0, Proof We note first that all the Haar determinants Δ(t1, . . . , tn ) (see Sect. 2.3) have the same sign for a ≤ t1 < . . . < tn ≤ b. Indeed, arguing by contradiction, assume that Δ(t1, . . . , tn ) < 0 < Δ(τ1, . . . , τn ). Then for some λ ∈ (0, 1), we have   Δ λt1 + (1 − λ)τ1, . . . , λtn + (1 − λ)τn = 0. Since the system {g1, . . . , gn } is a Haar system, it follows that for some distinct i and j, λti + (1 − λ)τi = λt j + (1 − λ)τj   (that is, some points from the definition of Δ λt1 + (1 − λ)τ1 , . . . , λtn + (1 − λ)τn are equal). This implies that ti − t j and τi − τj have opposite signs. ˆ  Next, the origin lies in the convex hull of the points λi ti if and only if the equation θ i λi tˆi = 0 has a positive solution (θ 1, . . . , θ n ). Writing this in the form tˆ0 =

n

−θ i λi i=1

θ 0 λ0

tˆi

and solving it by Cramer’s rule, we have −θ i λi Δ(t1, . . . , ti−1, t0, ti+1, . . . , tn ) . = θ 0 λ0 Δ(t1, . . . , tn ) Since it requires i−1 column interchanges in the numerator determinant to restore the natural order of the arguments (with monotone i’s), and since each such interchange alters the sign of a determinant, we see that sign(−θ i λi )(θ 0 λ0 ) = (−1)i−1 , whence sign λi = (−1)i sign λ0, the result required.



11.5 Characterization of Best Generalized Rational Approximation

221

Let us return to the proof of Theorem 11.11. Assume that r is an element of best approximation to f . In the subspace V + rW we can choose a Haar subspace M of dimension n := η(V + rW). Let, as before, Y := {τ | |e(τ)| = e}. By the characterization theorem, there do not exist ϕ ∈ M such that ϕ(t)e(t) > 0 on Y . If {ϕ1, . . . , ϕn } is a basis for M, then the system of inequalities

x ∈ Y, e(t) ci ϕi (t) > 0, is inconsistent. As a corollary, the origin of Rn lies in the convex hull of the set {e(t) tˆ | t ∈ Y }, where tˆ = [ϕ1 (t), . . . , ϕn (t)]. By Carathéodory’s theorem (see Chap. B) and the Haar condition, the origin lies in the convex hull of some set consisting of n + 1 such points e(ti )tˆi , where tˆi = (ϕ1 (ti ), . . . , ϕn (ti )). By Lemma 11.2, e(ti ) should alternate  if t1 < . . . < tn . Hence e has n + 1 points of alternant. The preceding theorem gives a complete characterization of best generalized rational approximations only when the two indices ν and η are the same for the subspace V + rW. Fortunately, this turns out to be true for ordinary rational approximations by the class Rn,m , in accordance with the following lemma (see [162, Chap. 5, Sect. 3]). Lemma 11.3 Let V = Pn , W = Pm , and let r = v/w, where v ∈ V, w ∈ W, w > 0 on [a, b], and the fraction u/v is irreducible. Then V + rW is a Haar subspace in C[a, b] of dimension dim(V + rW) = 1 + max{n + deg w, m + deg v}. Proof (of Lemma 11.3 ) We first show that the dimension of the subspace V + rW is k := 1 + max{n + deg w, m + deg v}. If r = 0, then by our convention, v = 0, w = 1, deg v = −∞, whence k = 1 + n, and this is the dimension of V. Now assume that R  0. We have dim(V + rW) = dim V + dim(rW) − dim(V ∩ rW). It is clear that dim V = n + 1, dim rW = m + 1. Then rW = {(v/w)w1 | deg w ≤ m}, and then the element (v/w)w1 also lies in V if and only if w1 divides w, leaving a quotient of degree ≤ n − deg v. In this case, w1 reads as ww2 , where deg w2 ≤ n − deg v. Since deg w1 ≤ m, we also have deg w2 ≤ m − deg w. Thus dim(V ∩ rW) = 1 + min{m − deg w, n − deg v} = m + n + 2 − k, whence dim(V + rW) = k. To show that V ∩ rW is a Haar subspace, we need to verify that each of its nontrivial elements has at most k − 1 zeros on [a, b]. Assume to the contrary that

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11 Rational Approximation

some element v1 + rw1 from V ∩ rW has k zeros on [a, b]. Then v1 w + vw1 also has k zeros. However, this is impossible, because the degree of this polynomial is at most max{n + deg w, m + deg v} =: k − 1. This completes the proof of Lemma 11.3.



Corollary 11.1 In order that an irreducible rational function v/w be an element of best uniform approximation to a function f from the class Rn,m , it is necessary and sufficient that the error function f − r have at least 2 + max{n + dim w, m + deg v} points of alternant.

11.6 Uniqueness of General Rational Approximation The uniqueness of rational approximants is known in two cases. The first case is approximation by the classical class of rational functions Rn,m [a, b], where the uniqueness was proved by N. I. Akhiezer. The second case is approximation by ratios of trigonometric polynomials in the space C(T) (see [409, p. 217]). The following result was proved by E. W. Cheney [162]. A similar assertion also holds in the space C(Q), where Q is a finite union of connected nonsingleton compact sets in Rd (see [475]). Theorem 11.13 (On uniqueness) Let r be an element of best approximation from the class RV,W [a, b] to f ∈ C[a, b]. If V + rW is a Haar subspace in C[a, b], then r is a unique element of best approximation to f . Corollary 11.2 (On uniqueness of Rn,m ) The class Rn,m is a set of uniqueness in C[a, b]. Corollary 11.2 follows from Theorem 11.13, because by Lemma 11.3, the subspace Pn + rPm is a Haar space. Thus a necessary condition that RV,W be a uniqueness set is that V +rW be a Haar space for every r ∈ RV,W . The converse result need not hold in this generality only because our approximating set has only one restriction: the denominator of w is positive on Q. We note the following result (Pinkus [475]). Here and in what follows in this section, Vn and Wm are finite-dimensional subspaces of dimension, respectively, n and m. Proposition 11.1 If Vn + rWm is a subspace in C(Q) for every r ∈ RVn,Wm , then Vn and Wm are also Haar subspaces. The converse result holds if n = 1 or m = 1. The converse result need not hold in general (see, for example, Cheney [162, p. 169]). To check this, it suffices to consider the subspaces V = span{1, t 2 }, W = span{1, t} and the function r = (1 + t 2 )/(1 − t) on the interval [0, 3]. In the next theorem, which we give without proof, Vn Wm := {vw | v ∈ Vn , w ∈ Wm } (see Pinkus [475]).

11.6 Uniqueness of General Rational Approximation

223

Theorem 11.14 (Pinkus unicity theorem) Let Q be a finite union of nonsingleton connected compact sets in Rd . Then if Vn, Wm are Haar subspaces in C(Q) and if dim(Vn Wm ) = n + m + 1, then RVn,Wm (Q) is a set of uniqueness in C(Q) (Vn Wm is a Haar subspace). To prove Theorem 11.13, we require the following auxiliary lemma. Lemma 11.4 Let r be an element of best approximation from RV,W [a, b] to f  RV,W and let V + rW be a Haar subspace. Then 0 is a unique element ϕ from V + rW with the property ϕ(τ)( f − r)(τ) ≥ 0 for every τ from the critical set   Y := τ | | f (τ) − r(τ)| =  f − r  . Proof (of Lemma 11.4) Let ϕ1, . . . , ϕn be a basis for the subspace V + rW. By Theorem 11.9, the zero element of Rn lies in the convex hull of the set {e(t) tˆ | t ∈ Y }, k where tˆ = (ϕ1 (t), . . . , ϕn (t)), e := f − r. Let 0 = i=0 θ i e(ti )tˆi , where ti ∈ Y , θ i > 0. By the Haar property, k ≥ n (if g1, . . . , gn is a Haar system, then 0 is a unique n ci gi with ≥ n zeros on [a, b]). If ϕ is an arbitrary nonzero function of the form i=1 k θ i e(ti )ϕ(ti ). By the Haar condition, no more element from V + rW, then 0 = i=0 than n − 1 terms e(ti )ϕ(ti ) in the sum can vanish. As a corollary, at least one such term is negative, and at least one is positive.  Proof (of Theorem 11.13) Assume to the contrary that r0 := v0 /w0 is another element of best approximation to f . The function ϕ = w0 (r0 − r) lies in V + rW, and hence setting ϕ := w0 (r0 − r) = w0 [( f − r) − ( f − r0 )]  and since [( f − r)2 − ( f − r)( f − r0 )](τ) ≥ 0 for all τ ∈ Y := τ | | f (τ) − r(τ)| =   f − r  , we see that at all critical points τ ∈ Y (of the f − r), we have the inequality ϕ(τ)( f − r)(τ) ≥ 0. Finally, f  RV,W , and hence ϕ = 0 by Lemma 11.4; that is,  r = r0 . Theorem 11.15 (Strong unicity theorem) Let r ∗ be an element of best approximation from the class RV,W [a, b] to f ∈ C[a, b]. If η(V + r ∗W) = dim V + dim W − 1, then there exists a constant γ > 0 such that for all r ∈ RV,W ,  f − r  ≥  f − r ∗  + γr − r ∗ .

(11.6)

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Corollary 11.3 (Strong unicity theorem for the class Rn,m ) Let r ∗ = v ∗ /w ∗ be an element of best approximation from the class Rn,m [a, b] to f ∈ C[a, b]. If min{n − deg v ∗, m − deg w ∗ } = 0, then there exists a constant γ > 0 such that for all r ∈ Rn,m [a, b],  f − r  ≥  f − r ∗  + γr − r ∗ . Let us give another variant of the strong unicity theorem [376]. Given an irreducible fraction r = p/q ∈ Rn,m [a, b], its defect is defined as follows:  min{n − deg p, m − deg q}, r  0; d(r) := m, r = 0. Clearly, a fraction r = p/q ∈ Rn,m is degenerate if and only if d(r) > 0. Moreover, d(r) is the greatest number such that r = p/q ∈ Rm−d(r),n−d(r) . Theorem 11.16 (Strong unicity theorem for the class Rn,m ) Let f ∈ C[a, b] \ Rn,m and let r ∗ = p∗ /q∗ be its best rational approximant from Rn,m . Then the following conditions are equivalent: (a) d(r ∗ ) = 0; (b) r ∗ is a strongly unique element of best approximation to f ; that is, there exists a constant γ > 0 such that  f − r  −  f − r ∗  ≥ γr − r ∗  for all r ∈ Rn,m ; (c) the metric projection operator f → PRn, m f is continuous at the point f . To prove Theorem 11.15, we require the following auxiliary lemma. Lemma 11.5 Let r ∗ = v ∗ /w ∗ ∈ RV,W [a, b] and let dim(V + v ∗W) = dim V + dim W − 1.

(11.7)

Suppose that v ∈ V, w ∈ W, v + w = v ∗  + w ∗ , v = r ∗ w, and w(t) ≥ 0 on [a, b]. Then v = v ∗ and w = w ∗ . Proof (of Lemma 11.5) If r ∗ = 0, then v ∗ = 0 and v = 0. Hence dim W = 1. Next, we have w = w ∗  and w(t) w ∗ (t) ≥ 0, and so w = w ∗ . Assume now that r ∗  0. From the equations v = r ∗ w, v ∗ = r ∗ w ∗ and the condition v + w = v ∗  + w ∗ , it follows that v, v ∗ are nonzero elements of the subspace V ∩ r ∗W; that is, dim(V + r ∗W) ≥ 1. In view of condition (11.7), the equality (11.8) dim(V + r ∗W) = dim(V) + dim(W) − dim(V ∩ r ∗W) shows that dim(V ∩r ∗W) ≤ 1, and hence, using the above equality, dim(V ∩r ∗W) = 1. As a result, v is v ∗ multiplied by a constant. Now by the remaining conditions we  see that v = v ∗ , w = w ∗ , the result required. Proof (of Theorem 11.15) The theorem is trivial in the case f ∈ RV,W . Correspondingly, we assume that f  RV,W . For r ∈ RV,W and r  r ∗ , we define γ(r) =

 f − r  −  f − r∗  . r − r ∗ 

11.6 Uniqueness of General Rational Approximation

225

We claim that the function γ(r) is bounded away from zero. Assume to the contrary that there is a sequence rk ∈ RV,W such that rk  r ∗ and γ(rk ) → 0. Let rk = vk /wk , where vk ∈ V, wk ∈ W. We can assume that vk  + wk  = 1. Similarly, if r ∗ = v ∗ /w ∗ , then we assume that v ∗  + w ∗  = 1. By compactness, passing to subsequences if necessary, we can assume that vk → v, wk → w uniformly on [a, b]. It is clear that v + w = 1. Next let us show that v = r ∗ w. Let σ(t) := sign( f − r ∗ )(t) and let τ be a critical point of the function f − r ∗ ; that is,    τ ∈ Y := t  |( f − r ∗ )(t)| =  f − r ∗  . Then γ(rk )r ∗ − rk  =  f − rk  −  f − r ∗ 

≥ σ(τ)( f − rk )(τ) − σ(τ)( f − r ∗ )(τ) = σ(τ)(r ∗ − r)(τ) σ(τ)(r ∗ wk − vk )(τ) . = wk (τ)

(11.9)

Using the fact that wk > 0 and wk  ≤ 1, we have, for all τ ∈ Y , σ(τ)(r ∗ wk − vk )(τ) ≤ γ(rk )r ∗ − rk wk (τ) ≤ γ(rk )r ∗ − rk .

(11.10)

Since γ(rk ) → 0, we have rk  ≤ μ for all k for some μ > 0. (If rkl  → ∞ for some subsequence (kl ), then γ(rkl ) → 1.) This shows that the right-hand side in (11.10) converges to zero as k → ∞, whence σ(τ)(r ∗ w − v)(τ) ≤ 0, r ∗ w;

τ ∈ Y.

∩ r ∗W.

By Lemma 11.4, v = that is, v ∈ V Since η(V + r ∗W) ≤ dim(V + r ∗W), from condition (11.6) and equality (11.7) we see that η(V + r ∗W) = dim(V + r ∗W); that is, V + r ∗W is a Haar subspace. Applying Lemma 11.5, we have v = v ∗ , w = w ∗ . Since w(t) = w ∗ (t) > 0 on [a, b], passing to a subsequence if necessary, we can assume that wk (t) ≥ ε for some ε > 0 and all k and t. We set c=

max σ(τ)ϕ(τ).

inf

ϕ ∈V +r ∗ W τ ∈Y ϕ =1

By Lemma 11.4, we have c > 0. Form the definition of c and inequality (11.9), it follows that there exists a point τ ∈ Y such that σ(τ)(r ∗ wk − vk )(τ) wk (τ) ∗ ≥ σ(τ)(r wk − vk )(τ)

γ(rk )rk − r ∗  ≥

≥ cr ∗ wk − vk  ≥ cεr ∗ − rk .

We have a contradiction, because rk  r ∗ , γ(rk ) → 0. Theorem 11.15 is proved. 

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11.7 Continuity of the Best Rational Approximation Operator H. Maehly and Ch. Witzgall were the first to notice that the operator of best rational approximation (associating with any f ∈ C[a, b] its fraction of best approximation) can be discontinuous even in the classical case of approximation by rational fractions from the class Rn,m . Consider the corresponding example (Figure 11.1). Setting λ (λ > 0), R0 (t) = 0, Rλ (t) = λ+t we consider Rλ as an element from the set R0,1 [0, 1]. It is clear that Rλ − R0  = 1 for λ > 0, because Rλ (0) = 1.

Fig. 11.1 Three alternant points for fλ − Rλ .

Now let us determine a continuous fλ for which Rλ is the best approximation in the class R0,1 . Let us find three alternant points for fλ − Rλ (by the alternation theorem, it will be enough to give two alternant points). For example, let fλ be the function defined by the dotted line in Fig. 11.1 ( fλ (0) = 1 − c, fλ (1/2) = Rλ (1/2) + c, fλ (1) = Rλ − c, 0 < c < 1/2, fλ is linear elsewhere). As λ → 0, the functions fλ converge uniformly to a continuous function f0 , for which the element of best approximation is R0 = 0. However, as we already noted, Rλ does not converge to R0 . This shows that the (single-valued) operator of best rational approximation is discontinuous at f0 . H. Maehly and Ch. Witzgall proved that a sufficient condition that the operator PRn, m be continuous at some f ∈ C[a, b] is that the degree of the numerator or denominator of the corresponding approximant be maximal for the given class (see also Theorem 11.16). H. Werner proved that this condition is also necessary (if, of course, f does not lie in the approximating class). Cheney (see [162]) extended the result of Maehly and Witzgall’s to generalized fractions, as follows (we recall that η(M) denotes the dimension of the Haar subspace from M).

11.7 Continuity of the Best Rational Approximation Operator

227

Theorem 11.17 (Continuity theorem) Let r0 ∈ PRV ,W f0 and let η(V + r0W) = dim(V) + dim(W) − 1. Then PRV ,W f   for every f from some neighbourhood of f0 , and the operator PRV ,W f0 is continuous at f0 in the following sense: there exists a number β > 0 such that r0 − r  < β f0 − f  for any r ∈ PRV ,W f . Proof (of Theorem 11.17) The search for a best approximation to f may clearly be confined to those r ∈ RV,W for which r − f  ≤ r0 − f . By Theorem 11.15 (the strong unicity theorem), such an r satisfies the inequalities γr − r0  ≤  f0 − r  −  f0 − r0  ≤  f0 − f  +  f − r  −  f0 − r0  ≤  f0 − f  +  f − r0  −  f0 − r0  ≤  f0 − f  +  f − f0 . Hence a best approximation r, if it exists, must satisfy r − r0 | ≤ 2 f − f0 /γ. So as the constant β in the theorem one may take 2/γ. Now it remains to show that f0 has a nearest element. Let r0 = v0 /w0 and assume that the normalization condition v0  + w0  = 1 is satisfied. The number 2ε1 := inf Q0 (t) is positive. We can select an ε2 > 0 such that v + w = 1,

r = v/w ∈ RV,W ,

r − r0  < ε2,

which gives w − w0  < ε2 . To verify the last claim, assume to the contrary that there exists a sequence rk = vk /wk ∈ RV,W such that vk  + wk  = 1, rk → r0 , and Q k − Q0  ≥ ε2 . By compactness, we may assume that vk → v, wk → w. Next, rk → r0 , and so v = r0 w. By Lemma 11.5, we have v = v0 , w = w0 , a contradiction. Now we assume that  f − f0  < γε2 /2. Then an element of best approximation r for f (if it exists) must satisfy the inequality r − r0  < ε2 . If we normalize r = v/w by setting v + w = 1, then it will follow that w − w0  < ε1 . Next, w0 (t) ≥ 2ε1 , and hence w(t) ≥ ε1 . So our search for r = v/w is confined to   v/w | v ∈ V, w ∈ W, v + w = 1, w(t) ≥ ε1 on [a, b] . It is clear that this set is compact, and it must therefore contain an element of best approximation r to f .  Exercise 11.4 Let ϕ be an element of an n-dimensional Haar subspace in C[a, b] and let a ≤ t0 < t1 < . . . < tn ≤ b. Assume that (−1)i ϕ(ti ) ≥ 0. Prove that ϕ = 0. Exercise 11.5 Show that if proof of Theorem 11.15).

 f −R k − f −R ∗   R k −R ∗ 

→ 0, then the sequence Rk  is bounded (see the

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11 Rational Approximation

Exercise 11.6 Show that if V, W are finite-dimensional subspaces in C[a, b], then the set   P/Q | P ∈ V, Q ∈ W,  P  + Q  = 1, Q(t) ≥ ε > 0 on [a, b] is compact in C[a, b] (see the proof of Theorem 11.17).

11.8 Notes on Algorithms of Rational Approximations The disadvantage of using polynomials for approximation is their tendency to oscillate. This often causes the error bounds in polynomial approximation to significantly exceed the average approximation error, because error bounds are determined by the maximum approximation error. In the methods involving rational functions, the approximation error is spread more evenly over the approximation interval. For methods of construction of elements of best rational approximation, we note that in general, this problem can be reduced to the problem of minimization of a linear function under nonlinear constraints and can possibly be attacked via general methods of minimization and methods of nonlinear programming. However, much more efficient are the specially designed algorithms that use the specifics of the problem. For example, more uniformly accurate rational-function approximations can be obtained using Chebyshev polynomials, which exhibit more uniform behavior. The Chebyshev method does not produce the best rational function approximation in the sense of the approximation whose maximum approximation error is minimal. The method can, however, be used as a starting point for an iterative method known as the second Remez algorithm, which converges to the best approximation (see, for example, Sect. 2.7 above). For algorithms of best uniform rational approximation of functions (the generalized Cheney–Loeb algorithm, the Rubinshtein algorithm), see the books by Berdyshev and Petrak [98], Petrushev and Popov [466], Collatz and Krabs [175], Rivlin [495], Watson [598], Powell [482, pp. 90–92], Trefethen [547], and Bogatyrev [110]. For the mean-square rational approximation, we mention, for example, the iteration algorithm for rational approximation based on the Levenberg–Marquardt method of approximation of functions of general form (see, for example, [330]). This method, which is one of the methods for dealing with nonlinear mean-square problems, uses the advantages of Newton minimization by taking into account the nonlinearity in the quadratic approximation of a minimized function without consideration of the second derivatives. This method is easy to implement. Despite the fact that the method may converge quite slowly on highly nonlinear problems, the experience of solution of applied problems of large dimension shows that in many cases, this algorithm gives a suitable approximation for a small number of iterations from a practical point of view. For more details, see [98], [547], [495].

Chapter 12

Haar Cones and Varisolvency

In this chapter, we consider the natural extension of the Haar property from Sect. 2.3 to special nonlinear sets (Haar cones). These classes of sets are frequently encountered in various extreme problems. Properties of Haar cones, as well as uniqueness and strong uniqueness of best approximation by Haar cone,s are discussed in Sect. 12.1. The alternation theorem for Haar cones is given in Sect. 12.2. Next in 12.3, we discuss the property of varisolvency, which is a generalization of the classical Haar condition. Haar cones were introduced by Braess [130], [131], [135] in solving the problem of characterization of elements of local approximation by γ-polynomials in terms of alternants. His studies were continued in particular by Peisker [464] (see also [137]) and Drozzhin [206], [207]. The latter also considered, in parallel with Haar cones, approximation by Chebyshev cones, quasi-Chebyshev cones, and cones with constraints. Many approximative properties of Chebyshev (Haar) subspaces can also be found in Haar cones, which are Chebyshev cones of special kind. Definition 12.1 Let 0 ≤ m ≤ n and let v1, v2, . . . , vn ∈ C[a, b]. If the functions {v j | j ∈ J} span a Haar (Chebyshev) system whenever {1, 2, . . . , m} ⊂ J ⊂ {1, 2, . . . , n}, then the set n     α j v j  α j ∈ R, j = 1, 2, . . . , m, α j ≥ 0, j = m + 1, . . . , n (12.1) M= j=1

is called a Haar cone of dimension n. If m < n, then a Haar cone M is said to be proper (or regular). The functions v j are called generators of a Haar cone. n αi (u)vi we associate the index With each u ∈ M which can be written as u = i=1 set  J(u) := {1, 2, . . . , m} { j | m + 1 ≤ j ≤ n, α j > 0}. To each Haar cone, its root number is assigned: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_12

229

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12 Haar Cones and Varisolvency

r = r(M) := sup{k ∈ N | there exists a nontrivial element u ∈ M with k − 1 zeros on [a, b]}. Example 12.1 The set of polynomials with nonnegative leading coefficient, n      M = u(t)  u(t) = α j t j , α j ∈ R, αn ≥ 0 , j=0

is a Haar cone with r = n + 1. Example 12.2 Let −1 < τ1 < τ2 < . . . < τn ≤ 1. Then n   M = u(t) = j=1

  αj   αj ≥ 0 1 − τj t

is a Haar cone in C[−1, 1] with r = 1 (see [137, p. 14]).

Remark 12.1 From the definition of a Haar cone and basic properties of Chebyshev systems it follows that for every {1, 2, . . . , m} ⊂ {l1, l2, . . . , lk } ⊂ {1, 2, . . . , n}, the determinants  k

l1, l2, . . . , lk U := det vli (t j ) i, j=1 t1, t2, . . . , tk have the same nonzero sign for arbitrary points a ≤ t1 < t2 < . . . < tk ≤ b.

12.1 Properties of Haar Cones. Uniqueness and Strong Uniqueness of Best Approximation Let us now formulate the basic properties of Haar cones, which will be used to derive a criterion for best approximation by Haar cones of continuous functions. We start with the following simple result (see [137, p. 14]). Theorem 12.1 Every Haar cone is a Chebyshev set in C[a, b]. Proof Let u1, u2 be two best approximations by a Haar cone M to a function f . Then clearly, u = 12 (u1 + u2 ) ∈ PM f . Note that u is a local element of best approximation from the subspace H := span{v j | j ∈ J(u)}. Since every local element of best approximation from a convex set (in this case, the subspace H) is an element of best approximation and since u1, u2 ∈ H and H is a Chebyshev set (by Theorem 2.6), we  have u1 = u = u2 . The next theorem (see [137, p. 14]) strengthens the strong uniqueness theorem of D. J. Newman, H. S. Shapiro, and N. G. Chebotarev to the Haar cone setting (Theorem 2.13). Theorem 12.2 Let M be a Haar cone in C[a, b]. Then the (unique) element of best approximation to f  M is a strongly unique best approximation from M.

12.1 Properties of Haar Cones. Uniqueness . . .

231

Proof It can be assumed without loss of generality that 0 = PM f , f  M. Setting P[ f ] := {t | | f (t) =  f }, we claim that for some c > 0, min u(t) sign f (t) ≤ −cu,

t ∈P[ f ]

u ∈ M.

(12.2)

The function ψ(u) := min f (t) u(t) t ∈P[ f ]

attains its maximum on the compact set M1 := {u ∈ M | u = 1} for some element u0 ∈ M1 . By the Kolmogorov criterion, ψ(u0 ) ≤ 0. If ψ(u0 ) < 0, we are done. Assume that ψ(u0 ) = 0. There are two cases to consider depending on whether 0 is a nearest element or not an element of best approximation to f in the subspace H := span{v j | j ∈ J(u0 )}. Case 1. 0 ∈ PH f . By the theorem on characterization of best uniform approximation from a finite-dimensional subspace (Theorem 5.9), there exist a number k, k ≤ dim H + 1, distinct points t1 < t2 < . . . < tk ∈ P[ f ], and numbers θ j > 0 such that k  θ j f (t j ) u0 (t j ) = 0. (12.3) j=1

The Haar condition implies that k = dim H + 1. Since u0  = 1, we see that u0 has at most dim H − 1 zeros, and so in (12.3), not all summands can vanish. Therefore, at least one summand is negative. This shows that ψ(u0 ) < 0. Case 2. 0  PH f . By the Kolmogorov criterion, k 

( f (t) − 0) u1 (t) > 0

(12.4)

j=1

for some u1 ∈ H. Note that uτ = u0 + τ(u1 − u0 ) ∈ M for sufficiently small τ. We have ψ(u0 ) = mint ∈P[ f ] f (t) u0 (t) = 0, and hence by (12.4), min f (t)uτ (t) > 0.

t ∈P[ f ]

But by the Kolmogorov criterion, 0 is not a nearest point to f from the convex set M. This is a contradiction. So (12.2) is satisfied for at least one u ∈ M, u = 1, and some c > 0. From (12.2), we have, for all u ∈ M,  f − u ≥ max | f (t) − u(t)| t ∈P[ f ]

≥ max [ f (t) − u(t)] · sign f (t) t ∈P[ f ]

=  f  − min u(t) sign f (t) ≥  f  + cu. t ∈P[ f ]

This shows that 0 is a strongly unique best approximation element.



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12 Haar Cones and Varisolvency

12.2 Alternation Theorem for Haar Cones Best approximations by Haar cones were characterized in terms of alternants by Peisker [464] and Drozzhin [206], [207] (see also Braess [137]). Here we consider only approximation by proper Haar cones; otherwise, the alternation results from linear Haar spaces can be applied (Theorem 2.8). Recall that m + 1 ≤ r(M) ≤ n, where r(M) is the root number of a Haar cone. For the following two results, see [137], [206]. Proposition 12.1 Let m + 1 ≤ l ≤ r(M). Then there exist an -dimensional Haar cone K ⊂ M and a nonzero function from K with exactly  − 1 zeros. Proof The proof makes use of a Carathéodory-type argument like the one from Chapter B. By definition, the set

N := v ∈ M | v  0, v has ≤  − 1 zeros on [a, b]  is nonempty. Choose a function u0 = α j v j ∈ N for which the cardinality k of the index set J(u0 ) is minimal. Denote the first  − 1 zeros of u0 by t1 < t2 < . . . < t−1 . We claim that k = . By definition of a Haar cone, span{v j | j ∈ J(u0 )} contains an -dimensional subspace H. There is a nontrivial function v ∈ H in H with zeros  t1, t2, . . . , t−1 . We set v = nj=1 b j v j . After replacing v by −v if necessary, we can assume that b j < 0 for at least one j > m. Then the function uτ := u0 + τv, τ ≥ 0, also vanishes at the given zeros. We set τ ∗ := min{|α j /b j | | j ≥ m + 1, b j < 0}. Then uτ ∗ ∈ M, but |J(uτ ∗ )| ≤ k − 1, which contradicts the minimality of |J(u0 )|. By  setting K := {u ∈ M | J(u) ⊂ J(u0 )}, we complete the proof. Proposition 12.2 Assume that a proper -dimensional Haar cone K contains a function with  − 1 zeros. Then there exists a nontrivial function from K with  − 1 prescribed zeros. Moreover, each element v ∈ K with  − 1 zeros on (a, b) has a fixed sign σ = σ(K ) at the right boundary point b of [a, b]. Proof Denote the prescribed zeros by t1 < t2 < . . . < x−1 . Assume that u0 ∈ K has zeros z1 < z2 < . . . < z−1 . For 0 ≤ τ ≤ 1, we set x j (τ) := z j + τ(t j − z j ) and let uτ be the solution of the following interpolation problem in the subspace H := span K : uτ (xk (τ)) = 0, j = 1, 2, . . . ,  − 1,  uτ 12 x1 (τ) + 12 x2 (τ) = u0 12 [z1 + z2 ] . Since each uτ has  − 1 zeros, none of its coefficients with index j ≥ m + 1, j ∈ J(u0 ), can vanish. The mapping τ → uτ is continuous, and the signs of the above coefficients above are independent of τ. Hence uτ ∈ K , 0 ≤ τ ≤ 1, and u1 is the solution of the interpolation problem. The statement on the sign follows from the fact that sign uτ (b) is constant if z−1 ,  t−1  b and since (−uτ )  K .

12.2 Alternation Theorem for Haar Cones

233

As a consequence of these two results we have the following result (Peisker [464]; see also Drozzhin [206, Proposition 1.5.4]). Lemma 12.1 All functions in a proper Haar cone M with precisely r(M) − 1 zeros on (a, b) have the same sign σ(M) at the point t = b. Conversely, for every m <  < r(M) and σ = +1 or σ = −1, there exists a cone K as stated in Proposition 12.1 with σ(K ) = σ. The number σ(M) is called the sign of the Haar cone M. Proof Note that the cones as specified in Proposition 12.1 are not unique. Let Kr , r ). Let v ∈ Kr , r be two such cones, where r = r(M). We claim that σ(Kr ) = σ(K K r be functions with the same set of zeros t1 < t2 < . . . < tr−1 < b. It can be  v∈K assumed without loss of generality that | v (b)| = |v(b)|. If the functions v and  v attain the opposite sign at the point b, then the function v +  v has an additional zero. Since (−v)  M, we have v +  v  0, which contradicts the maximality of r. If m <  < r, then there exists a function u0 ∈ K+1 with  zeros t1 < t2 < . . . < t−1 < t < b. We choose z1 , z2 such that t−1 < z1 < t < z2 . Then sign u0 (z1 ) = − sign u0 (z2 ). Next, choose functions in the ( + 1)-dimensional Haar subspace H containing K+1 and that have, respectively,  zeros t1, t2, . . . , t−1, z1 and t1, t2, . . . , t−1, z2 . Proceeding further as in the proof of Proposition 12.1, we get two  , for which σ(K ) = −σ(K  ).  cones K and K Theorem 12.3 (Alternation theorem for Haar cones [464]) The element 0 is an element of best approximation to a function f from a proper Haar cone M if and only if the difference f − 0 has an alternant of length r(M) with the sign σ(M) at the rightmost point of the alternant. Since every Haar cone M is a convex set, v ∈ M is an element of best approximation to f ∈ C[a, b] if and only if 0 is an element of best approximation from the cone KJ(v) to the function f − v. Hence in Theorem 12.3 we can, without loss of generality, study 0 as an approximant. Proof Assume that there is an alternant of length r = r(M) with the sign σ(M) to the right; that is, there exist points a ≤ t1 < t2 < . . . < tr ≤ b such that (−σ)(−1)r−i f (ti ) =  f ,

1 ≤ i ≤ r.

Assume that for f in M there exists a better approximant v. Then it follows from a de la Vallée Poussin-type argument that (−σ)(−1)r−i ( f − v)(ti ) ≤  f − v <  f  = (−σ)(−1)r−i f (ti ), whence

(−1)r−i (−σ) v(ti ) > 0,

i = 1, 2, . . . , r.

This shows that v ∈ M has r − 1 zeros in the interval (t1, tr ) and the ‘wrong’ sign to the right: sign u(tr ) = −σ(M). This contradiction shows that 0 ∈ PM f .

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Conversely, assume also that the sign at the rightmost point of the alternant is σ(M) for  = r (that is, there is no alternant of length r with the sign −σ(M) at the rightmost point). Let  ≤ r be the maximal length of the alternant t1 < t2 < . . . < t and let ε be its sign at the rightmost point. Then either  < r or  = r and ε = σ(M). By the second assertion of Lemma 12.1, there exists a Haar cone K such that σ(K ) = ε. Let H be the Haar subspace of dimension  that contains K . Since there is no alternant of length  + 1, by Haar’s theorem in H there is an element that approximates f better than 0; that is,  f − u0  <  f . As a result, (−1)−i σ(K ) u0 (ti ) > 0,

i = 1, 2, . . . , .

So u1 ∈ K is a function with the same  − 1 distinct zeros as u0 . Now, sign u0 (t ) = σ(K ), and hence after multiplying u1 by a positive number, we get u1 (x ) = u0 (x ). Hence u0 = u1 ∈ K ⊂ M, and thus 0 is not a best approximant from M to f . This contradiction proves the theorem. 

12.3 Varisolvency In the theory of nonlinear uniform approximation, the generalization of the Haar condition plays a central role. In the linear setting, the Haar condition can be defined in terms of the number of zeros of the underlying functions, or equivalently, in terms of existence of interpolating functions. In 1960, when introducing the concept of varisolvency, J. R. Rice observed that in the nonlinear setting, both such properties have to be postulated separately. It turned out that the interpolation property should be required only locally, while the property on the zeros (Property Z) should be stated in a global way. There is one more difference from linear families. In approximation by varisolvent families, it may well happen (and it is even necessary for a treatment of real and interesting examples) that the degree (that is, the number of interpolation conditions) is not constant in the family. For example, this is the case in approximation by rational functions and exponential sums. Varisolvent families were introduced by J. Rice as a generalization of a construction by L. Tornheim. The family of rational functions Rn,m serves as a classical example of a varisolvent family. Braess [129] noted that compact sets admitting varisolvent families are always homeomorphic to a subset of a circumference. In this connection, following [129], we assume in this chapter that Q = [α, β] ⊂ R, α < β. Definition 12.2 Let M ⊂ C[α, β], M  . (i) A set M is said to be locally solvent (or interpolation regular of degree m = m(u0 ) at a point u0 ∈ M if for every ε > 0 and m distinct points xi ∈ [α, β], i = 1, 2, . . . , m, there exists a number δ = δ(ε, u0, x1, . . . , xm ) such that |u0 (x) − yi | ≤ δ, i = 1, 2, . . . , m, implies the existence of a function u ∈ M satisfying u − u0  < ε and

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u(xi ) = y0,

i = 1, 2, . . . , m.

(12.5)

(ii) A set M has Property Z (or has the zero property) of degree n = n(u0 ) at a point u0 ∈ M if for every u ∈ M, the difference u − u0 either has at most ≤ n − 1 zeros or vanishes identically. (iii) M is a varisolvent family (or a superregular set) if at each point u ∈ M, both the local solvency condition and Property Z are defined and have the same degree m = m(u). (iv) A set M satisfies the density property if for every u0 ∈ M and ε > 0, there exist u1, u2 ∈ M such that ui − u0  < ε, i = 1, 2, and u1 (t) < u0 (t) < u2 (t)

for all t ∈ [α, β].

Obviously, each n-dimensional linear Haar (Chebyshev) subspace in C[α, β] is a varisolvent family of constant degree n. The example below [132] provides a simplest nontrivial family. Example 12.3 The family E1 = {u | u(t) = ae σ t , a, σ ∈ R} is varisolvent, and the degree is  2, u  0, (12.6) m(u) = 1, u = 0. Indeed, let u0  0. For all u ∈ M, the difference u − u0 = ae σ t − a0 e σ0 t = e σ0 t [ae σ−σ0 t − a0 ] has at most one zero or vanishes identically. This shows that Property Z is satisfied. Next, let y1, y2 be nonzero real numbers with the same sign, and let t1  t2 . The interpolation problem ae σ ti = yi , i = 1, 2, (12.7) can be solved explicitly,

y2 1 ln , a = y1 e−σ x1 , t2 − t1 y1 the parameters a, σ depending continuously on y1, y2 . This shows that E1 is locally solvent (interpolation regular) of degree 2 at u0  0. If u0 = 0, then the difference u − u0 has no zeros for u ∈ E1 , u  u0 . Since E1 contains the one-dimensional linear subspace of constant functions, we obtain local solvability of degree 1 at u0 = 0. σ=

The family E1 from Example 12.3 contains the simplest exponential sums. It elucidates how essential it is to postulate the interpolation feature only locally. For example, if y1 = −y2  0, then problem (12.7) has no solution. In general, it is easier to establish Property Z of the correct degree than to verify the solvability of the nonlinear interpolation problem (12.5). For families with continuous parametrization, use can be made of the following appropriate tools (see, for example, [137, p. 67]). We need the following classical result (see [528, Ch. 4, Sect. 7]). Theorem 12.4 (Brouwer’s theorem on the invariance of domain) Let A be an arbitrary subset of Rn and h a homeomorphism of A on another subset of Rn . Assume

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that x is an interior point of A. Then h(x) is an interior point of h(A). Further, if x is a boundary point of A, then h(x) is a boundary point of h(A). In particular, if A and C are homeomorphic subsets of Rn and A is open, then C is also open. Lemma 12.2 Let M ⊂ C[α, β], let A be an open subset of Rm , and let F : A → M be a continuous one-to-one mapping. Assume that the difference u1 − u2 of each pair u1, u2 ∈ F(A), u1  u2 , has at most m − 1 zeros. Then M is locally solvent of degree m at each point u ∈ F(A). Proof Given m distinct points t1, t2, . . . , tm ∈ [α, β], let R : C[α, β] → Rm be the restriction R f := ( f (t1 ), f (t2 ), . . . , f (tm )). The product map R ◦ F : A → R ◦ F(A) ⊂ Rm is continuous and one-to-one. Then R ◦ F is a homeomorphism. By Theorem 12.4, R ◦ F(A) is open in Rm . From this, the solvability follows immediately. 

12.3.1 Uniqueness of Best Approximation by Varisolvent Sets Definition 12.3 Let x0 be an interior point of the interval [α, β]. A function f ∈ C[α, β] is said to have a nonnodal zero in x0 if f (x) ≥ 0 (or f (x) ≤ 0) in some neighbourhood U of x0 and if f (x)  0 on the boundary of U. Nonnodal zeros are counted with multiplicity 2. All other zeros have multiplicity 1. The next lemma is required for the proof that Property Z remains true if the zeros are counted with their multiplicities. Lemma 12.3 (D. Braess) Let M be a varisolvent set, u, u1 ∈ M. Assume that there exist m + 1 = m(u) + 1 points t0 < t1 < . . . < tm such that (−1)i s [u1 (ti ) − u(ti )] ≥ 0,

i = 0, 1, . . . , m,

(12.8)

where s = +1 or s = −1. Then u1 = u. Proof By symmetry, it can be assumed that m(u1 ) ≥ m(u). The proof will be complete if we have equality in all inequalities in (12.8). Assume to the contrary that (−1)i s [u1 (ti ) − u(ti )] > 0 for some j ≤ m. Without loss of generality, we assume that s = 1. Because of the local solvability, there exists u2 ∈ M such that u2 (ti ) = u1 (ti ) + (−1)i δ,

0 ≤ i ≤ m, i  j.

Here δ is assumed to have been chosen such that u2 − u1  < ε := |(u1 − u)(t1 )|. We have i = 0, 1, . . . , m. (−1)i [u2 (ti ) − u(ti )] > 0, These inequalities show that u2 − u has at least m(u) zeros, which contradicts Property Z. 

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Lemma 12.4 (D. Braess) Let u, u1 lie in a varisolvent set M, u  u1 . Then u − u1 has at most m(u) − 1 zeros.1 Proof Assume that the difference u − u1 has at least m zeros (each zero counted with j its multiplicity). Let (ti )i=1 be the ordered set of points at which the difference u − u1 vanishes. This set can be augmented as follows: if ti is a nonnodal zero, then we add to this set a point from the interval (ti, ti+1 ) or from the interval (ti, β). We also supplement this set by another point, which is located to the left of the first nonnodal zero. The augmented set contains at least m + 1 points. After reordering, we have (−1)i s [u(ti ) − u1 (ti )] ≥ 0,

i = 1, 2, . . . , m + 1,

where s = 1 or s = −1. Now Lemma 12.3 gives u = u1 .



Remark 12.2 Often, the bound for the number of zeros can be reduced by 1 via a parity argument. Assume that (−1)m (u1 − u)(α) · (u1 − u)(β) > 0, where m = m(u). Then the number of zeros on the interval [α, β] must be of the parity of m. So the total number of zeros can be at most m − 2 (counting multiplicities). The first step of the characterization of best approximations in the nonlinear case is the following result [137], which can be looked upon as a generalization of de la Vallée Poussin’s theorem. Theorem 12.5 Let M ⊂ C[α, β] have Property Z of degree m at u0 and let ε0 := f − u0 . Assume that there exist m + 1 ordered points t0 < t1 < . . . < tm from [α, β] at which i = 1, 2, . . . , m. sign ε0 (ti ) = − sign ε0 (ti−1 ), Then for each u ∈ M,

 f − u ≥ min |ε0 (ti )|.

(12.9)

0≤i ≤m

Proof Denote the right-hand side of (12.9) by c and define s := sign ε0 (t0 ). By the hypothesis, s(−1)i ε0 (ti ) ≥ c. Assume that  f − u < c for some u ∈ M. Then s(−1)i [ f (ti ) − u(ti )] ≤  f − u < c ≤ s(−1)i [ f (ti ) − u0 (ti )],

i = 0, 1, . . . , m.

As a corollary, s(−1)i [u0 (ti ) − u(ti )] < 0, i = 0, 1, . . . , m, which contradicts Property Z.  The next result dates back to a result of J. Rice (see [137, p. 69]). Theorem 12.6 (alternation theorem) Let M ⊂ C[α, β] be a varisolvent set, u ∈ M, and let f ∈ C[α, β]. Assume that f − u is not constant. Then u ∈ PM f (is an element of best approximation from M to f ) if and only if f − u has an alternant of length m(u) + 1. 1 Recall that by Definition 12.3, the multiplicity of a nonnodal zero is 2.

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Proof If f − u has an alternant of length m + 1, then from de la Vallée Poussin’s theorem it follows that  f − u1  ≥  f − u for all u1 ∈ M; that is, u ∈ PM f . Assume now that ε := f − u is a nonconstant function and has an alternant of length exactly j ≤ m. Then the interval [α, β] can be split into j closed subintervals I1 = [α, t1 ], I2 = [t1, t2 ], . . . , I j = [t j−1, β] such that s(−1)i ε(t) ≥ −ε,

t ∈ Ii,

i = 1, 2, . . . , j,

where s = 1 or s = −1. A closed interval is compact, and hence s(−1)i ε(t) ≥ −ε + 2c,

t ∈ Ii,

i = 1, 2, . . . , j,

(12.10)

where 0 < c < ε/2. There are three cases to consider. Case 1. j = m. Since M is locally solvent at the point u, there exists an element u1 ∈ M such that i = 1, 2, . . . , m − 1, u1 (ti ) = u(ti ), (12.11) u1 (α) = u(α) − sδ(c, u, α, t1, . . . , tm−1 ). By Lemma 12.4, the difference u1 − u has m − 1 (above) zeros and has no other zeros. As a corollary, s(−1)i (u1 − u)(t) > 0,

t ∈ int Ii, i = 1, 2, . . . , j.

This together with the inequality u1 − u < c and (12.10) implies that ε ≥ s(−1)i ε(t) > s(−1)i ( f − u1 )(t) > −ε + c.

(12.12)

Since t1, . . . , tm−1 are not extremal points, from (12.11) it follows that  f − u1  <  f − u; that is, u cannot be an element of best approximation to f . Case 2. j = m − 1. Since M is locally solvent at u, there exists an element u1 ∈ M such that u1 (α) = u(α) − sδ, i = 1, 2, . . . , m − 2, u1 (ti ) = u(ti ), (12.13) u1 (β) = u(β) − s(−1)m δ,

where δ = δ(c, u, α, t1, . . . , tm−2, β). Using (12.13) to evaluate u1 − u at the endpoints of [α, β] and applying Remark 12.2, we see that the number of zeros inside [α, β] must be precisely m − 2. The rest of the proof proceeds as in Case 1. Case 3. j ≤ m − 2. We first put i = j. Since ε is not a constant function, we may i−1 and choose a pair of points ti < ti+1 such that [ti, ti+1 ] contains no point from (tν )ν=1 no point at which |ε(t)| ≥ ε − 2c. So we can replace i by i + 2. This procedure can be repeated until m − 1 or m − 2 points are constructed. Then we define u1 , respectively, by (12.11) or (12.13). Note that inequality (12.12) holds with the exception of the intervals [t j , t j+1 ], [t j+2, t j+3 ], . . . . Since u1 − u < c, the inequality | f − u1 | < ε also holds in the exceptional intervals. This shows that u1 is a better approximation to f than u.

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Thus, in each of the above three cases, we can find a better approximant from M that approximates f better than u ∈ PM f . This contradiction completes the proof of Theorem 12.6.  Remark 12.3 The condition that the difference f − u should not be constant was missed in Rice’s formulation of his theorem (cf. Theorem 12.6). This condition, which was overlooked by J. Rice in his proof of the alternation theorem, was noted in 1968 by Ch. Dunham, who pointed out that the possibility of existence of the so-called constant error curve ( f − u = const) was not considered. Later it was shown that the constant error curve cannot arise if the degree of a varisolvent set is 1, 2, or 3 (Barrar and Loeb [78]) or is maximal (Braess [133]). Meinardus and Taylor (see [407]) proved that if M is a varisolvent (superregular) family on [a, b] and if the interval [a, b] can be extended to [a1, b1 ] (where −∞ < a1 < a and (or) b < b1 < +∞), so that M is a restriction on [a, b] of some varisolvent set on [a1, b1 ], then M does not permit a constant error curve. We also note another result on the constant error curve [407]. Theorem 12.7 (Theorem on the constant error curve) Let M be a varisolvent set on [a, b] and let the set E(M) := { f ∈ C[a, b] | PM f  } be open in C[a, b]. Then M does not permit a constant error on [a, b]. Remark 12.4 Since by the alternation theorem (Theorem 12.6) and the theorem on the constant error curve (Theorem 12.6), the varisolvent sets of existence are characterized in terms of alternants, it follows from Theorem 2.3 of Sect. 2.2 that a varisolvent set of existence is a strict sun. The following result on uniqueness of approximation by varisolvent sets was obtained by J. Rice (see [137, p. 71]). Theorem 12.8 (Uniqueness theorem for varisolvent sets) Let M ⊂ C[α, β] be a varisolvent family with density property. Then M is a set of uniqueness (that is, for all f ∈ C[α, β] in M, there is at most one element of best approximation). Proof Let u ∈ PM f . By the density property, f − u is not a constant. Assume that there exists a different element of best approximation to f : u1 ∈ PM f , u1  u. By the alternation theorem, the difference f − u has an alternant t0 < t1 < . . . < tm(u) . Arguing as in the proof of de la Vallée Poussin’s theorem (Theorem 12.5), we get s(−1)i (u1 − u)(ti ) ≤ 0,

i = 0, 1, . . . , m(u),

whence u1 = u by Lemma 12.3.



12.3.2 Regular and Singular Points in Approximation by Varisolvent Sets As was already mentioned, of special interest in the study of varisolvent families is the case in which the degree varies in the family. In this context, we note that it was

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observed very early and proved later by H. Maehly and Ch. Witzgall in 1964 that in order for the metric projection PRn, m to be continuous at a point f ∈ C[a, b], it suffices that the degree of the numerator or the denominator of the corresponding fractions be maximal for the given class (see also Theorem 11.16). H. Werner showed that this condition is also necessary (of course, if f does not lie in the approximating class). Let us study this phenomenon for abstract varisolvent sets. Definition 12.4 An element u of a varisolvent set M ⊂ [α, β] is said to be regular [132] if one of the following equivalent properties holds: (1) the degree is constant in some degree neighbourhood of u; (2) there is a neighbourhood of u in M whose closure is compact; (3) there is a neighbourhood u in M that is a manifold (that is, that is homeomorphic to an open subset of Euclidean space). An element from M is singular if it is not a regular point. We first consider the definition of regularity by property (1), and then prove its equivalence to properties (2)–(3). Let M be varisolvent of degree m0 at u0 ∈ M. We claim that for every u ∈ M from some neighbourhood of u0 , its degree is at least m0 . Indeed, we choose m0 distinct points ti , i = 1, 2, . . . , m0 and define δ := δ(1, u0, t1, . . . , tm0 ) (where the number δ(1, u0, t1, . . . , tm0 ) is given in Definition 12.2). By property 1), no u ∈ M with u − u0  < δ is completely determined by its values at m0 − 1 points t2, t3, . . . , tm0 . Correspondingly, the degree of u must at least m0 to satisfy Property Z. As a corollary, all elements whose degree equals the maximal degree mmax := {max m(u) | u ∈ M } are regular. In the families of rational functions, these are the only regular points. However, this is not always true. Example 12.4 (D. Braess) We set

M0 = {0}, M1 = u(t) = a | a < 0 ,



M2 = u(t) = ae σ t | a > 0, σ ∈ R .

As in Example 12.3, one can show that M := M0 ∪ M1 ∪ M2 is varisolvent and that  2, u ∈ M2, m(u) = 1 otherwise. The subsets M1 , M2 contain regular points of degree 1 and 2, respectively, and the point u0 is singular.

More generally, given a varisolvent set M, consider its components Mn , n ≥ 1, consisting of regular points of M of degree n. Choose t1 < t2 < . . . < xn from [α, β]. The mapping

 R : Mn → Rn, R(u) = u(t1 ), u(t2 ), . . . , u(tn ) maps Mn bijectively onto an open set in Rn . From Definition 12.4, 1), it follows that the mapping R−1 is continuous. As a corollary, (R(Mn ), R−1 ) is an atlas for Mn that consists of only one chart. This shows that condition (1) implies condition (3), which, in turn, implies condition (2).

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A function f ∈ C[α, β] is said to be normal for M (or M-normal) if the best approximation to f from M exists and it is a regular point [132]. Theorem 12.9 (Continuity theorem for normal points) Let M be a varisolvent set in C[α, β]. Then: (1) the metric projection onto M is continuous at regular points; (2) if a singular point u lies in PM f for some f ∈ C[α, β] and if the difference f − u has an alternant of exact length m(u) + 1, then the metric projection operator PM onto M is discontinuous at f . We need an auxiliary lemma (see Braess [137, p. 56]). An element u ∈ M is called a strictly local element of best approximation to f ∈ X if u is a unique element of best approximation to f from some open neighbourhood O(u) ∩ M. Lemma 12.5 (Local continuity lemma) Let u be a strictly local element of best approximation from a locally compact set M to f . Then for every ε > 0, there exists δ > 0 such that for each x ∈ B( f , δ), there exists a strictly locally nearest element v to g from M such that v − u < ε. Proof (of Lemma 12.5) By the assumption, u is a unique nearest element for f from M from some neighbourhood U, and moreover, U ∩ M is compact. Let ε > 0. Reducing ε if necessary, it can be assumed that M ∩ B(u, ε) ⊂ int U. From the uniqueness and compactness we conclude that δ :=

 1 ˚ ε)} −  f − u > 0. inf{ f − v | v ∈ U \ B(u, 3

(12.14)

Let g ∈ B( f , δ). Then ρ(g, M ∩ B(u, ε)) ≤ g − u ≤  f − u + δ. For g in the compact set M ∩ B(u, ε) there is an element of best approximation v. By (12.14), it cannot be located on the boundary of this set. Hence v is a strictly local best approximation to g from M.  Proof (of Theorem 12.9) The first assertion is secured by Lemma 12.5. To prove the second assertion, we assume that in each neighbourhood of u there exists u1 ∈ M such that m(u1 ) ≥ m + 1 := m(u) + 1. We set f1 := f + u1 − u. Since f1 − u1 has an alternant of length exactly m + 1 = m(u1 ), the element u1 is not an element of best approximation to f1 from M. Next, by de la Vallée Poussin’s theorem,  f1 − u2  ≥  f1 − u1  for all u2 ∈ {v ∈ M | m(v) ≤ m}. As a corollary, each better approximation to f1 must have degree ≥ m + 1 and must be characterized by an  alternant of length ≥ m + 2. Hence it cannot lie closer than u1 . We note another property of varisolvent sets (see [137, p. 73]). Theorem 12.10 The density property holds at each regular point in a varisolvent set.

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Proof To verify the density property at u ∈ M, it suffices to show that u is not an element of best approximation to f (t) = u(t) + ε/2 (and, respectively, to g(t) = u(t) − ε/2). Hence by the alternation theorem, it suffices to construct u1 ∈ M, u1  u, such that (12.15) u(t) ≤ u1 (t) < u(t) + ε. Assume that in M there is no u1 , u1  u, satisfying (12.15). Then u is a strictly locally nearest element to f0 (t) = u(t) + 3ε from M. By the local continuity lemma (Lemma 12.5), for each f1 ∈ C[α, β],  f1 − f0  < δ, there exists a strictly locally nearest element u1 from M such that u1 − u < ε, provided that δ < ε is sufficiently small. Then ( f1 − u1 )(t) > ε, and hence f1 − u1 must be constant. We now fix f1 such that f1 − f alternates m + 2 times. Then either u1 − u has m zeros or (12.15) holds. In either case, we arrive at a contradiction.  Example 12.5 Bartke [81] (see also [137]) constructed an example of a varisolvent set without the density property. Let M = {0}∪ {u ∈ P5 | u have ≤ 3 zeros and u(x) > 0 for x ∈ [−1, +1]}, where the zeros of a polynomial are counted with their algebraic multiplicity. Then M is a varisolvent set on C[−1, 1] without the density property, and furthermore,  4, u = 0, m(u) = 6 otherwise.

It should also be noted that Haar subspaces are the only unique example of families that are simultaneously convex and varisolvent [132].

Chapter 13

Approximation of Vector-Valued Functions

The Haar property (see Sect. 2.1), which characterizes the Chebyshev subspace in C(Q), was first formulated for real-valued continuous functions. For real-valued functions, approximation by Chebyshev subspaces was found to be closely related to various problems in interpolation, uniqueness, and the number of zeros in nontrivial polynomials (the generalized Haar property). For vector-valued functions, the relation between such properties turned out to be less simple. In Sect. 13.1, we consider approximation of abstract (vector-valued) functions and study related interpolation and uniqueness problems. Uniqueness of best approximation in the mean for vector-valued functions is discussed in Sect. 13.2. The analogue of the classical Haar condition for vector-valued functions is studied in Sect. 13.3. Approximation of vector-valued functions by polynomials is considered in Sect. 13.4. Of special interest is the problem of approximation in spaces of continuous bounded functions on sets that are not even locally compact. Here we consider one natural extension of the space of polynomials of degree ≤ n that is in general not a finitedimensional space even for real-valued polynomials. Geometric approximation theory matured in the late 1940s and early 1950s. Among precursors of geometric approximation theory we mention, in addition to S. B. Stechkin, P. L. Chebyshev, A. Haar, L. N. H. Bunt, T. Motzkin, A. N. Kolmogorov, V. Klee, S. I. Zukhovitskii, and M. G. Krein. In these and later years, much attention was paid to problems of uniqueness, existence, and stability of best approximation, which are central problems of this theory. In this period, many principal problems arising from classical approximation theory were posed, and their solutions determined the contours of modern geometric approximation theory. Problems of existence, uniqueness, and characterization of polynomials of least deviation in sysN ⊂ C(Q, H) in the space C(Q, H), tems of continuous abstract functions F = { fi }i=1 where H is a complex Hilbert space and Q is a metrizable compact set, constitute an important subclass of such problems. Interest in such problems arose after the publications of A. N. Kolmogorov, S. I. Zukhovitskii, and M. G. Krein. This theory took a more or less finished form in joint works of Zukhovitskii and Stechkin [629], [630], [631]. Their studies opened a new world for analysis: approximation in abstract © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_13

243

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13 Approximation of Vector-Valued Functions

infinite-dimensional normed spaces. In this way, first results for abstract functions with values in an infinite-dimensional Hilbert space were obtained by Zukhovitskii and Stechkin [629]. In particular, they proved the existence of a polynomial of least deviation and characterized such polynomials (a generalized Kolmogorov test). They also characterized the systems F in which a polynomial of least deviation from an arbitrary continuous abstract function is unique. In [630], the existence theorem of polynomials of best deviation was extended to the setting of an arbitrary Banach space, and the uniqueness theorem was extended to strictly convex spaces. In [631], the authors considered the cases explored by Zukhovitskii and Krein in which the dimension of a system N is not a multiple of dim H < ∞. In later years, the vector-valued case was studied by A. L. Garkavi, V. A. Koshcheev [366], [367], and others. They studied topological properties of compact sets Q admitting Chebyshev n-systems. Tsar’kov (see [556]) proved an existence theorem for polynomials defined on the ball of an infinite-dimensional space.

13.1 Approximation of Abstract Functions. Interpolation and Uniqueness Let (X,  · ) be a real Banach space, Q a metrizable compact set. Consider the space C(Q, X) consisting of all abstract functions f : Q → X such that ϕ(t) − ϕ(t0 ) → 0 as t → t0 for all t0 ∈ Q. The norm on the space C(Q, X) is defined by ϕ(t)C = max ϕ(t). t ∈Q

In the space C(Q, X), one may consider the standard Chebyshev problem of best approximation of a given abstract function f ∈ C(Q, X) by polynomials in a finite system of linearly independent abstract functions fk ∈ C(Q, X), k = 1, 2, . . . , N. Namely, let N  αk fk (t), p(t) = pα (t) = k=1

where αk , k = 1, 2, . . . , N, are real numbers, be an arbitrary polynomial in the system { fk } and let E N ( f ) = inf  f (t) − pα (t)C = inf max  f (t) − pα (t)X α

α q ∈Q

be the best approximation of an abstract function f by polynomials p. The problem is to find whether there exists a best polynomial (a polynomial of best approximation) p∗ for which E N (ϕ) = ϕ(t) − p∗ (t)C and to study properties of these polynomials of best approximation (such as points of maximal deviation, uniqueness, and other characteristic properties).

13.1 Approximation of Abstract Functions. Interpolation and Uniqueness

245

A similar problem can be posed if X is a complex Banach space. The particular case of X a finite-dimensional unitary space or a complex Hilbert space H was considered in [627], [629]. An appeal to the general theorem on proximinality of finite-dimensional subspace (see Sect. 4.1) gives the following result. Theorem 13.1 (Existence theorem) For every abstract function ϕ ∈ C(Q, X), there exists at least one polynomial of best approximation p∗ ; that is, ϕ(t) − p∗ (t)C = max ϕ(t) − p∗ (t)X = E N (ϕ). t ∈Q

Let us now discuss the problem of points of maximal deviation. Let ϕ ∈ C(Q, X). We say that a point t0 ∈ Q is a point of maximal deviation of a polynomial p from a ϕ if ϕ(t0 ) − p(t0 )X = max ϕ(t) − p(t)X = ϕ(t) − p(t)C . t ∈Q

It is clear that each polynomial p has at least one point of maximal deviation. Consider the following problem: under what conditions on a system { fk } (k = 1, 2, . . . , N) does a polynomial of best approximation have sufficiently many points of maximal deviation? Following [629], [630], [631], consider two properties of systems { fk }: (Am ) for every abstract function ϕ ∈ C(Q, X) and polynomial p∗ of best approximation to ϕ, the number of points of maximal deviation is at least a given natural number m; 0 ) for every m distinct points t ∈ Q and m points x ∈ X, there exists at least (Im i i one polynomial p such that p(ti ) = xi , i = 1, 2, . . . , m (in other words, the corresponding interpolation problem is always solvable). Theorem 13.2 (On points of maximal deviation) A system { fk }, k = 1, 2, . . . , N, has 0 ). property (Am ) if and only if it has property (Im−1 So the number of points of maximal deviation of polynomials of best approximation is related to interpolation properties of the system { fk }. Let s be the dimension of the space X (1 ≤ s ≤ ∞). By compactness, if a system 0 ), then { fk } has property (Im m ≤ [N/s]. The following corollary holds. Corollary 13.1 Let X have dimension s. Then there exist an abstract function ϕ ∈ C(Q, X) and a polynomial p∗ of best approximation to ϕ such that p∗ has at most [N/s] + 1 points of maximal deviation. If N < s ≤ ∞, then it is clear that [N/s] + 1 = 1. In the case s = 1, it can be assumed that X = R, and we arrive at the case of a Chebyshev system in the space C(Q) (see Sect. 2.3).

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Corollary 13.2 A system of continuous real functions { fk }, k = 1, 2, . . . , N, has property (A N +1 ) if and only if it is a Chebyshev system. 0 ) is equivalent to the system { f } being Indeed, for real functions, property (I N k a Chebyshev system. Further, we study when a system of functions { fk } (k = 1, 2, . . . , N) has the following uniqueness property:

(U) for every abstract function ϕ ∈ C(Q, X), there exists a unique polynomial of best approximation p∗ in the system { fk }. We start with necessary conditions. Theorem 13.3 Let dim X = s and n ∈ N, (n − 1)s < N ≤ ns. Then a system { fk }, k = 1, 2, . . . , N, has property (U) if and only if it satisfies the following condition: (Tn−1 ) every polynomial p  0 in the system { fk } has at most n − 1 zeros on Q. Note that since { fk } is linearly independent, it follows from the hypotheses of the theorem that the compact set Q contains at least n points. We give an example showing that the hypotheses of this theorem are insufficient for a system { fk } to have property (U). Let X be the two-dimensional Euclidean space R2 (s = 2); a compact set Q consists of two points; N = 3, and hence n = 2. As { fk } we consider the system of functions f1 = {1, 0; 0, 0},

f2 = {0, 1; 0, 0},

f3 = {0, 0; 1, 0}.

Here f1 is the function that is equal to (1, 0) at the point t1 and assumes the value (0, 0) at t2 ( f2 and f3 are defined similarly). It is easily shown that this system satisfies condition (T1 ). Nevertheless, for the system ϕ = {0, 0; 0, 1}, a polynomial of best approximation is not unique: E3 (ϕ) = min max ϕ(t) − pα (t)R2 = 1, a

t ∈Q

and moreover, this value is attained for every polynomial pα (t) = α1 f1 (t) + α2 f2 (t) + α3 f3 (t) for which α3 = 0 and α12 + α22 ≤ 1. Theorem 13.4 Let dim X = s and let n be a natural number such that (n − 1)s < N ≤ ns. Then a system { fk } (k = 1, 2, . . . , N) has property (U) if and only if it satisfies condition (In−1 ). We need several auxiliary results. Lemma 13.1 Condition (Tn−1 ) is equivalent to the following condition: (In ) for every n distinct points ti ∈ Q and n points xi ∈ X, there exists at most one polynomial p in the system { fk } such that p(t j ) = x j

( j = 1, 2, . . . , n).

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247

Lemma 13.2 Let dim X = s and let n be a natural number such that (n − 1)s < N ≤ ns, and suppose that a system { fk } (k = 1, 2, . . . , N) satisfies condition (Tn−1 ). Then it also satisfies the following condition: 0 ): for every n − 1 distinct points t ∈ Q, there exists at least one polynomial (In−1 i p(0)  0 such that p(0) (t j ) = x j ( j = 1, 2, . . . , n − 1). The case N = ns is immaterial, because in that case, conditions (Tn−1 ) and (In0 ) are equivalent. 0 ) (taken together) are still not sufficient for In general, conditions (Tn−1 ) and (In−1 a system { fk } to have the property (U). This can easily be checked by considering as X the space of continuous functions C[0, 1]. Nevertheless, these two conditions are sufficient for strictly convex spaces X. The above example shows that condition (Tn−1 ) does not imply the property (U) even for a strictly convex X. Theorem 13.5 (on uniqueness) Let X be a strictly convex space of dimension s, and n a natural number such that (n−1)s < N ≤ ns. Then a system { fk } (k = 1, 2, . . . , N) 0 ). has the property (U) if and only if it satisfies conditions (Tn−1 ) and (In−1 The proof of this theorem depends on Theorem 13.2. 0 ) is secured by condition (T Note that if N ≤ s or N = ns, then condition (In−1 n−1 ) and hence can be discarded. In the case N = ns, we have a sharper result. Theorem 13.6 Let X be a strictly convex space of dimension s and let N = ns. Then a system { fk }, k = 1, 2, . . . , N, has property (U) if and only if it satisfies any of the equivalent conditions (Tn−1 ), (In0 ), (An+1 ). The equivalence of the properties (U) and (An+1 ) gives for N = ns a new characterization of systems of uniqueness in terms of the number of points of maximal deviation. Note that all results from this section (with the same formulations and proofs) can be carried out verbatim to the case in which X is a complex Banach space.

13.2 Uniqueness of Best Approximation in the Mean for Vector-Valued Functions In this section, we consider the problem of uniqueness of an element of best integral approximation of a continuous vector-valued function. We give a general criterion for a subspace in which there exists at most one element of best approximation in the mean for every such function. The principal result of this section is the extension of Jackson’s uniqueness theorem (see Theorem 2.17) to the case of functions with values in a strictly convex Banach space.

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Let X be a Banach space, dim X = q < ∞, Q = [0, 1] ⊂ R, C = C(Q, X) the space of functions f continuous on Q with values in X and norm  f  = sup  f (t)X . t ∈Q

On the space of all continuous function, consider the norm ∫  f  =  f (t)X dt.

(13.1)

(13.2)

Q

p∗ ( f , t)

p∗ ( f , t)

By and we denote elements of best approximation by some subspace M ⊂ C with respect to the norm (13.1) and (13.2), respectively. As M we consider the n-dimensional subspace M = M(Sn ) of polynomials in a system of functions Sn = {ϕ1 (t), . . . , ϕn (t)}, ϕi ∈ C(Q, X). By N(Sn ) we denote the greatest number of zeros that a nontrivial polynomial p in the system Sn can have. The problem of the uniqueness of a polynomial p∗ ( f ) ∈ M(Sn ) has been very fully treated. The following criterion was derived by Zukhovitskii and Stechkin [630]. Theorem 13.7 Let X be a strictly convex space. A polynomial p∗ ( f ) ∈ M(Sn ) is unique for every f ∈ C(Q, X) if and only if the following conditions are satisfied: A1) N(Sn ) < n/q; A2) every interpolation problem p(ti ) = xi (p ∈ M(Sn ), ti ∈ Q, xi ∈ X), i = 1, . . . , k for all k < n/q, is solvable (for integer n/q, condition A2) can be dropped). The properties of polynomials of best approximation p∗ ( f ) were also studied in [366]. Theorem 13.7 can be considered an extension of Chebyshev’s and Haar’s classical results on uniqueness of a polynomial p∗ ( f ) in a Chebyshev system of real functions, and so a system Sn satisfying conditions A1) and A2) can be naturally called a Chebyshev system of vector-valued functions. It is known that in the classical case (q = 1), condition N(Sn ) < n (which characterizes the classical Chebyshev systems) guarantees the uniqueness of a polynomial of best approximation p∗ ( f ) for every function f ∈ C(Q, R) (Jackson’s theorem, see Sect. 2.6). However, for a Chebyshev system of vector-valued functions (a system satisfying conditions A1) and A2)), a polynomial p∗ ( f ) may fail to be unique even for q = 2. A corresponding counterexample was constructed for complex-valued functions [373], that is, for X = C (in this case q = 2, the real dimension of M(Sn ) is 2n, and (2n)/q = n is an integer). We give the following extension of Jackson’s theorem, which holds for a broad class of strictly convex spaces X. The condition that follows guarantees also the uniqueness of a polynomial p∗ ( f ) of best uniform approximation for every function f ∈ C(Q, X). In what follows in this section, except Theorems 13.12 and 13.13, we assume that n > q. The following general criterion depends on the following proposition (Singer [519]).

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249

Theorem 13.8 A necessary and sufficient condition for any point y of a normed linear space Y to have at most one element of best approximation in a subspace M ⊂ Y is that there not exist a functional h from the dual space Y ∗ such that for some elements y 0 ∈ Y , p0 ∈ M \ {0}: (B1) h = 1, (B2) h(p) = 0 for all p ∈ M, (B3) h(y 0 ) =  y 0  =  y 0 − p0 . Let E = {e1, . . . , eq } be a basis for X. If x ∈ X, then by x1, . . . , xq we denote the coordinates of the vector x in the basis E; that is, x = x1 e1 + . . . + xq eq . By L 1 (Q) and L ∞ (Q) we denote, respectively, the space of real functions summable on Q and the dual space L 1 (Q)∗ of functions essentially bounded on Q. Theorem 13.9 (A. L. Garkavi [265]) A necessary and sufficient condition for a function f ∈ C(Q, X) to have in a subspace M ⊂ C(Q, X) at most one polynomial p∗ ( f ) of best mean approximation is that there not exist a function q q g(t) = i=1 gi (t)ei ∈ C(Q, X), points p0 (t) = i=1 p0i (t)ei ∈ M \ {0}, and functions αi (t) ∈ L ∞ (Q), i = 1, . . . q, such that: ∫  q 1 (a) sup αi (t) fi (t) dt = 1; f ∈C  f  Q i=1 ∫  q (b) αi (t)pi (t) dt = 0 for all p ∈ M; Q i=1

(c)

∫  q

αi (t)gi (t) dt = g = g − p0  ;

Q i=1 p0 (t) =

(d) 0 if g(t) = 0. If X is strictly convex, then condition (d) can be replaced by the following stronger condition: (d∗ ) p0i (t) = 0 if g(t) = 0 (1 < i < q). q Proof If f ∈ C(Q, X), then f (t) = i=1 fi (t)ei , where fi (t) ∈ C(Q, R). Hence C(Q, X) is the direct sum of q copies of the space C(Q) = C(Q, R). The space C(Q)∗ (with respect to the norm  ·  ) coincides with L 1 (Q)∗ = L ∞ (Q), and hence, taking into account the structure of the space dual to the direct sum, we ∫ have q the following representation of a linear functional h from C(Q, X)∗ : h( f ) = Q i=1 αi (t) fi (t)dt, q where f (t) = i=1 fi (t)ei and αi (t) ∈ L ∞ (Q). It follows that conditions (a), (b), (c) are particular cases of conditions of Theorem 13.8 with Y = C(Q, X) (with respect to the norm  ·  ). Let us show that among the tuples {g, p0, α1, . . . , αq } satisfying conditions (a), (b), (c), there exists a tuple that also satisfies condition (d). Conditions (a), (b), (c) mean that the elements p0 and p = 0 ∈ C deliver the best approximation [519] to g. Hence the zero element 0 ∈ C also delivers the best approximation E to the function gλ (t) = g(t) − λp0 (t) = λ[g(t) − p0 (t)] + (1 − λ)g(t), Hence gλ  = g = g − p0  = E, and so

0 < λ < 1.

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∫ E=

Q

g(t) − λp0 (t)X dt ∫

∫

≤λ

Q

g(t) − λp0 (t)X dt + (1 − λ)

g(t)X dt = E,

Q

which in turn gives g(t) − λp0 (t)X = λg(t) − p0 (t)X + (1 − λ)g(t)X

(13.3)

for all t ∈ Q. If gλ (t) ≡ g(t)−λp0 (t) = 0 ∈ X, then from (13.3) we have g(t)− p0 (t) = 0 ∈ X, g(t) = 0 ∈ X, and p0 (t) = 0 ∈ X. Thus the elements gλ and (1 − λ)p0 satisfy condition d); moreover, they satisfy conditions b) and c) with the same functions αi (t), the result required. Now let X be strictly convex. From (13.3) we have λ[g(t) − p0 (t)] = c(t)(1 − λ)g(t),

c(t)  0

for all t, where g(t)  0 ∈ X, and so g(t) − p0 (t) = K(t)g(t),

(g(t)  0 ∈ X),

(13.4)

where K(t) = (1−λ) λ c(t), 0  K(t) < ∞. It can be assumed that condition d) is satisfied for g and p0 . From (13.4) we have gi (t) − p0i (t) = K(t)gi (t),

(g(t)  0 ∈ X),

i = 1, . . . , q.

(13.5)

If gi = 0, g  0 ∈ X, then from (13.5) we get p0 = 0. If gi = 0, g = 0 ∈ X, then  by d) we have p0 = 0 ∈ X, and again p0i = 0, the result required. Definition 13.1 A space X with a basis E = {e1, . . . , eq } is said to be E-monotone if the conditions i = 1, . . . , q, (13.6) |ai |  |bi |, imply that

q     ai ei    i=1

X

q     bi ei  .  i=1

X

(13.7)

Moreover, if a strict inequality in (13.6) implies that (13.7) is also a strict inequality, then X is said to be strictly E-monotone. All classical spaces are E-monotone with respect to their canonical bases. Lemma 13.3 If a space X is E-monotone and strictly convex, then it is strictly E-monotone. q−1 q−1 Proof Let, for example, x = i=1 ai ei + aq eq , y = i=1 ai ei + bq eq , and |zaq | < |bq |. It can be assumed that ai  0, i = 1, . . . , q; bq > 0. Since X is E-monotone, we have  x   y. Assume that  x =  y. Since X is strictly convex, we have

13.2 Uniqueness of Best Approximation in the Mean for Vector-Valued Functions q−1 x + y  aq + bq     = eq  <  x. ai ei +    X 2 2 i=1

251

(13.8)

But

aq + bq , (13.9) 2  x+y  and so since X is E-monotone, we get  x   2 , which contradicts (13.8). The general case is thus reduced to one that has already been considered.  aq
0, we have sign gq (t) > 0, gq (t) = 0 for t ∈ U and  gq (t) = gq (t) for t ∈ Q \ U. Let and sign αq (t)  0. We set   g (t) =

q−1 

gi (t)ei +  gq (t)eq .

i=1

We have 0 =  gq (t) < gq (t) for t ∈ U, and ∫hence since X is∫strictly E-monotone, g (t)X dt < Q g(t)X dt = g . we have  g (t)X < g(t)X , t ∈ U, and so Q  Furthermore, αq (t)gq (t)  0 = αq (t) gq (t), t ∈ U. From the last two inequalities and condition c), it follows that ∫  q−1 Q



αi (t)gi (t) + αq (t) gq (t) dt

i=1

∫  q Q i=1

∫

αi (t)gi (t)dt = g >

Q

 g (t)X dt.

By Luzin’s theorem, the measurable bounded function  gq (t) can be replaced by a continuous function gq∗ (t) such that for the function g ∗ (t) =

q−1  i=1

gi (t)ei + gq∗ (t)eq ∈ C(Q, X),

we still have ∫  q−1 Q

i=1

∫ αi (t)gi (t) + αq (t)gq∗ (t) dt > g ∗ (t)X dt = g ∗  , Q

which contradicts condition a). Lemma 13.4 is proved.



Definition 13.2 A system Sn = {ϕ1 (t), . . . , ϕn (t)} (ϕi ∈ C(Q, X)) of vector-valued functions is called an E-Chebyshev system if for each nontrivial polynomial p in the system Sn , the total number of zeros of all its projections p1 (t), . . . , pq (t) over the basis E is smaller than n.

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It is easily seen that an E-Chebyshev system is a Chebyshev system (in the sense of Theorem 13.7), and for q = 1 it becomes a classical Chebyshev system. A zero t of the projection pi is said to be simple (double) if pi (t) changes (does not change) sign at t. A simple (double) zero is defined to be of multiplicity one (two). Multiple zeros t = 0 and t = 1 are assumed to be simple. The subspace M(Sn ) of polynomials p(t) in an E-Chebyshev system Sn is isomor Sn ) of real polynomials π(t) in a classical Chebyshev system phic to the subspace M( Sn ) defined on the union of Q(q) closed intervals Qi = [2i − 2, 2i − 1], i = 1, . . . , q. The correspondence p ↔ π is defined by pi (t) = π(2i − 2 + t), 0  t  1, and π(t) = pi (t − 2i + 2) for 2i − 2  t  2i − 1, i = 1, . . . , q. Such an isomorphism also establishes a one-to-one correspondence between the simple and double zeros of the polynomials p and π. It is clear that the properties of M(Sn ) can be studied in terms  Sn ). of M( Lemma 13.5 Let Sn be a Chebyshev system on the union Q(q) of closed intervals Qi = [ai, bi ], i = 1, . . . , q, a1 < b1 < a2 < b2 < . . . < aq < bq , and let π be a polynomial in Sn that has on Q k (1  k  q) only simple zeros, among which only one zero is on the boundary. Then there exists a polynomial π(t) such that sign π(t) = − sign π(t) on (ak , bk ) and sign π(t) = sign π(t) almost everywhere on [ai, bi ], i  k. Proof Assume first that π(t) has n − 1 simple zeros. In this case, π(t) can be written as the determinant

 ψ1 ψ2, . . . , ψn π(t) = c det (13.10) (ψ ∈ Sn ) t, t1, . . . , tn−1 (see [371], [216]), where c = const, t1 < . . . < tr are the zeros of π(t) lying in Q k (t1 = ak , tr < bk ); tr+1, . . . , tn−1 are the zeros lying in the remaining Qi (i  k). If the points t1, . . . , tr vary within [ak , bk ] and if t ∈ Q(q) is fixed, the determinant (13.10) preserves its sign, provided that the points t, t1, . . . , tr remain in increasing order (see also [371], [216]). If for two neighbouring points (and only for them), the order is reversed, then the determinant changes sign. We set t j (h) = t j + h(t j+1 − t j ),

j = 1, . . . , r;

tr+1 = bk ,

0  h  l.

Let π(h, t) be a polynomial (with the zeros t1 (h), . . . , tr (h), tr+1, . . . , tn−1 ) obtained by continuous deformation of the polynomial π(t) = π(0, t) with increasing h and with normalization π(h) = π. It is easily checked that as a polynomial π(t) one can take π(1, t). Indeed, if h is increased from 0 to 1, a change in the order of the points t, t1, . . . , tr occurs only for t ∈ (ak , bk ) (t  t j ) and only for a single pair of neighbouring points. Namely, if t ∈ (t j , t j+1 ), then instead t j < t, we get t < t j . Taking this into account, we conclude that the polynomial π(1, t) is the required one. The case t1 > ak , tr = bk is considered similarly. The case in which π(t) has double zeros (outside Q k ) and the case in which the total multiplicity of the zeros of π(t) is smaller than n − 1 can be reduced to the above case by standard arguments. In the  last case, π(t) can have more zeros than π(t)).

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253

Lemma 13.5 in terms of a system of vector-valued functions reads as follows. Corollary 13.3 Let Sn be an E-Chebyshev system on Q = [0, 1], let p(t) = (p1 (t), . . . , pq (t)) be a polynomial in Sn whose kth projection pk (t) has only simple zeros, and let pk (0) = 0, pk (1)  0 (or pk (0)  0, pk (1) = 0). Then there exists a polynomial p(t) = ( p1 (t), . . . , pq (t)) such that sign pi (t) = − sign pi (t) on (0, 1)

and

sign pi (t) = sign pi (t)

for i  k almost everywhere on [0, 1]. Corollary 13.4 Let Sn be an E-Chebyshev system, let α1 (t), . . . , αq (t) ∈ L ∞ (Q), let t1i , . . . , trii be the points of sign-change for the function αi (t), i = 1, . . . , q, and let r1 + . . . + rq = r < n − q. Then there exists a polynomial in the system Sn such that sign pi (t) = sign αi (t) a.e. on Q,

i = 1, . . . , q.

(13.11)

Proof Indeed, we construct a polynomial p(t) such that a) pi (t ij ) = 0, j = 1, . . . , ri ; i = 1, . . . , q; b) pi (0) = 0, i = 2, 3, . . . , q; c) p1 (t) has n − q − r double and boundary zeros (counting multiplicity) that differ from {t 1j }. Such a polynomial exists, since the total number of its zeros (counting multiplicities) is equal to r + (q − 1) + (n − q − r) = n − 1. It is obvious that pi (t) changes sign at the same points as αi (t) (i = 1, . . . , q). Multiplying, if necessary, p(t) by −1, we get sign p1 (t) = sign α1 (t) almost everywhere on [0, 1]. Let sign pi (t) = sign αi (t), i = 1, . . . , m − 1, but sign pm (t) = − sign αm (t) almost everywhere on [0, 1]. By Corollary 13.3, it is possible to construct a polynomial p (t) such that sign p i (t) = sign αi (t), i = 1, . . . , m. Continuing this procedure, we can construct a polynomial p(t) satisfying (13.11). This proves Corollary 13.4.  Theorem 13.10 (A. L. Garkavi [265]) Let X be a strictly convex E-monotone Banach space and let Sn be an E-Chebyshev system of functions from C(Q, X). Then a polynomial p∗ ( f ) ∈ M(Sn ) of least mean deviation from f is unique for all f ∈ C(Q, X). Proof We assume that the theorem is false. Then by Theorem 13.9 there exist a nontrivial polynomial p0 (t) and functions g(t), α1 (t), . . . , αq (t) satisfying conditions a), b), c), d∗ ) of this theorem. Since Sn is an E-Chebyshev system, p0i (t) has a finite number of zeros. By d∗ ), the same is true for gi (t). Hence by Lemma 13.4, αi (t) changes sign finitely many times. Let r be the total number of sign changes of the functions αi (t), i = 1, . . . , q. If r  n − q, then by Corollary 13.4, there exists a polynomial p(t) such that sign pi (t) = sign αi (t) almost everywhere, which contradicts condition b). Hence r  n − q + 1. Let {zij } be the points of sign change of αi (t) ( j = 1, . . . , ri ; r1 + . . . + rq = r). By Lemma 13.4, gi (t) changes sign and vanishes at these points. By condition d∗ ), p0i (t), i = 1, . . . , q, also vanishes at these points. Now we return to equality (13.4):

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13 Approximation of Vector-Valued Functions

g(t) − p0 (t) = K(t)g(t),

0 < K(t) < ∞ for g(t)  0 ∈ X.

The function K(t) is defined everywhere except at a finite number of points (the zeros of g(t)) and is continuous on its domain J . Since for K(t) = 1 (t ∈ J ) it follows from (13.4) that p0 (t) = 0 ∈ X, the difference K(t) − 1 can change sign only finitely many times, namely at the points of discontinuity or at points of J . We claim that such a point t ∈ Q actually exists. Indeed, if, for example, K(t)  1 on J , then it follows from what has been said that K(t) > 1 a.e. on Q, and hence, using (13.4), ∫ ∫ ∫ 0 0 g − p  = g(t) − p (t)X dt = K(t)g(t)X dt > g(t)X dt = g, Q

Q

Q

which contradicts condition c). Hence the point t exists. For the sake of definiteness, let K(t) < 1 for t − ε < t < t and K(t) > 1 for t < t < t + ε (ε > 0). For each i = 1, . . . , q, one of the following cases holds. (1) gi (t)  0. Then g(t)  0 ∈ X, t ∈ J , K(t) = 1, and from (13.4), (13.5) we get p0i (t) = 0. This zero of the projection p0i (t) is different from its other zeros {zij }, since gi (z ij ) = 0. (2) gi (t) = 0 and gi (t) does not change sign at t. Then by d∗ ), p0i (t) = 0, and the point t is again different from the points {z ij }, since gi (t) changes sign at those points. (3) gi (t) = 0 and gi (t) changes sign at t. In this case, by Lemma 13.4, the point t coincides with one of the points {z ij }. As in case 2), p0i (t) = 0. For the sake of definiteness, let gi (t) < 0 for t − ε < t < t and gi (t) > 0 for t < t < t + ε. Taking into account the sign of K(t) − 1 near t, it follows from (13.5) that gi (t) − pi (t) > gi (t) and pi (t)  0 for t − ε < t < t + ε, and hence t is a double zero of the projection of pi (t). We arrive at the same result for the other combinations of the signs of gi (t) and K(t) − 1 near t. Hence for every i = 1, . . . , q, either we get an additional zero of the projection pi (t) that is different from its zeros z1, . . . , zri , or one of these zeros is double. Therefore, the total number of zeros of the projections of {pi (t)} (counting multiplicity) turns out to be equal to r + q > (n − q + 1) + q = n + 1, which is impossible for a nontrivial polynomial in an E-Chebyshev system. Therefore, the case r  n − q + 1 is also impossible. Theorem 13.10 is proved.  Remark 13.1 The condition of strict convexity of X cannot be omitted. As a  with the norm  x  = q |xi | (x = counterexample we consider the space X X i=1 ∫  with the norm  f (t)X dt is isometric (x1, . . . , xq )). Indeed, the space C(Q, X) Q ∫ to C(Q(q), R) with the norm Q(q)  f (t)X dt (Q(q) is the union of q intervals), and in C(Q(q), R) there exist a Chebyshev system Sn and a function f ∈ C(Q(q), R) for which a polynomial of best mean approximation π∗ ( f ) is not unique (see [265, Remark 1]). Remark 13.2 Under the hypotheses of Theorem 13.10, a polynomial p∗ ( f ) (of best uniform approximation) is also unique for all f ∈ C(Q, X), because every EChebyshev system satisfies the hypotheses of Theorem 13.7.

13.2 Uniqueness of Best Approximation in the Mean for Vector-Valued Functions

255

We give another test for the uniqueness of the polynomial p∗ ( f ). By Mi we denote the subspace of projections of elements from M(Sn ) along the vector ei (Mi ∈ C(Q, R)). Theorem 13.11 (A. L. Garkavi [265]) Let X be a strictly convex E-monotone space and let M(Sn ) be the direct sum of the component subspaces of its elements, that is, q M(Sn ) = i=1 Mi ei . If the basis of every subspace Mi , i = 1, . . . , q, is a Chebyshev system in C(Q, R), then the polynomial p∗ ( f ) is unique for each f ∈ C(Q, X). Proof If the theorem were false, then by Theorem 13.9, there would exist g(t), p0 (t)  0 ∈ C, and α1 (t), . . . , αq (t) satisfying conditions a), b), c), d∗ ). Since p0 (t)  0 ∈ C, at least one of its projections, say p0k (t), is nontrivial, and by hypothesis, it has no more than nk −1 zeros, where nk = dim Mk . By property d∗ ), the same holds for gk (t), and by Lemma 13.4, αk (t) can change sign no more than nk − 1 times. Hence there exists pk (t) ∈ Mk ⊂ C(Q, X) such that sign pk (t) = sign αk (t) almost everywhere [371]. Since M(Sn ) is the direct sum of the subspaces Mi ei , the polynomial p(t) with the projections pi (t) ≡ 0, i  k, pk (t) = pk (t) belongs to  M(Sn ), but it does not satisfy condition b). Theorem 13.11 is proved. In the special case X = Rq , this theorem was obtained by Kroó [375]. Corollary 13.5 Let X be a strictly convex E-monotone space, let M(Sn ) be the direct sum of the subspaces Mi ei , i = 1, . . . , q, let n/q = k be an integer, and let N(Sn ) < k. Then a polynomial p∗ ( f ) is unique for each f ∈ C(Q, X). It is easy to verify that under the hypotheses of the corollary, dim Mi = k, i = 1, . . . , q, and each Mi has a Chebyshev system as a basis. This implies in particular the following theorem of Kripke and Rivlin [373] concerning complex-valued functions. Let X = C, Sn ∈ C(Q, C). If p ∈ M(Sn ) implies Re p ∈ M(Sn ) and if N(Sn ) < n, then a polynomial p∗ ( f ) is unique for each f ∈ C(Q, C). Remark 13.3 In the general case, the hypotheses of Theorem 13.11 are not sufficient for the uniqueness of a polynomial p∗ ( f ) of best uniform approximation. Nevertheless, under the hypotheses of the corollary, the polynomial p∗ ( f ) is unique for each f ∈ C(Q, X), since condition A1) of Theorem 13.7 is satisfied for the integer k = n/q. Up to this point, it has been assumed that dim X = q < n < ∞. Now we discuss the case n  q (q  ∞, n < ∞). Theorem 13.12 (A. L. Garkavi [265]) Let X be a strictly convex space of dimension q  ∞. If every polynomial p ∈ M(Sn ) \ {0} has no zeros on Q, then the polynomial p∗ ( f ) is unique for each f ∈ C(Q, X).

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13 Approximation of Vector-Valued Functions

Proof Otherwise, by Theorem 13.8, there would exist a function g(t) for which the origin 0 ∈ C and some nontrivial polynomial p0 (t) provided the best approximation. Just as in Theorem 13.9, we arrive at the relation g(t) − λp0 (t) = K(t)g(t) for g(t)  0 ∈ X

(13.12)

and the implication g(t) = 0 ∈ X → p0 (t) = 0 ∈ X. By the assumption, p0 (t)  0 ∈ X, and hence g(t)  0 ∈ X everywhere on Q. So the function K(t) is defined for all t ∈ Q. As in Theorem 13.9, we conclude that there exists a point t ∈ Q at which  K(t) = 1, but then from (13.12), we get p0 (t) = 0 ∈ X, which is impossible. Theorem 13.13 (A. L. Garkavi [265]) Let X be a strictly convex space, Sn ⊂ C(Q, X) an E-Chebyshev system, and let n  q = dim X  ∞, n < ∞. Then p∗ ( f ) is unique for every function f ∈ C(Q, X). This follows from Theorem 13.12. Indeed, if p ∈ M(Sn ) and p(t) = 0 ∈ X, then pi (t) = 0, i = 1, . . . , q, and q  n, which is impossible for an E-Chebyshev system. So for n  q, the condition N(Sn ) < n/q again turns out to be sufficient for the uniqueness of p∗ ( f ). Note that under the hypotheses of Theorems 13.12 and 13.13, the polynomial p∗ ( f ) of best uniform approximation is also unique for every function f ∈ C(Q, X) (see [630]). To complete this section, we point out that the above results remain true if in the definition of the norm (13.2), instead of the Lebesgue measure we consider a continuous σ-finite measure whose support is Q = [0, 1].

13.3 On the Haar Condition for Systems of Vector-Valued Functions Let Q be an interval [a, b] ⊂ R, let X be a Banach space of dimension q  ∞ with the norm  · , and let C(Q, X) be the space of continuous functions ϕ(t) : Q → X with the norm ϕC = max ϕ(t). Next, let SN = {ϕ1 (t), . . . , ϕ N (t)} be a system of N vector-valued functions (N < ∞), let ϕi ∈ C(Q, X), let L(SN ) be a subspace of polynomials in the system SN . From N and q one uniquely defines natural numbers k and m such that N = (k − 1)q + m, 1  m  q, 1  k < ∞ (for q = ∞ we set k = 1 and N = m). By Δ(ϕ) we denote the subspace of polynomials from L(SN ) that deliver the best approximation E(ϕ) = inf p ∈L(S N ) ϕ − pC = ρ(ϕ, L(SN )) to ϕ. A system SN is said to have the uniqueness property if every function ϕ ∈ C(Q, X) has a unique polynomial of best approximation (that is, dim Δ(ϕ) = 0). In [630], Zukhovitskii and Stechkin established that if a space X is strictly convex, then a system SN (N = (k − 1)q + m) has the uniqueness property if and only if (A) every nontrivial polynomial from L(SN ) has at most k − 1 zeros, and (B) for all t1, . . . , tk−1 from Q and x1, . . . , xk−1 from X, there exists a polynomial p ∈ L(SN )

13.3 On the Haar Condition for Systems of Vector-Valued Functions

257

such that p(ti ) = xi (i = 1, . . . , k − 1). For an integer N/q, that is, if N = kq, condition A) implies condition B); and for q = 1, it coincides with the Haar condition that characterizes the Chebyshev systems [630] in spaces of continuous functions (see Sect. 2.3). In this connection, conditions (A) and (B) can be naturally considered a (generalized) Haar condition for vector-valued functions. A system SN satisfying these conditions is called an H-system. In [366], Koshcheev extended a criterion from [630] (also assuming that a space X is strictly convex) and characterized the systems SN of a given Chebyshev rank (see below). This leads to the question whether H-systems lose (to any extent) the uniqueness property if a space X is not strictly convex. Let us consider this question. Definition 13.3 A space X is called r-strictly convex if the maximum of the dimensions of convex subsets of its unit sphere S is r. Clearly, the case r = 0 corresponds to a strictly convex norm. Definition 13.4 The Chebyshev rank of a system SN is defined as ν(SN ) = maxϕ ∈C(Q,X) dim Δ(ϕ) (this useful concept was introduced by G. Sh. Rubinshtein). Let RN (X) be the maximum of the Chebyshev ranks on a class of H-systems SN ⊂ C(Q, X) of order N. Theorem 13.14 Let a space X of dimension q  ∞ be r-strictly convex and N = (k − 1)q + m, 1  k < ∞, 1  m  q. Then RN (X) = (k − 1)r + min{r, m}. We give some particular cases of this theorem. (1) if r = 0, then RN (X) = 0 (see [630]); (2) if q = 1 (that is, X = R), then r = 0 and RN (R) = 0; (3) if N = kq, then RN (X) = kr = r N/q; (4) if N < q  ∞, then RN (X) = min{r, N }. So the uniqueness property of H-systems (which holds for strictly convex X) does not hold in general for a non-strictly convex space X. Moreover, R NN(X) → qr as N → ∞, q < ∞. We need some auxiliary results. Let {e1, . . . , eq } be a basis for X (dim X = q < ∞), and so if x ∈ X, then x = x 1 e1 + . . . + x q eq (x j ∈ R). Let π ⊂ X be a subspace with a basis {e j1 , . . . , e jm } (1  j1 < . . . < jm  q). By xπ we denote the canonical projection of x onto π; that is, xπ = x j1 e j1 + . . . + j x m e jm . Next, let SN = {ϕ1 (t), . . . , ϕ N (t)} be a system of functions ϕi ∈ C(Q, X). q j Then ϕi (t) = ϕi1 (t)e1 + . . . + ϕi (t)eq , where ϕi (t) ∈ C(Q, R). For the system SN and k points t1, . . . , tk from Q, consider the matrix

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13 Approximation of Vector-Valued Functions

ϕ1 (t ) . . . ϕ1N (t1 )

1 1  ................  q  q  ϕ1 (t1 ) . . . ϕ N (t1 )    A = .................  1  1  ϕ1 (tk ) . . . ϕ N (tk )    ................ q q  ϕ1 (tk ) . . . ϕ N (tk )  Lemma 13.6 Let SN = {ϕ1, . . . , ϕ N } be an H-system and let N = (k − 1)q + m. Then for every set of k distinct points t1, . . . , tk from Q, the rank of the matrix A is N, and this matrix has a nonzero minor of order N:  1   ϕ1 (t1 ) . . . ϕ1N (t1 )    ...................  q   ϕ1 (t1 ) . . . ϕqN (t1 )    ...................  1   ϕ1 (tk−1 ) . . . ϕ1N (tk−1 )   (13.13) MN =  . . . . . . . . . . . . . . . . . . .  ,  q  q  ϕ1 (tk−1 ) . . . ϕ N (tk−1 )   j1   ϕ (tk ) . . . ϕ j1 (tk )   1  N ...................  j   ϕ m (t ) . . . ϕ jm (t )  k k N 1 where 1  j1 < j2 < . . . < jm  q. Proof Indeed, from condition B) in the definition of an H-system, it follows that there exists a nonzero minor of the matrix A of order (k − 1)q consisting of elements of the first (k − 1)q rows of the matrix. Condition A) implies that the rank of the matrix A is N = (k − 1)q + m. Using the bordering method for finding the rank, we  get a minor MN  0 of the form (13.13). Corollary 13.6 For every set of k distinct points t1, . . . , tk from Q, every polynomial p in an H-system SN is uniquely determined by its values p(t1 ), . . . , p(tk−1 ) and the value pπ (tk ) of its projection pπ to the subspace π with the basis {e j1 , . . . , e jm }, where j1, . . . , jm are the same as in (13.13). We say that SN is a coordinate-type system if it consists of functions of the form i = 1, . . . , n j , j = 1, . . . , q < ∞, ϕi, j (t) = fi, j (t)e j , n1 + . . . + nq = N, where fi, j (t) ∈ C(Q, R).

(13.14)

Similar system were considered in [375], [265]. Lemma 13.7 Let SN be a coordinate-type system consisting of functions (13.14), and let N = (k − 1)q + m, 1  m  q < ∞. Then SN is an H-system in C(Q, X) if and only if each of the systems σ j = { f1, j , . . . , fn j , j }, j = 1, . . . , q, is an H-system in C(Q, R) and its order n j is1 or k − 1, or k. 1 In accordance with the above, a trivial system is an H-system of order zero.

13.3 On the Haar Condition for Systems of Vector-Valued Functions

259

Proof Indeed, if n j  k − 2, then the interpolation condition B) is not satisfied for the system σ j (in C(Q, R)), and hence is not satisfied for the system SN in C(Q, X). If n j  k + 1, then there exists a nontrivial polynomial h(t) in the system σ j that has k zeros. Hence the polynomial p(t) = h(t)e j lies in L(SN ), but this violates condition A). If n j = k −1 (n j = k) and if SN is not an H-system, then condition B) (respectively, A)) is again violated. On the other hand, a similar argument shows that under the  hypotheses of the lemma, the system SN satisfies conditions A) and B). Let us now prove Theorem 13.14. Proof We first assume that dim X = q < ∞. Let ϕ ∈ C(Q, X) and let SN be an Hsystem. From condition B) it follows (see [630]) that for each p ∈ Δ(ϕ), there exist at least k points of maximal deviation (points { t} at which ϕ( t) − p( t) = E(ϕ) = E). Hence there exist k points t1, . . . , tk of maximal deviation for all p ∈ Δ(ϕ). This i) with p ∈ Δ(ϕ) implies that for each ti (i = 1, . . . , k), the points ui (p) = ϕ(ti )−p(t E p) be a point from Ji that lies in form a convex subset Ji of the unit sphere S. Let ui ( p) and let Li = L(Di ) the relative interior of the affine hull of Ji . We set Di = Ji − ui ( be the subspace spanned by Di . Then for each p ∈ Δ(ϕ), p(ti ) = p(ti ) + x(p, ti ),

where x(p, ti ) ∈ Li,

i = 1, . . . , k.

(13.15)

By Corollary 13.6, equality (13.15) uniquely defines p ∈ Δ(ϕ). This is also true if instead of the last (the kth) equality in (13.15), one takes pπ (tk ) = pπ (tk ) +  x (p, tk ), where  x (p, tk ) lies in the canonical projection (Lk )π of the subspace Lk onto the subspace π with the basis {e j1 , . . . , e jm } chosen in accordance with Lemma 13.6 and Corollary 13.6. The dimension of the set Δ(ϕ) is clearly majorized by that of the product of the subspaces L1, L2, . . . , Lk−1, (Lk )π . But dim Li = dim Ji = r j  r (i = 1, . . . , k − 1) and dim(Lk )π  min{dim Lk , dim π}  min{r, m}. Hence dim Δ(ϕ)  r1 + . . . + rk−1 + min{r, m}  (k − 1)r + min{r, m} (ϕ ∈ C(Q, X)), and therefore, (13.16) RN (X)  (k − 1)r + min{r, m}. Let us prove that this estimate is sharp. Let J be a convex subset of the sphere S, dim J = r, let  z be a point from the relative interior of J , and let F ∈ S ∗ be a norm-one functional such that F( z ) = 1. Hence F(x) = 1 for x ∈ J . We set D = J − z , H = {x ∈ X | F(x) = 0}. It is clear that D ⊂ H. Let z1, . . . , zr be linearly independent points from D. We augment the system e1 = z1, . . . , er = zr with the vectors er+1, . . . , eq−1 to a basis for the subspace H and add to it another z . Now {e1, . . . , er , . . . , eq−1, eq } is a basis for X, and moreover, vector eq =  F(e j ) = 0 for j = 1, . . . , q − 1,

F(eq ) = eq  = 1,

(13.17)

e j ∈ D for j = 1, . . . , r.

(13.18)

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13 Approximation of Vector-Valued Functions

∗ in C(Q, X) consisting of the functions Consider the coordinate-type H-system SN

ϕi, j (t) = fi, j (t)e j , 

where κ=

i = 1, . . . , κ, j = 1, . . . , q,

k for j = 1, . . . , m, k − 1 for j = m + 1, . . . , q,

∗ and σ j = { f1, j , . . . , fκ, j } is an H-system in C(Q, X). The order of the system SN ∗ is km + (q − m)(k − 1) = (k − 1)q + m = N, and by Lemma 13.7, SN is an Hsystem in C(Q, X). Assume that the identically zero function is a polynomial of best approximation in the system σ q for a function g(t) ∈ C(Q, R), gC = 1. Hence the identically zero function is also a polynomial of best approximation ∗ for the function α(t) = g(t)e ∈ C(Q, X). Indeed, if p ∈ L(S ∗ ), then p(t) = in SN q N q−1 pq (t)eq + j=1 p j (t)e j , where p j (t) is a polynomial in the system σ j . Therefore, taking into account (13.17), we have q−1      p j (t)e j  α−pC = max α(t) − p(t) = maxg(t)eq − t t j=1   ≥ max F(g(t)eq − pq (t)eq ) t

= max |g(t) − pq (t)|  max |g(t)| = max α(t) = αC = 1. t

t

t

On the other hand, for each fi, j (t) (i = 1, . . . , κ, j = 1, . . . , r), there exists a number εi, j > 0 such that eq ± εi, j fi, j (t)e j ∈ J for all t. This follows from (13.18) and since z is a point from the relative interior of J =  z +D. Hence eq ±εi, j fi, j (t)e j  = 1, eq =  t ∈ Q (i = 1, . . . , κ, j = 1, . . . , r). We have |g(t)|  1, and therefore, assuming that |εi, j fi, j (t)| < 1, we have g(t)eq − εi, j fi, j (t)e j   1 for all t ∈ Q and j = 1, . . . , r. Thus the polynomials εi, j fi, j (t)e j , (κ = k for j  m

i = 1, . . . , κ, j = 1, . . . , r and κ = k − 1 for j > m)

lie in Δ(α). It is clear that they are linearly independent. Let us evaluate the number M of these polynomials. For r  m, we have M = kr = (k − 1)r + min{r, m}; for r > m, we have M = km + (r − m)(k − 1) = (k − 1)r + m = (k − 1)r + min{r, m}. Next, dim Δ(α)  M, and so RN (X)  (k − 1)r + min{r, m}, which proves the theorem in the case q < ∞. If dim X = q = ∞, then the proof is simpler, because in this case, we have k = 1, N = m, and condition B) becomes unnecessary. It is clear that ν(SN )  N. Moreover, as before, for every polynomial ϕ ∈ C(Q, X) there exists a point t1 ∈ Q for which the subset D1 = {p(t1 ) | p ∈ Δ(ϕ)} is convex, dim D1  r, and each p ∈ Δ(ϕ) is uniquely determined by its value p(t1 ). Hence ν(SN )  r and RN (X)  min{r, N }. The proof of the reverse inequality is carried out along the same lines as in the case q < ∞. The only difference is that (with the same notation) instead of the basis

13.4 Approximation of Vector-Valued Functions by Polynomials

261

{e1, . . . , er , . . . , eq−1, eq }, (e1, . . . , eq ∈ H, e1, . . . , er ∈ D), one should consider the z, where e1, . . . , e N ∈ H, system of linearly independent vectors e1, . . . , es, . . . , e N ,  and e1, . . . , es ∈ D, s = min{r, N }. ∗ is composed of the functions ϕ (t) = f (t)e ( j = 1, . . . , N, where The system SN j j j z and arguing as before, f j (t) is in C(Q, R) and has no zeros on Q. Putting α(t) ≡  we prove that dim Δ(α)  min{r, N }, and hence RN (X) = min{r, N } if q = ∞ and r  ∞. Theorem 13.14 is proved.  It is clear that an interval [a, b] in this theorem can be replaced by any compact set Q admitting classical H-systems (Chebyshev systems) in C(Q, R) of all orders n  N. It should also be noted that the statement and proof of the theorem can be carried over verbatim to the case that the space X and a subspace L(SN ) of X are considered over complex numbers and all dimensions are complex.

13.4 Approximation of Vector-Valued Functions by Polynomials Of special interest is the problem of approximation in spaces of continuous bounded functions on sets that are not even locally compact. Here we consider one natural extension of the space of polynomials of degree  n, which in general is not a finite-dimensional space even for real-valued polynomials. It should be noted that consideration of vector-valued polynomials in the case in which values are taken in a function space Y is a particular case of so-called quasipolynomials. By B(M, Y ) and C(M, Y ), where X and Y are real Banach spaces and M is a nonempty subset of X, we denote, respectively, the spaces of bounded and continuous bounded functions f : M → Y with the norm  f  = supt ∈M  f (t)Y . Let Pn = Pn (M, Y ) be the set of all polynomials of degree  n (these are the n bi (x, . . . , x), x ∈ M), where bi : X × · · · × X → Y functions p such that p(x) = i=0  i

are bounded i-linear mappings. For an arbitrary function f ∈ B(M, Y ) (or f ∈ C(M, Y )), by En ( f ) = ρ( f , P(M, Y )) we denote the best approximation of a function f by the class P(M, Y ); a polynomial p ∈ P(M, Y ) for which  f − p = En ( f ) is called a polynomial of best approximation. The next result shows that a polynomial of best approximation exists both in the space of bounded functions and in the space of bounded continuous functions. Theorem 13.15 (I. G. Tsar’kov) Let X, Y be real Banach spaces, n ∈ Z+ , and let M ⊂ X be a body.2 Assume that there exists a norm-one projector Y ∗∗ onto Y (this condition is satisfied in particular by all dual spaces). Then for each function f ∈ B(M, Y ) ( f ∈ C(M, Y )), there exists a polynomial p ∈ P(M, Y ) of best approximation of the function f by the space P(M, Y ). 2 A body is a set with nonempty interior.

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13 Approximation of Vector-Valued Functions

Theorem 13.16 is concerned with stability of best approximation. Theorem 13.16 Let X be a Efimov–Stechkin space.3 Then for every function f ∈ C([a, b], X) and minimizing sequence of polynomials (pk ) ⊂ P(M, X) (that is,  f − pk  → En ( f ), k → ∞), there exists a subsequence (pkn ) converging strongly to some polynomial p ∈ P(M, X) of best approximation.

13.5 Some Applications of Vector-Valued Approximation In this brief section, we give two examples showing the importance of approximation of vector-valued functions. First, uniform approximation of vector-valued functions from C(Q, Y ) in the case that Y = Y (Ω) is some function space, for example Y = C(Ω, R), leads to the problem of approximation of functions ϕ : Q × Ω → R by quasipolynomials n ak (y) fk (x), where ak (y) ∈ Y (Ω). Moreover, p(x, t) = k=1 n      ak ( · ) fk (x)  f (x) − k=1

C(Q,Y)

= ϕ(x, y) − p(x, y)C(Q×Ω,R),

where ϕ( · , y) = f ( · )[y] ∈ Y (Ω). Second, by finding best approximations of vector-valued functions, one can obtain lower estimates in the problem of approximation of real functions by a linear method on some class of functions W(R) ⊂ C(Q, R). In [555], it was shown that the quantity e(A) = sup f ∈W (R)  f − A f C(Q,R) for some linear operator A : C(Q, R) → N(R) is  = sup f ∈W (Y)  f − A  f C(Q,R), where A  : C(Q, Y ) → N(Y ) equal to the quantity e( A) is some natural linear operator, where Y is an arbitrary Banach space, W(Y ) : = { f ∈ C(Q, Y ) | y ∗ (ϕ) ∈ W(R) ∀y ∗ ∈ S ∗ }, N(Y ) : = {ϕ ∈ C(Q, Y ) | y ∗ (ϕ) ∈ N(R) ∀y ∗ ∈ Y ∗ }. Note in particular that if N(R) is the space of polynomials Pn (R), then N(Y ) is the space of vector-valued polynomials Pn (Y ). Using this method of vectorization of functions in the case of a linear method of approximation, one can obtain, in the problem of finding best approximation of vector-valued functions on the class W(Y ), lower estimates for approximation of real functions by a linear method on the class W(R). Exercise 13.1 Let f1 (t) = (1, t), f2 (t) = (t, t 2 ) ∈ C([−1, 1], R2 ). Check whether the system { f1, f2 } is a Chebyshev system. Exercise 13.2 Let f1 (t) = (1, t 2 ), f2 (t) = (2t − 2, 3t 2 − 1) ∈ C([−1, 1], R2 ). Check whether the system { f1, f2 } is a Chebyshev system. Exercise 13.3 Let f1 (t) = (1, t 2 ), f2 (t) = (t − 1, 1) ∈ C([−1, 1], R2 ). Check whether the system { f1, f2 } is a Chebyshev system. 3 For a definition and basic properties of Efimov–Stechkin spaces, see Sect. 9.1.

Chapter 14

The Jung Constant

The Jung constant appears in many problems in various fields of mathematics. In the present chapter, we will give examples of such problems and present the available exact values of the Jung constant for several classical spaces. In Sect. 14.1, we give the definition and discuss some  properties of the Jung conr(M)  M ⊂ X, diam M < ∞ (here r(M) is the Chebyshev stant J(X) := sup diam M radius of M; see Sect. 15.1). In Sect. 14.2, we discuss the relations between the measure of nonconvexity of a space and the Jung constant. Relations between the Jung constant and fixed points of condensing and nonexpansive maps are discussed in Sect. 14.3. The problem of approximate solution of the equation f (x) = x is considered in Sect. 14.4, in which the Jung constant in the Hilbert setting appears as a natural ingredient in the proofs. The Jung constant of the space n1 is discussed in Sect. 14.5. We formulate V. I. Ivanov’s conjecture to the effect that if there exists a Hadamard 1 ) = J( 1 ) = J( 1 ) = n/(n + 1). At present, matrix of order n + 1, then J(n+1 n+2 n+3 the value of the Jung constant for the space 51 is unknown. In Sect. 14.6, we discuss the relation between the Jung constant (a geometric characteristic of a space) and the Jackson constant (an approximative characteristic of a space). The relative Jung constant is considered in Sect. 14.7. We formulate a result obtained by S. V. Berdyshev, who found the relative Jung constant of the space n∞ , n ∈ N, and described the extremal subsets of n∞ , that is, the sets M with Js (n∞ ) = r M (M)/diam M. Note that in his result, as distinct from the Jung constant problem J(X), the exact value of the relative Jung constant Js (n∞ ) is given for all n. The Jung constant of a pair of spaces is discussed in Sect. 14.8. In Sect. 14.9, we give remarks on intersections of convex sets and discuss the relation of this problem to the Jung constant.

14.1 Definition of the Jung Constant The Jung constant of a normed linear space X is defined by © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_14

263

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14 The Jung Constant

 J(X) = sup

  r(M)  M ⊂ X, diam M < +∞ , diam M 

where as above, r(M) = inf{a  0 | M ⊂ B(x, a)} is the Chebyshev radius of a nonempty bounded set M ⊂ X, and diam M is the diameter of M. A point x0 ∈ X for which M ⊂ B(x0, r(M)) is called a Chebyshev centre of the set M. As before, by Z(M) we denote the set of all Chebyshev centres of a nonempty bounded set M (see Sect. 15.1 below). In other words, the Jung constant of a Banach space is the radius of a smallest ball that can cover every set of diameter 1. The inverse of the Jung constant is sometimes called the normal structure coefficient of a space. The term ‘Jung constant’ was introduced by Grünbaum in 1959. Jung constants play an important role in the geometry of Banach spaces. One should also mention the relation between Jung constants and Jackson inequalities, in which the best approximation of a function by finite-dimensional subspaces is estimated in terms of its modulus of continuity. We first note that for every normed space X, 1 ≤ J(X) ≤ 1 2 (both bounds can be attained). The first Jung constant was found in 1901 for a finite-dimensional Euclidean space by Jung himself [314]. Bohnenblust [111] (see also [398], Chap. 2, Sect. 11, and [30]) showed that J(Xn ) ≤ n/(n+1) for every n-dimensional Banach space Xn and proved that this inequality is sharp for all n. Leichtweiß (see [397] and Chap. 2, Sect. 11 in [398]) proved that in an n-dimensional space, the equality r(M)/ diam M = n/(n + 1) holds if M is the vertex set of an n-dimensional simplex Σ and B (the unit ball) is the difference body Σ + (−1)Σ of Σ. It can be easily shown that J(n∞ ) = 1/2 . For an infinite-dimensional Hilbert space √ H, Routledge [501] (and later V. I. Berdyshev [95]) established that J(H) = 1/ 2 . Further substantial advances in the study of Jung constants were made by Dol’nikov [201], V. I. Ivanov and Pichugov [471], [305], [306], Manokhin [420], and Ball [73]. The Jung constant for L p , 1 ≤ p < ∞, was found by Pichugov [471], [472] and independently by Ball [73]:  p  , 1≤p 0  conv A ⊂ B(a, r) = h(A, conv A), a∈A

where h(A, C) is the Hausdorff distance between sets A and C. The measure of nonconvexity η(A) is defined for every bounded set A. The following simple properties of the measure of nonconvexity η(A) are immediate (see, for example, [425]): (i) η(A) = 0 if and only if the set A is convex; (ii) η(αA) = |α|η(A), α ∈ R; (ii) η(A + C) ≤ η(A) + η(C); (iv) |η(A) − η(C)| ≤ η(A + (−C)); (v) η(A) = η(A); (vi) η(A) ≤ diam(A), where diam(A) is the diameter of the set A; (vii) |η(A) − η(C)| ≤ 2h(A, C). The measure of nonconvexity is nonmonotone in the sense that the inclusion A ⊂ C does not imply that η(A) ≤ η(C).

1 EL stands for Eisenfeld and Lakshmikantham, who in [225] introduced the measure of nonconvexity η(A) in analogy with the Kuratowski measure of noncompactness.

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Definition 14.1 Following [425], we say that the measure of nonconvexity ηX ( · ) has the Cantor property if every nested sequence (An )∞ n=1 of nonempty closed bounded (not necessarily convex) subsets of X has nonempty intersection, provided that η(An ) → 0. The measure of nonconvexity ηX ( · ) has the Cantor property in a set M ⊂ X if every nested sequence (An )∞ n=1 of nonempty closed bounded subsets of M has nonempty intersection, provided that η(An ) → 0. Remark 14.1 It is well known that every nested sequence of nonempty closed convex bounded subsets of a reflexive Banach space has nonempty intersection. Gulevich [282] (see Theorems 14.1 and 14.2 below) extended this result to the case of not necessarily convex sets. On the other hand, the following simple example shows that ηX ( · ) does not have the Cantor property in any nonreflexive space X. Indeed, in a nonreflexive space X, one can easily construct a nested sequence of nonempty closed convex bounded subsets with empty intersection: it suffices to take a functional f ∈ X ∗ that does not attain its norm on the unit ball B and to consider the sets {x ∈ B | f (x) ≥ 1 − 1/n}, n ∈ N. This example shows that ηX ( · ) does not have the Cantor property in any nonreflexive space X. Theorem 14.1 (N. M. Gulevich) Let (An )∞ n=1 be a nested sequence of nonempty closed bounded (not necessarily convex) subsets of a reflexive Banach space. If

η(An ) → 0 as n → ∞, then ∞ n=1 An is a nonempty closed convex set. We need the following well-known lemma. Lemma 14.1 Let (An )∞ n=1 be a decreasing sequence of nonempty bounded closed (not necessarily convex) subsets

∞ of a Banach space and let η(An ) → 0. We set A . Then A = A∞ = ∞ ∞ n=1 n n=1 conv An .

Proof (of Lemma 14.1) Since An ⊂ conv An , we have A∞ ⊂ ∞ n=1 conv An . Con ∞ versely, let x ∈ n=1 conv An . By the hypothesis, η(An ) → 0, and so for all ε > 0, there exists N ∈ N such that conv An ⊂ An + εB for all n ≥ N. Hence there exist a subsequence (nm )∞ n=1 and points anm ∈ Anm such that  x − anm  ≤ 1/m, m ∈ N. It is clear that anm → x, m → ∞. Moreover, since anm ∈ Anm ⊂ Ak for m, k ∈ N, nm ≥ k, and since Ak is closed, we have x ∈ Ak for all k ∈ N; that is, x ∈ A∞ .  We can now prove the following characterization theorem (see [282], [425]), which implies Theorem 14.1 as a corollary. Theorem 14.2 For a Banach space X, the following conditions are equivalent: (a) X is reflexive; (b) the measure of nonconvexity ηX ( · ) has the Cantor property; that is, every nested sequence (An )∞ n=1 of nonempty closed bounded subsets of X has nonempty intersection, provided that η(An ) → 0. Proof (a) ⇒ (b). Let X be a reflexive space and let (An )∞ n=1 be a decreasing sequence of nonempty bounded closed subsets of X, η(An ) → 0. By Lemma 14.1,

∞ A∞ = ∞ n=1 conv An , where A∞ := n=1 An . By James’s theorem (see Theorem 1.5 in Sect. 1.2), A∞  . The converse implication was established in Remark 14.1. 

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267

Theorem 14.3 (I. Marrero [425]) Let X be a Banach space, F a nonempty closed subset of X, and x0 ∈ X \ F. Next, let d = ρ(x0, F) and define 

 1 Fn [x0, F] := F ∩ x0 + d + B , n ∈ N. n Then the following conditions are equivalent: (a) X is reflexive (respectively, strictly convex); (b) if F ⊂ X is a nonempty closed set and η(Fn [x0, F]) → 0 for all x0  F, then F is a set of existence (a set of uniqueness). The next theorem characterizes the weakly compact subsets of a Banach space in terms of the Cantor property of the measure of nonconvexity of the space. Theorem 14.4 (I. Marrero [426]) Let X be a Banach space, let ηX ( · ) be the measure of nonconvexity, and let M be a nonempty weakly closed bounded subset of X. Then the following conditions are equivalent: (a) M is weakly compact; (b) the measure of nonconvexity ηX ( · ) has the Cantor property in the set convM.

14.3 The Jung Constant and Fixed Points of Condensing and Nonexpansive Maps The self Chebyshev radius of a bounded set M  is defined by (see Sect. 15.1 below) r M (M) := inf sup  x − y. y ∈M x ∈M

A classical result due to Klee (Theorem 1 in [341]) and Garkavi [258] asserts that in a normed linear space X, the equality r M (M) = r(M) holds for each convex bounded subset M ⊂ X if and only if either X is a Euclidean space or dim X = 2. This remark justifies the consideration of the relative Jung constant for convex sets2    r M (M)  M ⊂ X is convex, 0 < diam M < ∞ Jcv (X) := sup diam M  (in addition to the Jung constant J(X) and the relative Jung constant Js (X) of a space X; see Sect. 14.7 below). It is clear that Jcv (X) ≥ J(X). The corresponding definitions of a Banach space with normal structure and the coefficient of normal structure N(X) of a space are closely related to the Jung constant of a space.

2 Here the subscript cv stands for ‘convex’.

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Let X be a normed linear space and let  M ⊂ X be a bounded set. We define r cv (M) := rconv M M :=

inf

sup  x − y =

y ∈conv M x ∈M

inf

y ∈conv M

r(y, M)

and Zcv (M) := {y ∈ conv M | r(y, M) = rconv M M }, where as before, r(x, M) := inf{r ≥ 0 | M ⊂ B(x, r)} = supy ∈M  x − y. If M is convex, then we write r M (M) := r cv (M) and Zcv (M) = Z M (M) := Zsc M (M) := {x ∈ M | r(x, M) = r M (M)}. We mention several results on the constant Jcv (X). It is known (see Sect. 2 of [30]) that if X is a reflexive space, then Jcv (X) = sup{r cv (M) | M ⊂ X is finite, diam M = 1}. It is also known (see [30]) that if X is nonreflexive, then Jcv (X) = 1. Lim (see [404], [30]) showed that Jcv (X) = sup{r cv (M) | M ⊂ X is separable and diam M = 1} = max{Jcv (Y ) | Y is a separable subspace of X }. Amir [30] proved that if Jcv (X) < 1, then Jcv (X) = sup{r cv (M) | M ⊂ X is finite and diam M = 1} = sup{Jcv (Y ) | Y is a finite-dimensional subspace of X }. Remark 14.2 ([30]) A similar estimate for the (absolute) Jung constant J(X) in terms of the Jung constants of subspaces does not hold. For example, every space Y is a subspace of some space X =  ∞ (Γ); hence J(X) = 1/2, but J(Y ) ∈ [1/2, 1], and in particular, there exists Y such that J(Y ) > J(X) = 1/2. A lower estimate for J(X) in terms of the Chebyshev radii of finite subsets also does not hold. It is known that J(c0 ) = 1 (for a proof, it suffices to consider the set M = {(−1)n en | n ∈ N}). However, for every finite set M = {x (1), . . . , x (1) } ⊂ c0 , the point  1 max x (i) − min x (i) x := 1≤i ≤n 2 1≤i ≤n lies in c0 , which gives r(x, M) = (1/2) diam M. We consider another example from [30]. Let Γ be an uncountable set and let X = {x ∈  ∞ (Γ) | the set {γ ∈ Γ | x(γ)  0} is at most countable}. Thus, X is a closed subspace of  ∞ (Γ), and therefore is a Banach space. Every separable subset of X lies in a subspace of  ∞ (Γ0 ), where Γ0 ⊂ Γ is countable and  ∞ (Γ0 ) is a subspace of X isometric to  ∞ . Recall that J( ∞ ) = 1/2. On the other hand, Γ = Γ0  Γ1 , where Γ1 is uncountable. Let Mi := {x ∈ X | 0 ≤ x ≤ χΓi (x)}, i = 0, 1, where χA( · ) is the indicator function of a set A, and let M = M0 ∪ (−M1 ). It is easily seen that diam M = 1, but r(M) = 1. Hence J(X) = 1.

14.3 The Jung Constant and Fixed Points of Condensing and Nonexpansive Maps

269

Theorems 14.5–14.8 below are due to Gulevich [282]. Theorem 14.5 For every Banach space X, G(X) = Jcv (X). Theorem 14.6 For every Banach space X, Jcv (X) = sup{Jcv (L) | L is a finite-dimensional subspace of X }. Theorem 14.7 For every Banach space X, Jcv (X) = Jcv (X ∗∗ ). In particular, Jcv (X) = sup{r(M) | M ⊂ X is a finite set, diam M = 1} (see [282]). For the definition of the modulus of convexity δ(ε) of a space in the next theorem and below, see, for example, [481, Sect. 2.7]. Theorem 14.8 Let X be a Banach space and let δ(ε) be the modulus of convexity of X. Then Jcv (X) ≤ t0 , where t0 is the root of the equation t + 2δ(2t − 1) = 1 on the interval [1/2, 1]. Proof (of Theorem 14.8) Consider an arbitrary set M ⊂ X with diam M = 1. We can assume without loss of generality that 0 ∈ M. The function t → t + 2δ(2t − 1) is strictly increasing and continuous on the interval [1/2, 1], and hence the equation t + 2δ(2t − 1) = 1 has a unique root t0 ∈ [1/2, 1]. Assume to the contrary that η(M) > t0 . Then there exist a number t1 (t0 < t1 < η(M)) and a point b ∈ conv M such that M ∩ B(b, t1 ) = . Therefore, for all a ∈ M, we have             a − b  ≥  b − b  − a − b  =  a − b + b − 1 ≥ 2t1 − 1,       b b   because a − b ≥ t1 and 1 ≥ b ≥ t1 . Hence a + b/b  ≤ 2(1 − δ(2t1 − 1)). Let ϕ ∈ X ∗ be such that ϕ(b) = b and ϕ = 1. For all a ∈ M, we have 



  b b  b  −1 ϕ(a) = ϕ a + −ϕ ≤ a + b b b  ≤ 1 − 2δ(2t1 − 1) ≤ 1 − 2δ(2t0 − 1) = t0, which gives ϕ(b) ≤ t0 . But ϕ(b) = b > t0 . This contradiction shows that η(M) ≤ t0 .  By Theorem 14.5, we finally get Jcv (X) ≤ t0 . Amir [30] found some estimates for J( · ) and Jcv ( · ) in  p -spaces. For example,  1/p 2−1/p  (n − 1) + (n − 1) p ; n p 1/p−1 p 1/p−1 −1/p , Jcv ( ) ≥ max{2 ,2 }. J( ) ≥ 2 p

Jcv (n ) ≥

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√ We also note the following results. If X is infinite-dimensional, then Jcv (X) ≥ 1/ 2 , and if dim X ≤ n, then Jcv (X) ≤ n/(n + 1). To formulate another estimate of Jcv ( · ), we need the definition of the modulus of n-convexity of a space X, as given by F. Sullivan:  n   1    (n) δX (ε) := inf 1 − xi xi ∈ B(0, 1), i =0, . . . , n, n + 1  i=0    voln (x0, . . . , xn ) ≥ ε , where voln (x1, . . . , xn ) is the n-dimensional volume of the convex hull of a finite set x0, . . . , xn ; that is,   ⎧ ⎫  1 ... 1  ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎨  f1 (x0 ) . . . f1 (xn )  ⎪ ⎬ ⎪ ∗   : f . voln (x0, . . . , xn ) = sup  ∈ S , i = 1, . . . , n i  . . . . . . . . . . . . . . . ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪  fn (x0 ) . . . fn (xn ) ⎪ ⎩ ⎭ It is clear that δX(1) (ε) = δX (ε). Amir also showed that  1 2 Jcv (X) ≤ min max 1 − δX(2) (ε), ε + . ε 3 2 For X =  2 , Amir’s estimate gives Jcv ( 2 ) ≤ 0.805 (this estimate is√better than Bynum’s estimate Jcv (X) ≤ 2(1 − δ√X (1)), which implies that Jcv ( 2 ) ≤ 3/2). Note that the exact value of Jcv ( 2 ) is 2/2 . More results on estimates of Jcv ( · ) for  p -spaces with p > 2 can be found in [404], [30]. Some fixed-point results for nonexpansive maps can be formulated in terms of the normal structure of Banach spaces and the coefficient N(X) of the normal structure of a space (N(X) = 1/Jcv (X)). Definition 14.2 A nonempty closed convex subset M of a Banach space X is said to be diametral if diam M = r M (M). A Banach space X has normal structure (weak normal structure) if every nonempty closed bounded (respectively, weakly compact) convex diametral subset of X is a singleton. Remark 14.3 The spaces  p and L p (Ω) with 1 < p < ∞ have normal structure [50]. The space c0 does not have normal structure. Indeed, consider the set M := conv{en | n ∈ N}, where (en ) is the standard basis. We have diam M = 1 and r cv (M) = 1, because lim  x − en  ≥ 1 for all x ∈ c0 . Since (en ) is weakly null, the set M is weakly compact. Hence c0 also does not have weak normal structure. A similar example shows that the space  1 does not have normal structure. Theorem 3.3 of [50] shows that the space  1 (as well as every Banach space with the Schur property) has weak normal structure. A map f : X → X is said to be nonexpansive if  f (x) − f (y) ≤  x − y for all x, y ∈ X.

14.3 The Jung Constant and Fixed Points of Condensing and Nonexpansive Maps

271

Theorem 14.9 (W. A. Kirk, see [50]) Let X be a Banach space with weak normal structure, let C be a weakly compact convex subset of X, and let f : C → C be a nonexpansive map. Then f has a fixed point. Proof Let B be the family of all nonempty weakly compact convex subsets of C that are invariant under f . If these sets are partially ordered by inclusion, then it is easy to check that every net in B has a maximal element. Hence by Zorn’s lemma, the family B has a minimal element K. We have f (K) ⊂ K, and hence conv( f (K)) ⊂ K. Thus conv( f (K)) is a weakly compact subset of K that is invariant under f . The minimality of K implies that K = conv( f (K)). Since K is weakly compact and convex, the set ZK (K) is nonempty, being the intersection of the nested sequence of convex weakly compact sets ZεM (M) := {y ∈ K | r(y, K) ≤ rK (K) + ε}. Let x ∈ ZK (K); that is, r(x, K) = rK (K). For all y ∈ K, we have  f (x) − f (y) ≤  y − x ≤ rK (K). Therefore, f (K) is contained in the closed ball B( f (x), rK (K)), which implies that conv( f (K)) = K ⊂ B( f (x), rK (K)). Consequently, r( f (x), K) ≤ rK (K), which gives f (x) ∈ ZK (K). Thus ZK (K) is a nonempty convex weakly compact subset of K that is invariant under f . From the minimality of K we have ZK (K) = K. Since X has weak normal structure, diam K = 0; that is, K is a fixed point of f .  Theorem 14.9 can be used in establishing the existence of periodic solutions of differential equations [50]. Another important characteristic of a space is the normal structure coefficient    diam M  M⊂X , (14.1) N(X) := inf r M (M)  where the infimum is taken over all bounded convex subsets M ⊂ X with diam M > 0. It is clear that 1 ≤ N(X) ≤ 2. In contrast to the definition of the constant 1/J(X), in (14.1) we consider r M (M) rather than r(M), and the set M is assumed to be convex. Then N(X) = 1/Jcv (X). For basic properties of the constant N(X), see the above results on Jcv (X), and also [247]. From the definition it is clear that if N(X) > 1, then X has normal structure (Example 5 in [50] shows that the converse implication does not hold). Spaces X with the property N(X) > 1 are called spaces with uniformly normal structure. Bynum (see [50, Theorem 2.2 ]) showed that if δX ( · ) is the modulus of convexity is not sharp. For of a Banach space, then N(X) ≥ 1/(1 − δX (1)). This estimate  √ 2 example, for X =  , √ we have N(X) = 2 , and δ 2 (ε) = 1 − 1 − ε 2 /4, which gives the lower estimate 2/ 3. It is also known that if δ(3/2) > 1/4, then N(X) > 1. Prus showed that N(X) ≥ α − (α2 − 4)1/2 , where α := inf{ε/2 + 2 − δ(ε) | 1 ≤ ε ≤ 3/2}. We remark that N(L p (Ω)) = min{21−1/p, 21/p } for infinite-dimensional L p (Ω, Σ, μ)spaces, where μ is a σ-finite measure and 1 ≤ p < ∞. According to Theorem 2.6 of [50], if X is a Banach space and N(X) > 1, then X is reflexive. But if X is reflexive, then the infimum in (14.1) can be taken over√the convex hulls of finite subsets of X. We also note the general estimate N(X) ≤ 2 , where X is an infinite-dimensional Banach space.

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14 The Jung Constant

For nonexpansive maps we mention the following result, which is a corollary to Theorem 14.9: if N(X) > 1, then every nonexpansive map of a nonempty closed convex bounded subset of X has a fixed point. For further details, see [50] and the references cited there. The definitions and results that follow below in this section are due to Marrero [425], [426]. Definition 14.3 For a nonempty closed bounded subset Y of a Banach space X, we say that a map f : Y → Y has property (C) if limn→∞ η(Yn ) = 0, where (Yn ) is the nested sequence of nonempty closed bounded subsets of Y defined by Y1 = f (Y ),

Yn+1 = f (Yn ),

n∈N

(14.2)

(as before, η( · ) is the measure of nonconvexity). Proposition 14.1 Let Y  be a closed bounded subset of a reflexive space X and let f : Y → Y be a map with property (C). Then Y contains a nonempty closed convex set K such that f (K) ⊂ K. Proof (of Proposition 14.1) Let (Yn ) be given by (14.2). Then the set Y∞ := ∩∞ n=1Yn is closed and f (Y∞ ) ⊂ Y∞ . Since f has property (C), lim η(Yn ) = 0. Now Theorem  14.2 gives that Y∞ is nonempty and convex. Proposition 14.2 Let Y  be a weakly compact subset of a Banach space X and let f : Y → Y be a map with property (C). Then Y contains a nonempty closed convex (and hence weakly compact) set K such that f (K) ⊂ K. Proof (of Proposition 14.2) Let (Yn ) be the sequence from (14.2). Since f has property (C), we have lim η(Yn ) = 0. From Theorem 14.4 it follows that K := Y∞ := ∩∞ n=1Yn is nonempty, closed, and convex. It is clear that f (K) ⊂ K. Since a closed convex set is weakly closed, K is weakly compact.  Definition 14.4 Let Y be a nonempty bounded subset of a Banach space X. A map f : Y → Y is said to be δ-condensing if diam f (M) < diam M for every set M ⊂ Y such that f (M) ⊂ M and diam M > 0. A map f : Y → Y is said to be η-condensing if   η f (M) < η(M) for every set M ⊂ Y such that f (M) ⊂ M and η(M) > 0. The following two results, which follow from Propositions 14.1, 14.2 and results of [168], extend Ćirić’s fixed-point theorem [168] and Sadovskii’s fixed-point theorem [504], [53] to the case of not necessarily convex sets.

14.4 On an Approximate Solution of the Equation f (x) = x

273

Theorem 14.10 Let Y  be a closed bounded subset of a Banach space X such that conv Y is weakly compact. Suppose that f : Y → Y is a continuous η-condensing map with property (C). Then f has a fixed point. Theorem 14.11 Let Y  be a closed bounded subset of a reflexive space X, and let f : Y → Y be a δ- or η-condensing map with property (C). Then f has a fixed point. Definition 14.5 Let M ⊂ X be a nonempty bounded set. A point x ∈ M is said to be diametral if sup{ x − y | y ∈ M } = diam M. A convex set M is said to have normal structure if for every N ⊂ M containing more than one point, there exists a point x ∈ N that is not a diametral point of N (see also Definition 14.2). Sets with normal structure were introduced by Brodskii and Milman. For more details, see [274], Chap. 4. The next result (see [425], [426]) extends Kirk’s fixed-point theorem in [338]. Theorem 14.12 Let Y  be a closed bounded subset of a reflexive space X or a nonempty weakly compact subset of a Banach space X. Suppose that Y has normal structure and that f : Y → Y is a nonexpansive map with property (C). Then f has a fixed point. Proof (of Theorem 14.12) For weakly compact Y , this result follows from Proposition 14.2 and Theorem 4.1 of [274], according to which a nonexpansive mapping on a convex nonempty weakly compact subset of a Banach space X with normal structure has a fixed point. In the reflexive setting, one should employ Proposition 14.1 and Kirk’s fixed-point theorem for nonexpansive mappings (see [338], [274, p. 37]). 

14.4 On an Approximate Solution of the Equation f (x) = x In this section, the Jung constant in the Hilbert space setting appears as a natural ingredient in the proofs. In this section, X is a real Banach space and  M ⊂ X. Let N(X, M) (respectively, N(M, M)) be a class of maps from X (from M) to M. We set β0 = β0 (N, X, M) := β = β(N, X, M) :=

sup

inf  f (x) − x,

(14.3)

sup

inf  f (x) − x.

(14.4)

f ∈N(X, M) x ∈X

f ∈N(M, M) x ∈M

From the definition, it follows that β0 ≤ β. Babenko, Konyagin, and Tsar’kov [51] found estimates of β and β0 for classes of continuous (respectively, Lipschitz) maps. I. Let N(X, M) = C(X, M) (N(M, M) = C(M, M)). The Schauder fixed-point theorem shows that β0 = 0 (respectively, β = 0) if M is compact (respectively, if M

274

14 The Jung Constant

is convex and compact). According to a result of Dugundji (see, for example, [280], p. 108), if dim X = ∞, then for M := {x ∈ X |  x ≤ 1} there exists f ∈ C(M, M) such that f (x)  x for all x ∈ X (this implies that the unit sphere of an infinitedimensional space is contractible to a point). Klee (see [280]) extended this result to the case of convex subsets of metrizable locally convex spaces. The following theorem can be proved using Dugundji’s fixed-point theorem. Theorem 14.13 (See [51]) Let X be an infinite-dimensional Banach space and let M := {x ∈ X |  x ≤ 1}. Then for all ε > 0, there exists f ∈ C(X, M) such that  f (x) − x > 1 − ε

∀x ∈ X

(that is, β0 (M) = β(M) = 1). We set α0 = α0 (X, M) := inf sup ρ(y, T), T ⊂X y ∈M

where T ⊂ X is a finite set of points, ρ(y, T) := minx ∈T  y − x, and α0 is the radius of best covering of the set M by balls (the self radius of best covering of M by balls); α = α(X, M) := inf sup ρ(y, T), T ⊂M y ∈M

(α is the self radius of best covering of balls of the set M). It is clear that α/2 ≤ α0 ≤ α. Note that if M = B ⊂ X is the unit ball of an infinite-dimensional space, then α(B) = α0 (B) = 1, and hence by Theorem 14.13, β0 (B) = β(B) = α0 (B) = α(B), which shows that the estimate in the next theorem is sharp. Theorem 14.14 (See [51]) Let X be a Banach space and let M ⊂ X be a nonempty convex set. Then and β ≤ α. β0 ≤ α0 Proof (of Theorem 14.14) Let f ∈ C(X, M) (respectively, f ∈ C(M, M)). For each ε > 0, there exists a finite system of points T ⊂ X (T ⊂ M) such that for all x ∈ M, ρ(X, T) ≤ α0 + ε

(respectively ρ(X, T) ≤ α + ε).

Let L be the convex hull of a set T. Then the set L ⊂ X (L ⊂ M) is compact. Let, as before, PL x be the set of nearest points in L to x. The mapping ϕ := PL ◦ f | L has convex and compact values and is upper semicontinuous as a mapping from L into 2 L . By Kakutani’s fixed-point theorem (see, for example, [280, Sect. II.8.a]), there exists a point x0 ∈ L such that x0 ∈ ϕ(x0 ). Therefore,  f (x0 ) − x0  ≤ ρ( f (x0 ), L) ≤ α0 + ε

(respectively ≤ α + ε).

Letting ε → 0, we arrive at the conclusion of the theorem. In the next theorem, we estimate β and β0 from below. Let F (X) be the class of all nonempty closed subsets of a Banach space X.



14.4 On an Approximate Solution of the Equation f (x) = x

275

Theorem 14.15 (See [51]) Let X be an infinite-dimensional Banach space and let M ∈ F (X) be convex. Then: (a) if M is separable, then α0 /2 ≤ β0 , α/3 ≤ β, β0 ≤ α0 , and β ≤ α (the estimates are attained in the space of bounded sequences); (b) if M is not separable, then α0 /4 ≤ β0 and α/6 ≤ β. Sharper estimates can be obtained in the Hilbert space setting (the estimates are attained on some subset M of a Hilbert space. Theorem 14.16 (See [51]) Let M ∈ F (H) be convex. Then √ √ 2 2 α0 = α ≤ β ≤ β0 . 2 2 II. Let N(X, M) and N(M, M) be the classes of Lipschitz maps with fixed Lipschitz constant k ≥ 0. It is known that if k < 1, then β = β0 = 0 by the Banach contraction principle. Below, we assume that k ≥ 1. Theorem 14.17 (See [51]) Let X be a Banach space and let M ∈ F (X) be a convex set. Then k −1 k −1 α0 and β ≤ α β0 ≤ k k (the estimate is attained in X =  ∞ ). Proof (of Theorem 14.17) We fix an arbitrary ε > 0 and choose a finite system of points T ⊂ X (T ⊂ M) such that ρ(x, T) ≤ α0 + ε

(respectively ≤ α + ε)

∀ x ∈ M.

Let f ∈ N(X, M) ( f ∈ N(M, M)) and let L be the convex hull of T. We construct a mapping ϕ : L → M by associating with each point x ∈ L a point y ∈ X (y ∈ M) such that 1 ( f (y) − x) = y x+ k+ε 1 (such a point exists and is unique, because ψ(z) = x + k+ε ( f (z) − x) is a contraction from M into M). It is easily checked that ϕ ∈ C(L, M) and that χ := PL ◦ f ◦ ϕ is an upper semicontinuous mapping from L into 2 L with convex compact values. Hence by Kakutani’s fixed-point theorem, there exists a point x0 ∈ L such that x0 ∈ χ(x0 ). Therefore, (respectively ≤ α + ε). ( f ◦ ϕ)(x0 ) ≤ α0 + ε

276

14 The Jung Constant

By the construction, we have ( f ◦ ϕ)(x0 ) − x0 = (k + ε)(ϕ(x0 ) − x0 ), ( f ◦ ϕ)(x0 ) − ϕ(x0 ) = (k + ε − 1)(ϕ(x0 ) − x0 ), which gives ( f ◦ ϕ)(x0 ) − ϕ(x0 ) ≤ ≤

k +ε−1 (α0 + ε) k+ε

k +ε−1 ( f ◦ ϕ)(x0 ) − x0  k+ε   k +ε−1 (α + ε) . respectively ≤ k+ε

The proof concludes by letting ε → 0.



The next theorem is based on results of [405]. Theorem 14.18 (See [51]) There exists C > 0 such that C

k −1 α≤β k

for every infinite-dimensional Banach space X and convex set M ∈ F (X). A similar estimate also holds in a Hilbert space for α0 and β0 .

14.5 On the Jung Constant of the Space  n1 The general inequality for the Jung constant in an n-dimensional space implies that J(n1 ) ≤

n . n+1

Below, we show that this becomes an equality if and only if n is a Hadamard dimension (the definition is given below). By a Hadamard matrix A of order n we mean a square n × n matrix with all entries ±1 and with mutually orthogonal rows; that is, AAT = nE, where E is the identity matrix and AT is the transpose of A. It is known that if a Hadamard matrix of order n exists, then n either is a multiple of 4 or equals 1 or 2 (for more details, see [295]). There is a conjecture that this condition is also sufficient for the existence of a Hadamard matrix. In particular, Hadamard matrices have been constructed for all n = 2k and for all n = 4k < 668 (see, for example, [431], [176]). A dimension n is called a Hadamard dimension if there exists a Hadamard matrix of size n + 1. Let n be a Hadamard dimension. Consider a Hadamard matrix of size n + 1 in the normalized form: its first column consists only of ones. We reject the first column and interpret each column as a vector from Rn . We thus get the set Hn = {x 1, . . . , x n+1 }, which is called a Hadamard set.

14.5 On the Jung Constant of the Space n1

277

The main result of this section is the following. Theorem 14.19 (V. L. Dol’nikov [201]) The inequality n J(n1 ) ≤ n+1

(14.5)

(which holds for every n-dimensional space) becomes an equality if and only if there exists a Hadamard matrix of order n + 1. Before proceeding with the proof of Theorem 14.19, we mention some results on the Jung constant of the space n1 (see [201], [421]). 1 ) ≥ 2m−1 . Theorem 14.20 (1) J(2m 2m (2) If there exists a Hadamard matrix of size 2m, then 1 )= J(2m

2m − 1 . 2m

(3) Assume that there exists a Hadamard matrix of size n + 1. Then 1 )≥ J(n+1

n , n+1

1 J(n+2 )≥

n , n+1

1 J(n+3 )≥

n . n+1

V. I. Ivanov’s conjecture is that if there exists a Hadamard matrix of order n + 1, then 1 1 1 J(n+1 ) = J(n+2 ) = J(n+3 ) = n/(n + 1).

From a result of Dol’nikov (Theorem 2 of [201]) it follows that 3 J(41 ) = . 4 At present, the value of the Jung constant for the space 51 is unknown. The proof of Theorem 14.19 proceeds as in [307]. We set r(M) . Jn = J(n1 ) = sup J(M), where J(M) := diam M M ⊂X

(14.6)

From Helli’s theorem (see Appendix B) it follows that Jn =

sup

M={x0,...,x n }

J(M);

(14.7)

that is, in (14.6) it suffices to take the supremum only over (n + 1)-point subsets of n1 . Indeed, we define Jn by (14.7) and consider an arbitrary bounded set M. Let Bx = B(x, Jn diam M). Consider the family of balls {Bx | x ∈ M }. Each n + 1 balls B x0 , B x1 , . . . , B x n of this family have a common point c  ∈ Z({x0, . . . , xn }), because r({x0, . . . , xn }) ≤ Jn diam({x0, . . . , xn }) ≤ Jn diam M.

278

14 The Jung Constant

By Helli’s theorem, there is a point c lying in each ball Bx ; that is, for all x ∈ M,  x − c ≤ Jn diam M. Hence r(M) ≤ Jn diam M. Proposition 14.3 Let M = {x0, x1, . . . , x N } ⊂ n1 . Then J(M) ≤

N . N +1

Proof (of Proposition 14.3) We can assume without loss of generality that x0 + . . . + x N = 0;

(14.8)

that is, if xi = (xi(1), . . . , xi(n) ), i = 0, . . . , N, then N  i=0

(j)

xi = 0,

j = 1, . . . , n.

(14.9)

(j)

We set σi j = sign xi . Let 1 ≤ j ≤ n, 0 ≤ i ≤ N, 0 ≤ k ≤ N. We have   (j) (j) (j) (j) (j)  (j)  (j) (j) |xk − xi | = σi j |xk | − xi  =  |xk | − σk j xi  ≥ |x j | − σk j xi . Summing the inequality (j)

(j)

(j)

(j)

|xk − xi | ≥ |xk | − σk j xi

(14.10)

over to i and using (14.9), we find that N  i=0

(j)

(j)

(j)

|xk − xi | ≥ (N + 1)|xk |.

Dividing by N + 1 and discarding the zero term on the left (i = k) gives (j)

|xk | ≤

1  (j) (j) |x − xi |, N + 1 ik k

j = 1, . . . , n.

(14.11)

On summing the last inequality over j, we obtain  xk  ≤

1   xk − xi , N + 1 ik

Further, we have (for c ∈ Z(M))

k = 0, 1, . . . , N.

(14.12)

14.5 On the Jung Constant of the Space n1

279

r(M) = max  xk − c ≤ max  xk − 0 k

(14.13)

k

= xk0  ≤

1  N diam M,  xk0 − xi  ≤ N + 1 ik N +1 0



which proves Proposition 14.3. Using (14.7), it follows from Proposition 14.3 that Jn ≤

n . n+1

(14.14)

Let us see when estimate (14.14) becomes an equality. Let M = {x0, x1, . . . , x N } ⊂ n1 be a set on which estimate (14.14) is attained. We can assume without loss of generality that condition (14.8) holds and that diam M = n + 1, hence r(M) = n. Such sets will be called extremal sets. From (14.10)–(14.14), we get the following: (a) N = n; (b)  xk − xi  = n + 1 for each pair i, k, 0 ≤ i < k ≤ n; (c)  xk  = 1, k = 0, 1, . . . , n; (d) Z(M) = {0}. Assertions (b)–(d) follow from the fact that under the above assumptions, each of the inequalities in (14.13) is in fact an equality. Further, from (c) and (b) it follows that (14.11) (and hence (14.10)) is also an (j) (j) equality (for all possible j, i, k, i  k j). If xk and xi are of different signs, then the (j) (j) (j) (j) left- and right-hand sides of (14.10) are both equal to |xi | + |xk |. If xk and xi have the same sign, then the right-hand side of (14.10) is equal to (j)

(j)

(j)

(j)

(j)

(j)

|xk | − σk j xi = |xk | − σi j xi = |xk | − |xi |, (j)

(j)

and so it coincides with the left-hand side only if xi | ≤ |xk |. Swapping i and k, (j) (j) we get that xi | ≥ |xk |. Hence, we have: (e) among the coordinates with the same name in the vectors x0, . . . , xn , all positive coordinates are equal and all negative coordinates are equal. Remark 14.4 The zero coordinates can be looked upon as either positive (σi j = 1) or negative (σi j = −1). From the above it follows that if at least one of the coordinates is zero, then all coordinates with the same name are also zero. Clearly, such a set M can 1 . Hence by Proposition 14.3, J(M) ≤ n−1 . Therefore, be considered a subset of n−1 n assertion (e) can be augmented as follows: there are no zero coordinates. Further, we give a characterization of the Chebyshev centre and radius (see also Sect. 15.7 below) and also give another upper estimate. We define V n := {s = (σ1, . . . , σn ) | σj ∈ {−1, 1},

j = 1, . . . , n}.

280

14 The Jung Constant

Proposition 14.4 Let M = {x0, x1, . . . , x N } ⊂ n1 . Assume that there exist vectors si ∈ V n , i = 0, 1, . . . , N, and a vector A ∈ Λ N +1 := {α = (α0, . . . , α N ) ∈ R N +1 | αi ≥ 0, α1 + . . . + α N = 1} such that: (1) (si, xi ) =  xi  = r, i = 0, 1, . . . , N; N αi si = 0. (2) i=0 Then Z(M) = {0} and r(M) = r. Proof Condition (1) means that the Chebyshev radius of the set M with respect to the point 0 is r. We claim that for every other point c, it cannot be smaller than r. Using Hölder’s inequality, we have max  xi − c ≥ i

=

N  i=0 N 

αi  xi − c ≥

N 

αi (si, xi − c)

i=0

αi (si, x) −

i=0

N 

αi (si, c) = r −

i=0

N 

 αi si, c = r,

i=0



proving Proposition 14.4. The converse result is also true. We give only a particular case.

Proposition 14.5 Let M = {x0, x1, . . . , x N } ⊂ n1 . Assume that Z(M) = {0}, r(M) = r =  xi , i = 0, 1, . . . , n, and that the vectors xi have no zero coordinates. Then there exist vectors si ∈ V n , i = 0, 1, . . . , n, and a vector A = (α0, α1, . . . , αn ) ∈ n+1 βi = 1} such that: Λn+1 := {(β1, . . . , βn+1 ) ∈ Rn+1 | βi ≥ 0, i=1 , x ) =  x  = r, i = 0, 1, . . . , n; (1) (s i i i n αi si = 0. (2) i=0 (j)

Proof Let xi = (xi(1), . . . , xi(n) ), σi j = sign xi , i = 0, . . . , N, j = 1, . . . , n. Clearly, the vectors si = (σi1, . . . , σin ), i = 0, 1, . . . , n), satisfy condition 1), and so we need to verify only condition 2), that is, to show that the point 0 lies in the convex hull (the simplex) of the vectors {s0, . . . , sn }. If this were not so, then by the separation theorem for convex sets, there would exist a hyperplane separating 0 and si . Hence, for some vector m = (μ1, . . . , μn ) and a number δ > 0, we would have (si, n) ≥ δ,

i = 0, 1, . . . , n,

whereas (0, m) = 0 < δ. For the point εm, we have  xi − εm :=

n  j=1

(j)

|xi − ε μ j | =

n n      (j)  σi j |x (j) | − ε μ j  =  |x | − ε μ j σi j . j=1

i

j=1

i

(j)

We choose ε > 0 to be so small so as to have the inequality |xi | > ε μ j for all i and j (we recall that all coordinates are nonzero). Hence

14.5 On the Jung Constant of the Space n1

 xi − εm =

n   j=1

281

n n    (j) (j) |xi | − ε μ j σi j = |xi | − ε μ j σi j j=1

j=1

=r − ε(si, m) ≤ r − εδ. But this contradicts the assumption r(M) = r.



Under the hypotheses of Proposition 14.5, we have the following estimate for the Jung constant: i = 0, 1, . . . , n. (14.15) J(M) ≤ 1 − αi, Indeed, diam M(1 − αi ) ≥ =

n  k=0 n 

αk |xk − xi | ≥ αk (sk , xk ) −

k=0

n 

αk (sk , xk − xi )

k=0 n 

αk (sk , xi ) = r(M).

k=0

This allows one to refine condition e) for attainability of estimate (14.5). Indeed, from (14.15) it follows that if this estimate is attained, then αi = 1/(n + 1) for all i, and hence n  si = 0. i=0

Hence among the jth coordinates of the vectors forming an extremal set, one half are positive and the other half are negative. By (14.8), the absolute values of the positive and negative coordinates are equal. Hence we reach the following conclusion. Proposition 14.6 Let M = {x0, x1, . . . , xn } ⊂ n1 be an extremal set satisfying (14.8). Then there exist positive numbers λ1, . . . , λn such that the jth coordinate of half of the vectors xi is λ j , and for the other half, it is −λ j . We next show that each extremal set of diameter n + 1 satisfying condition (14.8) coincides with a Hadamard set Hn . Let M = {x0, . . . , xn } be such a set. We have xi = (xi(1), . . . , xi(n) ), (j) xi

=

i = 0, 1, . . . , n,

σi j |xi(i) |

= σi j λ j ,

j = 1, . . . , n,

(14.16)

since by Proposition 14.6, the absolute values of the jth coordinates are equal. From property c) of attainability of estimate (14.5), we have  xi  =

n 

λ j = n.

j=1

Using property b), we have, for all i, k, i  k,

(14.17)

282

14 The Jung Constant n 

n + 1 =  xi − xk  =

j=1

Hence

n 

(j)

(j)

|xi − xk | =

n 

λ j (1 − σi j σk j ).

j=1

λ j σi j σk j = −1.

(14.18)

j=1

Consider two square matrices of size n + 1, H = (hi j ),

hi0 = 1,

hi j = σi j ,

1 ≤ j ≤ n,

and consider the matrix G obtained from the matrix of coordinates of the vectors by adding the first column consisting of ones: G = (gi j ),

gi0 = 1,

(j)

gi j = xi ,

1 ≤ j ≤ n.

Let us find the product P = GHT . If i  k, then by (14.18), pik = 1 +

n  j=1

(j)

xi σk j = 1 +

n 

λ j σi j σk j = 0.

j=1

For i = k, using (14.17), we have pii = 1 +

n  j=1

Therefore,

(j)

xi σi j = 1 +

n 

λ j = 1 + n.

j=1

GHT = (n + 1)I,

where I is the identity matrix of size n + 1. From (14.16), we have G = HΛ,

(14.19)

(14.20)

where Λ = diag(1, λ1, . . . , λn ). Substituting (14.20) into (14.19) and dividing by (n + 1), we have HΛHT = I, where Λ :=

1  n+1 Λ .

Consider the matrix Λ1/2 =

We have

(14.21)

 1  1/2 diag(1, λ11/2, . . . , λn1/2 ). n+1 HΛ1/2 (HΛ1/2 )T = I,

and hence HΛ1/2 is orthogonal. Since the length of each column of an orthogonal matrix is 1, we have λ1 = . . . = λn = 1,

14.6 The Jung Constant and the Jackson Constant

283

and now (14.21) assumes the form HHT = (n + 1)I.

(14.22)

Relation (14.22) is one of the definitions of a Hadamard matrix. So every extremal set is a Hadamard set.

14.6 The Jung Constant and the Jackson Constant Stechkin and V. I. Berdyshev (see [95], and also [307, Chap. 4, Sect. 5]) established a link between the Jung constant (a geometric characteristic of a space) and the sharp constant in Jackson’s theorem on best approximation of functions by constants (an approximative characteristic of a space). Namely, let X be the Banach space of integrable 2π-periodic functions f : R → Y (Y is a normed linear space) with a norm satisfying the conditions  f (t + u) =  f (t),

∀t, |u| ≤ π,

lim  f (u + t) − f (u) = 0,

t→0

f ∈ X.

(14.23)

We assume that X contains the constant functions and that the curve M f := { f ( · +t) | |t| ≤ π} is compact for every function f ∈ X. In particular, the above properties are satisfied by the uniform norm on the space of continuous 2π-periodic functions. Let E0 ( f ) = E0 ( f )X := inf{ f ( · ) − cX | c ∈ Y } be the best approximation of a function f ( · ) by constants c ∈ Y , and let ω( f , δ, X) := sup |t | ≤δ  f ( · + t) − f ( · )X be the (first) modulus of continuity of the function f ( · ). The Jackson constant is defined as E0 ( f )X . K(X) := sup f ∈X ω( f , π, X) We also define the Jung constant for the class of compact sets: J ∗ (X) := sup

r(M) , diam M

where the supremum is taken over all compact sets M ⊂ X. It is clear that J ∗ (X) ≤ J(X). In the next theorem (see [25]), the case Y = R was obtained by Stechkin [95]. Theorem 14.21 (S. B. Stechkin [95]) Let X be a Banach space of 2π-periodic integrable functions f : R → Y with a norm satisfying conditions (14.23), and suppose that X contains the constant functions. Then K(X) ≤ J ∗ (X). Proof Let f ∈ X, f  0. By (14.23), for a function ϕ ∈ X, u=t

 f (τ + t) − ϕ(τ) =  f (τ − u + t) − ϕ(τ − u) =  f (τ) − ϕ(τ − t).

284

14 The Jung Constant

Hence 1 2π

∫

∫ π 1  f (τ + t) − ϕ(τ) dt =  f (τ) − ϕ(τ − t) dt 2π −π −π ∫ π   ∫ π  ∫ π   1  1 1    [ f (τ) − ϕ(τ − t)] dt  =  f (τ) dt − ϕ(τ − t) dt  ≥  2π −π 2π −π 2π −π     ∫ τ+π ∫ π     1 1    =  f (τ) − ϕ(τ) dτ  =  f (τ) − ϕ(τ) dτ  ≥ E0 ( f ), 2π τ−π 2π π π

and therefore, r(M f )X := sup  f (τ + t) − ϕ(τ) ≥ |t | ≤π

1 2π

∫

π

−π

 f (τ + t) − ϕ(τ) dt ≥ E0 ( f ),

where M f := { f ( · + t) | |t| ≤ π}. On the other hand, diam M f = sup  f (τ + t1 ) − f (τ + t2 ) = sup  f (τ + t) − f (τ) = ω( f , π, X), t1,t2

|t | ≤π

and since M f is compact, we have

r(M f ) E0 ( f ) ≤ ≤ J ∗ (X). ω( f , π, X) diam M f



The next result follows from Theorem 14.21 and a result of V. I. Berdyshev [94]. Corollary 14.1 In the space L p [−π, π] with 1 ≤ p < ∞, max{2(1−p)/p, 2−1/p } ≤ K(L p ) ≤ J ∗ (L p ) ≤ J(L p )

(14.24)

(and for p = 1, 2, all the inequalities in (14.24) become equalities). Note that (see [95]) J ∗ (C[−π, π]) = K(C[−π, π]) = 1/2,

(14.25)

J(C[−π, π]) = 1.

(14.26)

but To prove (14.26), it suffices to consider the following example given by Stechkin (see [95]). Consider the set M = ( fk (t))∞ k=1 from the space C[−π, π] of continuous 2π-periodic functions. We set ⎧ ⎪ 1, ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ fk (t) = −1, ⎪

⎪ ⎪ π ⎪ ⎪ −t , ⎪k ⎩ 2

π 1 0≤t≤ − , 2 k π 1 + ≤ t ≤ π, 2 k π 1 π 1 − 0 and point x ∈ n∞ , the set x + aM is also extremal. So it suffices to describe the extremal convex nonsingleton sets M with relative Chebyshev radius 1 and relative Chebyshev centre (see (15.2)) lying at the origin. In this section, by faces of the unit ball B of the space n∞ we shall mean the sets of the form   B j (ε) = x = (ξ (i), . . . , ξ (n) ) : ξ (j) = ε, ξ (i) ∈ [−1, 1] ∀ i  j , where ε = ±1. Faces B j (ε) and B j (−ε) will be called opposite. Let E = (ε1, . . . , εn ), where ε j = ±1, and let M(E) = conv (y1 (E), . . . , yn (E)) be the convex hull of the points yk (E) = (ηk(1), . . . , ηk(n) ), whose coordinates for k = 1, . . . , n are given by

(j)

ηk

ηk(k) = ηk(k) (E) = εk , εj (j) , j = 1, . . . , n, = ηk (E) = − n−1

j  k.

n We have k=1 yk (E) = 0, and hence 0 ∈ M(E). By y(E) we denote the point with εn ε1 , . . . , − n−1 , and by M(E) we denote the convex hull coordinates − n−1 M(E) = conv (y(E), y1 (E), . . . , yn (E)). If all coordinates ε j in a tuple E are equal to 1, then for simplicity, instead of M(E), yk (E), y(E), M(E), we shall write, respectively, M ∗, yk∗, y ∗ and M ∗ . So  1  1 , . . ., − , M ∗ = M(1, . . . , 1), y ∗ = (η∗ (1), . . . , η∗ (n) ) = − n−1 n−1 where the kth coordinate ηk∗ (k) of the point yk∗ is 1, and the remaining coordinates 1 ; M ∗ is a convex combination of the ηk∗ (j) , j = 1, . . . , n, j  k, are equal to − n−1 ∗ ∗ ∗ points y , y1, . . . , yn . Recall that for a nonempty bounded subset M of X and a nonempty set Y ⊂ X, the quantity rY (M) = inf y ∈Y r(y, M), where r(x, M) := inf{r ≥ 0 | M ⊂ B(x, r)}, is called the relative Chebyshev radius. The set of relative Chebyshev centres is defined as ZY (M) := {y ∈ Y | r(y, M) = rY (M)}. Theorem 14.22 (S. V. Berdyshev) The following equality holds: Js (n∞ ) =

n−1 , n

n ≥ 2.

14.7 The Relative Jung Constant

287

Moreover, for n ≥ 3, a set M with unit self Chebyshev radius and whose relative Chebyshev centre is at the origin is extremal if and only if there exists a tuple E = (ε1, . . . , εn ) with the property M(E) ⊂ M ⊂ M(E). Note that in Theorem 14.22, as distinct from the Jung constant problem J(X), the exact value of the relative Jung constant Js (n∞ ) is given for all n. Proof By V(X) we denote the class of all convex bounded closed nonsingleton subsets of X. We first note that for spaces of dimension 2, the Chebyshev and the self Chebyshev radii of a bounded convex set are equal (see [262], and also p. 290). Hence by Lemma 14.2, the equality Js (2∞ ) = 12 is clear. The same lemma implies that in this case, every set M ∈ V(2∞ ) is extremal. Now let n ≥ 3. We denote by V1 (n∞ ) the class of sets M ∈ V(n∞ ) with unit self Chebyshev radius whose self Chebyshev centre is at the origin. We also define Δn = inf {diam M | M ∈ V1 (n∞ )}. It is clear that in the definition of Js (n∞ ), one can consider only sets M from V1 (n∞ ). Hence the conclusion of the theorem is equivalent to the equality n . (14.28) Δn = n−1 We first prove an upper estimate for Δn . Consider the sets M ∗ , M ∗ and an arbitrary set M satisfying M ∗ ⊂ M ⊂ M¯ ∗ . Using Lemma 14.3, one can readily check that diam M ∗ = diam M = diam M ∗ =

n . n−1

(14.29)

Let us verify that the self Chebyshev radius of the set M is 1. As was noted above, the origin lies in the set M ∗ , and hence in the set M. Hence r M (M) ≤ 1. Consider an arbitrary nonzero point y = (η(1), . . . , η(n) ) ∈ M. Since the set M lies in the half space 1n η(j) ≤ 0, the point y has a negative coordinate η(j0 ) . Hence  y − y ∗j0  ≥ |η(j0 ) − 1| > 1, and therefore, (14.30) r M (M) = 1, r M (M) 0 is a proper Chebyshev centre of M, and diam M = ∗ ∗ M ⊂ M ⊂ M¯ . From this and (14.29), we have the inequality

Δn ≤

n , n−1

n−1 n

for all M with the property

(14.31)

n ) ≥ n−1 . or, what is the same, Js (∞ n Let us prove the reverse inequality. By (14.29) and (14.30), the set M ∗ lies in n . Hence in the definition of Δn , it suffices to consider V1 (n∞ ), and diam M ∗ = n−1 ∞ a family V = V(n ) of sets M ∈ V1 (n∞ ) with the property

diam M ≤

n . n−1

288

14 The Jung Constant

Let us estimate diam M from below for sets M ∈ V (for n ≥ 3). n < 2, there is no pair of points from M that lie Let M ∈ V. Since diam M ≤ n−1 in two opposite faces of B. We claim that there exist n pairwise nonopposite faces of the ball B that contain points from M. Indeed, assume that the points from M lie in only k (where k < n) pairwise nonopposite faces. It can be assumed without loss of generality that these points x¯i , i = 1, . . . , k, lie on the faces B1 (1), . . . , Bk (1), respectively. In coordinate form, the points x¯i read as x¯1 = (1, ξ1(2), . . . , ξ1(k), . . . , ξ1(n) ), . . . , x¯k = (ξk(1), ξk(2), . . . , 1, . . . , ξk(n) ). Let 0 < δ < k1 . Then the point s = δ x¯1 + . . . + δ x¯k = (σ (1), . . . , σ (n) ) lies in the convex hull conv(0, x¯1, . . . , x¯k ) of the points 0, x¯1, . . . , x¯k , and hence is contained in the set M. n and  x¯i  ≤ 1, 1 ≤ i, j ≤ k. Hence for all Since M ∈ V, we have  x¯i − x¯ j  ≤ n−1 (j) the first k coordinates ξi of the points x¯1, . . . , x¯k , we have −

1 (j) ≤ ξi ≤ 1, n−1

1 ≤ i, j ≤ k.

(14.32)

Therefore, the first k coordinates of the point s are such that 0 0 for i = 1, . . . , k. Under the above assumptions on the position of the set M and the faces of the unit ball, the distances ρ(M, B j (ε)) between the set M and the faces B j (ε) for k + 1 ≤ j ≤ n with ε = ±1, and also for 1 ≤ j ≤ k with ε = −1, are positive. Let ρ > 0 be the minimum of these distances:   ρ = min min ρ(M, B j (ε)), min ρ(M, B j (−1)) . k+1≤ j ≤n ε=±1

1≤ j ≤k

So the coordinates of an arbitrary point x = (ξ (1), . . . , ξ (n) ) from M satisfy the inequalities − 1 + ρ ≤ ξ (i) ≤ 1, |ξ | ≤ 1 − ρ, (j)

i = 1, . . . , k, j = k + 1, . . . , n.

(14.33) (14.34)

We choose δ, 0 < δ < k1 , so as to have |σ (i) | < ρ, i = 1, . . . , n.

(14.35)

14.7 The Relative Jung Constant

289

Since σ (i) > 0 for i = 1, . . . , k, it follows from (14.33)–(14.35) that for every point x = (ξ (1), . . . , ξ (n) ) ∈ M, we have   |σ (i) − ξ (i) | ≤ max σ (i) + 1 − ρ, 1 − σ (i) < 1, i = 1, . . . , k. Moreover, from (14.35) we get |σ (i) − ξ (i) | ≤ |σ (i) | + |ξ (i) | < ρ + 1 − ρ = 1, i = 1, . . . , n. So the set M lies in the ball with centre at the point s ∈ conv (0, x¯1, . . . , x¯k ) ⊂ M whose radius is strictly less than 1. But this contradicts the assumption r M (M) = 1. So n pairwise nonopposite faces of B (of Bi (1), i = 1, . . . , n, without loss of generality) contain the points x¯i , i = 1, . . . , n, from the set M. We have M ∈ V, and n , hence for an arbitrary point x = (ξ (1), . . . , ξ (n) ) from M, we have  x − x¯i  ≤ n−1 i = 1, . . . , n, and  x ≤ 1. It follows that −

1 ≤ ξ (j) ≤ 1, n−1

j = 1, . . . , n.

(14.36)

In coordinate form, the points x¯i , i = 1, . . . , n, read as x¯1 = (1, ξ1(2), . . . , ξ1(n) ), . . . , x¯n = (ξn(1), ξn(2), . . . , 1). We now claim that x¯i = yi∗ , and hence M ∗ ⊂ M. To verify this claim, it suffices to  show that all points x¯i lie in the hyperplane nj=1 η(j) = 0. Assume that this is not so. Then one of the points x¯i does not lie in this hyperplane. Hence by (14.36), the sum of its coordinates is positive, and hence one of its non-1 coordinates is greater than 1 . We can assume without loss of generality that the first coordinate ξn(1) of the − n−1 n 1 . The first coordinate of the point i=1 x¯i is positive, point x¯n is greater than − n−1 and the remaining coordinates are negative. Next, for t ∈ (0, 1), consider the point n x¯i . If the parameter t is sufficiently close to 1, then all coordinates of z = t x¯1 + i=2 the point z are positive. This is clear for the first coordinate. For the ith coordinate, i = 2, . . . , n, there are two cases to consider: 1) ξ1(i) < 0 and 2) ξ1(i) ≥ 0. Case 1) is trivial. In the second case, the ith coordinate of the point z is estimated from below (i) (i) 1 + 1 + ξi+1 + . . . + ξn(i) ≥ n−1 . For the point z in terms of the sum ξ2(i) + . . . + ξi−1 1 with positive coordinates, the point 2n z lies in the convex hull conv (0, x¯1, . . . , x¯n ), 1 z < 1, and hence is contained in the set M. Using (14.36), we get supx ∈M  x − 2n which contradicts the equality r M (M) = 1. n n , and therefore, Δn = n−1 . To complete the proof, So, M ∗ ⊂ M, diam M = n−1 it suffices to show that the coordinates of every point x = (ξ (1), . . . , ξ (n) ) from M  are such that nj=1 ξ (j) ≤ 0. Assume to the contrary that this is not so. Indeed, if n (j) > 0, then the set A = conv (x, x¯1, . . . , x¯n ) lies in M and contains the vector j=1 ξ y = (ε, . . . , ε) for some ε > 0. Hence supx ∈M  x − y < 1, which contradicts the equality r M (M) = 1. Using (14.36), we find that M ⊂ M¯ ∗ and M∗ ⊂ M ⊂ M∗. Theorem 14.22 is proved.



290

14 The Jung Constant

Consider the quantity γ(X) = sup

r M (M) , r(M)

where the supremum is taken over all closed convex nonempty subsets of X. The constant γ appeared in Arestov’s paper [41] on the recovery of operators. It is easily seen that for all M ⊂ X, r(M) ≤ r M (M) ≤ diam M ≤ 2 r(M). Hence using the clear inequality diam M ≤ 2 r(M) ≤ 2 diam M (see (15.5) below), we get that max{1, Js (X)} ≤ γ(X) ≤ 2Js (X). Klee [341] and independently Garkavi [258] (see Theorem 15.10 below) showed that at least one Chebyshev centre of each bounded nonempty subset of X lies in the convex hull of this set if and only if X is a Hilbert space or the dimension of X is at most two. In [99] it was noted that these two conditions are equivalent to saying that γ(X) = 1. Corollary 14.2 The following equality holds: γ(n∞ ) = 2

n−1 . n

This result is a clear corollary of Theorem 14.22 and Lemma 14.2. For more on the relation between the Jung and Jackson constants in L p -spaces, see the book [307].

14.8 The Jung Constant of a Pair of Spaces Let X1 and X2 be normed linear spaces with X2 ⊂ X1 , and let M ⊂ X2 be a bounded set. The Jung constant of M ⊂ X2 ⊂ X1 for the pair of spaces (X1, X2 ) is defined by J(M, X1, X2 ) =

r(M)X1 diam(M)X2

(see [307]), where r(M)X1 is the Chebyshev radius of M in X1 , and diam(M)X2 is the diameter of M in the space X2 . The relative Jung constant of M for a pair of spaces (X1, X2 ) is defined by r M (M)X1 . Js (M, X1, X2 ) = diam(M)X2

14.8 The Jung Constant of a Pair of Spaces

291

By the Jung constant of a pair of spaces (X1, X2 ) we mean (see [307]) J(X1, X2 ) = sup J(M, X1, X2 ),

(14.37)

M ⊂X2

and by the relative Jung constant of a pair of spaces, Js (X1, X2 ) = sup Js (M, X1, X2 ).

(14.38)

M ⊂X2

It can be easily checked that r(M)X1 = r(conv M)X1 = r(convM)X1 , diam(M)X2 = diam(conv M)X2 = diam(convM)X2 . Indeed, if x1, . . . , xn ∈ M, y1, . . . , ym ∈ M, α1 + · · · + αn = 1, αi ≥ 0, and β1 + · · · + βm = 1, β j ≥ 0, then     m m   n   n    αi xi −    β y = α β (x − y ) j j i j i j    i=1

X2

j=1

≤

i=1 j=1 n m  

αi

i=1

X2

β j max  xi − y j  ≤ diam(M)X2 .

j=1

i, j

Hence the set M in (14.37) and (14.38) can be assumed to be closed and convex. Consider the class of spaces    p nk = x = (x (1), . . . , x (n) )  x (i) ∈ Rk , i = 1, . . . , n with the norm  x p =

n 1 

n

|x (i) | p

 1/p

n  k 1 

=

n

i=1

 x∞ = max |x (i) | = max i

1 q

i

|xs(i) | 2

 p/2  1/p

,

1 ≤ p < ∞,

i=1 s=1

k 

|xs(i) |

 1/2 ,

p = ∞.

i=1

In Theorems 14.23 and 14.24, q  is the conjugate exponent of 1 ≤ q ≤ ∞, + q1 = 1.

Theorem 14.23 (See [307, Chap. 4]) Let 1 ≤ p, q ≤ ∞, n, k ∈ N. Then

292

14 The Jung Constant p q J(nk , nk ) p

≤

q

J(nk , nk ) ≤ q

J(n1, n ) ≤ p

J(n , n∞ ) = q

J(n∞, n ) ≤



1/q nk if 1 ≤ p ≤ q ≤ 2;  21/q nk + 1

1/q nk n1/q−1/p if 2 ≤ q ≤ p < ∞; nk + 1 21/q

1/q n 1 if 1 ≤ q ≤ ∞, k = 1; 21/q n + 1 1 if 1 ≤ p ≤ ∞, k = 1; 2 n1/q if 1 ≤ q ≤ ∞, k = 1. 2 1

Theorem 14.24 (See [307, Chap. 4]) Let 1 ≤ p, q < ∞, n, k ∈ N. Then:

1/q nk 1 p q Js (nk , nk ) ≤ 1/q if 1 ≤ p ≤ q ≤ 2; nk + 1 2

1/q nk 1 p q Js (nk , nk ) ≤ 1/q if 2 < q = p < ∞; nk + 1 2

1/4

1/4 k nk 8(k + 1) p 4 Js (nk , nk ) ≤ ; if 1 ≤ p ≤ 4(k + 1) nk + 1 3k + 4

1/4 nk (p − 2)1/2−1/p (k(4 − p))1/p−1/4 p 4 Js (nk , nk )≤ nk + 1 (2k + 4)1/p−1/4 p1/4 8(k + 1) ≤ p ≤ 4; if 3k + 4 3/2 ≤ p ≤ 2, q = 3,

1/q n 1 p q 2 < q ≤ 7/2, Js (nk , nk ) ≤ 1/q if 1 ≤ p ≤ q/2, n+1 2 1 ≤ p ≤ 2, q > 7/2; p

q

Js (nk , nk ) ≤

1 21/q



n n+1

1/q if

3q/2 ≤ p ≤ 2q/3, q/2 ≤ p ≤ 4q/7, 2 ≤ p ≤ 4q/7, 2 ≤ p ≤ q/2,

2 < q ≤ 3, 3 < q ≤ 7/2, 7/2 < q ≤ 4, q > 4.

14.9 Some Remarks on Intersections of Convex Sets. Relation to the Jung Constant Every closed bounded subset of a finite-dimensional Banach space is compact. Hence, the intersection of every nested sequence of closed bounded subsets of

14.9 Some Remarks on Intersections of Convex Sets. Relation to the Jung Constant

293

a finite-dimensional Banach space is nonempty. The converse also holds (which is an easy consequence of the characterization of finite-dimensional spaces in terms of compactness of the unit ball): if the intersection of every sequence of nested closed bounded sets in a Banach space is nonempty, then the space is finite-dimensional. Questions related to intersections of nested families of closed bounded subsets of infinite-dimensional Banach spaces have been considered by Vakhania, Kartsivadze, Chelidze, Papini, Jachymski, and others (see [161], [311] and the references therein). An example of a nested sequence of balls with empty intersection can be constructed in a complete linear metric space. However, according to Vakhania and Kartsivadze, in a (real or complex) Banach space, the intersection of every nested sequence of closed balls is always nonempty (even if the radii do not tend to zero). By an admissible sequence of sets (An ) we shall mean a sequence of nested closed bounded sets A1 ⊃ A2 ⊃ · · · ⊃ An ⊃ · · · in a normed linear space X. To study intersections of sets more general than balls, we recall a definition from [161]), in which one associates with a bounded subset of a Banach space a certain number characterizing (in a sense) the deviation of this set from a ball. Definition 14.6 Let M be a bounded subset of a Banach space, and let Rx (M) := sup  x − y y ∈M

(=: r(x, M)) and rx (M) := inf  x − y. y ∈X\M

We exclude the trivial cases that the set M is empty or is a singleton, and consider the quantity rx (M) χ(M) := sup . x ∈M Rx (M) It is known (see [160] that if (An ) is an admissible sequence of sets in a Banach space and if lim χ(An ) > 1/2, then the sequence (An ) has nonempty intersection. On the other hand, there exists a Banach space X (for example, X = c) such that for every χ < 1/2, there exists an admissible sequence (An ) of closed convex bounded

subsets of X such that χ(An ) = χ for n = 1, 2, . . . and ∞ n=1 An = (see [161]). In the particular case that the space is reflexive and all the closed bounded sets An

in the nested sequence are convex, the intersection ∞ A n=1 n is always nonempty regardless of the behaviour of the sequence χ(An ). It is also known (see [160]) that in the  p -spaces with 1 ≤ p < ∞, the intersection of an admissible sequence (An ) is nonempty if lim χ(An ) ≥ 1/(1 + 21/p ). The definition of the critical value of a Banach space was given in [160]. Definition 14.7 A number α ∈ R+ is called a critical value for a Banach space X (notation: α = cv(X)) if: (a) every admissible sequence (An ) has nonempty intersection, provided that lim χ(An ) > α; (b) for every

ε > 0, there exists an admissible sequence (An ) such that lim χ(An ) > α − ε and ∞ n=1 An = . The values of cv(X) are known for some spaces. In particular, cv(X) ∈ [1/3, 1/2] and cv( p ) = 1/(1+21/p ) for 1 ≤ p < ∞ (so that for every α in the interval (1/3, 1/2),

294

14 The Jung Constant

there exists a reflexive space for which α is a critical value). An example of a reflexive space X with critical value 1/2 was constructed in [161] (X is a subspace of  1 ). It is unknown whether there exists a reflexive space with critical value 1/3. A set A is said to be nontrivial if it contains at least two points. For an admissible sequence consisting of nontrivial sets (An ), we define (see [161]) Δ(An ) := lim

n→∞

r (An ) , diam An

where r (An ) := sup rx (An ) = sup rx (An ). x ∈X

x ∈ An

Definition 14.8 A number α ∈ R+ is said to be CV-critical for X (notation: α = CV(X)) if: a) every admissible sequence of nontrivial sets (An ) has nonempty intersection if Δ(An ) > α; b) for every ε > 0, there exists an admissible sequence (An ) of nontrivial sets

such that Δ(An ) > α − ε and ∞ n=1 An = . Chelidze and Papini [161] showed that for every Banach space X, CV(X) =

cv(X) , 1 + cv(X)

and thus CV(X) ∈ [1/4, 1/3]. Following [161], for every closed bounded set A, we define χ(A) =

r (A) , r(A)

where r (A) := sup{rx (A) | x ∈ A}.

It is clear that χ(A) ≤ χ(A) ≤ 1 for every A. Moreover, the intersection of an admissible sequence (An ) is nonempty if lim χ(An ) > 1/2 . In connection with these questions, we mention a result on the Jung constant (see [161]). Theorem 14.25 The intersection of an admissible sequence (An ) is nonempty if lim χ(An ) >

J(X) . 4

Consequently, CV(X) ≤ J(X)/2 and cv(X) ≤ J(X)/(2 − J(X)).

Chapter 15

Chebyshev Centre of a Set. The Problem of Simultaneous Approximation of a Class by a Singleton Set

In earlier chapters we have been mostly concentred with the problem of best approximation of an element of a normed space by a given set from that space. In this chapter, we consider the problem of approximating a set by a class of sets. In this problem, it is not only the evaluation of the approximation that is important, but also the set that best approximates this class (an optimal set). In particular, for the class of singletons, we arrive at the Chebyshev centre problem and the Chebyshev radius problem. The Chebyshev net problem appears if we consider the class consisting of nets of cardinality n (n-nets). For the class of all n-dimensional planes, the problem becomes the Kolmogorov width problem. In Sect. 15.1, we define the Chebyshev centre Z M ( · ) and the Chebyshev radius r(M) of a set M and discuss its basic properties. The relation between Chebyshev centres and spans is discussed in Sect. 15.2. Using the machinery of spans, we refine and give simple proofs of some classical results on the Chebyshev centre (see, for example, Theorem 15.2). The existence of a Chebyshev centre in normed spaces is discussed in Sect. 15.4. In Sect. 15.4.1, we give some recent results by Veselý, who used the concept of quasi-uniformly convex spaces to extend some of classical results on the existence of Chebyshev centres. The uniqueness of a Chebyshev centre is considered in Sect. 15.5. The stability of the Chebyshev-centre map Z M ( · ) is studied in Sect. 15.6. In particular, we present some results of Veselý (see Sect. 15.6.2) and Tsar’kov (see Sect. 15.6.3) on stability of the Chebyshev-centre map Z M ( · ) in quasi-uniformly convex spaces and in finite-dimensional polyhedral spaces. Stability of the Chebyshev-centre map in C(Q)-spaces is discussed in Sect. 15.6.4. The same problem for Hilbert and uniformly convex spaces is considered in Sect. 15.6.5. Stability of the self Chebyshev-centre map Zsc M is studied in Sect. 15.6.6. Upper semicontinuity of the Chebyshev-centre map and the Chebyshev-near-centre map ZVδ (M) is examined in Sect. 15.6.7. Lipschitz selections of the Chebyshev-centre map are discussed in Sect. 15.6.8. Discontinuity properties of the Chebyshev-centre map are considered in Sect. 15.6.9. Characterization theorems of a Chebyshev centre are given in Sect. 15.7. Some results on Chebyshev centres that are not farthest © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_15

295

296

15 ChebyshevCentre of a Set . . .

points are presented in Sect. 15.8. Results on smooth and continuous selections of the Chebyshev near-centre map are given in Sect. 15.9. Algorithms and applied problems connected with Chebyshev centres are discussed in Sect. 15.10.

15.1 Chebyshev Centre of a Set We shall be mostly concerned with the Chebyshev centre problem. In this problem, one searches for a point that best approximates (represents) a given set. Such a problem is naturally related to practical situations when optimal estimates are required, for example, in settings in which in a mathematical model of a physical process is represented by some unknown point x in a space X. Suppose that from certain experiments or observations, some estimates (possibly with errors) of a point x are known (for example, x is known to lie in a set M). The available information is usually insufficient for precise determination of the point x. In this setting, an element  x that best approximates (represents) the set M is called a Chebyshev centre of the set M. The Chebyshev centre problem is sometimes called the problem of best simultaneous approximation. It is plain that in this problem, the set M is bounded. Definition 15.1 For a nonempty bounded subset M of a metric space (X, ), the quantity diam M = supx,y ∈M (x, y) is called the diameter of M, and r M := r(M) := inf {a  0 | x ∈ X, M ⊂ B(x, a)} is called the Chebyshev radius of M. A point x0 ∈ X for which M ⊂ B(x0, r(M)) is called a Chebyshev centre of M. (Here and below, B(x, r) is the ball with centre x ˚ r) is the open ball.) and radius r, and B(x, Thus, a Chebyshev centre of a bounded subset of a normed linear space is the centre of a ball of smallest radius containing this set; in other words, a Chebyshev centre is a point in the space that ‘best approximates’ the entire set. The radius of this ball is called the Chebyshev radius of this set. Chebyshev centres are sometimes called best simultaneous approximations. In general, a Chebyshev centre is not unique. By Z(M) we denote the set of all Chebyshev centres1 of a bounded set M. The (set-valued) operator M −→ Z(M)

(15.1)

is called the Chebyshev-centre map or simply the centre map. Example 15.1 The Chebyshev centre of an acute-angled triangle in the Euclidean plane R2 is unique and lies at the centre of the circumscribed circle. The Chebyshev centre of an obtuse-angled triangle in R2 lies at the middle of the largest edge (see Fig. 15.1). Example 15.2 In the plane 2∞ with the max-norm, the set of Chebyshev centre of the set M := {(λ, 0) | |λ| ≤ 1} and that of the set M := {(λ, 0) | |λ| = 1} is the closed interval {(0, μ) | |μ | ≤ 1} (see Fig. 15.2). 1 The use of the letter Z is traditional, and comes from the German Zentrum for centre.

15.1 Chebyshev Centre of a Set

297

Fig. 15.1 The Chebyshev centre of an acute-angled triangle M (on the left) and the Chebyshev centre of an obtuse-angled triangle M (on the right) in the Euclidean plane R2 .

Fig. 15.2 The Chebyshev centre of a two-point set M (on the left) and of a triangle M (on the right) in the plane with the max-norm.

It is clear that the set Z(M) is bounded, closed, and convex (see Proposition 15.1 below); moreover, Z(M) has no interior points. In some practical cases, the Chebyshev centre problem has to be solved under constraints on the centres of the balls under consideration (for example, the centres are frequently supposed to lie in a subspace, or in a more general setting, in a convex set). This leads to the relative Chebyshev centre problem. Throughout this chapter, X is a normed linear space. Definition 15.2 Given a nonempty bounded subset M of X and a nonempty set Y ⊂ X, the quantity rY (M) = inf r(y, M), y ∈Y

where r(x, M) := inf{r ≥ 0 | M ⊂ B(x, r)} = sup x − y , y ∈M

is called the relative Chebyshev radius (of the set M with respect to Y ). The set of relative Chebyshev centres is defined by ZY (M) := {y ∈ Y | r(y, M) = rY (M)}.

(15.2)

Thus, the set of relative Chebyshev centres ZY (M) consists of the points y in Y such that every ball of smallest possible radius rY (M) with centre at y contains M. For Y = X, we get the definitions of a Chebyshev centre and the Chebyshev radius. If M = {y}, then r(x, M) = x − y , rY (M) = ρ(y, Y ) is the distance from y to Y , and the relative-Chebyshev-centre map ZY (M) is the metric projection of the point y on Y .

298

15 ChebyshevCentre of a Set . . .

It is clear that the set of Chebyshev centres ZY (M) is closed in Y . Moreover, rY (M) = rY (M),

ZY (M) = ZY (M).

The self Chebyshev radius of a set M, sc r M (M) := r M := inf r(x, M), x ∈M

(15.3)

is a particular case of the relative Chebyshev radius.2 The set of self centres of M is defined by sc (15.4) Zsc M (M) := Z M := {x ∈ M | r(x, M) = r M (M)}. The following inequalities are straightforward: diam M ≤ 2r(M) ≤ 2 diam M, |r(x, M) − r(y, M)| ≤ x − y ≤ r(x, M) + r(y, M)

(15.5)

for all  M ⊂ X and x, y ∈ X. Next, if cM ∈ Z(M) and c N ∈ Z(N), then cM − c N ≤ r(M) + r(N) + d(M, N),

(15.6)

where M, N ⊂ X and d(M, N) := supx ∈M ρ(x, N) is the deviation of the set M from the set N. It is easily checked that these inequalities are sharp. Further, it can easily be verified that (15.7) r M (M) ≤ diam M ≤ 2 r(M) for every M ⊂ X. In general, the sets Z(M) and Z M (M) can be ‘quite large’: one can construct (see Example 15.3 below) a nonsingleton set M such that diam(Z(M)) = 2 r(M),

diam(Z M (M)) = diam M.

The following properties follow from the continuity and convexity of the distance function y → y − x : r(x, M) = r(x, M) = r(x, convM) = r(x, conv M),

(15.8)

rY (x, M) = rY (x, M) = rY (x, convM) = rY (x, conv M), Z(M) = Z(M) = Z(convM) = Z(conv M), ZY (M) = ZY (M) = ZY (convM) = ZY (conv M),  Π; Z(M) = Z(br(M)), where br(M) =

(15.9)

M ⊂Π Π is a closed span

 here a span is a set N such that [[x, y]] ⊂ N if x, y ∈ N, where [[x, y]] := z ∈  X | min{ϕ(x), ϕ(y)} ≤ ϕ(z) ≤ max{ϕ(x), ϕ(y)} ∀ ϕ ∈ ext S ∗ , and here and below, 2 The superscript ‘sc’ refers to ‘self centre’.

15.1 Chebyshev Centre of a Set

299

ext S ∗ is the set of extreme points of the unit sphere S ∗ of the dual space X ∗ (see Sect. 7.7). Similarly,  Z(M) = Z(m(M)), where m(M) = B(y, r) M ⊂B(y,r)

(m(M) is the Banach–Mazur hull of a set M). In the (relative) Chebyshev centre problem one can thus assume without loss of generality that M is nonempty, closed, and convex. It is easily checked that M ⊂ conv M ⊂ conv M ⊂ br(M) ⊂ m(M);

(15.10)

and moreover, Z(M) = Z(conv M) = Z(convM) = Z(br(M)) = Z(m(M)).

(15.11)

If Y is a subspace, then ZY (M + y) = ZY (M) + y,

y ∈ Y;

ZY (x, αM) = |α|ZY (x, M); if x0 ∈ Y, then {x0 } = ZY (M) ⇐⇒ {0} = ZY (M − x0 ); if α ≥ 0, then {0} = ZY (M) ⇐⇒ {0} = ZY (αM). It is clear that Z(M) =



(15.12) (15.13) (15.14) (15.15)

B(x, r(M))

(15.16)

B(x, rV (M)) ∩ V

(15.17)

x ∈M

and ZV (M) =

 x ∈M

(V is an arbitrary nonempty subset of X). Taking into account that a closed ball is a closed bounded span (see [24], Sect. 9,1) and that an intersection of spans is a span, we find that Z(M) is a closed bounded span,

(15.18)

ZV (M) is the intersection of a closed bounded span with V . From (15.16) it follows that m(Z(M)) = Z(M). From (15.16)–(15.18) we have the following corollary. Proposition 15.1 If Y is a convex set in X and M ⊂ X is a nonempty bounded set, then the set ZY (M) is convex.

300

15 ChebyshevCentre of a Set . . .

The relation between Chebyshev centres and spans will be considered in more detail in Sect. 15.2. Another important and useful generalization of the definition of a Chebyshev centre is the concept of a Chebyshev net of cardinality n (a Chebyshev n-net, a best n-net) and the related concept of the entropy of a set. Definition 15.3 Let n ∈ N. By a net of cardinality n we mean a system Nn = {y1, . . . , yn } of n (not necessarily distinct) points in X; the covering radius of a set M  by a net Nn is defined by R(M, Nn ) := sup min x − yi . x ∈M 1≤i ≤n

By a Chebyshev (or a best) net of cardinality n for a set M we mean a net N∗n = {y1∗, . . . , yn∗ } of cardinality n such that R(M, N∗n ) = inf R(M, Nn ), where the infimum is taken over all possible nets of cardinality n in X. Thus, R = R(M, N∗n ) is the radius of best covering of the set M by n balls of equal radius R. Of course, the Chebyshev centre problem (that is, the problem of covering a set by a ball of smallest possible radius) is a particular case of the Chebyshev n-net problem. The phrase ‘best net of cardinality n’ was used in an oral discussion of Kolmogorov’s 1936 paper [346] on widths. The Chebyshev n-net problem is a more involved problem than the Chebyshev centre problem even in the Euclidean setting — cf. the well-known spherical design problem (the problem of best distribution of a finite number of points on a sphere) [611], [610]. In contrast to the Chebyshev centre problem, a Chebyshev n-net may not be unique even in a Euclidean space; moreover, some points of this net may lie outside the closed convex hull of the set under consideration. For some sets, best n-nets can be found analytically for small n (see [577]). Note that if a sequence (Mn ) of nonempty sets converges in the Hausdorff metric to a set M and if Rn is the best covering radius of Mn , n = 1, 2, . . . , then lim Rn = R, where R is the radius of best covering of M. Another extension of the definition of a Chebyshev centre is the concept of a Chebyshev point (Beloborov [89]). Definition 15.4 Given a normed linear space X, consider a system of sets {Gα | α ∈ A} for which the quantity R(y) := supα∈ A ρ(y, Gα ) is finite for all y ∈ X. A point y ∗ ∈ X for which R(y ∗ ) := inf y ∈X R(y) is called a Chebyshev point for this system of sets. A Chebyshev centre of a set can be looked upon as a Chebyshev point of a system of singletons. Further, if x ∗ ∈ X ∗ , x ∗  0, c ∈ R, and H := (x ∗ )−1 (c) = {x ∈ X | x ∗ (x) = c} is a closed hyperplane, then for all x ∈ X, the distance from x to H

15.1 Chebyshev Centre of a Set

301

is given by the well-known formula ρ(x, H) = |x ∗ (x) − c|/ x ∗ (see, for example, Sect. 1.3). This observation shows that the concept of a Chebyshev point of a system of sets is an extension of the concept of a Chebyshev point of an (inconsistent) system of linear equations (see Zukhovitskii [632]). Belobrov [89], [92] solved some problems on the existence of a Chebyshev point of a system of hyperplanes of arbitrary cardinality in a finite-dimensional space and on the uniqueness of a Chebyshev point for a system of balls. Below in this chapter we shall be mainly concerned with the Chebyshev centre problem. Klee [341] and, independently, Garkavi [258] proved that (some) Chebyshev centre of every bounded subset M of a space X lies in the closed convex hull of the subset if and only if X either is a Hilbert space or has dimension at most two. Garkavi showed that this result also holds if one considers only three-point sets. For every closed convex bounded subset of a Hilbert space, a Chebyshev centre exists, is unique, and lies in that subset (Theorems 15.10 and 15.16 below). Garkavi [256] showed that each bounded subset of a Banach space X has at most one Chebyshev centre if and only if X is uniformly convex (or uniformly rotund) in every direction (that is, for all z ∈ X and ε > 0, there exists δ = δ(z, ε) > 0 such that if x1 = x2 = 1, x1 − x2 = λz, and x1 + x2 > 2 − δ, then |λ| < ε). Belobrov [91] extended this condition by considering spaces in which the set of Chebyshev centres of every set has dimension at most r < ∞. Let us dwell on these results in more detail. The case of the space Rn will be considered separately. Remark 15.1 In the finite-dimensional Euclidean space Rn , a Chebyshev centre (of a bounded nonempty set) exists, is unique, and lies in the convex hull of the set (see Theorem 15.10 below). Theorem √ 15.1 The Chebyshev radius of a set M ⊂ Rn of diameter  2 is bounded above by 2n/(n + 1). This estimate is attained if and only if the closure of M contains the vertices of a regular n-dimensional simplex with edge length 2. Proof (of Theorem 15.1) Without loss of generality, it can be assumed that the set M is closed (see (15.8)). By Helly’s classical theorem (see, for example, Appendix B), if for every set of n + 1 points in M, the balls of radius r = r(M) with centres at these points have nonempty intersection, then the balls of the same radius with centres at all the points of M also have nonempty intersection. It thus suffices to consider the case in √ which M consists of at most n + 1 points. Let us show that r = r(M)  R := 2n/(n + 1). In Rn , the (unique) Chebyshev centre y lies in the convex hull of M. In addition, we claim that the point y lies in the convex hull of the set M0 := {x ∈ M | x − y = r } = M ∩ S(y, r). If this were not so, then it would be possible to strictly separate the point y and the set M0 by a closed hyperplane. We drop the perpendicular [y, y0 ] to this hyperplane. The distance from a point z ∈ (y, y0 ] to the set M0 is less than r. Moreover, the distance from y to every point in the set M \ M0 is also less than r. Hence the interval (y, y0 ) has a point z at a distance less than r from M. But this contradicts the choice of r. By Carathéodory’s theorem m ⊂ M such that x − y = r (m  n), (see Appendix B), there exist points {xi }i=0 0 i

302

15 ChebyshevCentre of a Set . . .

m ∈ R such that m α = 1 and m α x = y. Note and there exist numbers {αi }i=0 + i=0 i i=0 i i that ai j = xi − x j  2 . We can assume without loss of generality that

y=0=

m 

αi xi .

(15.19)

i=0

By the law of cosines, ai2j = 2r 2 − 2(xi, x j ).

(15.20)

Hence for each j, 1 − αj =

 ij

αi 

m α a2  i ij i=0

4

(15.20)

=

m r2 1  − αi xi, x j 2 2 i=0

(15.19)

=

r2 . 2

m Summing these equalities over j = 0, . . . , m and using i=0 αi = 1, we obtain



(m+1)r 2 2m 2n , and hence r  m+1  n+1 = R. m 2 m form The case r = R implies that m = n and ai j = 2, and so the points {xi }i=0 the vertices of a regular n-dimensional simplex with edge length 2. Theorem 15.1 is proved.  Corollary 15.1 The Jung constant of the space Rn is  n J(Rn ) = . 2(n + 1)

15.2 Chebyshev Centres and Spans We recall (see Sect. 7.7.2) the definition of a segment [[x, y]] and a span in a normed linear space X: A segment [[x, y]] in a normed linear space X is defined as (see Sect. 7.7.2 and [24])   [[x, y]] := z ∈ X | min{ϕ(x), ϕ(y)} ≤ ϕ(z) ≤ max{ϕ(x), ϕ(y)} ∀ ϕ ∈ ext S ∗   = z | ϕ(z) ∈ [ϕ(x), ϕ(y)] , (15.21) where ext S ∗ is the set of extreme points of the unit sphere S ∗ of the dual space X ∗ . In fact, if a subset A ⊂ ext S ∗ is such that A ∪ (−A ) = ext S ∗ , then   [[x, y]] = z ∈ X | f (z) ∈ [ f (x), f (y)] ∀ f ∈ A . For example, for X = C(Q), it is convenient to take the point evaluation functionals (ϕ → ϕ(t)) as A . In this case, for all ϕ, ψ ∈ C(Q), we have   [[ϕ, ψ]] = g ∈ C(Q) | g(t) ∈ [ϕ(t), ψ(t)] ∀t ∈ Q .

15.2 Chebyshev Centres and Spans

303

With a set A ⊂ ext S ∗ for which A ∪ (−A ) = ext S ∗ , one can associate the linear map c : X → c(X) that takes every x ∈ X to the function c(x) : A → R defined by x(x ∗ ) = x ∗ (x) for x ∗ ∈ A . Thus c(X) is the space of continuous functions on A that are the restrictions to A of the continuous linear functionals from X acting on X ∗ . We equip c(X) with the uniform norm: c(x) = sup |x(x ∗ )|. x ∗ ∈A

As a result, the map c : X → c(X) is an isometry. It will be convenient to identify a point x with its image c(x), and a set E ⊂ X with c(E). Moreover, the ball B(x, R) is identified with the set b(x, R) := c(B(x, R)). Since B(x, R) := {y ∈ X | |x ∗ (y − x)| ≤ R for all x ∗ ∈ ext S ∗ }, we have b(x, R) = [[x − R, x + R]] = {y ∈ X | x(x ∗ ) − R ≤ y(x ∗ ) ≤ x(x ∗ ) + R, x ∗ ∈ ext S ∗ } = {y ∈ X | x(x ∗ ) − R ≤ y(x ∗ ) ≤ x(x ∗ ) + R, x ∗ ∈ A } = {c(y) | c(x) − R ≤ c(y) ≤ c(x) + R}. The set m(E) := E ⊂B(x,R) B(x, R) (the Banach–Mazur hull of the set E) is identified with the set  b(x, R). m(c(E)) = c(E)⊂b(x,R)

Recall that a set E with  E ⊂ X is called a span (see [24]) if [[x, y]] ⊂ E

for all x, y ∈ E.

It is clear that every closed ball is a closed span. Definition 15.5 Recall that a function f : X → R ∪ {+∞} is lower semicontinuous if it satisfies any one of the following equivalent conditions: (a) the set {x ∈ X | f (x) ≤ α} is closed in X for all α ∈ R; (b) the set {x ∈ X | f (x) > α} is open in X for all α ∈ R; (c) the epigraph epi f of f is closed in X × R. A map F : X → 2Y is said to be lower semicontinuous at a point x0 if for every neighbourhood O(y) of every y ∈ F(x0 ), there exists a neighbourhood O(x0 ) such that F(x) ∩ O(y)  for every point x ∈ O(x0 ). As usual, F is lower semicontinuous on X if it is lower semicontinuous at every point x0 ∈ X. Remark 15.2 It is well known that if f (t0 ) is finite, then a function f is lower semicontinuous at the point t0 if and only if for all ε > 0, there exists δ > 0 such that f (t0 ) − ε < f (t) if |t − t0 | < δ, t ∈ [a, b]. Remark 15.3 Vasil’eva (see [581], [582]) showed that a set Π ⊂ C(Q), where Q is a compact Hausdorff space, is a nonempty closed span if and only if Π can be written as a generalized segment

304

15 ChebyshevCentre of a Set . . .

  [[ f1, f2 ]] = f ∈ C(Q) | f (t) ∈ [ f1 (t), f2 (t)] ∀t ∈ Q ,

(15.22)

where f1, f2 : Q → R, f1 ≤ f2 , f1 is upper semicontinuous on Q, and f2 is lower semicontinuous (in the definition of [[ f1, f2 ]], the functions f1 and f2 need not lie in C(Q)). By the Katětov–Tong separation theorem for semicontinuous functions (see, for example, [252]), the set [[ f1, f2 ]] is nonempty in C(Q). Note that the generalized segment [[ f1, f2 ]] is a singleton if and only if f1 = f2 (in this case, f1 and f2 are continuous functions). Vasil’eva also showed that the metric projection onto a closed span has a continuous 1-Lipschitz selection if and only if the span is a segment [[ f1, f2 ]] with f1, f2 ∈ C(Q). For some properties of generalized segments, see also [241]. Let V  and let r := rV (M) be the relative Chebyshev radius of a set M with respect to V. On A we set m∗ ( · ) = sup y( · ) y ∈M

and

m∗ ( · ) = inf y( · ). y ∈M

(15.23)

Here y is regarded as c(y) (that is, as a function on A ), and the functions m∗ ( · ) and m∗ ( · ) are lower and upper semicontinuous, respectively. Therefore, Π := [[m∗ − r, m∗ + r]] is a generalized segment. Given x ∗ ∈ A , consider the strip   Πx ∗ := x ∈ X | m∗ (x ∗ ) − r ≤ x(x ∗ ) ≤ m∗ (x ∗ ) + r . It is easily seen that Πx ∗ consists precisely of the points x ∈ X such that M ⊂ {u ∈ X | |x ∗ (u − x)| ≤ r } =: Πx ∗ (x). Since x ∗ ∈A Πx ∗ (x) = B(x, r), the following conditions are equivalent: (a) v ∈ ZV (M); ; (b) v ∈ V ∩ Πx ∗ for every x ∗ ∈ A (c) v ∈ V ∩ Π, where Π = Πr := x ∗ ∈A Πx ∗ . So ZV (M) = V ∩ Π.

(15.24)

From (15.24), it follows in particular that the set ZV (M) of relative Chebyshev centres is convex for a convex set V ⊂ X (see Proposition 15.1). It is also clear that diam M, diam Π ≤ 2r(M). The set of Chebyshev centres of a bounded set  M ⊂ X forms the closed segment   Π := x ∈ X | m∗ ( · ) − r ≤ x( · ) ≤ m∗ ( · ) + r , (15.25) where r = r(M) is the Chebyshev radius of M. Consider the functions N( · ) :=

inf

x ∈X: m∗ −r ≤x

x( · )

and

n( · ) :=

sup

x ∈X: x ≤m∗ +r

x( · ).

(15.26)

15.2 Chebyshev Centres and Spans

305

These functions are upper and lower semicontinuous, respectively. By the construction, m∗ ( · ) − r ≤ N( · ) and n( · ) ≤ m∗ ( · ) + r. Now from (15.25) and (15.26) we have the equality of generalized segments (cf. (15.22)): Π = [[m∗ ( · ) − r, m∗ ( · ) + r]] = [[ N( · ), n( · )]]. If X is a Banach space such that c(X) contains the constants (for example, X = C(Q)), then replacing x in (15.26) by y − r and y + r, we get that N( · ) = N( · ) − r,

n( · ) = n( · ) + r,

where N( · ) = inf{y( · ) | m∗ ( · ) ≤ y( · )}, n( · ) = sup{y( · ) | m∗ ( · ) ≥ y( · )}. Hence if X is such that c(X) contains the constants, then Π = [[N( · ) − r, n( · ) + r]].

(15.27)

Let B be the set of bars (see [24], Sect. 8.4) of the form Π = (x ∗ )−1 [a, b], where −∞  a  b  +∞. Note that the ‘bar hull’  br(M) := {Π ∈ B | Π ⊃ M } coincides with the generalized segment [[m∗ ( · ), m∗ ( · )]]. We also note that in the space C(Q),  br(M) = br(M) := {Π | Π ⊃ M, Π a closed span}. Recall that the hull of the set b(x, R) coincides with the generalized segment [[x( · ) − R, x( · ) + R]], and moreover,  m(M) = m(br(M)) = b(x, R). br(M)⊂b(x,R)

Hence for all x ∈ X and R such that br(M) ⊂ b(x, R), we have m∗ ( · ) ≤ x( · ) + R,

m∗ ( · ) ≥ x( · ) − R,

and moreover, if X is a Banach space such that c(X) contains the constants, then [[n( · ), N( · )]] ⊂ b(x, R)

provided that br(M) ⊂ b(x, R).

306

15 ChebyshevCentre of a Set . . .

Therefore, [[n( · ), N( · )]] ⊂ m(br(M)) = m(M). In the case of a topological Q, by the space C(Q) we mean the space of continuous bounded functions on Q with the norm f = supt ∈Q | f (t)|. The following result was established by Tsar’kov [567], [25]. Theorem 15.2 Let X = C(Q), where Q is a normal topological space, and let M be a nonempty bounded subset of X with a unique Chebyshev centre. Then m(M) = B(z, r), where z is the Chebyshev centre of M. Moreover, z = (N( · ) + n( · ))/2. Proof (of Theorem 15.2) Let r = r(M). We have {z} = [[N( · ) − r, n( · ) + r]]. Therefore,  1 N( · ) − r + n( · ) + r = N( · ) − r = n( · ) + r. z= 2 Since the function N( · ) is upper semicontinuous and n( · ) is lower semicontinuous, the equality N( · ) = n( · ) + 2r implies that N( · ) and n( · ) are continuous functions. Hence [[n( · ), N( · )]] is a segment, which is the ball B(z, r). Hence B(z, r) = [[n( · ), N( · )]] ⊂ m(M) ⊂ B(z, r). 

The theorem is proved.

15.3 Chebyshev Centre in the Space C(Q) In this section, we recall and prove some results on the existence of Chebyshev centres in the space C(Q). Many results here can be proved using the results in Sect. 15.2 on representation of the set of Chebyshev centres Z( · ) as a closed span. Let M be a bounded subset of C[a, b]. Consider two functions m∗ (t) := inf x(t), x ∈M

m∗ (t) := sup x(t),

(15.28)

N(t) := lim m∗ (τ) ∈ R.

(15.29)

x ∈M

and define the functions n(t) := lim m∗ (τ), τ→t

τ→t

We claim that the functions m∗ (t) and n(t) are lower semicontinuous, and N(t) and m∗ (t) are upper semicontinuous. We use Remark 15.2. There exists a point x0 ∈ M such that x0 (t0 ) > m∗ (t0 ) − ε/2 and there exists δ > 0 such that |x0 (t) − x0 (t0 )| < ε/2 for all t, |t − t0 | < δ. So m∗ (t) ≥ x0 (t) ≥ x0 (t0 ) −

ε > m∗ (t0 ) − ε. 2

15.3 Chebyshev Centre in the Space C(Q)

307

Let us show that n(t) is lower semicontinuous. We need to show that lim n(τ) ≥ n(t).

(15.30)

τ→t

By definition of the lower limit, there exists a sequence (τk ) from [a, b], τk → t, such that lim n(τ) = lim n(τk ) = lim k→∞

τ→t

lim m∗ (w)

k→∞ w k →τk l

= lim lim m∗ (wlk ). k→∞ l→∞

Then there exists a subsequence k(l) such that wlk(l) → t,

l → ∞,

which proves (15.30). Note that for all x ∈ M, n(t) ≤ x(t) ≤ N(t),

t ∈ [a, b].

(15.31)

Consider the difference N(t) − n(t). Since it is upper semicontinuous, there exists a point t0 ∈ [a, b] at which this difference attains its maximum value: max{N(t) − n(t)} = N(t0 ) − n(t0 ) = 2 r ≥ 0. t

(15.32)

Remark 15.4 The function N(t) − n(t) is upper semicontinuous. As a corollary, there exists a point t0 ∈ [a, b] such that max{N(t) − n(t)} = N(t0 ) − n(t0 ) =: 2 r ≥ 0. t

(15.33)

Remark 15.5 If M is a compact set, then since a compact set in C[a, b] is equicontinuous, it follows that the regularizations m∗ (t), m∗ (t), t ∈ [a, b], are continuous and equal, respectively, to N(t) and n(t). We need two auxiliary lemmas. Lemma 15.1 For every function z ∈ C[a, b], sup x − z ≥ r,

x ∈M

(15.34)

where r is defined by (15.32). Proof (of Lemma 15.1) Assume that equality (15.33) holds for a point t0 ∈ [a, b]. Given ε > 0, consider the neighbourhood (t0 −δ, t0 +δ) of t0 so small that the variation of the function z(t) in this neighbourhood is smaller than ε/2. By inequality (15.32), N(t0 ) − z(t0 ) ≥ r

or

z(t0 ) − n(t0 ) ≥ r.

(15.35)

Assume without loss of generality that the first inequality of (15.35) is satisfied. Since the function m∗ (t) is lower semicontinuous, from the definition of N(t) (see

308

15 ChebyshevCentre of a Set . . .

(15.29)) we have that in every neighbourhood of an arbitrary point of the graph of N(t) there exists a point of the graph of m∗ (t), and hence by (15.28), in every neighbourhood of an arbitrary point of the graph of the function N(t) there exists a point of the graph of some function x ∈ M. This means that (for such x ∈ M) there exists a point t1 ∈ [a, b] such that |t1 − t0 | < δ

and

|x(t1 ) − N(t0 )|
0, y(0) = 0, t = 0, −1 ≤ y(t) ≤ 0, t < 0. 

It is clear that

0, m (t) = 1, ∗

t ≤ 0, t > 0,



Hence

0, N (t) = 1,

t < 0, t ≥ 0,

 m∗ (t) =  n(t) =

−1, 0,

−1, 0,

t < 0, t ≥ 0. t ≤ 0, t > 0.

We have 2 r = sup N (t) − n(t) = 2 = N (0) − n(0), t ∈[a, b]

which shows that the Chebyshev radius r M of the set M is 1. The functions n(t)+r M and N (t)−r M coincide with m∗ (t) and m∗ (t), respectively, and now Theorem 15.4 gives us that the set of all Chebyshev centres of M coincides with M itself. Hence the set M thus constructed provides an example of a nontrivial set coinciding with the set Z(M) of its Chebyshev centres

From Theorem 15.4 and (15.27) we have the following. Theorem 15.5 Let  M ⊂ C(Q) be a bounded set and let Q be a topological space. Then Z(M) is a closed span. (15.39) For compact sets M, Theorem 15.4 assumes the following more manageable form. Theorem 15.6 Let M be a compact set in C(Q), where Q is a topological space, and let r = r(M) be the Chebyshev radius of M. Then the set Z(M) of all Chebyshev centres of M can be written as   Z(M) = y ∈ C(Q) | m∗ (t) − r ≤ y(t) ≤ m∗ (t) + r  , where

m∗ (t) = max x(t), m∗ (t) = min x(t), r = m∗ − m∗ /2. x ∈M

In particular,

x ∈M

Z(M) = Z(m∗, m∗ ) := Z({m∗, m∗ }).

310

15 ChebyshevCentre of a Set . . .

Corollary 15.3 In the space C(Q), where Q is a topological space, the Chebyshevcentre map Z( · ) has a 1-Lipschitz selection on the class of nonempty compact sets M ⊂ C(Q) with respect to the Hausdorff metric. As a 1-Lipschitz selection in Corollary 15.3 one can consider the map x → (m∗ ( · ) + m∗ ( · ))/2 (see Sect. 15.2, and also [334]). Remark 15.7 For vector-valued analogues of Theorems 15.4–15.6, see [597], [28], [622], and also Theorem 15.41 below. For example, Ward [597] established that if Q is a paracompact Hausdorff space and X is a finite-dimensional space, then in C(Q, X), every bounded set has a Chebyshev centre. A similar result also holds in the space C(Q, H), where Q is a normal space and H is a Hilbert space. Amir [28] (and independently Ka-Sing Lau) extended Ward’s results to the case that X is a uniformly convex space, Q is an arbitrary topological space, and C(Q, X) is the space of bounded continuous functions on Q with values in X (see Theorem 15.41). Zamyatin and Shishkin [622] extended this result by showing that a Chebyshev centre exists for every bounded subset of C(Q, X), where Q is an arbitrary topological space and X is a KB-linear space of bounded elements (for the definition of a KB-linear space, see [382]). Some other generalizations may also be found in [497], [486], [167].

15.4 Existence of a Chebyshev Centre in Normed Spaces In this section, we formulate more results on the existence of Chebyshev centres in normed linear spaces (see, for example, [256, p. 102], [293, pp. 184–187], [36], [35], [356], [586], [587], [588], [383], [163, Chap. 27]). Theorem 15.7 (A. L. Garkavi) Suppose that the image of a space X under the canonical embedding in the second dual X ∗∗ is norm-1 complemented (that is, there exists a norm-1 projection of X ∗∗ onto X). Then every nonempty bounded subset of X has a Chebyshev centre. In particular, the hypotheses of Theorem 15.7 are satisfied for the L 1 -spaces and for the dual spaces (the first dual X ∗ is always 1-complemented in the third dual X ∗∗∗ ; see Proposition 15.2 below). An analogue of Theorem 15.7 and its corollary also hold for Chebyshev n-nets (see [256, Sect. 2]). Let us consider in more detail the important property of dual spaces related to the Chebyshev centre problem. A set M ⊂ X can be regarded as a subset of the second dual X ∗∗ . However, the radius of a smallest ball for M in the space X ∗∗ can be smaller than r(M) = inf y ∈X supx ∈M x − y . Nevertheless, if X is a dual space, then these quantities are equal for every M ⊂ X. Remark 15.8 The condition of Theorem 15.7 is not necessary for each bounded subset of the space to have a Chebyshev centre. For example, Garkavi [256, Sect. 2]

15.4 Existence of a Chebyshev Centre in Normed Spaces

311

showed that in the space c0 , every bounded set has a Chebyshev centre. However, it is well known that c0 is not even complemented in the second dual: there is not even a bounded projection from  ∞ onto c0 (Phillips–Sobczyk theorem, for example; see [4, Theorem 2.5.5]). The spaces c and C[0, 1] are also not complemented in their second duals. The space  ∞ is complemented in every space that contains it. We need the following auxiliary result. Proposition 15.2 Let X be a normed linear space. Then X ∗ is complemented in X ∗∗∗ . Proof (of Proposition 15.2) Let j : X → X ∗∗ be the canonical embedding defined by (15.40) ( j x, x ∗ ) = (x ∗, x). Note that j is a linear isometry. Indeed, j(αx + βy)(x ∗ ) = x ∗ (αx + βy) = αx ∗ (x) + βx ∗ (x) = [α j(x) + β j(x)](x ∗ ). (15.41) Further, for x ∈ X, we have j(x) = sup |x ∗ (x)| ≤ sup x ∗ · x ≤ x x ∗ ∈B∗

x ∗ ∈B∗

(recall that B∗ is the unit ball of the dual space). Let a functional x0∗ ∈ S ∗ be such that x0∗ (x) = x . Hence by (15.41), j(x) ≥ |x0∗ (x)| = x . As a corollary, j(x) = x . Under this natural embedding, x ∈ X is identified with its image j x ∈ X ∗∗ ; we also identify X and j X. In particular, for x ∈ X, x ∗ ∈ X ∗ , we have x(x ∗ ) = x ∗ (x). We similarly define (15.42) (J x ∗, x ∗∗ ) = (x ∗∗, x ∗ ). Consider the operator j ∗ : X ∗∗∗ → X ∗ defined by ( j ∗ x ∗∗∗, x) = (x ∗∗∗, j x). We have ( j ∗ J x ∗, x)

(15.43)

=

(J x ∗, j x)

(15.42)

=

( j x, x ∗ )

(15.43) (15.40)

=

(x ∗, x),

that is, j ∗ J acts identically on X ∗ . Since X ∗ is identified with J X ∗ , j ∗ acts identically on X ∗ . Now j ∗ = 1, and so j ∗ is the required norm-1 projection from X ∗∗∗ onto X ∗ .  Proof (of Theorem 15.7) Assume first that we know that in every dual space X (X = Y ∗ ), a nonempty bounded set has a Chebyshev centre. We claim that every nonempty bounded set M ⊂ X admits a Chebyshev centre in X ∗∗ . Let us show that the image of this centre under the norm-1 projection j ∗ from X ∗∗ onto X (the projection exists by Proposition 15.2) is a Chebyshev centre of the set M. Indeed, we have r( j M) :=

inf

sup x ∗∗ − y ∗∗ ≤ inf sup j x − y ∗∗ ≤ r(M).

x ∗∗ ∈X ∗∗ y ∗∗ ∈ j M

x ∈X y ∗∗ ∈ j M

(15.44)

312

15 ChebyshevCentre of a Set . . .

Let z ∗∗ be a Chebyshev centre from X ∗∗ for the set j M. By definition of a Chebyshev centre, z ∗∗ − y ∗∗ ≤ r( j M) for every y ∗∗ ∈ j M. Let z := j ∗ (z ∗∗ ). We have j ∗ = 1, and so for every y ∈ M, z − y = j ∗ (z ∗∗ − y ∗∗ ) ≤ z ∗∗ − y ∗∗ ≤ r( j M)

(15.44)

≤

r(M),

that is, z is a Chebyshev centre for M. So it remains to show that a nonempty bounded subset M of a dual space X has a Chebyshev centre. Given n ∈ N, there exists a sequence (xn ) ⊂ X such that 1 sup xn − y ≤ r(M) + . n

y ∈M

(15.45)

Let x0 be its w ∗ -limit. The norm · is w ∗ -continuous (if a sequence (zn ) weakly converges to a point z, then z ≤ limn→∞ zn ), and hence by (15.45), we have  x0 − y ≤ r(M) for all y ∈ M; that is, x0 is a Chebyshev centre of the set M. We mention the following simple result. Theorem 15.8 Suppose that every nonempty bounded set in a space X has a Chebyshev centre. Then in every 1-complemented affine subspace L ⊂ X, every bounded set has a Chebyshev centre. Proof Indeed, let π be the norm-1 projection from X onto L. Then for every nonempty bounded set M ⊂ L and its Chebyshev centre x ∈ X, the point π(x) is a Chebyshev centre for M. This follows from the inequality y − π(x) = π(y − x) ≤ y − x , which holds for every y ∈ M.



The Kolmogorov width of a nonempty bounded set M is defined by dn (M, X) =

inf

L ∈Aff n (X)

d(M, L),

where Aff n (X) is the class of all affine subspaces of dimension  n in the space X, and d(M, L) = sup ρ(x, L) x ∈M

is the deviation of M from an affine subspace L. For n = 0, the Kolmogorov width problem becomes the Chebyshev centre problem. For more details on widths, see Chap. 16. In the study of widths it is not only the values or order estimates for the widths (quantitative characteristics of the approximation) that are important, but also the concrete form of extreme subspaces. (A subspace L0 ∈ Aff n (X) is said to be extreme (or best) if d(M, L0 ) = dn (M, X).)

15.4 Existence of a Chebyshev Centre in Normed Spaces

313

Theorem 15.9 (A. L. Garkavi) Assume that the image of a space X under the canonical embedding in the second dual X ∗∗ is 1-complemented. Then for every nonempty bounded set M ⊂ X and n ∈ N, there exist a Chebyshev net of cardinality n and an extreme subspace L0 ∈ Aff n (X). In particular, this is true for X = Y ∗ . Proof (of Theorem 15.9) Let   n R := inf r > 0 | ∃ an r-net {xi }i=1 for M ,

X n :=

n 

X.

k=1

On the space X n , consider the norm (x1, . . . , xn ) n :=

n 

xk 2

 12

.

k=1

n We have (X n )∗∗ = k=1 X ∗∗ . Consider a family xi = (x1i , . . . , xni ) that is an (R + n1 )net for M. Arguing as in the proof of Theorem 15.7, we see that there exists a limit point (family) x∗∗ = (x1∗∗, . . . , xn∗∗ ) ∈ (X n )∗∗ (with respect to the w ∗ -star topology in (X n )∗ ) that is an R-net of the set M in the space X ∗∗ . Hence the family x = (P(x1∗∗ ), . . . , P(xn∗∗ )) is an R-net of the set M in the space X. Now let Lk ∈ Aff n (X) be planes such that d(M, Lk ) < dn (M, X) + k1 =: αk (k ∈ N). It can be assumed without loss of generality that the dimension of each plane Lk is n. Let xk = (x0k , x1k , . . . , xnk ) be an orthogonal basis for Lk (k ∈ N) whose origin (the point x0k ) is taken from the αk -neighbourhood of M and the basis vectors (x1k , . . . , xnk ) have unit length. There exists a limit point (family) x∗∗ = (x0∗∗, x1∗∗, . . . , xn∗∗ ) ∈ (X n+1 )∗∗ that defines the plane L∗∗ ⊂ X ∗∗ passing through the point x0∗∗ and is parallel to the vectors from the family (x1∗∗, . . . , xn∗∗ ). It is easily checked that d(M, L) = d(M, L∗∗ )  dn (M, X), where L = P(L∗∗ ) ∈ Aff n (X). The theorem is proved.  Theorem 15.10 (A. L. Garkavi) A Chebyshev centre of a nonempty bounded subset of a Hilbert space is unique and lies in the closure of the convex hull of this set. Proof (of Theorem 15.10) Assume that some bounded set M has a Chebyshev centre x0 not lying in K := conv M. By the geometric form of the Hahn–Banach theorem, there exists a functional x ∗ ∈ S ∗ strictly separating the point x0 and the set K. Without loss of generality, we can assume that x ∗ (x0 ) < 0 < inf x ∗ (z). We z ∈K

set L = Ker x ∗ and let π be a norm-1 projection onto the hyperplane L, x 0 = π(x0 ). For an arbitrary point z ∈ M, consider the point y := L ∩ [x0, z]. We have y − x0  y − x 0 = π(y − x0 ) , z − x 0  z − y + y − x 0  z − y + y − x0 = z − x0 . Hence x 0 is another Chebyshev centre of M, which contradicts the uniqueness of a Chebyshev centre in a Hilbert space (see Theorem 15.26 below). 

314

15 ChebyshevCentre of a Set . . .

Theorem 15.11 (A. L. Garkavi) In every two-dimensional space, each bounded nonempty set has a Chebyshev centre lying in the closure of its convex hull. Moreover, if a two-dimensional space is strictly convex, then such a centre is unique and lies in the closure of the convex hull of this set. Proof For a strictly convex space, we argue as in the proof of Theorem 15.10 and use the fact that in every two-dimensional space, for each one-dimensional subspace  there exists a norm-1 projection onto this line (this is the projection along a support line to the unit sphere S at one of the two points of intersection of  and S). The second assertion follows from the first one after renormalization · n := · + n1 ·  (n ∈ N) and making n → ∞, where ·  is some Euclidean norm on the plane.  The following two results are auxiliary (see, for example, [181], [490]). Theorem 15.12 (M. Fréchet) A Banach space of dimension ≥ 3 is a Hilbert space if and only if all of its three-dimensional subspaces are Euclidean. Theorem 15.13 (W. Blaschke–S. Kakutani) A three-dimensional space X is a Hilbert space if and only if there exists a norm-1 projection of X onto each two-dimensional subspace. Garkavi [258] obtained the following characterization of Hilbert spaces in terms of the existence of Chebyshev centres of three-point subsets. Theorem 15.14 (A. L. Garkavi) Let X be a Banach space of dimension at least three. If every set of three points in X admit a Chebyshev centre lying in their affine hull, then X is a Hilbert space. Proof By Fréchet’s theorem, it suffices to consider the case that X is threedimensional. Let L be an arbitrary two-dimensional subspace of X and let z0  L. We fix a sufficiently large n  n0 so that the sets Dn = {x ∈ L | z0 − x  n}

and

Γn = {x ∈ L | z0 − x = n}

are nonempty. Let y1 , y2 , y3 be points from Γn . By the assumption, there exists a Chebyshev centre x  for these three points lying in the subspace L. It is clear that y1 − x   yi − z0 = n,

i = 1, 2, 3.

(15.46) 3

Let S(y) = {x ∈ L | y − x ≤ n}. By (15.46), the intersection i=1 S(yi ) is nonempty for every choice of y1 , y2 , y3 from Γn . Further, since for y ∈ Γn , S(y) is a closed bounded subset of a two-dimensional subspace, by Helli’s theorem (see Appendix B) the set Sn = {S(y) | y ∈ Γn } is nonempty. Let xn ∈ Sn . Then y − x n  y − z0 = n

(15.47)

x − x n  x − z0

(15.48)

for all y ∈ Γn . We claim that

15.4 Existence of a Chebyshev Centre in Normed Spaces

315

for all x ∈ Dn \Γn . The set Dn \Γn is a convex two-dimensional set with boundary Γn . Therefore, for every point x ∈ Dn \ Γn , x  xn , and for every point xn in the plane L, the interval [xn, x] can be extended to its intersection with the contour Γn in such a way that the point x will be an interior point of the interval [xn, y], where y ∈ Γn . But in this case, xn − x = λ(xn − y), where λ ∈ (0, 1). Hence z0 − x = (z0 − xn ) + (xn − x) = z0 − xn + λ(xn − y), and so inequality (15.48) can be written as (15.49) (z0 − xn ) + λ(xn − y)  λ xn − y . Assuming that (z0 − xn ) + λ(xn − y) < λ xn − y , we have z0 − y = (z0 − xn ) + λ(xn − y) + (1 − λ)(xn − y)  (z0 − xn ) + λ(xn − y) + (1 − λ)(xn − y) < λ xn − y + (1 − λ) xn − y = xn − y , which contradicts (15.47). So inequality (15.48) holds for all x ∈ Dn . Consider now a sequence of increasing sets {Dn }nn0 . For each Dn , there exists a point xn ∈ L such that x − xn  x − z0 for all x ∈ Dn . If as x we take some fixed point from Dn0 , then xn ∈ B(x, x − z0 ); that is, the sequence (xn ) is bounded,  and hence (xn ) has a cluster point x 0 ∈ L. Moreover, L = ∞ n=n0 Dn , and hence x − x 0  x − z0

(15.50)

for all x ∈ L. We now represent each point z ∈ X in the form z = az0 + x (x ∈ L). The operator P(z) = P(az0 + x) := ax 0 + x projects the space X onto L. Moreover, by (15.50),   x  x    P(z) = ax 0 + x = |a| x 0 +   |a| z0 +  = az0 + x = z . a a Therefore, P = 1. Now by Theorem 15.13 (Blaschke–Kakutani) we conclude that X is a Hilbert space.  The proof of Theorem 15.14 shows that it can be strengthened as follows. Theorem 15.15 (A. L. Garkavi) Let X be a Banach space and let dim X ≥ 3. If every set of three points of the unit sphere of X can be covered by a ball of radius 1 with centre in their affine hull, then X is a Hilbert space. In [35, Corollary 2.9] it was established that if X is a Hilbert space, Y ⊂ X is a convex closed set, and K ⊂ X is a convex compact set, then ZY (K) ⊂ PY (K). For a closed convex set K and Y ⊂ X, this conclusion ceases to hold. A corresponding example was constructed by Benítez (cf. [59]): let X =  2 , let (en ) be the standard basis for  2 , and let Y = span n−1 en and K = conv {n(n + 1)−1 en | n ∈ N}. Then ZY (K) = {0},

but

0  PY K.

The following result is a direct consequence of Theorems 15.9–15.14.

316

15 ChebyshevCentre of a Set . . .

Theorem 15.16 (A. L. Garkavi) A necessary and sufficient condition that each nonempty bounded subset of a Banach space have a Chebyshev centre lying in the closure of its convex hull is that the space be a Hilbert space or have dimension at most two. In the two-dimensional case, Theorem 15.16 (see (15.51) below) can be refined as follows. Proposition 15.3 If every Chebyshev centre of two arbitrarily chosen points in a Banach space X (of any dimension) lies on the line passing through these points, then X is a strictly convex (rotund) space. Indeed, if the unit sphere S contains a nondegenerate interval [a, b] (and hence the interval [−a, −b]), then the interval [(a − b)/2, (b − a)/2] consists of Chebyshev centres of the two-point set M := {(a+b)/2, −(b+a)/2}; that is, the set of Chebyshev centres of the set M does not lie on the line connecting the points (a + b)/2 and −(b + a)/2. Thus from Theorem 15.11 and Proposition 15.3, it follows that a two-dimensional space X is strictly convex if and only if every Chebyshev centre of every bounded set  M ⊂ X lies in its convex hull. (15.51) Given ε > 0 and x ∗ ∈ S ∗ , we set A(x ∗, ε) := {x ∈ S | x ∗ (x)  1 − ε} and denote by R(x ∗, ε) the Chebyshev radius of the set A(x ∗, ε). Consider the function f(ε) := sup R(x ∗, ε). x ∗ ∈S ∗

The following result holds (see [258], and also [59]). Theorem 15.17 (A. L. Garkavi) A necessary and sufficient condition that X be a Hilbert space is that f(ε) tend strictly monotonically to zero as ε → 0 Among negative results, we mention the following ones. Garkavi [255], [256] constructed an example of a Banach space that contains three points without a Chebyshev centre (see also Veselý [586]). The number three in this result is smallest possible, because two points always admit a Chebyshev centre. For similar results for Chebyshev nets of cardinality n, see [256, Sect. 2], and also Example 3.1 in [36]. Konyagin [356] showed that every nonreflexive Banach space X can be (equivalently) renormed so that the resulting space contains a three-point set without a Chebyshev centre. The following question is natural. Assume that in a Banach space, every finite (or compact) set has a Chebyshev centre. Is it true that in such a space, every bounded set has a Chebyshev centre? Veselý [587], [587] answered this question

15.4 Existence of a Chebyshev Centre in Normed Spaces

317

in the negative for compact sets: he constructed a space of the form3 X = c0 (E) in which all compact sets admit Chebyshev centres but that contains a bounded set without a Chebyshev centre. Earlier, Smith and Ward [525] presented an example of a proximinal hyperplane H in C[0, 1] and a bounded set M ⊂ H without a relative Chebyshev centre in H. See also Rao [489]. The following example [588], which extends Garkavi’s construction, gives an example of a set without a Chebyshev centre in a closed hyperplane of c0 . Example Let f = ( f (i) ) ∈ (c0 )∗ be a functional with infinite support, let f (1) = f (2) = f (3) = 1, 15.4 (i) and let ∞ f = 1. Then the set M that consists of the three points i=1 (−1, 1, 1, 0, 0, . . . ), lies in the hyperplane

f −1 (1)

(1, −1, 1, 0, 0, . . . ),

(1, 1, −1, 0, 0, . . . )

and has no Chebyshev centre there.

Veselý [588] also proved the following more general result. Theorem 15.18 Let f = ( f (i) ) ∈ (c0 )∗ be a functional with infinite support and let 2 f ∞ < f 1 . Then there exist σ ∈ R and a three-point set M = {u, v, w} ⊂ f −1 (σ) that has no Chebyshev centre in f −1 (σ). We note another result from [588] (for a similar result for hyperplanes in C(Q), see Zamyatin [620]). Theorem 15.19 Let f = ( f (i) ) ∈ (c0 )∗ and H = f −1 (0). Then the following conditions are equivalent: (a) f has finite support or 2 f ∞ ≥ f 1 ; (b) the hyperplane H is proximinal or is norm-1 complemented in c0 ; (c) every nonempty bounded subset of H has a Chebyshev centre in H; (d) every finite subset of H has a Chebyshev centre in H; (e) every three-point subset of H has a Chebyshev centre in H. The next result follows from the Garkavi–Klee characterization of the spaces in which a Chebyshev centre of a set lies in its convex hull (Theorems 15.10 and 15.11) and from Belobrov’s results [89] on best nets. Theorem 15.20 Let X be a Banach space. Then the following conditions are equivalent: (a) X is either a two-dimensional space or a Hilbert space; (b) there exists n ∈ N such that for every nonempty set M ⊂ X, there exists a Chebyshev net of cardinality n lying in the convex hull of M; (c) for every n ∈ N and nonempty set M ⊂ X, there exists a Chebyshev net of cardinality n lying in the convex hull of M.

3 In his example, X = c0 (E) is the Banach space of null sequences in E with the norm x ∞ = max{ x(n) | n ∈ N}, where E is a three-dimensional Banach space.

318

15 ChebyshevCentre of a Set . . .

15.4.1 Quasi-uniform Convexity and Existence of Chebyshev Centres In this section, we extend some of the above theorems on the existence of Chebyshev centres. Quasi-uniformly convex spaces were introduced in 1973 in [158] (see also [589], [36], [383]). Definition 15.6 A Banach space X is said to be quasi-uniformly convex (X ∈ (QUC)) if for all ε > 0, there exists δ > 0 such that for all x ∈ X, there exists y ∈ B(0, ε) with B(0, 1 + δ) ∩ B(x, 1) ⊂ B(y, 1). We mention the following results. Proposition 15.4 (Veselý [589]) Let X be a Banach space. Then the following conditions are equivalent: (a) X ∈ (QUC); (b) for all ε > 0, there exists δ > 0 such that for all x ∈ X and β > 0, there exists y ∈ B(0, ε) with B(0, 1 + δ) ∩ B(x, 1) ⊂ B(y, 1 + β); ∞(c) there exist sequences of positive numbers (εn ) and (δn ) such that δn → 0, n=1 εn < ∞, and for each n ∈ N and x ∈ X, there exists yn ∈ B(0, εn ) with B(0, 1 + δn ) ∩ B(x, 1) ⊂ B(yn, 1 + δn ). Proposition 15.5 (See [36], [158]) A Banach space is uniformly convex if and only if it is both quasi-uniformly convex and strictly convex. Remark 15.9 (a) The spaces  ∞ , c0 , c, and C[a, b] lie in the class (QUC) (see [158]). (b) If X is uniformly convex, then C(Q, X) ∈ (QUC), where Q is a compact Hausdorff space (see [36]). (c) If L 1 (μ) is infinite-dimensional, then L 1 (μ)  (QUC) (see [36]). In the following theorem, the existence of a Chebyshev centre was proved in [158], and its uniform continuity was proved in the Hausdorff semimetric in [36]. Theorem 15.21 If X ∈ (QUC) is a Banach space, then every nonempty bounded subset M of X has a Chebyshev centre. Moreover, the Chebyshev-centre map Z( · ) is uniformly continuous in the Hausdorff semimetric on the class of nonempty subsets of X with uniformly bounded Chebyshev radii. For further results on the stability of the Chebyshev-centre map in quasi-uniformly convex spaces, see Sect. 15.6.1 below.

15.5 Uniqueness of a Chebyshev Centre The problem of uniqueness of a Chebyshev centre has been studied by Golomb [275], Garkavi [256], [262], Laurent and Pham-Dinh-Tuan [392], Rozema and Smith [500], Lambert and Milman [384], Smith and Ward [525], Amir and Ziegler [37], [38],

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Amir [29], Li and Watson [402], Laurent and Pai [391], and Peng and Li [465] (this list is by no means complete). Belobrov [91] considered spaces in which the set of Chebyshev centres of every set has dimension at most r < ∞.

15.5.1 Uniqueness of a Chebyshev Centre of a Compact Set We first consider the problem of uniqueness of a Chebyshev centre for compact sets. Theorem 15.22 (A. L. Garkavi [255], [256]) Each compact set M in a space X has at most one Chebyshev centre if and only if X is a strictly convex space. Amir and Ziegler [37] and later Amir [31] (see also [434]) extended Theorem 15.22 as follows. First we need a definition. Definition 15.7 A space X is said to be strictly convex (rotund) in every direction in a convex subset Y (X ∈ (RED-Y )) if the unit sphere of X contains no nondegenerate interval parallel to an interval in Y . This is equivalent to the following condition:     x + y   = 1, x − y ∈ Y − Y =⇒ x = y. (15.52) x = y =  2  If Y is a subspace, then for brevity we say that X is strictly convex with respect to Y instead of saying that X is strictly convex in every direction in Y . A space X is strictly convex (rotund) if and only if it is strictly convex with respect to Y = X. If X is strictly convex in every direction in a subspace Y , then clearly X is strictly convex in every direction in every subspace Z ⊂ Y . In this case, every subspace X0 with Y ⊂ X0 ⊂ X is strictly convex with respect to every direction in Y , and in particular, Y is strictly convex. It is also clear that if X is strictly convex with respect to every 1-dimensional subspace, then X itself is strictly convex. We recall the following definition. Definition 15.8 Let x ∈ X and  M ⊂ X. A point y0 ∈ M is a farthest point in the set M from the point x if x − y0 = sup{ x − y | y ∈ M } = r(x, M). Theorem 15.23 Let X be a normed linear space and let Y ⊂ X be a convex set. Then the following conditions are equivalent: (a) X is strictly convex in every direction in Y (that is, the unit sphere S does not contain a nontrivial interval parallel to some interval in Y ); (b) |ZY (K)| ≤ 1 for every compact set K ⊂ X; (c) |ZY (K)| ≤ 1 for every set K ⊂ X such that every y ∈ Y has a farthest point in K; (d) |ZY ({x, y})| ≤ 1 for every x, y ∈ X; (e) if u = v = u + v /2 and u − v ∈ Y − Y , then u = v; (f) every closed interval in Y is a Chebyshev set in X.

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Proof (a) ⇒ (b). It can be assumed without loss of generality that rY (K) = 1. If z, w ∈ ZY (K), then since ZY (K) is convex (see Proposition 15.1), we have (z +w)/2 ∈ ZY (K). Since K is compact, there exists a point x ∈ K such that  z + w    = 1. x − 2 Further, we have rY (K) = 1, and hence x−z = x−w = 1. Next, (x−z)−(x−w) = w − z ∈ Y − Y , and since X is strictly convex in every direction from Y , we have w = z. The implications (b) ⇒ (d), (a) ⇔ (e), (a) ⇔ (f) are clear. Let us prove (d) ⇒ (a). Assume that (a) does not hold. Correspondingly, for some x, y ∈ X, we have x  y, x = y = (x + y)/2 = 1, x − y ∈ Y − Y . But hence (x − y)/2, (y − x)/2 ∈ ZY ({−(x + y)/2, (x + y)/2}); that is,  x + y x + y  , is nonsingleton. the set ZY − 2 2 For the implication (c) ⇔ (a), see Theorem 15.30 and Remark 15.11 below. Theorem 15.23 is proved.  Definition 15.9 A subspace Y of dimension n in a normed linear space X is called an interpolating subspace [49] if no nontrivial linear combination of n linearly independent extreme points of the dual ball B∗ annihilates Y . This definition is a natural extension of the definition of a Haar (Chebyshev) subspace (see Sect. 2.3). Theorem 15.24 (D. Amir [29]) Let Y be an interpolating subspace of a normed linear space X and let M ⊂ X be a compact set such that r(M) < rY (M). Then the set ZY (M) is a singleton. Remark 15.10 The conclusion of Theorem 15.24 is not true for bounded sets (this was pointed out in [29], where the following counterexample was constructed, disproving the corresponding erroneous assertion from [385]). Let X = {x ∈ C[−1, 1] | x(0) = (x(−1) + x(1))/2} (with the Chebyshev norm). We set y0 (t) = t,

Y := span y0,

M := {x ∈ X | 0 ≤ x(t) ≤ 1 − |t|}.

Since ext B∗ = {±et | 0 < |t| ≤ 1}, where et (x) := x(t), it follows that Y is an interpolating subspace, but r(M) = r(2−1, M) = 2−1 < 1 = rY (M) = r(αy0, M) for |α| ≤ 1. Let Q be a topological space and let L be a locally compact topological space. We denote by C0 (L, X) the closed subspace of C(L, X) consisting of all functions x that vanish at infinity (this means that for every ε > 0, the set {t ∈ L | |x(t)| ≥ ε} is compact).

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We note the following simple fact. Proposition 15.6 There is no subspace of the space C0 (Q) of dimension ≥ 2 with respect to which C0 (Q) is strictly convex. Proof (of Proposition 15.6) If dim Y ≥ 2, then some z  0 from Y has a zero on Q. Hence by Haar’s theorem (see Sect. 2.3), the subspace span z is not a Chebyshev subspace. It remains to apply the above implication a) ⇒ f).  Proposition 15.7 If μ is a measure, then there is no subspace of dimension ≥ 2 in L 1 (μ) with respect to which L 1 (μ) is strictly convex. If the measure μ is atomless, then in L 1 (μ) there is no subspace with respect to which L 1 (μ) is strictly convex. Proof (of Proposition 15.7) A trivial corollary of Phelps’s characterization of finitedimensional Chebyshev subspaces in the∫ space L 1 (Ω, ∫ μ) (see [470]) is that span v is a Chebyshev subspace if and only if A v dμ  Ω\A v dμ for every measurable set A. Let Y have dimension ≥ 2, let v and w be two linearly independent points, and let A be a fixed set. By the intermediate value theorem, there exist numbers ∫ ∫ α and β with α2 + β2 = 1 such that A(αv + βw) dμ = Ω\A(αv + βw) dμ. Hence the 1-dimensional subspace spanned by the vector z = αv + βw is not a Chebyshev subspace, and therefore the space L 1 (μ) is not strictly convex with respect to Y by the implication a) ⇒ f) of Theorem 15.23. If μ is atomless, then the required assertion follows from the following well-known result: there are no finite-dimensional Chebyshev subspaces in the space L 1 (μ).  A similar analysis (see [37]) shows that the space (C[a, b], · 1 ) of continuous functions with the L 1 -norm is not strictly convex with respect to every subspace Y of finite dimension ≥ 2. To conclude this section, we mention another result on uniqueness of relative Chebyshev centres. We need the following definition. Definition 15.10 A space X is said to be weakly uniformly convex if the conditions w xn ≤ 1, yn ≤ 1, xn + yn → 2 imply that xn − yn → 0. Theorem 15.25 (J.-Zh. Xiao and X.-H. Zhu [605]) Let X be a Banach space with dim X ≥ 2, and let M  be a convex weakly compact set in X. Assume that one of the following conditions is satisfied: (a) X is weakly uniformly convex; (b) X is locally uniformly convex and M is compact. Then Zconv M (M) is a singleton that lies in M.

15.5.2 Uniqueness of a Chebyshev Centre of a Bounded Set Definition 15.11 Following Garkavi, we say that a space X is uniformly convex (or uniformly rotund) in every direction (X ∈ (URED)) if for every z ∈ X and every ε > 0, there exists δ = δ(z, ε) > 0 such that if

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15 ChebyshevCentre of a Set . . .

x1 = x2 = 1,

x1 − x2 = λz,

and

x1 + x2 > 2 − δ,

then |λ| ≤ ε. Note that X ∈ (URED) if and only if xn − yn → 0 whenever (xn ), (yn ) ⊂ S, (xn + yn )/2 → 1, and xn − yn ∈ span v for some v ∈ S and all n ∈ N. Equivalently, X ∈ (URED) if and only if for every z ∈ S and all bounded sequences (xn ), (yn ) such that 2 xn 2 + 2 yn 2 − xn + yn 2 → 0 and xn − yn = λn z for some λn , we have λn → 0 (see [230], p. 456). The next theorem [255], [256] characterizes the spaces in which every bounded set has at most one Chebyshev centre. Theorem 15.26 (A. L. Garkavi) A necessary and sufficient condition that each bounded subset of a Banach space X have at most one Chebyshev centre is that X be uniformly convex in every direction (X ∈ (URED)). Note that the condition X ∈ (URED) is weaker than the condition that X be a uniformly convex space. Garkavi [255], [256] constructed an example of an (incomplete) normed space that is uniformly convex in every direction but is not an (incomplete) uniformly convex space. Let us give some extensions of Theorem 15.26. Definition 15.12 Let Y be a convex set in X. A space X is uniformly convex (or uniformly rotund) in every direction in Y (X ∈ (URED-Y )) if for every z  0 with z ∈ Y − Y and every ε > 0, there exists δ = δ(z, ε) > 0 such that     x + y  > 1−δ =⇒ |λ| < ε. (15.53) x = y = 1, x − y = λz,  2  Note that if X ∈ (URED-Y ), then X is strictly convex in every direction in Y . A number of properties of spaces that are uniformly convex in every direction can be found in [182]. The next result [37], [31] is a direct extension of Theorem 15.26. Theorem 15.27 Let Y be a convex subset of a Banach space X. Then the following conditions are equivalent: (a) |ZY (M)| ≤ 1 for every bounded set M ⊂ X; (b) X is uniformly convex in every direction in Y (X ∈ (URED-Y )); (c) if un , vn → 1, un + vn → 2, and if un − vn = λn z  0 for some λn and z ∈ Y − Y , then λn → 0. Proof Let y1, y2 be two distinct elements from ZY (M). Since the set of Chebyshev centres is convex (Proposition 15.1), we have y0 := (y1 + y2 )/2 ∈ ZY (M). We choose a sequence (xn ) ⊂ M such that y0 − xn → rY (M). Then yi − xn → rY (M), i = 1, 2. It can be assumed that y1 − xn ≥ y2 − xn . We set zn := y1 + tn (y2 − y1 ), where tn ≥ 1 is such that zn − xn = y1 − xn . Hence for the points un :=

y1 − xn , y1 − xn

vn :=

zn − xn , zn − xn

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we have un + vn → 2,

un − vn ∈ Y,

but un − vn → 0; that is, X  (URED-Y ). Conversely, assume that X  (URED-Y ). Then there exist z ∈ Y and two sequences (xn ), (yn ) such that xn = yn = 1, xn − yn = λn z, |λn | ≥ λ > 0,

xn + yn → 1. 2

Let un := (xn + yn )/2, M := {±un | n = 1, 2, . . . }. It can be assumed without loss of generality that [a, b] ⊂ Y , 0 = (a + b)/2 and z = b − a. We have un → 1, and hence rY (M) = 1 and 0 ∈ ZY (M). However, we also have ±λz/2 ∈ ZY (M), which gives  1 1  1   λz    1  xn + yn  ≤ 1. ± un ±  =  ± 2 2 2λn 2 2λn This shows that the set of relative Chebyshev centres ZY (M) has more than two points.  We mention some further results (see also Theorem 15.58 below). Theorem 15.28 (Amir [28], [36]) A Banach space X is uniformly convex if and only if for every nonempty bounded set M ⊂ X, the set Z(M) of Chebyshev centres is a singleton and the Chebyshev-centre map M → Z(M) is locally uniformly continuous. Proof (of Theorem 15.28) Assume that X is uniformly convex. Since Z is reflexive, Z(M)  for every nonempty bounded M ⊂ X. By Theorem 15.26, a Chebyshev centre of the set M is unique. It is easily seen that if x , y ≤ c and x − y ≥ ε, then  x + y  1 − δ(ε/c)   . ≤  2 c Let N ⊂ X be a nonempty bounded set such that Z(M) = {z}, Z(N) = {w}, r(M) < R, and h(M, N) < η < 1. Hence r(N) ≤ r(z, N) < r(z, M) + η = r(M) + η, which gives r(M) ≤ r(w, M) < r(N) + η < r(M) + 2η. Therefore, for every u ∈ M, we have u − z ≤ r(M), u − w < r(M) + 2η, and   ε  z + w   ((u − z) + (u − w))    (r(M) + 2η). =  ≤ 1−δ u − 2 2 R+2 However, for some u ∈ M, we have u−(z+w)/2 ≥ r(M), and hence if z−w ≥ ε, then ε r(M) δ( R+2 ) r(M)  ε  η≥ ≥ δ . ε 2(1 − δ R+2 ) 2 R+2 Hence, if η < r(M) d(ε/(r +2))/2, then z−w < ε. Putting η = r(M) d(ε/(r +2))/4, we have that either r(M) ≥ ε/2 and hence z − w < ε or r(M) < ε/2 and hence z − w ≤ z − u + u − w < r(M) + r(M) + 2η < ε for an arbitrary u ∈ M.

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15 ChebyshevCentre of a Set . . .

Conversely, assume that X is not uniformly convex. Then there exist points xn, yn ∈ X such that xn = yn = 1, xn − yn = ε, xn + yn → 2. Setting   xn + yn xn + yn xn + yn , Mn := conv xn, ,− , −yn , zn := xn + yn 2 2 Nn := conv{zn, xn, −zn, −yn }, we have h(Mn, Nn ) → 0, but (xn − yn )/4  Z(Mn ), 0 ∈ Z(Nn ). By the hypothesis, |Z(Mn )| = 1, |Z(Nn )| = 1, and hence h(Z(Mn ), Z(Nn )) = (xn − yn )/4 = ε/4.  Definition 15.13 For a given subspace Y of a Banach space X, an equivalent definition of X as being uniformly convex in every direction in Y (X ∈ (URED-Y )) is that    x + y  x = y = 1, x − y ≥ ε, x − y ∈ Y > 0 (15.54) δY (ε) := inf 1 −  2 for all ε > 0. Theorem 15.29 (D. Amir [28], [36]) Let Y be a subspace of a Banach space X. Then X ∈ (URED-Y ) if and only if the relative-Chebyshev-centre map M → ZY (M) is single-valued and locally uniformly continuous on the class of bounded subsets of X. We note the following result (see [434]). Theorem 15.30 Let X be strictly convex and let Y ⊂ X be closed and convex. Then the set of Chebyshev centres ZY (M) is at most a singleton for every uniquely remotal set M ⊂ X. Proof (of Theorem 15.30) Let  M ⊂ X be a remotal set (that is, for every x ∈ X in M, there exist farthest points for x; see Sect. 6.2.2 and Sect. 17.10 below). We can assume without loss of generality that rY (M) = 1. If y1, y2 ∈ ZY (M), then by Proposition 15.1, (y1 + y2 )/2 ∈ ZY (M). Since M is remotal, there exists 2 M is the metric max-projection (or antiprojection)). y ∈ F M ( y1 +y 2 ) (here F : X → 2 We have  y + y   (y − y ) + (y − y )  1 1  1 2 1 2  = 1=r  ≤ y − y1 + y − y2 2 2 2 2 1 1 ≤ r(y1, M) + r(y2, M) = rY (M) = 1. 2 2 2) = 1, and since X is strictly As a corollary, y − y1 = y − y2 = (y−y1 )−(y−y 2  convex, we have y1 = y2 .

Remark 15.11 In fact, the conclusion of Theorem 15.30 holds if X is strictly convex (rotund) in every direction in Y (X ∈ (RED-Y )). To conclude this section, we briefly discuss the problem of uniqueness of the absolute and relative Chebyshev centres in C(Q).

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The subsets of the space C(Q) (where Q is a complete metric space) with unique Chebyshev centre were characterized by Smith and Ward [525] in terms of the behaviour of the functions m∗ (t) := inf x ∈M x(t) and m∗ (t) := supx ∈M x(t). Their result was extended to not necessarily metrizable spaces Q by Zamyatin and Shishkin ([622, Theorem 9]). In the more convenient language of the Banach–Mazur hull, the result on the uniqueness of a Chebyshev centre in C(Q) (Q a normal space) is given in Theorem 15.2 (see also [25]). We recall (see Theorem 15.23) that the single-valuedness of the relativeChebyshev-centre map ZY ( · ), where Y is a linear subspace, implies that the original space is strictly convex with respect to Y . As a corollary, in the space C(Q) (Q a compact Hausdorff space), for every finite-dimensional subspace Y with dim Y ≥ 2, the relative-Chebyshev-centre map ZY ( · ) is not single-valued (see Proposition 15.6). Under the additional condition r(M) < rY (M), the problem of uniqueness of a relative Chebyshev centre of M was investigated by Amir [29] for a certain special class of subspaces Y ⊂ C(Q). Definition 15.14 A subspace Y = span{y1, . . . , yn } of dimension n of a normed linear space X is said to be strictly interpolating [29] if no nontrivial linear combination ∗ of n linearly independent functionals in the w ∗ -closure extw B∗ annihilates Y . Each of the following conditions is equivalent to the strict interpolation condition [29]: (i) det[ fi (y j )]  0 for every set of n linearly independent functionals f1, . . . , fn ∈ extB∗ ; (ii) for every linearly independent f1, . . . , fn ∈ extB∗ and every set of numbers c1, . . . , cn , there exists a unique y ∈ Y such that fi (y) = ci for each i = 1, . . . , n; (iii) X ∗ = Y ⊥ ⊕span{ f1, . . . , fn } for every linearly independent f1, . . . , fn ∈ extB∗ . It is easily seen that the interpolating subspaces of X are strictly interpolating if ext B∗ is w ∗ -closed (this is so, for example, in the spaces C(Q), where Q is a compact topological space, and in L 1 (μ)) or if ext B∗ ∪ {0} is w ∗ -closed (this condition is satisfied, for example, in the spaces C0 (Q) with locally compact Q). Theorem 15.31 (D. Amir [29]) Let Y be a strictly interpolating subspace of a normed linear space X, and let  M ⊂ X be a bounded set such that r(M) < rY (M). Then ZY (M) is a singleton. Theorem 15.32 (D. Amir [29]) Let Y be a subspace of a normed space X such that for every y0 ∈ X, z ∈ X, and ε > 0, there exists y1 ∈ Y with inf{ f (y1 ) | f ∈ ext B∗, | f (z − y0 )| > ε} > 0. Let  M ⊂ X be a bounded set such that r(M) < rY (M). Then ZY (M) is a singleton. Remark 15.12 An example of a space X and an infinite-dimensional subspace Y ⊂ X satisfying the hypotheses of Theorem 15.32 was constructed in [29]: X = {x ∈ C[−1, 1] | x(0) = 0}, Y = {x ∈ X | x|[0,1] is a polynomial of degree ≤ n}.

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15 ChebyshevCentre of a Set . . .

15.6 Stability of the Chebyshev-Centre Map The problem of stability of the Chebyshev-centre map Z( · ) in dependence on the properties of a space X and a subset of X has been extensively studied. We mention Belobrov [88], [89], [90], Ward [596], Rozema and Smith [500], Bosznay [125], Mach [411], [410], [412], Szeptycki and Van Vleck [537], Prolla, Chiacchio, and Roversi [486], Amir, Mach, and Saatkamp [36], Amir and Mach [35], Beer and Pai [86], Chiacchio, Prolla, and Roversi [167], Tsar’kov [557], Al’brecht [7], Li and Lopez [399], Balashov and Polovinkin [481], Baronti and Papini [77], Alvoni and Papini [27], Balashov and Repovš [72], Balashov and G. E. Ivanov [69], G. E. Ivanov [302], Xiao and Zhu [605], Druzhinin [208], and Lalithambigai et al. [383]. (This list is by no means complete.)

15.6.1 Stability of the Chebyshev-Centre Map in Arbitrary Normed Spaces Let M and N be nonempty bounded subsets of a normed linear space X, let z M ∈ Z(M) and z N ∈ Z(N) be (some) Chebyshev centres of the sets M and N, and let  ∈ Zsc and z  ∈ Zsc be (some) self Chebyshev centres of M and N (see (15.4)). zM N M N Here and in what follows, h(M, N) is the Hausdorff distance between sets M and N. To begin with, we note that each of the numbers h(M, N)

and

z M − z N

  (or z M − zN )

can be greater than, equal to, or less than the other number (even if the Chebyshev centre is unique). The following simple result holds (see, for example, [86], [77]). Proposition 15.8 Let M and N be nonempty closed bounded subsets of a normed linear space X, and let x, y ∈ X. Then |r(x, M) − r(y, M)| ≤ x − y ,

(15.55)

|r(x, M) − r(y, N)| ≤ h(M, N) + x − y , |r(M) − r(N)| ≤ h(M, N).

(15.56) (15.57)

Proof Let us prove (15.56). Consider a ∈ M, ε > 0. We choose a point bε ∈ N such that a − bε < h(M, N) + ε. Then for every x ∈ X, a − x ≤ a − bε + bε − x < h(M, N) + ε + r(x, N). As a corollary, r(x, M) ≤ g(M, N) + r(x, N). Swapping M and N gives (15.56). Further, from (15.56) we have r(M) := inf r(x, M) ≤ h(M, N) + inf r(x, N) = h(M, N) + r(N), x ∈X

which gives (15.57).

x ∈X



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327

Remark 15.13 An analogue of (15.57) does not hold for self centres Zsc (M) (see [77]). Indeed, let X = C[0, 1] and 0 < ε < 1/2, and let   M := f ∈ X | 0 ≤ f (t) ≤ 1 − ε ∀t ∈ [0, 1], f (0) = 0 ,   N := g ∈ X | 2−1 − ε ≤ g(t) ≤ 2−1 ∀t ∈ [0, 1], g(0) = 2−1 − ε . Then r M (M) = 1 − ε and r N (N) = ε, but h(M, N) = 2−1 − ε < 1 − 2ε = r M (M) − r N (N). For self centres, there is a weaker estimate (see [77]): |r M (M) − r N (N)| ≤ 2h(M, N).

(15.58)

Nevertheless, this estimate is sharp: it suffices to let ε → 0 in the example in Remark 15.13. Let us prove (15.58). Let ε > 0. Consider a point bε ∈ N such that r(bε, N) < r N (N) + ε and choose aε ∈ M so as to have aε − bε < h(M, N) + ε. Using (15.5) and (15.56), this gives r(aε, M) ≤ r(bε, N) + aε − bε < h(M, N) + r(bε, N) + h(M, N) + ε < 2h(M, N) + r N (N) + 2ε, from which we have r M (M)−r N (N) ≤ 2h(M, N). Now (15.58) follows by symmetry. For self centres, the following analogue of the inequality (15.6) holds (see [77]).  ∈ Zsc (M), z  ∈ Zsc (N), then Proposition 15.9 If z M N

   2   z M − z N ≤ h(M, N) + r M · h(M, N) + r N , r  + r N   . − zN ≤ h(M, N) + M z M 2

(15.59) (15.60)

Proof (of Proposition 15.9) From (15.55) we have     z M − zN ≤ r(z N , M) ≤ r(z N , N) + h(M, N) = r N + h(M, N);  − z  ≤ r  + h(M, N). Now the required result a similar analysis shows that z N M M follows by multiplication of the above inequalities. 

Remark 15.14 The estimates in Proposition 15.9 are sharp: it suffices to consider the sets M = {(x, y) | 0 ≤ x ≤ 1, |y| ≤ 1} and N = {(x, y) | −1 ≤ x ≤ 0, |y| ≤ 1} in the space 2∞ .  − z  , where Remark 15.15 Each of the numbers h(M, N), z M − z N (or z M N    z M ∈ Z M (M), z N ∈ Z N (N) , can be greater than, equal to, or less than the other, also in the case of uniqueness.

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15 ChebyshevCentre of a Set . . .

15.6.2 Quasi-uniform Convexity and Stability of the Chebyshev-Centre Map Quasi-uniformly convex (QUC) spaces were discussed above in Sect. 15.4.1. We recall (Theorem 15.21) that if X is a quasi-uniformly convex Banach space (X ∈ (QUC)), then every nonempty bounded set M ⊂ X admits a Chebyshev centre, and the Chebyshev-centre map Z( · ) is uniformly continuous on the class of nonempty subsets of X with uniformly bounded Chebyshev radii. Below in this section, B H (X) is the semimetric space of all nonempty bounded subsets of a normed linear space X equipped with the Hausdorff semimetric. We also require the following definitions from [589]. Definition 15.15 Given A ∈ B H (X) and r ≥ r(A), we set Zr (A) := {x ∈ X | r(x, A) ≤ r }, where r(x, A) := sup{ x − a | a ∈ A}. The definitions of a Chebyshev centre and the Chebyshev radius have the following analogues for bounded nets in B H (X). Definition 15.16 Given a bounded decreasing (with respect to set inclusion) net A := (Ai )i ∈I from BH (X) and x ∈ X, we define ϕ(A, x) := lim r(x, Ai ) = inf r(x, Ai ), i ∈I

i ∈I

r(A) := inf ϕ(A, x) = inf inf r(x, Ai ), x ∈X

x ∈X i ∈I

Zr (A) := {x ∈ X | ϕ(A, x) ≤ r } (r(A))

Z(A) := Z

∀r ≥ r(A),

(A).

The nonnegative number r(A) is called the asymptotic radius of the net A, and the (possibly empty) set Z(A) is called the set of asymptotic centres of A . We also recall the classical definitions of the asymptotic radius of a set and the set of asymptotic centres of a sequence. Definition 15.17 Given a bounded decreasing (with respect to set inclusion) net A := (Ai )i ∈I in B H (X) and x ∈ X, we define ϕ(A , x) := lim r(x, Ai ) = inf r(x, Ai ), i ∈I

i ∈I

r(A ) := inf ϕ(A , x) = inf inf r(x, Ai ), x ∈X

x ∈X i ∈I

Zr (A ) := {x ∈ X | ϕ(A , x) ≤ r } r(A )

Z(A ) := Z

∀r ≥ r(A ),

(A ).

The nonnegative number r(A ) is called the asymptotic radius of the net A , and the (possibly empty) set Z(A ) is known as the set of asymptotic centres of A .

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Definitions 15.15 and 15.17 are particular cases of Definition 15.16. Theorem 15.33 (L. Veselý [589]) For a Banach space X, the following conditions are equivalent: (a) X ∈ (QUC); (b) for every bounded net A in B H (X), the set Z(A ) is nonempty, and the map r → Zr (A ) is continuous on [r(A ), ∞) and uniformly continuous on the class of nets with uniformly bounded asymptotic radii; (c) every set A ∈ B H (X) admits a Chebyshev centre, and the map r → Zr (A) is continuous on [r(A), ∞) and uniformly continuous on the class of sets with uniformly bounded Chebyshev radii; (d) for every (some) r0 > 0, the map (A, r) → Zr (A) has values in B H (X) and is uniformly continuous on the set   (A, r) ∈ B H (X) × (0, ∞) | r(A) ≤ r ≤ r0 ; (e) for every (some) r0 > 0, the map Zr0 has values in B H (X) and is uniformly continuous on the set { A ∈ B H (X) | r(A) ≤ r0 }; (f) for every (some) r0 > 0, the Chebyshev-centre map Z( · ) has nonempty values and is uniformly continuous on the set { A ∈ B H (X) | r(A) = r0 }. Any one of the conditions a)–f) implies the following property: (g) every bounded sequence (an ) ⊂ X has an asymptotic centre, and the map r → Zr (an ) is continuous on [r(an ), ∞) and uniformly continuous on the class of sequences with uniformly bounded asymptotic radii. Moreover, if X is separable, then conditions a)–g) are equivalent. We mention some results regarding the space of bounded vector-valued continuous functions with values in a Banach space X ∈ (QUC) (see [589]). Definition 15.18 Let Q be a topological space. As before, let C(Q, X) be the space of all bounded continuous functions x on Q with values in X equipped with the norm x = supt ∈Q x(t) . The next result from [589] extends assertion b) in Remark 15.9. Theorem 15.34 (L. Veselý) If X ∈ (QUC) is a Banach space and Q is a topological space, then C(Q, X) ∈ (QUC). Definition 15.19 Let Q be a topological space, Q0 ⊂ Q a closed subset, L a locally compact topological space, and Γ a nonempty set with the discrete topology. We consider the following spaces: C(Q, Q0, X) is the closed subspace of C(Q, X) consisting of all functions that vanish on Q0 ;

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15 ChebyshevCentre of a Set . . .

C0 (L, X) is the closed subspace of C(L, X) consisting of all functions x that vanish at infinity (this means that the set {t ∈ L | |x(t)| ≥ ε} is compact for every ε > 0);  ∞ (Γ, X) := C(Γ, X) and c0 (Γ, X) := C0 (Γ, X), where Γ is equipped with the discrete topology. Theorem 15.35 (L. Veselý) Let X be a Banach space. Then the following conditions are equivalent: (a) X ∈ (QUC); (b) C(Q, X) ∈ (QUC) for every topological space Q; (c) C(Q, Q0, X) ∈ (QUC) for every topological space Q and every closed Q0 ⊂ Q; (d) C0 (L, X) ∈ (QUC) for every locally compact space L; (e) c0 (Γ, X) ∈ (QUC) for every Γ; (f)  ∞ (Γ, X) ∈ (QUC) for every Γ. The next result follows from Theorems 15.33 and 15.35. Theorem 15.36 (L. Veselý) Let X ∈ (QUC) be a Banach space and let Y = C(Q, X) (or Y is any other space in Theorem 15.35). Then the Chebyshev-centre map Z( · ) is uniformly continuous on the class of sets { A ∈ B H (Y ) | r(A) ≤ r0 } with every given r0 > 0. To conclude this section, we mention one sufficient result from [589]. Theorem 15.37 (L. Veselý) Let X be a finite-dimensional Banach space that is either polyhedral or two-dimensional. Then X ∈ (QUC), and therefore C(Q, X) ∈ (QUC).

15.6.3 Stability of the Chebyshev-Centre Map in Finite-Dimensional Polyhedral Spaces By n∞ we denote the space of all n-dimensional real vectors with the max-norm. m of coordinate nonzero Let V be a subspace of n∞ . Consider a family L := {Li }i=1 subspaces L1 ⊃ L2 ⊃ . . . ⊃ Lm such that V1 := V ⊂ L1 , Vl+1 = Vl ∩ Ll , l = 1, . . . , m − 1, where Ll is the nonzero coordinate subspace of minimal dimension that contains Vl , l = 1, . . . , m. Such a family of subspaces will be called admissible for V. For each index i = 1, . . . , m, by Ai we denote the set of all vectors e ∈ Vi such that ρ(e, Li+1 ) > 0. We set βi (L ) := sup

e ∈Ai

ρ(e, Vi+1 ) , ρ(e, Li+1 )

β(L , V) :=

Further, let α(V) := sup β(L , V), L

max

i=1,...,m−1

βi (L ).

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where the supremum is taken over all admissible systems L for V. Note that for each isometry τ, that is a permutation of coordinates of vectors in n∞ , we have α(τ(V)) = α(V). In the next result, we assume that a subspace V does not contain nonzero coordinate subspaces. n [ai, bi ], V ∩ Π  , and let y ∈ V, Lemma 15.2 (I. G. Tsar’kov) Let Π := i=1 ρ(y, Π) ≤ δ. Then there exists a point z ∈ V ∩ Π such that y − z ≤ (α + 1)n δ,

where

α = α(V).

Proof Let L1 be the minimal coordinate subspace in n∞ that contains V1 := V, dim L1 = n1 . We set Π1 = Π ∩ L1 . Relabeling the coordinates of vectors if required, n1 [ai, bi ]. Translating the origin if necessary, it can be we can assume that Π1 = i=1 supposed that 0 ∈ Π1 ∩ V1 . Let y1 := y. Then δ1 := ρ(y1, Π1 ) ≤ δ. Induction step. We assume that at the kth step, we constructed a coordinate subspace Lk , a subspace Vk ⊂ Lk , where nkLk is the coordinate subspace of minimal [ai, bi ], 0 ∈ Πk ∩ Vk , δk := ρ(yk , Πk ) ≤ dimension nk that contains Vk , Πk = i=1 (α + 1)k−1 δ. There exists a coordinate hyperplane πk in Ll passing through some face Πk (which intersects the interval [0, yk ]) and separating the point yk and the parallelepiped Πk . It is clear that πk is a translation of some coordinate hyperplane. Moreover, ρ(yk , πk )  δk . We set Vk+1 = Vk ∩ πk . Then there exists a point yk+1 ∈ Vk+1 such that yk+1 − yk ≤ αδk . Hence ρ(yk+1, Πk ) ≤ yk+1 − yk + ρ(yk , Πk ) ≤ (α + 1)δk ≤ (α + 1)k δ. Translating if necessary, we can assume that 0 ∈ Πk ∩ Vk+1 . By Lk+1 we denote the coordinate subspace in Lk+1 of minimal dimension nk+1 that contains Vk+1 . Rearranging coordinates if necessary, nk+1we can assume, to continue the induction [ai, bi ]. The induction terminates at the step process, that Πk+1 = Πk ∩ Lk+1 = i=1 k = m if nk+1 = 0. By construction,  and y − z  δ(1 + α)k < δ(1 + α)n, z = ym ∈ V ∩ Π k 0.

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15 ChebyshevCentre of a Set . . .

Proof To begin with, we note that the cases dim V = 0 and dim V = n are trivial. Then the case in which V contains a nonzero coordinate subspace can be reduced to the case of a subspace V0 of smaller dimension that does not contain nonzero coordinate subspaces. Indeed, let L be a coordinate subspace of maximal dimension that is contained in V. Let V0 := V ∩ L ⊥ , where L ⊥ is the orthogonal complement of L (V0 is a coordinate subspace in n∞ ). The parallelepiped Π(t) can be written as Π0 (t) × Π1 (t), where Π0 (t) (respectively Π1 (t)) is the orthogonal projection of Π(t) onto L ⊥ (onto L) parallel to L (parallel to L ⊥ ). In this case, to prove that the mapping Φ(t) := Π(t) ∩ V = (Π0 (t) ∩ V0 ) × Π1 (t) is Lipschitz, one should verify that Φ0 (t) := Π0 (t) ∩ V0 is Lipschitz continuous, but in this setting, V0 does not contain nonzero coordinate subspaces, and Φ0 (t)  for all t ∈ M . So to prove the theorem, it suffices to study the case of subspaces V that do not contain nonzero coordinate subspaces. By the hypothesis of the theorem, |ai − ai0 |, |bi − b0i | ≤ c θ(t, t0 ) =: δ for every t, t0 ∈ M and ai = ai (t), bi = bi (t), ai0 = ai (t0 ), b0i = bi (t0 ). By Lemma 15.2, for every point y ∈ Π(t0 ) ∩ V, there exists a point z ∈ Π(t) ∩ V such that z − y ≤ (α(V) + 1)n δ. Hence d(Φ(t0 ), Φ(t)) ≤ c(α(V) + 1)n θ(t, t0 ). A similar analysis shows that d(Φ(t), Φ(t0 )) ≤ c(α(V) + 1)n θ(t, t0 ). 

Theorem 15.38 is proved.

Choosing as (M , θ) in Theorem 15.38 the semimetric space of all nonempty bounded subsets of the space n∞ equipped with the Hausdorff semimetric and setting ai (N) = inf z (i) + r, z ∈N

bi (N) = sup z (i) − r, z ∈N

where r := rV (N), V is a subspace in n∞ , we arrive at the following result. Theorem 15.39 (I. G. Tsar’kov) Let V be a subspace in n∞ . Then there exists a number c = c(V) > 0 such that h(ZV (M), ZV (N)) ≤ c h(M, N) for every pair of nonempty bounded subsets M, N ⊂ n∞ . Corollary 15.4 (I. G. Tsar’kov) Let V be a subspace in n∞ . Then the relativeChebyshev-centre map ZV ( · ) admits a Lipschitz selection. Proof With each bounded subset M of V we associate its Steiner centre ϕ(M) (see (15.88) below). This mapping is Lipschitz continuous on the set of all nonempty bounded subsets of V equipped with the Hausdorff semimetric. It is easy to see that  ϕ ◦ ZV is the required Lipschitz selection.

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Corollary 15.4 implies one Druzhinin’s result (see Theorem 15.64 below) on the existence of a Lipschitz selection of the Chebyshev-centre map in finite-dimensional polyhedral spaces, because every such space can be isometrically embedded into n∞ for sufficiently large n. From Theorem 15.39 we have following result. Corollary 15.5 (I. G. Tsar’kov [25]) In a finite-dimensional polyhedral space, the metric projection onto a subspace is globally Lipschitz. Remark 15.16 In connection with Corollary 15.5, we mention some earlier results. Cline [170, Theorem 4] and, independently, V. I. Berdyshev [96, Theorem 2] (see also Bartelt [79] and Finzel [236, Sect. 5]) showed that in n∞ , the metric projection onto a Chebyshev subspace is globally Lipschitz (uniformly continuous) on the entire space. This result follows from Theorem 15.39. Let Xn be an n-dimensional polyhedral Banach space (that is, the unit ball of Xn is the convex hull of a finite number of points in Xn ). Let K ⊂ Xn be a convex polyhedral set (that is, K is the intersection of finitely many closed half-spaces in Xn ). In the polyhedral space Xn , the metric projection onto a polyhedral set is globally Lipschitz continuous (Li [403]) and has a Lipschitz selection (Finzel and Li [237, Theorem 6.1]). The next theorem extends these results of Finzel and Li. Theorem 15.40 (I. G. Tsar’kov [25]) Let V be a nonempty polyhedral subset of a finite-dimensional polyhedral Banach space X. Then the relative-Chebyshev-centre map

 M ⊂ X, M → ZV (M), is globally Lipschitz continuous on X and admits a Lipschitz selection.

15.6.4 Stability of the Chebyshev-Centre Map in C(Q)-Spaces It is known that the Chebyshev-centre map Z( · ) (which associates with a nonempty bounded set the set of its Chebyshev centres) is uniformly continuous in some class of spaces containing the uniformly convex spaces (see Sect. 15.6.5) and spaces of type C0 (Q) (in particular, the space C(Q), where Q is a compact Hausdorff space); see [89], [28], [36], [31], [383], and also Theorem 15.28. The next theorem ([28, Theorem 2], and [36], Corollary 5.2) is one of the most general results on the existence and stability of the Chebyshev-centre map in C(Q, X). Theorem 15.41 (D. Amir and Ka-Sing Lau) Let Q be an arbitrary topological space and X a uniformly convex Banach space. Then every bounded set in the space C(Q, X) admits a Chebyshev centre, and the map Z( · ) is uniformly continuous on the class of nonempty bounded subsets with uniformly bounded diameters.

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15 ChebyshevCentre of a Set . . .

Proof Since X is a uniformly convex space, it follows by definition that for every ε > 0, there exists δ(ε) > 0 such that if x, y ∈ S, x ∗ ∈ S ∗ , x ∗ (y) = 1, and x ∗ (x) > 1 − δ(ε), then x − y < ε. Note that by Lemma 1 in [28], one can always assume that δ(ε) < ε/2 and δ(ε/2) < ε/4. Let M ⊂ C(Q, X) be a nonempty bounded set. We can assume without loss of generality that r(M) = 1. For every ε > 0, we choose f0 ∈ C(Q, X) such that r( f0, M) ≤ 1 + δ(ε). We assert that there exists a function f1 ∈ C(Q, X) such that r( f1, M) ≤ 1 + δ(ε/2) and f1 − f0 ≤ 2ε. Indeed, consider an arbitrary function g ∈ C(Q, X) such that r(g, M) ≤ 1 + δ(ε/2), and define β(t) =

⎧ ⎪ ⎨ 1, ⎪

g(t) − f0 (t) ≤ 2ε,

2ε ⎪ ⎪ g(t) − f (t) , 0 ⎩

g(t) − f0 (t) > 2ε

and f1 (t) = f0 (t) + β(t)(g(t) − f0 (t)). It is clear that f1 ∈ C(Q, X) and f1 − f0 ≤ 2ε. Moreover, if g − f0 > 2ε, then f1 − f0 = 2ε. Consider an arbitrary point a ∈ M. We need to show that f1 (t) − a(t) ≤ 1 + δ(ε/2). This inequality is clear if β(t) = 1, since in this case, f1 (t) = g(t), and also if β(t) < 1 and g(t) − a(t) ≥ f0 (t) − a(t) , since in this case f1 (t) ∈ [ f0 (t), g(t)]. Thus we can always assume that 1 + δ(ε) ≥ f0 (t) − a(t) > g(t) − a(t) . Setting u := f0 (t) − a(t) and v := g(t) − a(t), we have v ≤ 1 + δ(ε/2) and 1 + δ(ε) ≥ u > v . Let us show that if we move a distance 2ε away from the point u in the direction v, then we land in the ball B(0, 1 + δ(ε/2)). Since δ(ε) < ε/2 and δ(ε/2) < ε/4, the required result holds for v = 0, and so it suffices to consider the case v = 1 + δ(ε/2). In the two-dimensional space spanned by the points 0, u, and v, consider a point z lying on the sphere S(0, v ) on the same side as v of the line passing through 0 and u and such that the line uz supports the ball. We extend this line to a hyperplane H := ψ −1 (1) that supports the sphere S(0, v ) in the space X. It is clear that ψ = 1/ v . Let ϕ := v ψ, x := u/ u , and y := z/ z . Then ϕ = ϕ(y) = 1 = y = x and ϕ(x) = v / u ≥ 1/ u ≥ 1/(1 + δ(ε)) > 1 − δ(ε). Since the space is uniformly convex, we then have x − y < ε and u − z < ε + u − x + z − y ≤ ε + δ(ε) + δ(ε/2) < 2ε, which proves the assertion, because the distance from u to the ball B(0, v ) in the direction v is smaller than the maximum of the distances in the direction x (which is at most δ(ε)) and in the direction z (which is less than 2ε). Arguing by induction, we can find a function fn+1 such that fn+1 − fn ≤ 2ε/2n and r( fn+1, M) ≤ 1 + δ(ε/2n+1 ). The space C(Q, X) is complete, and hence the Cauchy sequence ( fn ) converges to some function f such that f − f0 ≤ 4ε

and r( f , M) ≤ lim r( fn, M) ≤ 1.

This shows that r( f , M) = 1 and that f is a Chebyshev centre of the set M.

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Thus we have considered a point f0 in the set of Chebyshev near-centres of the set M, and we have shown that there exists a Chebyshev centre f of M that lies sufficiently close to f0 . Hence, taking f0 to be a Chebyshev centre of a set N that is close to M (in the Hausdorff metric), we get that f0 (a Chebyshev centre of N) is close to some Chebyshev centre f of M. Therefore, the directed (one-sided) Hausdorff distance (and hence the Hausdorff distance) between the corresponding sets of Chebyshev centres of these close sets is small. Consequently, the Chebyshevcentre map Z( · ) is uniformly continuous (see also [28, Corollary 3]). Theorem 15.41 is proved.  Amir [28] also showed that if X = C(Q) and Y is a closed linear sublattice4 of X, then every nonempty bounded set M ⊂ X admits a Chebyshev centre, and the Chebyshev-centre map M → ZY (M) is uniformly continuous on bounded subsets of the space of nonempty bounded sets in X (cf. Theorem 15.28). The next result can be proved using Amir’s arguments for the case of C(Q, R). Theorem 15.42 Let M and N be nonempty bounded subsets of C(Q), where Q is an arbitrary topological space. Then for every δ ≥ 0, h(Zδ (M), Zδ (N)) ≤ 2h(M, N). Zamyatin and Kadec (see, for example, [621]) established the result of Theorem 15.42 with Q = [a, b] and δ = 0. Theorem 15.43 (I. G. Tsar’kov [25]) Let Q be a topological space. Then for every nonempty closed span Y ⊂ C(Q) and every nonempty bounded set M ⊂ C(Q), ZY (M)  . Moreover, for every δ ≥ 0 and arbitrary nonempty bounded sets M, N ⊂ C(Q), h(ZYδ (M), ZYδ (N)) ≤ 2h(M, N). Definition 15.20 Let V be a nonempty closed subset of a normed space X, and let F be a subfamily of the family of all nonempty closed bounded subsets of X. The triple (X, V, F ) is said to have property (R1 ) (see [457]) if the conditions x ∈ V, M ∈ F , r1 > 0, r2 > 0, V ∩ Or1 (M)  (where Or (M) := {x ∈ X | ρ(x, M) ≤ r }), and ρ(x, M) < r1 + r2 imply that V ∩ B(x, r1 ) ∩ Or2 (M)  . According to [457, Theorem 2.2], if V is a nonempty closed subset of a Banach space X, F is a family of nonempty closed bounded subsets of X, and the triple (X, V, F ) has property (R1 ), then ZV (M)  for every M ∈ F . 4 A linear sublattice is a subspace L with the property that f ∈ L ⇒ | f | ∈ L.

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15 ChebyshevCentre of a Set . . .

Pai and Nowroji [457] constructed several examples of triples (X, V, F ) with property (R1 ). In particular, they showed that if Q is a compact Hausdorff space, X = C(Q, R), and V is a closed subalgebra of X, then the triple (X, V, K (X)) (where K (X) is the family of compact subsets of X) has property (R1 ). The following result partially strengthens the Zamyatin–Kadec result on stability of the Chebyshev-centre map in C[a, b]. Theorem 15.44 (Pai and Nowroji [457]) Let V be a nonempty closed subset of a normed linear space X, and let the triple (X, V, F ) have property (R1 ). Then the Chebyshev-centre map ZV ( · ) is Lipschitz continuous in the Hausdorff metric: h(ZV (M), ZV (N)) ≤ 2h(M, N)

(15.61)

for every pair of nonempty closed bounded sets M, N ⊂ X. The constant 2 in inequality (15.61) is best possible. For a similar result in  ∞ (Γ), see [167]. The next result follows from Michael’s selection theorem (see Theorem 16.9 below) and Theorem 15.43. Corollary 15.6 In C(Q), where Q is a topological space, the Chebyshev-centre map Z( · ) has a continuous selection. Using Amir’s results (see Theorem 15.41), it can be shown that in the space C(Q, X), where Q is a topological space and X is a uniformly convex Banach space, the Chebyshev-centre map Z( · ) has a continuous selection. Remark 15.17 In C(Q), where Q is an infinite compact Hausdorff space, the relativeChebyshev-centre map ZV ( · ), where V is a finite-dimensional Chebyshev subspace of dimension ≥ 2, is not Lipschitz continuous (uniformly continuous). This result is a consequence of the fact that the restriction of the operator ZV ( · ) to singletons is the metric projection and since, according to a result of Cline, for every5 finitedimensional Chebyshev subspace V of dimension ≥ 2, there exist functions x, y ∈ C[−1, 1] with x = y = 1 such that x − y < ε but PV x − PV y ≥ 1. Cline’s theorem also shows that the metric projection operator P is not uniformly continuous on the unit ball of the space C(Q). In the particular case of the subspaces Pn generated by the algebraic polynomials of degree ≤ n with n ≥ 2 in the space C[a, b], this result dates back to a construction of S. N. Bernstein. For n = 1, Stechkin [251] showed that the metric projection operator to the subspace of linear functions in C[a, b] is not uniformly continuous.

5 By Mairhuber’s theorem (see Sect. 2.5), the space C(Q), where Q is a compact metric space, contains a Chebyshev subspace of every finite dimension n = 2, 3, . . . if and only if Q is homeomorphic to an infinite closed subset of the closed unit interval [0, 1].

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15.6.5 Stability of the Chebyshev-Centre Map in Hilbert and Uniformly Convex Spaces We first consider the Hilbert space setting. As mentioned above (see Theorem 15.10 or Lemmas 2.1.1 and 2.1.2 in [481]), for every closed convex bounded subset of a Hilbert space, a Chebyshev centre exists, is unique, and lies in the subset. By z M we denote the (unique) Chebyshev centre of a bounded subset M of a uniformly convex space or a Hilbert space, and we denote by r M = r(M) its Chebyshev radius (see Sect. 14.1). The following result is due to Ward [596] and Alvoni and Papini [27]. Proposition 15.10 Let H be a Hilbert space, let M = {x1, . . . , xn } ⊂ H, N = {y1, . . . , yn } ⊂ H, xi ≤ R, yi ≤ R, and xi − yi ≤ h, i = 1, . . . , n. Then " z M − z N ≤ 2 hR + h2 . (15.62) Moreover, if r M ≤ r N , then " z M − z N ≤ h + 5h2 + 2hr M + 2hr N .

(15.63)

Proof Let us prove that (15.62) implies (15.63) (see [27]). We set α := r M , β := r N , γ := z M − z N . We assume without loss of generality that z M = 0 and let z N = c. Case 1. Let γ ≤ β − α. Then " c − xi ≤ γ + α ≤ β, c − yi ≤ β. As a corollary, from (15.62) we have γ ≤ 2 hβ + h2 , which gives (15.63) (here we can use the fact that 2hβ ≤ 2hα + 2h2 ). Case 2. Let γ > β − α. Setting p := γ−α+β 2γ c, we have, for i = 1, . . . , n, γ+α+β γ−α+β +α = , 2 2 γ+α+β γ−α+β p − yi ≤ p − c + c − yi ≤ +β= . 2 2 p − xi ≤ p + xi ≤

Using (15.62) (we recall that γ = c ≥ 0), this establishes  

γ + α + β γ≤ h + h2 = 2h(γ + α + β) + 4h2, 2 γ 2 − 2hγ − (2hα + 2hβ + 4h2 ) ≤ 0,

γ ≤ h + h2 + 2hα + 2hβ + 4h2, which proves (15.63).



Consider now the case of arbitrary bounded sets. The following result holds (for compact sets, see [537, Theorem 1], and for the general case, [77], [27]). Theorem 15.45 Let M and N be nonempty bounded subsets of a Hilbert space, and let z M and z N be their Chebyshev centres. Then

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15 ChebyshevCentre of a Set . . .

z M − z N ≤

" (h(M, N) + r M + r N ) h(M, N).

(15.64)

Remark 15.18 The estimate (15.64) is sharp: there exist pairs of sets (with the same Chebyshev radius) for which inequality (15.64) becomes an equality. It is clear that equality in (15.64) is also attained for singletons, but (15.64) ceases to be true in spaces in which the Chebyshev-centre map is not single-valued. Proof (of Theorem 15.45) Let h := h(M, N). Let L be the affine manifold passing through z M perpendicularly to the interval between z M and z N . The following result is known (see Theorem 15.66 below): a point y is a Chebyshev centre of a nonempty bounded subset A of a Hilbert space (y = z A) if and only if there exists ε0 > 0 such that   y ∈ conv a ∈ A | y − a ≥ r(y, A) − ε ∀ 0 < ε < ε0 . (15.65) By (15.65), for every n ∈ N there exists a point yn ∈ M such that z M − yn > r M − n1 , yn lies in the half-space with boundary L and not containing z N . As a corollary,  1 2 + z M − z N 2 . yn − z N 2 ≥ r M − n On the other hand, yn − z N ≤ r(z N , M) ≤ r N + h (see (15.56)), which gives that   1 2 1 2 ≤ (r N + h)2 − r M − . z M − z N 2 ≤ yn − z N 2 − r M − n n Letting n → ∞, we find that 2 z M − z N 2 ≤ (h + r N )2 − r M .

(15.66)

Swapping M and N, we get 2 z M − z N 2 ≤ (h + r M )2 − r N .

Now the required result follows if we add (15.66) and (15.67).

(15.67) 

The following strengthening of estimate (15.64) was obtained in [27] with the use of a technique in [553]. Theorem 15.46 Let M and N be nonempty bounded subsets of a Hilbert space, and let z M and z N be their Chebyshev centres. Suppose that r M ≤ r N . Then 2 z M − z N 2 ≤ (r M + h(M, N))2 − r N .

From (15.68) it follows that (see [27, Remark 3]) " z M − z N ≤ 2h(M, N) r N + h2 (M, N).

(15.68)

(15.69)

Remark 15.19 Estimate (15.68) can be seen as an estimate for the variation of the radius when the distance between the centres is known. For example (see [27, p. 431]), 2 ≤ r 2 + 2h(M, N) r , which implies that if z M − z N ≥ h(M, N), then r N M M

15.6 Stability of the Chebyshev-Centre Map

rM ≥



339

2 − h(M, N). h2 (M, N) − r N

Some other estimates for the distance between Chebyshev centres in a Hilbert space can be found in [481, Theorem 2.1.1], [69, Sect. 5], [302, Lemma 3.5.3]. Remark 15.20 Examples showing that the estimate (15.64) cannot be significantly improved have been known for a long time (see, for example, [481]). Namely, for every r > 0 and every ε ∈ (0, r), there exist closed convex sets A1, A2 ⊂ B(0, r) such that " ε = c1 − c2 ≥ 2r h(A1, A2 ), where ci is the Chebyshev centre of the set Ai , i = 1, 2. Indeed, let r > 0 and ε ∈ (0, r). In the Euclidean plane R2 (with Cartesian coordinates (x (1), x (2) )), consider two sets A1 and A2 defined as follows:   A1 := x ∈ R2 | 0 ≤ x (2) ≤ ε, (x (1) )2 + (x (2) )2 ≤ r 2 and the set A2 is symmetric to the set A1 relative to the line x (2) = ε/2. It is easily centres of A1 and seen that the points c1 = {(0, 0)}, c2 = {(0, ε)} are the Chebyshev √ A2 , respectively, and that the Hausdorff distance h(A1, A2 ) = r 2 + ε 2 −r 2 is at most ε 2 /(2r). As a result, " c1 − c2 ≥ 2r h(A1, A2 ). Theorem 15.46 shows that in a Hilbert space, the Chebyshev-centre map is Hölder continuous with exponent 1/2 uniformly on sets of fixed diameter. Simple examples can be constructed to show that this map is not Lipschitz continuous. We note the following result (see [27, Proposition 4], and also [205, Theorem 1]). Proposition 15.11 Let M ⊂ H be a nonempty bounded subset of a Hilbert space and let z be its Chebyshev centre. Then 2 + x − z 2 ≤ r 2 (x, M) for every x ∈ X rM

(here we recall that r(x, M) := inf{r ≥ 0 | M ⊂ B(x, r)}). For uniformly convex spaces we mention the following results. Theorem 15.47 (See [27], [77]) Let M and N be nonempty bounded subsets of a uniformly convex space X. 1. If r M ≤ r N , then   z M − z N r N ≤ (r M + h(M, N)) 1 − δ . (15.70) r M + h(M, N) 2. The following inequality holds:  z M − z N max{r M , r N } δ ≤ 1− h(M, N) + min{r M , r N } min{r M , r N } + h(M, N)  h(M, N) ≤ . min{r M , r N } + h(M, N)

(15.71)

340

15 ChebyshevCentre of a Set . . .

Here and below, δ(ε) is the modulus of convexity of a space (see Sect. 9.2). Another variant of estimate (15.71) was established in [553, Lemma 4]. " 2 For Hilbert spaces (for which δ(ε) = 1 − 1 − ε /4, 0 ≤ ε ≤ 2), Theorem 15.47 gives the estimate

2, z M − z N ≤ 2 (r M + h(M, N))2 − r N which is weaker than (15.68). The following result [27] strengthens (15.70). Theorem 15.48 (See [27]) Let M and N be nonempty bounded subsets of a uniformly convex space X, and let r M ≤ r N . Then   z M − z N + r M − h(M, N) − r N r N ≤ (r M + h(M, N)) 1 − δ . (15.72) r M + h(M, N) Proof (See [27]) Let γ := z M − z N , z M = 0, c := z N . We set c  := c + η cc , where η := r M − h(M, N) − r N (then 0 − c  = γ + η). Thus N lies in the ball B(0, r M + h(M, N)) and in the ball B(c , r N + η) = B(c , r M + h(M, N)). So if x ∈ M, then the points 0 and c  lie in the ball B(x, r M + h(M, N)). Since 0 − c  = γ + η and since the space is uniformly convex, we have    x − (0 + c ) γ+η ≤ (r M + h(M, N)) 1 − δ . 2 r N + h(M, N) 

 

γ+η As a corollary, N ⊂ B 0+c 2 , r M + h(M, N) 1−δ r N + h(M, N ) , the result required. For analogues of the estimate in Theorem 15.47 and for its sharpness in puniformly convex spaces, see [595], [401]. We mention another result (see [77]). Theorem 15.49 Let X be a uniformly convex space and let M and N be bounded subsets of X. Let c := z M − z N , where z M and z N are the Chebyshev centres of the sets M and N, respectively, and let γ := max{r M + h, r N + h}, where h := h(M, N). Suppose that 0  c ≤ 2γ. Then c ≤h (15.73) γδ γ and c ≤ γε

h γ

In Theorem 15.49,

.

   x + y  ε(δ) := sup x − y  x ≤ 1, y ≤ 1, ≥ 1−δ 2 

is the inverse of the modulus of convexity δ(ε).

(15.74)

15.6 Stability of the Chebyshev-Centre Map

341

Remark 15.21 The estimates in Theorem 15.49 are quite rough in Hilbert spaces. For example, if z M − z N ≤ 2h, then the equality in (15.74) is impossible (see [77, Remark 4.2]). It is unknown whether the estimates (15.73) and (15.74) are sharp in the class of uniformly convex spaces. For L q -spaces with 1 < q < ∞, Prus and Smarzewski (see Theorem 4.1 in [488]) established the following result. Theorem 15.50 Suppose that for some q ≥ 2, there exists a positive constant C such that (0 < ε ≤ 2). δX (ε) ≥ C ε q Let M ⊂ X and Z M = {x} (or Zsc M = {x}). Then r(x, M) ≤ r(y, M) − k x − y q

for every y ∈ M,

where the constant k depends only on q and C. We also mention the following characterization of Hilbert spaces from [27]. Theorem 15.51 Let X be a Banach space. Then the norm on X is Euclidean if and only if the inequality 2 z M − z N 2 ≤ (r M + h(M, N))2 − r N

for r M ≤ r N

(15.75)

is satisfied for every pair of sets M, N ⊂ X that admit Chebyshev centres. To prove Theorem 15.51, we require the following auxiliary result (see, for example, [32, Proposition 6.9]). Proposition 15.12 The norm on a Banach space X is Euclidean if and only if 

x − y 2 + x + y 2 ≥ 2 x 2 + y 2 (15.76) for every x, y ∈ X such that x − y = x + y . Proof (of Theorem 15.51) It suffices to consider the case of a strictly convex space X (since otherwise, a Chebyshev centre is not unique and (15.75) has no sense). For strictly convex spaces, Theorem 15.22 shows that every finite set has at most one Chebyshev centre. If the norm on X is not Euclidean, then there exist two points x, y such that x − y = x + y , but (15.76) is not satisfied; that is,

 x − y 2 + x + y 2 > 2 x 2 + y 2 . Replacing if necessary x, y by t x, t y, we can assume that x ≤ y = 1. Let x + y √ = x − y =: σ. We have√2 x 2 > 2σ 2 − 2 y 2 = 2(σ 2 − 1), and hence x > σ 2 − 1, which givesσ < 2.  x−y  x−y and N := x+y Consider the sets M := −y, y, x, x+y 2 , 2 2 , 2 , x, x −  σ σ σ y, x + y . It is clear that r = 1, Z(M) = {0} =: {z }, r = M M N 2 2 2 (≤ r M ), Z(B) = {x} =: {z N }. We have

342

15 ChebyshevCentre of a Set . . .



 σ σ 2 + z M − z N = x > −1= −1 2 2

2

2 , ≥ r N + h(M, N) − r M "

σ2

which completes the proof of Theorem 15.51.



15.6.6 Stability of the Self-Chebyshev-Centre Map The stability problem for the self-Chebyshev-centre map was first examined by Borwein and Keener [124].  and r  (M) the self Chebyshev Given a nonempty bounded set M, we denote by z M centre and self Chebyshev radius of M, respectively (see (15.4), (15.3)). Let M be the class of nonempty closed convex bounded sets with the property ˚  , r  ) = for every distinct M, N ∈ M with Zsc = {z  } and ˚  , r  ) ∩ B(z that B(z M M N N M M sc   Z N = {z N } (here, as before, Zsc M = {z M } is the set of self Chebyshev centres of a set M; see (15.4)). Theorem 15.52 (J. Borwein and L. Keener [124]) Let X be a Hilbert space and M, N ∈ M . Then √ 1+ 5   h(M, N). z M − z N ≤ 2

15.6.7 Upper Semicontinuity of the Chebyshev-Centre Map and the Chebyshev-Near-Centre Map Recall that a map F : X → 2Y is upper semicontinuous at a point x0 if for every open set V ⊂ Y with F(x0 ) ⊂ V, there exists a neighbourhood O(x0 ) such that F(x) ⊂ V for every x ∈ O(x0 ). A map F is upper semicontinuous on X if it is upper semicontinuous at every point x0 ∈ X. Theorem 15.53 (P. K. Belobrov [89]) Let X be a Efimov–Stechkin space,6 and let the sequence of sets Mn ⊂ X converge to a compact set M in the Hausdorff metric. Then for every ε > 0, Z Mn ⊂ Oε (Z M ) for n starting from some N ∈ N. For extensions of Theorem 15.53, see Theorem 15.55 below. 6 A Banach space X is a Efimov–Stechkin space if for every x n ∈ S and f ∈ S ∗ such that f (x n ) → 1, the sequence (x n ) has a convergent subsequence (see Sect. 9.1 and [24]). Efimov–Stechkin spaces are also called reflexive Kadec–Klee spaces.

15.6 Stability of the Chebyshev-Centre Map

343

Remark 15.22 In the proof of Theorem 15.53, Belobrov [89] also showed that in a Efimov–Stechkin space, the set of Chebyshev centres of a compact set is compact. Remark 15.23 The relation Z M ⊂ Oε (Z Mn ), which is the reverse of the inclusion in Theorem 15.53, does not hold in general even in finite-dimensional Banach spaces. In this regard, consider the following example (see [89], [383, example 3.2]). Let a1 and a2 be opposite points on a circle O in R3 . By h1 and h2 we denote closed intervals of equal length having midpoints a1 and a2 and being perpendicular to the plane of the circle O. Let B be the convex hull of O and of the intervals h1 and h2 . Consider the sets Mn = {xn , xn }, where xn and xn are opposite points on the circle O such that xn → a1 and xn → a2 in the norm of the space with unit ball B, xn  a1 , xn  a2 , n = 1, 2, . . . . Each set Mn , n = 1, 2, . . . , has a unique Chebyshev centre (the centre of the circle O). On the other hand, every point on the diameter of B parallel to the interval h1 (or h2 ) is a Chebyshev centre of M. Definition 15.21 Given δ > 0, a bounded set M ⊂ X, and a nonempty closed subset V of X, we define the set of relative Chebyshev δ-centres of M by   ZVδ (M) := y ∈ V | r(y, M) ≤ rV (M) + δ . (15.77) Definition 15.22 Let V ⊂ X be a nonempty closed convex set, and let F (X) be a family of closed bounded subsets of X with nonempty sets of relative Chebyshev centres: ZV (M)  for every M ∈ F . Following [412], [383], we say that a pair (V, F ) has the property (P1 ) if for every M ∈ F and every ε > 0, there exists δ > 0 such that ZVδ (M) ⊂ ZV (M) + εB. Property (P1 ) was introduced by Mach [412] and further studied in [383]. Remark 15.24 Mach [412] found some pairs (V, F ) with property (P1 ). For example, a pair (V, F ) (where F is the class of closed bounded subsets of X) has property (P1 ) in the following cases: (1) X is a Banach space and V ⊂ X is a finite-dimensional subspace; (2) X =  1 and V is a w ∗ -closed convex subset of X; (3) X is a uniformly convex space and V is a closed bounded convex subset of X; (4) X is a Lindenstrauss space (that is, a space predual to L 1 (μ)), V is an M-ideal in X, and F is the class of nonempty compact sets in X. According to [383, Theorem 2.5], if X ∈ (CLUR),7 V is a closed convex bounded subset of X, and F = K (X) is the class of nonempty compact sets in X, then the pair (V, F ) has property (P1 ). Remark 15.25 Note that property (P1 ) does not in general imply the continuity of the operator ZV ( · ). Indeed, finite-dimensional subspaces with not lower semicontinuous metric projection can be constructed in many (even finite-dimensional) spaces. Let V be such a subspace and let F p be the class of singletons in X. According to 7 X ∈ (CLUR) if the conditions x, x n ∈ S(X) and x + x n /2 → 1 imply that (xk ) has a convergent subsequence.

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15 ChebyshevCentre of a Set . . .

Remark 15.24, the pair (V, F p ) has property (P1 ), but the (relative) Chebyshevcentre map {x} → ZV ({x}) = PV (x) is not continuous. Mach [412, Theorem 5] proved the following result. Theorem 15.54 (J. Mach) Let X be a Banach space and let (V, F ) be a pair with the property (P1 ). Then the map M −→ ZV (M),

M ∈ F,

is upper semicontinuous in the Hausdorff metric. The next result (see [383, Theorem 2.3]) extends Theorem 15.53. Theorem 15.55 The following conditions on a Banach space X are equivalent: (a) X is a Efimov–Stechkin space; (b) if V is a nonempty closed convex subset of X and F = K (X) is the class of nonempty compact sets in X, then the pair (V, F ) has property (P1 ). Proof (of Theorem 15.55) (a) ⇒ (b). Assume that property (P1 ) does not hold for some pair (V, K (X)). Then there exist ε > 0, M ∈ K (X), δn > 0, δn → 0, and xn ∈ ZVδn (M) such that ρ(xn, ZV (M)) > ε for all n ∈ N. Note that r(xn, M) ≤ rV (M) + δn and xn ∈ V for all n. Since the sequence (xn ) is bounded and since X is reflexive, (xn ) contains a weakly convergent subsequence (xnk ). Let x ∈ V be a w-cluster point of the subsequence. Then for every y ∈ M, x − y ≤ lim xnk − y ≤ lim r(xnk , M) = rV (M). k→∞

k→∞

This implies that r(x, M) ≤ rV (M) and hence x ∈ ZV (M). Let a point y0 ∈ M be such that x − y0 = r(x, M) = rV (M). It is clear that xnk − y0 → x − y0 . We have rV (M) = x − y0 ≤ lim xnk − y0 ≤ lim xnk − y0 ≤ rV (M). k→∞

k→∞

As a corollary, xnk − y0 → x − y0 . Since by the hypothesis, X is a Kadec–Klee space (see Sect. 9.1), we have xnk → x. Next, x ∈ ZV (M), and so xnk → ZV (M)+εB for large n, a contradiction. (b) ⇒ (a). If property (b) holds, then for every x ∈ X, a convex closed V ⊂ X, and ε > 0, there exists δ > 0 such that ZVδ ({x}) ⊂ ZV ({x}) + εB. This implies that PVδ (x) := {y ∈ V | x − y ≤ ρ(x, V) + δ} ⊂ PV x + εB. Now from Theorem 1.4 of [211] it follows that X is a Efimov–Stechkin space.  The relative Chebyshev δ-centre map ZVδ (see (15.77)) can be shown to have stronger continuity properties without assumptions on the space structure or on the properties of the pair (V, F ). The following result holds [567], [25].

15.6 Stability of the Chebyshev-Centre Map

345

Theorem 15.56 (I. G. Tsar’kov) If V  is a closed convex subset of a Banach space X and δ > 0, then the Chebyshev δr(M)-centre map M → ZVδ r(M) (M) is Lipschitz continuous on the class of nonempty bounded subsets of X. Corollary 15.7 Let X be a finite-dimensional Banach space, let V  be a closed convex bounded subset of X, and let δ > 0. Then the Chebyshev δr(M)-centre map M → ZVδ r(M) (M) has a continuous Lipschitz selection. The following property (P2 ) (see [412], [383]), which is stronger than (P1 ), gives a sufficient condition for the continuity of the Chebyshev-centre map. Definition 15.23 Let V ⊂ X be a nonempty closed convex set, and let F (X) be a family of closed bounded subsets of X with nonempty sets of relative Chebyshev centres: ZV (M)  for every M ∈ F . A pair (V, F ) is said to have property (P2 ) if for every ε > 0, there exists δ > 0 such that ZVδ (M) ⊂ ZV (M) + εB

(15.78)

for every M ∈ F . Theorem 15.57 (see [412], [383]) Let V be a closed bounded subset of X, and let F be a family of nonempty closed bounded subsets of X. Suppose that the pair (V, F ) has property (P2 ). Then the Chebyshev-centre map M → ZV (M) is uniformly continuous in the Hausdorff metric on the family F . The next two results (see [383]) extend Theorems 15.28 and 15.29. Theorem 15.58 The following conditions on a Banach space X are equivalent: (a) X is uniformly convex; (b) for every subspace V ⊂ X, every number α > 0, and the family F of nonempty closed bounded sets M ⊂ X with rV (M) ≤ α, the pair (V, F ) has property (P2 ), and the relative-Chebyshev-centre map ZV (M) is single-valued for every M ∈ F ; (c) for every subspace V ⊂ X the relative-Chebyshev-centre map M → ZV (M) is single-valued and uniformly continuous on the class of nonempty bounded subsets of the space X; (d) the Chebyshev-centre map M → Z(M) is single-valued and uniformly continuous on the class of nonempty bounded subsets of X. Theorem 15.59 Let V be a subspace of a Banach space X. Then the following conditions are equivalent:

346

15 ChebyshevCentre of a Set . . .

(a) X is uniformly convex with respect to V (see (15.54)); (b) for every α > 0, the pair (V, F ) has property (P2 ), where F consists of all nonempty closed bounded subsets M ⊂ X with rV (M) ≤ α, and the relativeChebyshev-centre map ZV (M) is single-valued for every M ∈ F . The next theorem describes the uniformly convex Banach spaces in terms of the uniform approximative stability of the Chebyshev near-centre map on the class of sets of Chebyshev radius 1. Theorem 15.60 (see [383]) The following conditions on a Banach space X are equivalent: a) X is uniformly convex; b) for every ε > 0, there exists δ > 0 such that ZVδ (M) ⊂ ZV (M) + εB for every closed subspace V ⊂ X and a bounded closed set M ⊂ X with rV (M) = 1. Let us briefly discuss the problem of convergence of Chebyshev nets. The following result holds. Theorem 15.61 (P. K. Belobrov) Suppose that a sequence (Mn )∞ n=1 of nonempty convex subsets of a Hilbert space H converges in the Hausdorff metric to a compact set M ⊂ H. Then there exists a sequence (Sn∗ )∞ n=1 of best N-nets for the sets Mn that contains a subsequence converging in the Hausdorff metric to some best N-net for the set M. Proof (of Theorem 15.61) Let S N = {y 1, . . . , y N } be a best net of cardinality N for a closed convex set L ⊂ H and let R be the radius of best covering of the set L by Nnets. We set Li = L ∩ B(y i, R). Let yˆi be a Chebyshev centre of Li . Since L is a subset of a Hilbert space H, we have yˆi ∈ Li . It is clear that the net SN = { yˆ1, . . . , yˆ N } is a best net for the set L and lies in this set. ∗ (n) = {y (n), . . . , y (n) } be a best N-net for the set M ⊂ H lying in this set. Let SN n N 1 Since the set M is compact and since the sequence (Mn ) converges in the Hausdorff metric to this set, every sequences (yi(n) )∞ n=1 , i = 1, . . . , N, is compact. (nk ) Now let a subsequence (yi ) of the sequence (yi(n) ), i = 1, . . . , N, be such that limnk →∞ yi(nk ) =: yi∗ exists. We claim that the net {y1∗, . . . , y ∗N } is a best net for M. For an arbitrary x ∈ M, there exists a point yx(k) such that x − yx(k) ≤ αnk . Hence we have min x − yi∗ ≤ αnk + sup

1≤i ≤ N

min ξ − yi(k) + yi∗ − yi(k)

ξ ∈Mn k 1≤i ≤ N

= αnk + Rnk + yi∗ − yi(k) , sup min x − yi∗ ≤ αnk + Rnk + yi∗ − yi(k) .

x ∈M 1≤i ≤ N

15.6 Stability of the Chebyshev-Centre Map

347

Fig. 15.3 The set of ‘hourglasses’ converges to the coordinate cross. Each set in this sequence has a Chebyshev 2-net (indicated by stars). But for the coordinate cross, a Chebyshev 2-net is a singleton (the origin).

Letting k → ∞ in the last inequality, we get sup min x − yi∗ ≤ R,

x ∈M 1≤i ≤ N

the result required.



Knowing that in a Euclidean space there is no continuous selection from the set of Chebyshev 2-nets, many who study the properties of Chebyshev nets were satisfied with this state of affairs and did not consider the problem of the existence of continuous selections from the set of Chebyshev nets. Druzhinin [209] investigated this problem for various spaces and showed that every selection from the set of Chebyshev n-nets for n ≥ 2 is discontinuous in every non-strictly convex Banach space. In addition, he proved the absence of a Lipschitz selection in an arbitrary Banach space of dimension ≥ 2 that has an exposed smooth point on the unit sphere.

Fig. 15.4 Example (similar to Fig. 15.3) showing the absence of a continuous selection of the set of Chebyshev 2-nets. The sequence of ‘coaxial propellers’ (‘Maltese crosses’) converges to the coordinate cross.

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15 ChebyshevCentre of a Set . . .

15.6.8 Lipschitz Selection of the Chebyshev-Centre Map As before, by B H (X) we denote the class of nonempty bounded subsets of a space X equipped with the Hausdorff semimetric. A map ϕ : B H (X) → X is called a selection of the Chebyshev-centre map Z( · ) if ϕ(M) ∈ Z(M) for every M ∈ B H (X). We first give the following simple result. Theorem 15.62 In X =  ∞ (Γ), the Chebyshev radius r(M) of every nonempty bounded set M ⊂ X is half the diameter of M. Moreover, the Chebyshev-centre map Z( · ) admits a 1-Lipschitz selection. Proof Let M ∈ B H (X). For each t ∈ Γ, we set  1 m(t) := inf x(t) + sup x(t) . 2 x ∈M x ∈M It is easily checked that the point m is a Chebyshev centre of the set M. Furthermore,   1 diam M = sup sup x(t) − inf x(t) . r(M) = diam M, 2 x ∈M t ∈Γ x ∈M The mapping ϕ(M) := m is the required 1-Lipschitz selection.



The problem of the existence of a Lipschitz selection of the Chebyshev-centre map has been investigated by Amir [31, Sect. 6.4], Amir, Mach, and Saatkamp [36, Sect. 4], Pai and Nowroji [457], Druzhinin [208], and others (see also the book [434]). We say that the Chebyshev-centre map Z admits a Lipschitz selection with constant θ if there exists a single-valued operator T that associates with every bounded set some (single) Chebyshev centre of this set and is such that T(M) − T(N) ≤ θ · h(M, N) for some θ > 0 and all nonempty bounded sets M and N, where as before,   h(M, N) := max sup inf x − y , sup inf x − y (15.79) x ∈M y ∈N

y ∈N x ∈M

is the Hausdorff distance between sets M and N. Above (see Remark 15.20), it was shown that in a Hilbert space, the (singlevalued) Chebyshev-centre map is not Lipschitz continuous. Recall that a point s of the unit sphere S of a normed space X is called a smooth point if the support hyperplane to S at this point is unique; a point s is an exposed point of the sphere S (or of the unit ball B) if there exists a hyperplane H supporting B at s such that H ∩ B = {s}. Theorem 15.63 (Yu. Yu. Druzhinin) If the unit sphere of a Banach space X has a smooth exposed point, then the Chebyshev-centre map Z( · ) does not have a Lipschitz selection.

15.6 Stability of the Chebyshev-Centre Map

349

Theorem 15.64 (Yu. Yu. Druzhinin) If X is a finite-dimensional Banach space, then the Chebyshev-centre map Z( · ) admits a Lipschitz selection if and only if X is polyhedral. For the existence of a Lipschitz selection in polyhedral spaces, see also Sect. 15.6.3. The problem of existence of a Lipschitz selection of the Chebyshev-centre map Z( · ) in C(Q), where Q is a compact Hausdorff space, has not been solved in the general case (cf. Corollary 15.6). However, if the Chebyshev-centre map Z( · ) is restricted to compact sets, then the existence of a Lipschitz selection of the map Z( · ) is fairly clear. Druzhinin [208] obtained the following partial answer in the problem of the existence of a Lipschitz selection of the Chebyshev-centre map Z( · ). Theorem 15.65 Let Q be a compact Hausdorff space with finitely many limit points. Then in the space C(Q), the Chebyshev-centre map Z( · ) has a Lipschitz selection.

15.6.9 Discontinuity of the Chebyshev-Centre Map For infinite-dimensional L 1 -spaces, the Chebyshev-centre map Z( · ) (see (15.1)) is not lower semicontinuous even on the class of two-point sets (as a consequence, Z( · ) does not have a continuous selection) [31]. It is well known that in a finite-dimensional space X, the metric projection onto a finite-dimensional subspace need not be continuous. Let V be such a subspace. According to Remark 15.25, the (relative) Chebyshev-centre map {x} → ZV ({x}) = PV (x) is not continuous; here we associate with a singleton {x} the set of relative (with respect to V) Chebyshev centres. A similar example of a discontinuous single-valued relative-Chebyshev-centre map ZV : X → V can be constructed in every space X containing a Chebyshev subspace V with discontinuous metric projection (and in particular, in  1 ; see [412, Sect. 2]). Note that in a finite-dimensional space, the Chebyshev-centre map is always upper semicontinuous (see Theorem 15.53). One can easily construct an example of a threedimensional space in which the Chebyshev-centre map is not lower semicontinuous (see [36], Example 2.5). Let B be the (symmetric) convex hull of the circle {e1 cos t + e2 sin t | 0 ≤ t ≤ 2π} and the closed interval {se1 + e3 | −1 ≤ s ≤ 1}, and let · be the norm generated by the body B. In other words, "  (1) (2) (3)  (x (2) )2 + (|x (1) | − |x (3) |)2 + |x (3) |, |x (1) | ≥ |x (3) |, (x , x , x ) = |x (1) | ≤ |x (3) |. |x (2) | + |x (3) |,

350

15 ChebyshevCentre of a Set . . .

We have Z({−e1, e1 }) = [−e3, e3 ] and Z({−e1 − ηe2, e1 + ηe2 }) = {0} for η  0, so the map M → Z(M) is not lower semicontinuous, and thus not continuous (see also Remark 15.23).

15.7 Characterization of a Chebyshev Centre. Decomposition Theorem We first mention one simple result for Hilbert spaces (see, for example, [35, Corollary 2.5], [178, Lemma 4], [447, Lemma 2], [27, Lemma 0]). Theorem 15.66 Let M be a bounded subset of a Hilbert space. Then a point z is a Chebyshev centre of M if and only if  z∈ conv {y ∈ M | y − z ≥ r(M) − ε}. ε>0

Remark 15.26 A similar characterization also holds in an arbitrary two-dimensional strictly convex space. We also mention two more characterizations of the relative Chebyshev centre in a Hilbert space [59]. Theorem 15.67 (V. S. Balaganskii) Let X be a Hilbert space, let Y ⊂ X be a nonempty closed convex set, let K ⊂ X be a nonempty closed convex bounded set, and let y ∈ Y and r := r(y, K). Then: (1) {y} = ZY (K) ⇐⇒ y ∈ PY (conv (K \ B(y, t))) ∀ 0 < t < r; (2) {y} = ZY (K) ⇐⇒ y ∈ PY (conv (K ∩ S(y, r))). The following result (Pichugov [471]) characterizes the Chebyshev centres and the Chebyshev radii in finite-dimensional Banach spaces. Theorem 15.68 (S. A. Pichugov) Let M be a closed convex subset of a finitedimensional Banach space of dimension n, and let r(M) = r. Then a point y is a Chebyshev centre of M (y ∈ Z(M)) if and only if there exists a natural number N ≤ n + 1 such that: (a) there are points xi in M, i = 1, . . . , N, such that xi − y = r; (b) there are functionals fi in (Xn )∗ , i = 1, . . . , N, such that ( fi, xi − y) = xi − y ,

fi = 1; n (c) there exist numbers αi , i = 1, . . . , n, with i=1 αi = 1 and αi ≥ 0 such that N i=1 αi fi = 0. Proof The convex function F(x) = max{ x − z | z ∈ M } has a minimum at a point y if and only if 0 ∈ ∂F(y) (see [299, p. 89, Proposition 1]), where ∂F(y) is the subdifferential of F at y. Now the required result follows from the finitedimensional decomposition theorem (see, for example, Appendix B.2). 

15.8 Chebyshev Centres That Are Not Farthest Points

351

15.8 Chebyshev Centres That Are Not Farthest Points Sain, Kadets, Paul, and Ray [505] considered the problem of when a Chebyshev centre of a bounded subset of a Banach space can be a farthest point of the subset. Following [505], we say that a set M is nontrivial if |M | ≥ 2. We first recall the definition of a farthest point of a set. Let x ∈ X and  M ⊂ X. A point y0 ∈ M is called a farthest point of M from the point x if x − y0 = sup{ x − y | y ∈ M } = r(x, M); that is, x − y0 ≥ x − y for every point y ∈ M. The set of all farthest points in M from a point x (the metric antiprojection, the max-projection) is denoted by F(x, M): F(x, M) = {y ∈ M | x − y = r(x, M)}. By Far M we denote the set of all points in M on which the supremum in the definition of r(x, M) is attained at some x ∈ X; that is, # Far M = F(x, M). x ∈X

Baronti and Papini (see Proposition 15.11 above) showed that if M ⊂ H is a nonempty bounded subset of a Hilbert space and z M is a Chebyshev centre of M, then 2 + x − z M 2 ≤ r 2 (x, M) for every x ∈ X. rM Hence in a Hilbert space, we have r(x, M) > x − z M , which implies that z M  Far M, where z M is the unique Chebyshev centre of a nontrivial bounded subset M of H. Remark 15.27 It is clear that if X is not strictly convex, that is, the unit sphere of X contains a nontrivial interval M, then all points in M lie at distance 1 from the origin, and hence M = Far M. In particular, the Chebyshev centre (u + v)/2 of M belongs to Far M. This observation led Sain to the following question: in a strictly convex Banach space, can a Chebyshev centre of a bounded nontrivial set be a farthest point of this set from a point (cf. Remark 15.27)? The answer to this question turns out to be positive. Definition 15.24 Following [505], we say that M is a CCF-set if there is a Chebyshev centre for M lying in Far M.8 Correspondingly, M is an NCCF-set if M  (CCF). A space X is said to lie in the class (CCF) if it contains a nontrivial CCF-set; X lies in (NCCF) if X  (CCF), that is, every nontrivial subset of X is an NCCF-set. 8 ‘CCF’ is an abbreviation for ‘a Chebyshev centre lies in Far M’.

352

15 ChebyshevCentre of a Set . . .

From Remark 15.27 it follows that if X ∈ (NCCF), then X is strictly convex. Theorem 15.70 gives the converse result: each two-dimensional strictly convex Banach space lies in the class (NCCF). This ceases to be true if the dimension of the space is greater than two (see Example 15.5 in the finite-dimensional case and Example 15.6 in the infinite-dimensional setting). Sain, Kadets, Paul, and Ray [505] characterized the NCCF-spaces as follows. In the next theorem, rt,z denotes the Chebyshev radius of the set At,z := B ∩ B(z, t). Theorem 15.69 Let X be a Banach space. Then the following three conditions are equivalent: (a) X ∈ (NCCF); (b) the inequality rt,z < t holds for every z ∈ S and t ∈ (0, 1]; (c) for every ε ∈ (0, 1], there exists t0 ∈ (0, ε) such that rt,z < t for all z ∈ S and t ∈ (0, t0 ]. The proof of Theorem 15.69 requires the following three lemmas from [505]. Lemma 15.3 Let M be a nontrivial bounded subset of a Banach space X, x ∈ Far M. Then for every R > 0, there exists a point y ∈ X such that x is a farthest point of M from y and x − y > R. Proof (of Lemma 15.3) By definition of Far M, there exists a point z ∈ X such that x − z ≥ a − z for all a ∈ M. We claim that for every t > 1, the point x is a farthest point of M from every point in the set tz + (1 − t)x. Indeed, for every a ∈ M, (tz + (1 − t)x) − a ≤ tz + (1 − t)x − z + z − a ≤ (t − 1) z − x + z − x = t z − x = (tz + (1 − t)x) − x . Note that tz + (1 − t)x = t(z − x) + x ≥ t z − x − x → ∞ as t → ∞. So for sufficiently large t, the point y = tz + (1 − t)x is the required one.  Lemma 15.4 Let M be a nontrivial bounded subset of a Banach space X. Assume that some point z M ∈ M is a farthest point of M from some y ∈ X. Let r be the Chebyshev radius of M and let R := z M − y . Then r ≤ R, and the set U := B(z M , r) ∩ B(y, R) has the following properties: (a) M ⊂ U; (b) the Chebyshev radius of the set U is r; (c) z M is a Chebyshev centre of the set U; (d) z M is a farthest point of U from y. Proof The embeddings M ⊂ B(z M , r) and M ⊂ B(y, R)

(15.80)

follow, respectively, from the definition of a Chebyshev centre and a farthest point. As a corollary, (a) holds by (15.80); the Chebyshev radius r of the set M cannot exceed R. From (a) we have r(U) ≥ r, and the embedding

15.8 Chebyshev Centres That Are Not Farthest Points

353

U ⊂ B(z M , r)

(15.81)

implies the converse embedding, which proves (b). Using (b), we see that (15.81) implies (c). Finally, (d) is secured by the embeddings z M ∈ M ⊂ U and U ⊂ B(y, R).  The next result [505] shows that in the two-dimensional setting, the class (NCCF) coincides with the class of strictly convex Banach spaces. This ceases to be true for dimension ≥ 3. Theorem 15.70 A two-dimensional Banach space X is strictly convex if and only if every nontrivial bounded set M ⊂ X that contains some Chebyshev centre of M is an NCCF-set. Example 15.5 (see [505]) On the space X = (R n, · ), n ≥ 3, consider the norm $ % n n  1  (i) 2 (1) (2) (n) (i) (x , x , . . . , x = |x | + |x | . 2 i=1 i=1 This norm is strictly convex. By Theorem 15.22, every bounded set has a unique Chebyshev centre. Let {e1, . . . , en } be the standard basis for R n and let θ := (0, 0, . . . , 0). Consider the set M := {θ, e1, e2, . . . , en } and put z = (1, 1, . . . , 1) ∈ √R n . We claim that θ ∈ Z(M) and that θ is a farthest point for M. We have z − ek = (n − 1) + (n−1) for all k = 1, . . . , n. However, 2 √

n > (n − 1) + z − θ = n + 2

√

(n − 1) , 2

that is, θ is a farthest point from M to z. Now let us show that {θ} = Z(M). If (a(1), a(2), . . . , a(n) ) ∈ R n is a Chebyshev centre for M, then by symmetry, all cyclic permutations of coordinates (a(2), a(3), . . . , a(n), a(1) ), . . . , (a(n), a(1), . . . , a n−1 ) also give a Chebyshev centre for M. Since Z(M) is convex by Proposition 15.1, the point (α, α, . . . , α) is also a Chebyshev centre for M, where α = (a(1) + a(2) + . . . + a(n) )n−1 . The uniqueness of a Chebyshev centre implies that the set of Chebyshev centres of M reads as sz, s ∈ R. We have a − θ = 3/2 for all a ∈ M \ {θ}, and so it suffices to show that for every s ∈ R, there exists p ∈ M such that sz − p ≥ 3/2. If s ≥ 1 or s < 0, then the required assertion holds, respectively, with p = θ and p = e1 . Let 0 < s < 1. Setting p = e1 , we have 2s > s ⇒ (1 − 2s) < (1 − s)

" ⇒ 1 − (2s + 2s − 2s) < (1 − s) ≤ (1 − s)2 + s 2 + . . . + s 2 " ⇒ (s + s − s) + (1 − s)2 + s 2 + . . . + s 2 /2 > 1/2 " ⇒ (1 − s) + s + . . . + s + (1 − s)2 + s 2 + . . . + s 2 /2 > 3/2 ⇒ sz − e1 > 3/2,

that is, θ is a Chebyshev centre of the set M.

354

15 ChebyshevCentre of a Set . . .

Remark 15.28 When applied to the set M from Example 15.5, Lemma 15.4 shows that there exists a finite-dimensional strictly convex normed space X containing a nontrivial compact convex CCF-subset. Definition 15.25 A set M is said to be centrable if r(M) =

1 diam M . 2

We give a similar example (see [505]) of a centrable subset of an infinite-dimensional strictly convex space. Theorem 15.71 (see below) shows that such an example is impossible in the finitedimensional case. Example 15.6 As X, consider the space c0 with the norm $ %∞  1 (k) x = max |x | + |x (k) | 2 . k 4k k=1

(15.82)

It is easily checked that such a norm is strictly convex. We set θ = (0, 0, . . . , 0, . . .), en = (0, 0, . . . , 0, 1, 0 . . .), where 1 is at the nth place. We define x n :=

 1 1 e1 + 1 − e n, n n

yn =

 1 1 e1 − 1 − en n n

and consider the set M := {θ} ∪ {x n | n = 2, 3, . . . } ∪ {yn | n = 2, 3, . . . }. We claim that M lies in the unit ball, which implies that r(M) ≤ 1. Indeed, we have   1 1 1  1 2 + . 1 − + x n = yn = 1 − n n 4n2 4 n Since

1 4n

≤

1 4n 2

for all n = 2, 3, 4, . . ., we have   1 1 1 2 + x n = yn ≤ 1 − < 1 − + = 1. n n n 4n2

So M ⊂ B(0, 1). Further, we have   1 1 lim x n − yn = lim 2 1 − en ≥ lim 2 1 − = 2. n→∞ n→∞ n→∞ n n As a corollary, diam M ≥ 2. Further, since r(M) ≥ 12 diam M, we have r(M) = 1. So θ is a Chebyshev centre of the set M. Now let us show that θ is a farthest point of M for the point u = e1 . We have e1 − θ = e1 = 32 . On the other hand,  1 e1 − x n = e1 − yn = 1 − e1 ± en n   1  1 1  1+ + n = 1− n 4 4  3 3 1  3 1 1 1 3 = + n − < , + − ≤ 1− n 2 2n 2 2 2n n2 n 2 that is, θ is a farthest point from the set M to the point e1 . Theorem 15.71 (See [505]) Let X be a uniformly convex Banach space, and let M be a nontrivial bounded centrable subset of X containing a Chebyshev centre z M of itself. Then M is an NCCF-set.

15.8 Chebyshev Centres That Are Not Farthest Points

355

Proof Let r > 0 be the Chebyshev radius of the set M. By definition of a centrable set, there exist points un, vn ∈ M, n = 1, 2, . . ., such that lim un − vn = 2r .

n→∞

Let

(15.83)

1 1 (un − z M ), yn = (z M − vn ). r r x n + yn = 2, and hence since X is uniformly convex, we have xn =

Then x n, yn ∈ B, lim n→∞

lim x n − yn = 0,

n→∞

which implies lim n→∞ un + vn − 2z M = 0. In other words, u n + vn → 2 z M . Let z M be a farthest point of M for some point y ∈ X. We set R := z M − y and define x˜ n =

1 (un − y), R

y˜ n =

1 (vn − y). R

We have x˜ n, y˜ n ∈ B, lim x˜ n + y˜ n =

n→∞

1 1 lim un + vn − 2y = 2z M − 2y = 2. R n→∞ R

Since X is uniformly convex, we see that x˜ n − y˜ n → 0, that is, un −vn → 0, which contradicts (15.83). Theorem 15.71 is proved.  Remark 15.29 Example 15.6 shows that the uniform convexity condition in Theorem 15.71 cannot be replaced by the strict convexity condition. Definition 15.26 (see [505]) Let x ∈ X. A sequence (a n ) ⊂ M is said to be maximizing (minimizing) if x − a n → r(x, M) (respectively, x − a n → ρ(x, M)). A point x ∈ X is a point of max-approximative compactness (a point of min-approximative compactness) of a set  M ⊂ X if every maximizing (minimizing) sequence in M contains a subsequence converging to a point in M. A set M is said to be max-approximatively compact (min-approximatively compact) if every point x ∈ X is a point of max-approximative compactness (min-approximative compactness) for the set M. In these terms, min-approximative compactness is the classical approximative compactness (see Sect. 4.1 and [24]). Theorem 15.72 (see [505]) Let X be a strictly convex Banach space, and let M ⊂ X be a nontrivial bounded centrable max-approximatively compact set containing a Chebyshev centre z M of itself. Then M is an NCCF-set. Remark 15.30 Example 15.6 shows that the max-approximative compactness assumption in Theorem 15.72 cannot be dropped. Lemma 15.5 (See [505]) Let X be a Banach space and let M be a nontrivial bounded centrable max-approximatively compact set containing (some) Chebyshev centre z M of itself. Then M attains its diameter. Proof (of Lemma 15.5) Since diam M = sup a − b = 2 r(M), a, b∈ M

356

15 ChebyshevCentre of a Set . . .

there exist sequences (x n ), (yn ) from the set M such that x n − yn → 2 r(M). We claim that the sequence (x n ) is a maximizing sequence for z M . Indeed, if this were not true, then there would exist ε0 > 0 and a subsequence (x n k ) such that z M − x n k ≤ r(M) − ε0 . Hence x n k − yn k = (x n k − z M ) + (z M − yn k ) ≤ (x n k − z M ) + (z M − yn k ) ≤ r(M) − ε0 + r(M) = 2 r(M) − ε0, which contradicts the fact that x n − yn → 2 r(M). A similar analysis shows that (yn ) is a maximizing sequence from M for z M . Since M is max-approximatively compact, there exist a subsequence (nk ) ⊂ N and points x, ˜ y˜ ∈ M such that x n k → x˜ and yn k → y. ˜ We have diam M = lim x n − yn = lim x n k − yn k = x˜ − y˜ . n→∞

k→∞



So the set M attains its diameter.

Proof (of Theorem 15.72) We assume to the contrary that M is a CCF-set. Then there exists a Chebyshev centre z M ∈ Far M. By definition, there exists a point x ∈ X such that z M ∈ F(x, M). We set R = x − z M = sup x − a . a∈ M

By Lemma 15.5, M attains its diameter, and further, since M is centrable, diam M = 2 r(M). This means that there exist a1, a2 ∈ M such that a1 − a2 = sup a − b = 2 r(M). a, b∈ M

(15.84)

We claim that z M − a1 = z M − a2 = r(M). It is clear that z M − a1 ≤ r(M) and z M − a2 ≤ r(M). Moreover, the assumption that both these quantities are strictly smaller than r(M) leads to a contradiction: 2 r(M)= a1 −a2 = a1 −z M +z M −a2 ≤ a1 −z M + a2 −z M 2 lies in the class (CCF). Since the spaces L1 and L∞ are not strictly convex, by the above, they do not lie in the class (CCF). The next result is auxiliary. Let e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1). p

Proposition 15.13 (See [505]) For the set A0 = {e1, e2, e3 } ⊂ 3 , the set of its Chebyshev centres is the point x p = (s p , s p , s p ), where s p = 1/(1 + 21/(p−1) ). p

Proof Since 3 is uniformly convex, the set A0 has a unique Chebyshev centre. By symmetry, it has the form (s, s, s). It remains to minimize the quantity f (s) = ek − (s, s, s) p = |1 − s | p + 2|s | p ,

s ∈ R.

It is clear that the minimum is attained on the interval (0, 1) (for otherwise, we would have f (s) ≥ 1), where f  (s) = 2ps p−1 − p(1 − s) p−1 ; moreover, s p is a unique root of the equation f  (s) = 0.  p

As before, we set A p = {e1, e2, e3, x p } ⊂ 3 (where x p is the point from the statement of Proposition 15.13). Proposition 15.14 (See [505]) For p ∈ (1, 2) ∪ (2, ∞), the set A p is a CCF-set. As a consequence, p 3 ∈ (CCF). Proof The set A p is the union of A0 with its Chebyshev centre x p , so that x p is a Chebyshev centre also for the set A p . It remains to show that x p ∈ Far A p . There are two cases to consider. Case 1: p ∈ (1, 2). In this case, we have 0 < sp
1, the point x p is a farthest point in A p for y = (t, t, t). The distance from y to each of the vectors ek is ((t − 1) p + 2t p )1/ p , y − x p = 31/ p (t − s p ), and so it suffices to verify that the inequality p

(t − 1) p + 2t p < 3 t − s p holds for large t. Dividing by t p and setting τ = 1t , we get (1 − τ) p + 2 < 3(1 − s p τ) p

(15.86)

for sufficiently small positive τ. At the point τ = 0, the left-hand side of (15.86) is equal to the right-hand side. So to verify (15.86) for τ that are close to zero, it suffices to check the inequality f1 (0) < f2 (0) for f1 (τ) = (1 − τ) p + 2, f2 (τ) = 3(1 − s p τ) p . This inequality is a consequence of the inequality −p < −3ps p , which follows from Corollary (15.87). Case 2: p ∈ (2, ∞). In this case,

358

15 ChebyshevCentre of a Set . . .

1 (15.87) . 3 We claim that for sufficiently large t > 0, the point x p is a farthest point in A p for y = (−t, −t, −t). The distance from y to each of the points ek is ((t + 1) p + 2t p )1/ p , y − x p = 31/ p (t + s p ). Hence we need to show that the inequality p

(t + 1) p + 2t p < 3 t + s p sp >

is satisfied for all sufficiently large t. As before, we need to verify that (1 + τ) p + 2 < 3(1 + s p τ) p for small τ > 0. Let g1 (τ) = (1+τ) p +2, g2 (τ) = 3(1+s p τ) p . We need to show that g1 (0) < g2 (0), that is, that p < 3ps p . But this inequality follows from (15.87).  Theorem 15.75 (See [505]) Let (Ω, Σ, μ) be a finite or σ-finite measure space containing a disjoint 3 triple {Δi }i=1 ⊂ Σ of subsets of finite positive measure. Then L p = L p (Ω, Σ, μ) ∈ (CCF) for every p ∈ (1, 2) ∪ (2, ∞). 3 Proof We set fi = 1Δi / 1Δi , i = 1, 2, 3, E = lin { fi }i=1 ⊂ L p . It is well known (and can be p p checked easily) that E is isometric to 3 , where the corresponding isometry T : 3 → E is as follows: T (x1, x2, x3 ) = x1 f1 + x2 f2 + x3 f3 .

It is also well known that E is 1-complemented in L p , where the corresponding projection P : L p → E is given by ∫ 3  f dμ Δi 1Δi Pf = μ(Δ i) i=1 (the equality P = 1 follows from Hölder’s inequality). Let A p be the set from Proposition 15.14. If we consider T (A p ) as a subset of E, then T x p is its Chebyshev centre, because T is an isometry. Since E is 1-complemented in L p , T x p is also p a Chebyshev centre of the set T (A p ), where T (A p ) is considered as a subset of L p . Let y ∈ 3 be a point for which x p ∈ F(y, A p ). Since T is an isometry, T x p is a farthest point of T (A p ) from T y. This means that the Chebyshev centre T x p of the set T (A p ) ⊂ L p is a farthest point. Theorem 15.75 is proved. 

15.9 Smooth and Continuous Selections of the Chebyshev-Near-Centre Map Let V ⊂ X be a nonempty convex set, and let  M ⊂ X be a bounded set. Consider the following sets of relative Chebyshev near-centres: ε ZV (M) := {y ∈ V | r(y, M) ≤ r(M) + ε }, ε Z˚ V (M) := {y ∈ V | r(y, M) < r(M) + ε }

(the quantities r(y, M) and r(M) were defined in Sect. 15.1). In the proof of the next result, Z ε (M) := Z εX (M) and Z˚ ε (M) := Z˚ εX (M) (M ⊂ X). Theorem 15.76 The map M → Z ε (M) admits a continuous selection for every ε > 0.

15.9 Smooth and Continuous Selections of the Chebyshev-Near-Centre Map

359

Proof We set

δ(y) := ε + r(M) − r(y, M). It is easily seen that δ(y) > 0 for y ∈ Z˚ ε (M). We claim that the mapping M → Z˚ ε (M) is lower semicontinuous. To verify this claim, it suffices to show that if Mn → M in the Hausdorff metric, then r(y, Mn ) < r(M) + ε. Indeed, if Mn → M and y ∈ Z˚ ε (M), then there exists N ∈ N such that y ∈ Z˚ ε (Mn ) for every n ≥ N . Hence M → Z˚ ε (M) is lower semicontinuous. Since Z˚ ε (M), Z ε (M) are convex sets, by the classical Michael’s selection theorem (see, for example, [481, Sect. 2.9]), the mapping M → Z˚ ε (M) has a continuous selection as a lower semicontinuous mapping on the metric space of bounded sets with Hausdorff metric. Since Z˚ ε (M) is a body, the mapping M → Z ε (M) also has a continuous selection.  ε (M) is lower semiRemark 15.31 If V is a nonempty closed convex set, then the map M → ZV continuous and admits a continuous selection for every ε > 0 (see also Theorem 15.56 above).

The next result shows that the set of Chebyshev near-centres has a Lipschitz selection in the space C(Q). Theorem 15.77 Let X = C(Q), where Q is a compact metric space. Then there exists a 2-Lipschitz map ϕ : B H (X) → X such that M ⊂ B(ϕ(M), diam M)

for every M ∈ B H (X).

Proof It is known that C(Q) can be isometrically embedded in the space  ∞ (Q). By Theorem 15.62, there is a 1-Lipschitz selection ψ of the Chebyshev-centre map in the space Y =  ∞ (Q). According to the Lindenstrauss–Kalton theorem (see [316], Theorem 3.5), there is a 2-Lipschitz retraction π from  ∞ (Q) onto C(Q), where Q is a compact metric space. It is easily seen that ϕ = π ◦ ψ is the required map.  Tsar’kov [557] investigated the existence of smooth selections of the Chebyshev near-centre map. Let X be a Banach space. For an arbitrary body M ⊂ X, consider the quantity q(M) := sup{a ≥ 0 | B(x, a) ⊂ M }. Given τ > 1, we denote by N(X, τ) the metric space of all closed convex bodies M ⊂ X such that diam M = 1

and

r(M) ≤ τ. q(M)

The space N(X, τ) ⊂ H (X) is equipped with the standard Hausdorff metric. For metric spaces (X1, ρ1 ) and (X2, ρ2 ) and a map ϕ : X1 → X2 , the modulus of continuity is defined by   ω(ϕ, δ) := sup ρ2 (ϕ(x), ϕ(y)) | ρ1 (x, y) ≤ δ , δ ≥ 0. Given γ ∈ [0, 1], we let UCγ denote the class of all maps ϕ : N(X, τ) → X such that ω(ϕ, δ) = o(δ γ ) for δ → 0+. The theorems in this section are due to Tsar’kov [557]. The following result shows that better smoothness for a selection of the Chebyshev near-centre map cannot be achieved if Chebyshev near-centres are considered near the corresponding sets. Theorem 15.78 Let X be an infinite-dimensional Banach space. Then for every τ > 1 and ε ∈ (1/τ, 1), the class UC1/2 does not contain a map ϕ : N(X, τ) → X such that ρ(ϕ(M), M) < r(M) (1 − ε)

∀M ∈ N(X, τ).

We recall (see [230], p. 51) that a Banach space X is of type p if there exists a constant C such that for all x1, . . . , x n ∈ X,

360

15 ChebyshevCentre of a Set . . .  n   1/ p n  1   p  ε x x . ≤ C i i i 2 n ε =±1 i=1 i=1 i

Theorem 15.79 Let X be an infinite-dimensional Banach space of type p > 1. Then for every τ > 1, the class UC1/2 does not contain a map ϕ : N(X, τ) → X such that   sup ρ(ϕ(M), M) | M ∈ N(X, τ) < ∞. The next result establishes a link between smoothness of selections of the Chebyshev near-centre map and smoothness of operators of nonlinear projection onto a subspace. Theorem 15.80 Let X be an infinite-dimensional Banach space. Suppose that there exist numbers τ > 1 and γ ≥ 1/2 and a map ϕ ∈ UCγ such that   sup ρ(ϕ(M), M) | M ∈ N(X, τ) < ∞. Then there exists a subspace L ⊂ X such that no uniformly continuous projection onto L exists in any uniform neighbourhood of L. The Steiner centre of a compact convex set M ⊂ R n is defined as ∫ 1 s(p, M) dp, s(M) = vn S(0,1)

(15.88)

where vn denotes the volume of the unit ball in R n (see [481, Sect. 2.1], [441, Sect. 12.4], [508]), s(p, M) := sup{ p, x  | x ∈ M }, and p ∈ X ∗ is a support function of the set M. The Steiner-centre map s( · ) is Lipschitz continuous in the standard Hausdorff metric as a function of compact convex sets in R n , namely, s(A1 ) − s(A2 ) ≤ L n h(A1, A2 ), where

(15.89)

n 2 Γ( 2 + 1) L n := √ π Γ( n+1 2 )

for every pair of compact convex sets A1 and A2 in R n , where 2 Γ(n/2 + 1) L n := √ , π Γ((n + 1)/2) and the Lipschitz constant L n √in (15.89) is best possible (see √ Sect. 2.1 of [481]). We note that L n is bounded from above by n and behaves roughly like n with increasing n. Hence as the dimension n of the space increases, the Steiner-centre map cannot be extended as a Lipschitz selection to every infinite-dimensional space (or even to a Hilbert space). We also note that on the class of compact convex subsets of R n one can consider different metrics (not equivalent to the Hausdorff distance), in which the Steiner-centre map (as a function of compact convex sets) provides a Lipschitz selection with Lipschitz constant 1 (see Theorem 2.1.3 in [481]). In a real Banach space (X, · ), the set of Steiner points for every given n-tuple {x1, . . . , x n }, n ≥ 3, of points in X is defined by     n   n St n (x1, . . . , x n ) = s ∈ X  xk − s = inf xk − x . x ∈X k=1

k=1

Steiner points are also called Fermat points, Lamé points, or medians. The corresponding Steiner map St n : X n → X of the space X n = {(x1, . . . , x n ) | xk ∈ X } with norm (x1, . . . , x n ) n = x1 + · · · + x n to X is, in general, set-valued, and its domain may not be the whole of X n .

15.10 Algorithms and Applied Problems Connected with Chebyshev Centres

361

For a Hilbert space X and n = 3, a Steiner point s(x1, x2, x3 ) exists and is unique — it lies in the plane spanned by the points x1 , x2 , x3 and either coincides with one of these points (if one of the angles in the triangle x1 x2 x3 is at least 120◦ ) or coincides with the Torricelli point (at which each side of the triangle subtends an angle of 120◦ ); see [85]. Steiner points need not exist even for three-point subsets M3 of a normed linear space. The first example of such a space X and a set M3 was constructed by Garkavi [266] (for other examples, see [585], [76], [458], [116]). Veselý [585] proved that every nonreflexive Banach space X can be equivalently renormed so that some triple of points M3 ⊂ X has no Steiner point in the new norm (cf. also also [356]). Kadets [315] proved this result using a different method. At the same time, in every Banach space X that is 1-complemented in its second dual (in particular, in every reflexive space, as well as in every space L 1 ), the set St n (x1, . . . , x n ) is nonempty for every n-tuple of points xk and every natural number n. Bednov, Borodin, and Chesnokova [85] investigated the problem of the existence of Lipschitz selections of the Steiner map St n (which associates with every n points of a Banach space X the set of their Steiner points) in dependence on the geometric properties of the unit sphere S of X, the dimension of X, and the number n. For n ≥ 4, they obtained general conditions. For a finite-dimensional X, they showed that if n ≥ 4 is even, then the mapping St n has a Lipschitz selection if and only if the unit sphere S of X is a finite polytope; this is not true for odd n ≥ 3.

15.10 Algorithms and Applied Problems Connected with Chebyshev Centres The problem of approximation of geometrically complicated sets M by more convenient sets (balls of radius r(M) in the Chebyshev centre problem or by sets of balls of fixed covering radius in the best n-net problem) is a classical problem in computational geometry [485], and it is interesting both from the theoretical point of view and in relation to multiple applications to problems of cellular [626] and space communication [268], logistics [157], construction of reachability sets for control systems [284], and also optimization problems [309], [310] and approximation of optimal packings [578]. For applications of Chebyshev centres to problems of optimal recovery of linear operators, see [546], [41]–[43], [163], [196]. As a recent application of the Chebyshev centre machinery, we mention the construction of space-filling designs for computer experiments based on an extension of Lloyd’s clustering algorithm [487]. A Chebyshev centre can also be naturally regarded as a centre of an information set in control problems with uncertain disturbances and errors in the state information. The problem of constructing an approximation of an information set (which characterizes the uncertainty in the evaluation of the state vector from observations) as a system of linear inequalities was considered in [575]. This choice of the class of approximating regions and the method of approximation is superior to the vertex representation in terms of memory space and is better than approximation by ellipsoids in terms of accuracy. To be able to control an object, one has to know a point estimate rather than a guaranteed estimate, which leads to the problem of finding a point estimate from an information set. If a guaranteed state estimate (that is, an information set) is known, then as a point estimate one can choose a point in this information set. For a minimax filtration problem, we again choose one point (a Chebyshev centre of the information set) from the whole set of points. The problem of constructing a Chebyshev centre of a given finite point set {a1, . . . , a N } in a finite-dimensional space is a classical optimization problem, for which many algorithms are available (for a historical survey, see [226], and for more references, see [609], and also [126], [296], [326], [577], [512]). The earliest such algorithms had been known long before the maturation of the theory of Chebyshev centres.

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15 ChebyshevCentre of a Set . . .

Exact algorithms for constructing Chebyshev centres of finite sets are based on various linear programming methods (see, for instance, [126], [267], [601]); approximate algorithms can deliver (1 + εk )-approximations of a Chebyshev centre as εk → 0. Exact algorithms depend exponentially on the dimension n of the space, and hence can be used only for small dimensions (say, for n  20). Approximate algorithms can work for large n to produce a (1 + ε)-approximation of a Chebyshev centre with O(N n/ε) arithmetic operations (see [609], [169]). An efficient algorithm for finding a conditional Chebyshev centre in the space  1 was constructed in [159]. The problem of finding a Chebyshev centre for a set M is an NP-hard problem, except for a few simple cases (for example [606], when M is finite and the metric is Euclidean, when M is polyhedral in n∞ , or when M ⊂ C is given by two ellipsoids [84]; see [606]). The problem of approximation of objects by subsets of a finite point set was considered in [394], [396] in the Euclidean plane setting, in [327] for a sphere in a Euclidean space, and in [327], [328] for a plane with inhomogeneous metric.9 An algorithm constructing best nets of cardinality N in general metric spaces is given in [326]. In the Euclidean plane setting, the algorithm for finding best N -nets involves a partition of a given set into subsets with the subsequent step of finding their Chebyshev centres (see [326]); for small N , this method is optimal from the viewpoint of computational operations. For n = 2 and n = 3, numerical algorithms for constructing best n-nets were developed in [393], [394]. Little is known about the construction of the set of Chebyshev centres in the non-Euclidean case (although, of course, general optimization arguments can also be applied to this setting). We give an algorithm of Botkin and Turova-Botkina [126] for finding the Chebyshev centre of a finite set A := {a1, . . . , a m } ⊂ R n . We set     E(x) := a ∈ A  x − a = max x − w . w∈ A

Let  x be the (unique) Chebyshev centre of Z(A) in

Rn .

Step 0. Choose an initial point x0 ∈ conv A and set k = 0. x = xk ); else go to Step 2. Step 1. Find E(xk ). If E(xk ) = A, then stop (with the answer  Step 2. Let yk be the nearest point in conv E(xk ) to xk . If yk = xk , then stop (with the answer  x = xk ); else go to Step 3. Step 3. Evaluate αk := min− i∈I k

2 (x ) ai − xk 2 − dmax k , 2(yk − xk , ai − yk )

where Ik− := {i | ai ∈ A \ E(xk ), (yk − xk , ai − yk ) < 0}, and dmax (x) := max a∈ A x − a (we formally set αk = +∞ if Ik− = ). If αk ≥ 1, then stop (with the answer  x = yk ); else go to Step 4. Step 4. Let xk+1 := xk + αk (yk − xk ), k := k + 1. Go to Step 1.

This algorithm includes the nontrivial operation of finding a nearest point in a polyhedron, and for this, the recursive algorithm from [512] can be applied. Note (see [126]) that the point yk in Step 2 is the Chebyshev centre of the set E(x k ). A similar algorithm for finding the relative Chebyshev centre in R n can be found in [296].

∫ 9 The inhomogeneous metric is defined as ρ f (a, b) := Γ∈Γ(a, b) ( f (x, y))−1 dΓ, where 0 < f (x, y) < c is a piecewise-continuous function, and Γ(a, b) is the class of continuous paths between points a and b. If f ≡ 1, then the metric ρ f ( · , · ) is Euclidean. Such metrics appear in transport and infrastructure logistics problems, for example in optimal placement of objects within a fixed number of logistics centres.

15.10 Algorithms and Applied Problems Connected with Chebyshev Centres

363

One can also mention several algorithms for finding a Chebyshev point (see Definition 15.4) for a system of sets or hyperplanes. In finite-dimensional spaces, such algorithms were designed by Zukhovitskiy [632]; the existence of a Chebyshev point in an arbitrary Banach space for a finite number of sets was established by Garkavi. Chebyshev points proved useful in constructing iteration processes for solving convex programming problems, for solving inconsistent systems of linear equations, in the study of dynamical systems, and also in optimal control problems and in problems of optimal quality and reliability provision for complex systems (see [93], [153], [381], [628], [632, Chap. 5]).

Chapter 16

Width. Approximation by a Family of Sets

A significant part of approximation theory is devoted to various problems related to approximations by classes of sets. Such classes include, for example, classes of finitedimensional subspaces (nested or not nested), classes of nonlinear objects defined by a certain parameter or by a set of parameters. In particular, this problem includes the classical Bernstein’s problem of approximation of an element by a fixed family of nested planes or the generalization of this problem to rational approximations by a family of rational functions. The most studied area of approximation theory, which is related to problems of this type, is the theory of widths. In this chapter, we will consider different types of widths and evaluate them explicitly or approximately. The authors wish to express their special gratitude to V. M. Tikhomirov and A. S. Kochurov for a useful discussion of the material presented in this chapter. In Sect. 16.1, we discuss problems in recovery and approximation leading to widths. Definitions of various widths are given in Sect. 16.2. Fundamental properties of widths are presented in Sect. 16.3. Widths of  p -ellipsoids are evaluated in Sect. 16.4. Dranishnikov–Shchepin widths and their relation to the CE-problem are discussed in Sect. 16.5. The Bernstein widths in the spaces L ∞ [0, 1] are studied in Sect. 16.6. Widths of function classes are examined in Sect. 16.7. The relation between the Jung constant and widths of sets is discussed in Sect. 16.8. We conclude this chapter by considering in Sect. 16.9 general properties of the sequence of best approximations by an infinite system of expanding subspaces.

16.1 Problems in Recovery and Approximation Leading to Widths One usually has to deal with approximation and width in some quantitative problems in which an exact answer is either impossible or infeasible. As a simplest problem of this sort we mention the problem of evaluation of a function F : M → R at a certain point x ∈ M or evaluation of this function at a set of points. Usually, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_16

365

366

16 Width. Approximation by a Family of Sets

in such problems there are constraints on the number of operations, the size of computer memory, or the execution time. In other problems, for calculations it is required to replace elements x of a set M by new elements  x that are completely determined from a finite number of n parameters (each of which has length m in  = M(n,  m). This set will be used in calculations decimal notation) of a new set M instead of x. It is also useful to replace the evaluation of F : M → R by a finite : M  → R. family of standard computer operations, that is, to evaluate the function F As a result, instead of evaluating F(x) at a point x ∈ M, we get a different number  x ): F( F

M −−−−−−→ R  ⏐ τ⏐  ⏐ id ⏐ 

 −−−−F−−→ R M This substitution can be quantitatively estimated, for example, as follows:  sup |F(x) − F(τ(x))|.

x ∈M

(16.1)

 in place of F) is If this quantity is small, then the corresponding change ((τ, F) satisfactory; if not, then this change is unfit. So the solution of (16.1) can be used to  evaluate the change of F by (τ, F). A different (slightly simplified) variant of estimation is the approach whereby  ⊂ M, F  : M → R, and problem (16.1) splits into two parts. It is assumed that M  the quality of the method (τ, F) depends on two characteristics: sup  x − τ(x),

(16.2)

 sup |F(x) − F(x)|,

(16.3)

x ∈M x ∈M

 and the stability of the mapping F or the mapping F,    − F(τ(x))|  sup |F(x) − F(τ(x))| ≤ sup |F(x) − F(x)| + sup | F(x)

x ∈M

x ∈M

x ∈M

(16.4)

or   sup |F(x) − F(τ(x))| ≤ sup |F(x) − F(τ(x))| + sup |F(τ(x)) − F(τ(x))|. (16.5)

x ∈M

x ∈M

x ∈M

 are used, for example, in Moreover, the assumptions on the stability of F or F obtaining estimates in the form of the inequalities  − F(τ(x))|  |F(x) − F(τ(x))| ≤ C1 ·  x − τ(x) or | F(x) ≤ C2 ·  x − τ(x). Clearly, under this approach it is assumed that x and τ(x) can be compared to each other in terms of the norm  x − τ(x) (M is a subset of a normed linear space (X,  · ).

16.1 Problems in Recovery and Approximation Leading to Widths

367

Separation of problem (16.1) into two parts (as in (16.4) and in (16.5)) is quite convenient, because it allows one to reduce the number of model problems of the form  · )), one should separately (16.1) under study: instead of comparing F( · ) with F(τ(  · ) and compare the identity mapping with τ( · ). The quantity compare F( · ) with F( sup  x − τ(x)X

(16.6)

x ∈M

is fundamental for posing approximation theory problems. Let us now discuss problem (16.6) in more detail. In this problem, M is a subset of a normed linear space X, τ(M) ⊂ X is a set representing an ‘n-parameter family’, which can be parametrized by elements of Rn : Rn ←→ τ(M) ⊂ M,

n ∈ Z+ .

Consider some classes of n-parameter subsets of X frequently used in approximation theory. (1) L n = L n (X) is the class of all linear subspaces of dimension ≤ n. This family of n-parameter subsets of X will be called the family of linear subspaces of dimension ≤ n. In particular, L1 = L1 (X) is the set of all lines in X passing through the origin. (2) Aff = Aff(X) is the class of all linear subspaces of dimension at most n and all possible translations of such subspaces by elements of X. This family of n-parameter subsets of X will be called the family of affine subspaces of dimension ≤ n. In particular, Aff(X) is the set of all lines in X. (3) L n = L n (X) is the class of all linear subspaces of codimension ≤ n. This family of n-parametric subsets of X will be called the family of all linear subspaces of codimension ≤ n. In particular, L 1 (X) is the set of all hyperplanes in X. (4) Ξ n = Ξn (X) is the class of all subsets of X of cardinality at most n ∈ N. It is convenient to consider this family as an n-parametric family. Note that the family Ξ1 (X) can be conveniently identified with X. k Mi | Mi ∈ Aff(X), i = 1, . . . , k} is the class of all (5) Cn = Cn (X) = { i=1 finite unions of linear affine subspaces of X of dimension at most n ∈ Z+ . From the definition it follows that C0 (X) = ∪∞ k=1 Ξk (X). Let us again consider problem (16.2). Of course, an algorithm τ : M → τ(M) ⊂ M can be chosen in many different ways (including a set M and a mapping τ( · )). Note that not every pair (τ, M) is appropriate. Given a fixed set M, we consider the  is appropriate. Hence one should problem of whether a replacement of M by M consider a ‘reference’ or ‘best replacement’: inf sup  x − τ(x). τ x ∈M

(16.7)

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16 Width. Approximation by a Family of Sets

Having defined this quantity, we can now define the width of a set. However, before giving the formal definition, we discuss the problem of the infimum over τ  in the ‘complexity’. Of in (16.7). An important feature in the replacement of M by M  course, given an n-parameter set M, one can wish to define a mapping τ : M → M by associating with each x ∈ M an element of its best approximation from the set  M: τ(x) = arg{ inf  x − z}.  z ∈M

Remark 16.1 In approximation theory, one frequently deals with boundedly compact  (the intersection of such a set with a closed ball in X is compact; see sets M  always exists: if Sect. 4.1). For such sets, an element of best approximation from M  e(x) := ρ(x, M) = inf  x − z, then  z ∈M

  ∩ B(x, 1 + e(x))} . τ(x) = arg inf{ x − z : z ∈ M However, finding an element of best approximation from a boundedly compact  is usually a computationally involved procedure. To reduce the number of set M calculations, one places additional constraints on an algorithm (τ, M).  ∈ Aff(X) as follows: In the first four examples, we choose n ∈ Z+ and M  E(M, M)  is the class of all mappings from M into M  (no (1) τ ∈ E(M, M), constraints on τ);  C(M, M)  is the class of all continuous mappings from M (2) τ ∈ C(M, M),  into M;  A(M, M)  is the class of all continuous affine mappings from M (3) τ ∈ A(M, M),  The notation τ ∈ A(M, M)  means that into M. τ(x) = l(x) + z0,

x ∈ X,

 − z0 ) of X, z0 is l( · ) is a linear continuous mapping from X into the subspace ( M  a fixed element from M. In addition to the class of affine mappings, one also considers the class τ ∈  of all continuous linear mappings from X into a subspace M  ∈ L n (X). L(X, M)  P(X, M)  is the class of all continuous affine projections from X (4) τ ∈ P(X, M), onto M (these are continuous affine mappings from X onto M each of which is the sum of a continuous linear projection and a constant mapping).   ∈ Ξn , n ∈ N, one usually considers only the class E(X, M). (5) In the case M A reason for this is that for sets M that are encountered in approximation theory  ∈ Ξn , the image τ(M) is very and for continuous mappings τ of these sets into M frequently a singleton (this case is immaterial).  ∈ Cn (X), one usually considers only the class C(X, M).  (6) In the case n ∈ Z+ , M The above examples show that one additional and frequently useful constraint on the choice of a mapping τ( · ) is that this mapping should be defined and have certain properties not only for elements x ∈ M, but also for all x ∈ X. In what follows, we shall assume that this constraint is satisfied.

16.2 Definitions of Widths

369

16.2 Definitions of Widths Let us now give the definition of an n-width of a set M in a normed linear space X, n ∈ Z+ (for a slightly more general setting of a seminormed or a linear metric space X, the definition is similar). Definition 16.1 The Kolmogorov width of order n of a set M in the space X is defined as inf sup  x − τ(x) = inf sup ρ(z, L). dn (M, X) = inf L ∈Aff(X) τ ∈ E(X, L) x ∈M

L ∈Aff(X) x ∈M

 of Here, among all possible replacements of the set M by a linear subspace L = M dimension ≤ n (or its translation), we choose the one that delivers the smallest error. Different constraints on the choice of a mapping τ( · ) lead to different widths. The linear width of order n of a set M in a space X is defined as λn (M, X) =

inf

inf

sup  x − τ(x).

L ∈Aff(X) τ ∈A(X, L) x ∈M

The projection width of order n of a set M in a space X is defined as πn (M, X) =

inf

inf

sup  x − τ(x).

inf

inf

sup  x − τ(x)

L ∈Aff(X) τ ∈ P(X, L) x ∈M

The width γn (M, X) =

L ∈Aff(X) τ ∈ C(X, L) x ∈M

will be shown to coincide with the width dn (M, X). The entropy width of order n of a set M in a space X is defined as εn (M, X) =

inf

inf

sup  x − τ(x).

Σ ∈Ξ n (X) τ ∈ E(X,Σ) x ∈M

Let us discuss the above definitions using two examples, but first we recall Hausdorff’s compactness criterion and give necessary definitions. A set A is called an ε-net for a set M in a space X = (X, ρ) if for every point y ∈ M, there exists a point x ∈ A such that ρ(x, y) ≤ ε. A set M is called totally n in the space X bounded in X if for every ε > 0, there exists an ε-net A = {xi }i=1 consisting of a finite number of points. Theorem 16.1 (Hausdorff’s compactness test) A subset of a complete metric space is compact if and only if it is totally bounded. Corollary 16.1 Let X = (X,  · ) be a Banach space, M ⊂ X. Then lim εn (M, X) = 0 ⇔ M is a compact subset of X.

n→∞

Since dn (M, X) ≤ εn (M, X), it follows that if the closure of M is compact, then lim dn (M, X) = 0.

n→∞

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16 Width. Approximation by a Family of Sets

Note that there exist a normed space X and a compact set M ⊂ X such that λn (M, X) does not tend to zero as n → ∞. The Alexandrov width of order n of a set M in a space X is defined as an (M, X) =

inf

inf

sup  x − τ(x).

inf

sup  x − τ(x)

N ∈ Cn (X) τ ∈ C(X, N ) x ∈M

However, the approximative characteristic a˜n (M, X) =

inf

N ∈ Cn (X) τ ∈ E(X, N ) x ∈M

is unsuccessful. Exercise 16.1 Show that a˜ n (M, X) = 0 for every compact set M ⊂ X.

The Fourier width of order n of a set M is defined as follows. Definition 16.2 Given an inner product normed space X, assume that the orthogonal projection operator τL onto every subspace L ∈ Aff(X) is continuous. The Fourier width of order n of a set M in the space X is defined as ϕn (M, X) =

sup  x − τM (x).

inf

L ∈Aff(X) x ∈M

Example 16.1 (a) Let X = 22 be the two-dimensional Euclidean plane, M = B(0, r) the disc of radius r > 0 with centre at the origin. If n = 0, then d0 (M, X) =

inf

sup ρ(x, L) = inf sup x − z .

L∈Aff 0 (X ) x ∈ M

z ∈X x ∈ M

Hence d0 (M, X) ≤ sup x  = r . x∈M

On the other hand, d0 (M, X) =

inf

sup x − z  ≥

z ∈X, z0 x ∈ M

that is, d0 (M, X) = r. (b) if n = 1, then

d1 (M, X) =

z ∈X,

inf

z

z

− z  ·

= r,

−r · z  z  z0

inf

sup ρ(x, L).

L∈Aff(X ) x ∈ M

By definition, d1 (M, X) ≤ d0 (M, X) = r . On the other hand, d1 (M, X) ≥

inf

sup

L∈Aff(X ) x ∈ M ∩L ⊥

ρ(x, L) =

inf

sup

L∈Aff(X ) x ∈ M ∩L ⊥

x − z ⊥ L ,

where L ⊥ is the line passing through the origin orthogonally to the line L, z ⊥ L is the intersection of ⊥ the lines L and L ⊥ . Moreover, M ∩ L ⊥ is an interval of length 2 r, z ⊥ L is a point (on the line L ). Now sup x − z ⊥ L  ≥ r, x ∈ M ∩L ⊥

and so d1 (M, X) = r.

16.2 Definitions of Widths

371

If n = 2, then d2 (M, X) =

inf

sup inf x − z  = sup inf x − z  = 0.

M ∈Aff 2 (X ) x ∈ M z ∈ M

x ∈ M z ∈X

We give without proof the following theorem about the width of a ball. Theorem 16.2 (V. M. Tikhomirov) Let (X,  · ) be a normed linear space, n ∈ Z+ , L a subspace of X of dimension (n + 1), M = B(0, r) the ball of radius r > 0 with centre at 0 in the space X. Then dn (M ∩ L, X) = r.

 n 2 2 Example 16.2 Let X = n2 , let M = {(x1, . . . , x n )  i=1 xi /αi ≤ 1} be an ellipsoid in a Euclidean space, α1 ≥ α2 ≥ · · · ≥ αn > 0. Then: (a) d0 (M, n2 ) = α1 (the major semiaxis of the ellipsoid); (b) dn (M, n2 ) = 0; (c) moreover, dm (M, n2 ) = αm+1 if m = 0, 1, . . . , n − 1. Let us first derive the upper estimate. Let Pk : (x1, . . . , x n ) → (x1, . . . , x m, 0, . . . , 0) be the orthogonal projection of the space n2 onto the subspace L = {(x1, . . . , x m, 0, . . . , 0) ∈ R n | x1, . . . , x m ∈ R} (this projection is the metric projection of n2 onto L). Then dm (M, n2 )2 ≤ sup x − P(x) 22 = sup x∈M

n

x ∈ M i=m+1

xi2 = sup

n

xi2

x ∈ M i=m+1

αi2

2 · αi2 = αm+1 .

Hence dm (M, n2 ) ≤ αm+1 . On the other hand, let

 L = {(x1, . . . , x m+1, 0, . . . , 0) ∈ R n  x1, . . . , x m+1 ∈ R}

(the dimension of L is (m + 1)). Then the set

x2  m+1 i M ∩ L = {(x1, . . . , x m+1, 0, . . . , 0)  ≤ 1} αi2 i=1 contains αm+1 · B2n ∩ L. Using Theorem 16.2 (the ball width theorem), we get dm (M, n2 ) ≥ dm (αm+1 · B2n ∩ L, n2 ) = αm+1 . Example 16.3 Let X = L ∞ [0, 1] be the space of essentially bounded functions, M = { f : [0, 1] →   R | f (x) − f (y)| ≤ |x − y | } the class of 1-Lipschitz functions on the interval [0, 1]. Then dm (M, X) =

1 , 2m

m ∈ Z+ .

Indeed, d0 (M, X) = ∞, because M is not bounded in X. Now let n ∈ N. To estimate the width from above, we set m   L= ci · χi, m | ci ∈ R, i = 1, . . . m i=1

and let χi, m ( · ) be the characteristic function of the interval Δi := [ i−1 m , τ : X → M defined by τ( f ) =

m

i=1

∫ ci · χi, m,

ci = m

Δi

f (x) dx.

i m ]. Consider the mapping

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16 Width. Approximation by a Family of Sets

Then dm (M, X) ≤ sup  f − τ( f ) ∞ f ∈M

∫  ∫   = m · sup max sup  ( f (t) − f (x)) dx  ≤ m· f ∈ M i=1, m t ∈Δi

Δi

0

1 m

x dx =

1 . 2m

Now let us estimate the width from below. Let ϕ k be the restriction to the interval [0, 1] of the k+1 1 k continuous function that vanishes outside the interval [ k−1 2m , 2m ], is equal to 2m at the point 2m , k−1 k k k+1 and is linear on the intervals [ 2m , 2m ] and [ 2m , 2m ], k = 0, 1, . . . , m; next, let L be the span of m , dim L = (m + 1). We have 2m · M ∩ L = B ∩ L, and hence by Theorem 16.2 (on the {ϕk } k=0 X ball width), dm (M, L ∞ [0, 1]) ≥ dm (M ∩ L, L ∞ [0, 1]) =

1 1 · dm (B X ∩ L, B X ) = . 2m 2m

The Bernstein width bn (M, X) is defined as bn (M, X) =

sup

L ∈ L n+1 (X)

sup{r  0 | B(x, r) ∩ L ⊂ M }.

There exist other definitions of widths. For more details, see [103], [360], [344], [419], [474], [454], [534], [415], [514], [540]–[542].

16.3 Fundamental Properties of Widths Given an arbitrary set A ⊂ X, we denote by s(A) the union of A and all closed intervals connecting points from A. As before, conv A is the convex hull of A. Note that from Carathéodory’s theorem (see Appendix B), it follows that for every set A ⊂ Rk , k ∈ N, we have conv(A) = s(s(. . . s(A) . . . )), where the operation s is applied k times. Let n ∈ Z+ , let pn (M1, X) be any of the above widths: dn (M1, X), λn (M1, X), πn (M1, X), ϕn (M1, X), γn (M1, X), an (M1, X), εn (M1, X). Then: (1) p0 (M1, X) ≥ p1 (M1, X) ≥ p2 (M1, X) ≥ p3 (M1, X) ≥ . . . (monotonicity with respect to n ∈ Z+ ); (2) pn (M1, X) ≤ pn (M2, X), n ∈ Z+ if M1 ⊂ M2 ⊂ X (monotonicity with respect to the set M); (3) dn (M1, X) ≤ γn (M1, X) ≤ λn (M1, X) ≤ πn (M1, X) ≤ ϕn (M1, X) and an (M1, X) ≤ γn (M1, X), dn (M1, X) ≤ εn (M1, X). Each of the above widths will also be denoted by pn (M1, X) =: pn (M1, BX ); we emphasize in this notation the unit ball of the normed linear space X.

16.3 Fundamental Properties of Widths

373

In the next two properties of widths it is assumed that a linear space X contains sets B1 , B2 that can play the role of a unit ball of X. For example, a set B1 is the unit ball of some norm if and only if B1 is centrally symmetric, convex, and 0 ∈ X lies in the kernel of B1 : each line  ∈ L1 (X) passes B1 in a nondegenerate interval containing 0 in its relative interior. (4) Let α > 0, B2 = α · B1 , M ⊂ X. Then pn (M, B1 ) = α · pn (M, B2 ). This property follows from the definition of a width and that of the Minkowski functional and the following equivalences:  xB2 ≤ 1 ⇐⇒ x ∈ B2 = α · B1 ⇐⇒

x ∈ B1 ⇐⇒  xB1 ≤ α. α

(5) Let B1 ⊂ B2 , M ⊂ X. Then pn (M, B2 ) ≤ pn (M, B1 ), because  xB2 ≤  xB1 ,

x ∈ X.

(6) Let α ∈ R, M ⊂ X. Then pn (α · M, BX ) = |α| · pn (M, BX ). Let us verify this equality for the Kolmogorov width (for other widths, the argument is similar). The result is clear if α = 0 and dn (M, BX )  ∞. If α = 0 and dn (M, BX ) = ∞, then the equality under consideration is satisfied by definition. Let α  0. Then dn (αM, BX ) = inf

sup ρ(z, M) =inf sup inf αy − z

L ∈Aff(X) x ∈αM

= |α| inf sup

L ∈Aff y ∈M z ∈L

inf

L ∈Aff y ∈M w ∈α−1 L

 y − w = |α| · dn (M, BX ).

(7) For all widths pn (M, X) except the Alexandrov and the entropy widths, we have pn (M, X) = pn (conv(M), X). Let us discuss the Kolmogorov and linear widths. The inequality pn (M, X) ≤ pn (conv(M), X) holds for each of the above widths (see property (2) above). Let us prove the converse inequality. If the width of a set M is infinite, then so is the width of its convex hull, and so property (7) holds. Let d = dn (M, X) < ∞. Given an arbitrary ε > 0, we choose a subspace L0 ∈ Aff(X) so as to have dn (M, X) = inf

sup ρ(z, M) ≤ sup inf  x − z ≤ d + ε,

L ∈Aff(X) x ∈M

x ∈M z ∈L0

that is, for all x ∈ M, there exists y ∈ L0 for which  x − y ≤ d + ε. Consider the convex set  = {x ∈ X | ∃ y ∈ L0 ,  x − y ≤ d + ε}. M  and hence We have M ⊂ conv(M) ⊂ M,  X) ≤ d + ε. dn (conv(M), X) ≤ dn ( M, Since ε > 0 is arbitrary, we have dn (conv(M), X) ≤ d, which shows that the Kolmogorov width satisfies property (7).

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16 Width. Approximation by a Family of Sets

Property (7) for the linear width is proved in the same way: if the width is finite, we can choose a subspace L0 ∈ Aff(X) and a mapping τ0 ∈ A(X, L0 ) such that λ = λn (M, X) =

inf

inf

sup  x − τ(x)

L ∈Aff(X) τ ∈A(X, L) x ∈M

≤ sup  x − τ0 (x) ≤ λ + ε, x ∈M

that is,  x − τ0 (x) ≤ λ + ε for all x ∈ M. Consider the convex set  = {x ∈ X |  x − τ0 (x) ≤ λ + ε}. M  and hence We have M ⊂ conv(M) ⊂ M  X) ≤ λ + ε. λn (conv(M), X) ≤ λn ( M, This proves property (7) for linear widths. (8) For every width pn (M, X), we have pn (M, X) = pn (M, X). We prove this equality for the Kolmogorov and Alexandrov widths. The estimate pn (M, X) ≤ pn (M, X) holds by the monotonicity property of widths (see property (2) above). Let us prove the converse inequality. If the width of a set M is infinite, then so is the width of its closure, and so property 8) holds. Let d = dn (M, X) < ∞. For an arbitrary ε > 0, we choose a subspace L0 ∈ Aff(X) so as to have dn (M, X) = inf

sup ρ(z, M) ≤ sup inf  x − z ≤ d + ε,

M ∈Aff(X) x ∈M

x ∈M z ∈L0

that is, for all x ∈ M, there exists y ∈ L0 for which  x − y ≤ d + ε. Consider the closed set  = {x ∈ X | ∃ y ∈ L0 ,  x − y ≤ d + ε}. M  and hence We have M ⊂ M ⊂ M,  X) ≤ d + ε. dn (M, X) ≤ dn ( M, Since ε > 0 is arbitrary, we have dn (M, X) ≤ d, proving property (8) for the Kolmogorov width. The same argument proves property (8) for the Alexandrov width: if the width is finite, we can choose N ∈ Cn (X) and a mapping τ0 ∈ C(X, N) so as to have a = an (M, X) = inf

inf

sup  x − τ(x)

C ∈ Cn (X) τ ∈ C(X,C) x ∈M

≤ sup  x − τ0 (x) ≤ a + ε, x ∈M

16.3 Fundamental Properties of Widths

375

that is,  x − τ0 (x) ≤ a + ε for all x ∈ M. Consider the closed set  = {x ∈ X |  x − τ0 (x) ≤ a + ε}. M  and hence For this set, we have M ⊂ M ⊂ M,  X) ≤ a + ε. an (M, X) ≤ an ( M, This proves property (8) for Alexandrov widths. Remark 16.2 Properties (7) and (8) show that the exact values and estimates of widths can be found if one considers closed and convex sets only. (9) Let M = (−M) be a centrally symmetric subset of a space X. Then dn (M, X) = inf

sup inf  x − z =: d.

L ∈ L n (X) x ∈M z ∈L

Indeed, the inequality dn (M, X) ≤ d follows by definition. Conversely, let ε > 0, and let a subspace L ∈ L n (X) and a point a ∈ X be such that sup inf  x − z ≤ d + ε.

x ∈M z ∈L+a

Consider x ∈ M and choose l1 , l2 from L so as to have  x − l1 − a ≤ d + ε ,

 − x − l2 − a ≤ d + ε .

Next, 2x − l1 + l2  =  x − l1 − a − (−x − l2 − a) ≤ 2d + 2ε,

l1 − l2



≤ d + ε.

x − 2

A similar result also holds for the linear and projection widths. Thus if M = (−M), then πn (M, X) = inf inf sup  x − π(x) =: α. L ∈ Af fn (X) π ∈ P(X, L) x ∈M

We argue as for the Kolmogorov widths: the inequality πn (M, X) ≤ α follows from the definition. Let ε > 0, L ∈ Aff(X), and let a projection τ ∈ P(X, L) be such that sup  x − τ(z) ≤ α + ε.

x ∈M

Consider the linear operator τ1 (x) = τ(x) − τ(−x). This operator is a liner projection, and its image lies in the subspace (L − L) of dimension ≤ n. If τ(x) = l1 + a, τ(−x) = l2 + a, l1 = −l2 , then

l1 − l2



2x − l1 + l2  =  x − l1 − a − (−x − l2 − a) ≤ 2α + 2ε, x −

≤ α + ε. 2 We now need some topological results.

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16 Width. Approximation by a Family of Sets

Theorem 16.3 (K. Borsuk) Let n ∈ N, and let f be an odd continuous mapping of the unit sphere S n−1 of Rn in the space Rn−1 . Then there exists x ∈ S n−1 such that f (x) = 0. This theorem has several equivalent formulations. We recall some of them. Theorem 16.4 There exists no odd continuous mapping of the sphere S n ⊂ Rn+1 into the sphere S n−1 ⊂ Rn , n ∈ N. Proof Let f : S n → S n−1 be odd and continuous. Then f : S n → Rn is odd and continuous, and hence by Borsuk’s theorem, there exists x ∈ S n such that f (x) = 0, a contradiction to f (x) ∈ S n−1 . Conversely. Let f : S n → Rn be odd, continuous, and assume that f does not vanish on S n . Then the mapping g(x) :=

f (x)  f (x) 

is odd and continuous from S n into S n−1 , a contradiction.

Theorem 16.5 For a continuous mapping f : S n → Rn , n ∈ N, there exists a point x ∈ S n such that f (x) = f (−x). Proof If a mapping f : S n → Rn is continuous, then the mapping g(x) := f (x) − f (−x) is odd and continuous from S n into Rn . By Borsuk’s theorem, there exists x0 ∈ S n such that g(x0 ) = 0. Conversely. Let f : S n → Rn be odd and continuous. Then for some point x0 ∈ S n , we have f (x0 ) = f (−x0 ). Since f (−x0 ) = − f (x0 ), we get that f (x0 ) = 0.  Theorem 16.6 (Lyusternik–Shnirel’man covering theorem) If m, n ∈ N, m ≤ n + 1, and if open sets U1, . . . , Um cover the sphere S n , then at least one of the sets Ui , i = 1, . . . , m, contains two opposite points of S n . To prove that Borsuk’s and Lyusternik–Shnirel’man’s theorems are equivalent, we need the definition of the nerves of covers and the simplest properties of a canonical map into the nerve. Let K be a metrizable compact set with metric , let U1, . . . , Um be open sets covering K, m ∈ N, and let (X,  · ) be a normed linear space. Consider points m ⊂ X in general position (this means that no k of the points (k ∈ N, 2 ≤ k ≤ {ei }i=1 m lie in the same affine plane of dimension (k − 1). Given dim(X) + 1) from {ei }i=1 k = 1, . . . , m, we set ϕk (x) = min{ (x, y) | y ∈ K \ Uk },

μk (x) = ϕk (x) ·

m 

ϕi (x)

 −1

i=1

We note the following properties of the functions ϕk , μk , k = 1, . . . , m: (1) ϕk , μk are continuous; (2) ϕk (x) ≥ 0, μk (x) ≥ 0, x ∈ K, ϕk (x) > 0, μk (x) > 0, x ∈ Uk ;

.

16.3 Fundamental Properties of Widths

377

(3) ϕk (x) = μk (x) = 0, x  Uk ; m μk (x) ≡ 1. (4) k=1 A family of functions μk satisfying properties (1)–(4) is said to form a partition of unity subordinated to the cover U1, . . . , Um . Let F(x) =

m

μk (x) · ek .

k=1

The range of K under the mapping F( · ) is called the nerve of K generated by the m and is denoted by N(e , . . . , e ); the mapping F( · ) is a mapping of K set {ei }i=1 1 m into the nerve N(e1, . . . , em ). By definition, a nerve is a polyhedron. Let us show that Lyusternik–Shnirel’man’s and Borsuk’s theorems are equivalent. m ⊂ Rn be a family of points in general position, F( · ) a mapping Proof Let {ei }i=1 n from S into the nerve N(e1, . . . , em ) generated by the cover U1, . . . , Um . A mapping F( · ) is continuous from S n into Rn . Hence by Borsuk’s theorem, there exists a point . , m} for which x ∈ S n such that F(x) = F(−x). Let {i1, . . . , i p } be all i ∈ {1, . .  p μi (x) > 0. From the properties of the functions μi , we get x, −x ∈ k=1 Uik . Conversely, Let η(x) = (η1 (x), . . . , ηn (x)) be an odd vector field. Let us show that for some point x0 , we have η(x0 ) = 0

(this is equivalent to Borsuk’s theorem). Given an arbitrary δ > 0, we set Ai (δ) := {x ∈ S n | ηi (x) ≥ δ},

i = 1, . . . , n.

The open sets int { Ai (δ)},

i = 1, . . . , n,

Sn \

n 

Ai (2δ)

i=1

cover S n . Further, the sets Ai (δ), i = 1, . . . , n, do not contain diametrically opposite n A (2δ) conpoints, and hence by Lyusternik–Shnirel’man’s theorem, the set S n \ i=1 i n Ai (2δ); tains such points. Hence there exists a point x(δ) such that ± x(δ) ∈ S n \ i=1 i = 1, . . . , n. that is, |ηi (x(δ))| < 2δ, Since the sphere S n is compact, we see on letting δ → 0 that the above inequalities  also hold for δ = 0 for some x(0) ∈ S n . Theorem 16.7 Let n ∈ Z+ and let C be a nonempty subset of a normed linear space X. Then bn (C, X) ≤ γn (C, X) = dn (C, X). Proof We fix a subspace L ∈ L n+1 (M) and an arbitrary α ∈ R+ , x ∈ X, such that α · (BX ∩ L) + x ⊂ C. We have γn (C, X) ≥ γn ((BX ∩ L) + x, X) = α · γn (BX ∩ L, X).

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16 Width. Approximation by a Family of Sets

We claim that γn (BX ∩ L, X) = 1. The widths are monotone in n, and hence γn (BX ∩ L, X) ≤ 1. Let us prove the converse inequality. Consider an arbitrary plane L0 ∈ Aff n (X) and a continuous mapping of the space X (and hence of the relative boundary ∂(BX ∩ L) of the set (BX ∩ L)) into L0 . Since dim L0 ≤ n, the continuous mapping f : ∂(BX ∩ L) → L0 can be naturally identified with a continuous mapping g : S n → Rn of the sphere S n ⊂ Rn+1 into Rn . By Borsuk’s theorem, for g there exists a point s ∈ S n such that g(s) = g(−s). Hence for f there also exists a point x0 ∈ ∂(BX ∩ L) such that f (x0 ) = f (−x0 ) ∈ L0 . Therefore, 2 =  x0 − (−x0 ) =  x0 − f (x0 ) − ((−x0 ) − f (−x0 )) ≤  x0 − f (x0 ) + ((−x0 ) − f (−x0 )); that is, we have either  x0 − f (x0 ) ≥ 1 or (−x0 ) − f (−x0 ) ≥ 1. So sup  x − f (x) ≥ 1.

x ∈B∩L

Since L0 ∈ Aff n (X) and f ∈ C(X, L0 ) are arbitrary, we have γn (BX ∩ L, X) =

inf

inf

sup

L ∈Aff n (X) f ∈C(X, L) x ∈B X ∩L

 x − f (x) ≥ 1. 

Theorem 16.7 is proved.

Corollary 16.2 (The ball width theorem) Let X be a normed linear space, L ∈ L n+1 (X) \ L n (X), n ∈ Z+ . Then bn (BX ∩ L, X) = dn (BX ∩ L, X) = γn (BX ∩ L, X) = λn (BX ∩ L, X) = πn (BX ∩ L, X) = 1. Proof Setting M := BX ∩ L, we have 1 ≤ bn (M, X) ≤ γn (M, X) = dn (M, X) ≤ λn (M, X) ≤ πn (M, X) ≤ π0 (M, X) = 1, the result required.



The next lemma is required for the extension of some topological results from Euclidean finite-dimensional spaces to arbitrary normed linear spaces. Lemma 16.1 Let V ⊂ Rm+1 be a bounded convex closed centrally symmetric body and let ∂V be its boundary. Then there exists an odd homeomorphism between ∂V and S m . Proof Since V is a convex closed centrally symmetric body, 0 ∈ Rm+1 is an interior point of V and every ray emanating from the origin and passing through a point x ∈ S m intersects the set V in a nondegenerate interval

16.3 Fundamental Properties of Widths

379

[0, rV (x) · x],

rV (x) > 0.

Let us check that f ( · ) : S m → Rm+1,

f (x) = rV (x) · x,

x ∈ S m,

is the required mapping. By definition, f ( · ) : S m → ∂V is odd and one-to-one. It remains to verify that it is continuous. Assume that a sequence (xn ) ⊂ S m converges to x ∈ S m . Using properties of V, we find that [0, r] · x ⊂ V,

r := lim sup rV (xn ),

and so rV (x) ≥ r. Conversely, let L be a plane passing through 0 perpendicular to the interval [−x, x], and let U ⊂ V be a Euclidean ball with centre at the origin. We also set W = conv{U ∩ L, rV (x) · x, −rV (x) · x} ⊂ V . By definition, rV (x) = rW (x),

rV (y) ≥ rW (y),

y ∈ Sm .

From the definition it also follows that rW is a continuous mapping (for W, the mapping rW (y) can be written down explicitly in terms of y; from this formula it follows that rW is continuous). Hence rV (xn ) ≥ rW (xn ) → rW (x) = rV (x) ≥ r

as n → ∞,

and in particular, lim inf rV (xn ) ≥ r = lim sup rV (xn ), 

proving the continuity of rV .

Corollary 16.3 Let V ⊂ Rm+1 be a bounded convex closed centrally symmetric body with boundary ∂V and let F : ∂V → Rm , m ∈ Z+ , be continuous. Then there exists a point  x ∈ ∂V such that F( x ) = F(− x ). Theorem 16.8 (Yu. I. Makovoz, 1972) Let X be a normed linear space, V ⊂ Rn+1 a bounded convex closed centrally symmetric body, n ∈ Z+ , and let F : ∂V → X be an odd continuous mapping from ∂V into X. Then   dn F(V), X ≥ ρ := min  x. x ∈F(∂V )





Proof It suffices to verify that γn F(V), X ≥ ρ. Let L ∈ Aff n (x) be an arbitrary affine subspace and let τ : X → L be a continuous mapping. Since L is homeomorphic to Rn , by Corollary 16.3 for the mapping (τ ◦ F) : ∂V → L there exists a point λ ∈ ∂V such that (τ ◦ F)(λ) = (τ ◦ F)(−λ). Hence

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16 Width. Approximation by a Family of Sets

 1 F(λ) − (τ ◦ F)(λ) + F(−λ) − (τ ◦ F)(−λ) 2

F(λ) − (τ ◦ F)(λ) F(−λ) − (τ ◦ F)(−λ)



≥

−

= F(λ) ≥ ρ . 2 2

sup  x − τ(x) ≥

x ∈F(∂V )



Theorem 16.8 is proved. Let us now consider the equality dn (A, M) = γn (A, M),

A ⊂ M,

n ∈ Z+,

where X is a normed linear space. The proof of this equality depends substantially on the classical Michael’s selection theorem for set-valued mappings (see, for example, [481, Sect. 2.9]). Before formulating this theorem (not in the most general setting), we give some definitions and prove some auxiliary results. Definition 16.3 Let M be a subset of a metric space X = (X, ), let Y be a normed linear space, and let F : X → 2Y be a set-valued mapping. A continuous selection (or simply a selection) of a set-valued mapping F is a continuous single-valued mapping f : M → Y such that for all x ∈ M, f (x) ∈ ϕ(x). Theorem 16.9 (E. Michael) Let (Y, ) be a metric space, (X,  · ) a Banach space, and F : M → 2Y a lower semicontinuous set-valued mapping with convex and closed values. Then F has a continuous selection; that is, ∃ f ∈ C(Y, X) :

f (x) ∈ F(x) (x ∈ Y ).

Theorem 16.10 Let X be a normed linear space and let M be a nonempty subset of X, n ∈ Z+ . Then dn (M, X) = γn (M, X). Proof The inequality dn (M, X) ≤ γn (M, X) follows from the definition of the widths dn (M, X), γn (M, X). Let us verify the converse inequality: dn (M, X) ≥ γn (M, X). If dn (M, X) = ∞, then the required inequality holds. Let dn (M, X) < +∞, let ε > 0 be fixed, and let L ∈ Aff n (X) be such that dn (M, X) + ε ≥ sup inf  x − yX . x ∈X y ∈L

Let L = a + Y,

Y ∈ L n,

a ∈ X.

We define a set-valued mapping F := Fε,Y from a normed linear space X into a finite-dimensional Banach space Y by

16.3 Fundamental Properties of Widths

381

˚ ρ(x, Y ) + ε), F(x) = Y ∩ B(x,

x ∈ X.

Let us check that F is lower semicontinuous. Consider an arbitrary point x0 ∈ X and set d = ρ(x0, Y ). Next, consider an arbitrary Y -open set V and a point y0 ∈ F(x0 ) ∩V. Then  x0 − y0  < d + ε. We choose δ > 0 so as to have  x0 − y0  < d + ε − δ. Now if  x − x0 X < δ/2, then δ ρ(x, Y ) ≥ ρ(x0, Y ) −  x0 − xX = d − , 2 δ δ  x − y0 X ≤  x0 − y0 X +  x0 − xX < d + ε − δ + = d + ε − ; 2 2 that is, the distance from the point y0 ∈ V to x is smaller than d + ε − δ2 . Hence y0 lies in the open ball with centre at x and radius δ ρ(x, Y )X + ε ≥ d + ε − . 2 So for each point x,  x − x0  < δ/2, we have y ∈ F(x),

that is,

F(x) ∩ V  ,

which proves that F is lower semicontinuous. It is easily checked that the mapping H(x) = F(x) = Y ∩ B(x, ρ(x, Y ) + ε),

x∈X

is also lower semicontinuous. By Michael’s selection theorem, there exists a singlevalued function f ∈ C(X, Y ) such that f (x) ∈ H(x), for every point x ∈ X,  x − f (x)X ≤ ρ(x, Y ) + ε. Hence γn (M, X) ≤

inf

sup  x − τ(x)X ≤ sup (x − a) − f (x − a)X

τ ∈C(X,a+Y) x ∈M

x ∈M

≤ sup ρ(x − a, Y ) + ε = sup ρ(x, L) + ε ≤ dn (M, X) + 2ε. x ∈M

x ∈M



Theorem 16.10 is proved.

Remark 16.3 The use of Michael’s selection theorem in the proof of the inequality γn (M, X) ≤ dn (M, X) is closely related to the problem of continuity of the best approximation operator by a plane L ∈ Aff n (M): y(x) ∈ PL (x).

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16 Width. Approximation by a Family of Sets

Since L is finite-dimensional, PL (x) is nonempty for all x (see Sect. 4.1). However, in general, PL is not single-valued, and the mapping y can fail to be continuous. However, if L is a finite-dimensional Chebyshev subspace of X, then the mapping y : M → L is continuous. In particular, if a finite-dimensional space X is strictly convex, then every subspace of X is a Chebyshev subspace.

16.4 Evaluation of Widths of  p -Ellipsoids In this section, we shall require the theorem on ε-displacement. We first recall one variant of Brouwer’s fixed-point theorem (see Theorem 6.4). Theorem 16.11 Let M be homeomorphic to the ball B n and let F ∈ C(M, M). Then F has a fixed point x ∈ M (that is, F(x) = x). Definition 16.4 A mapping F of a metric space (Y, ρ) into itself is called an ε-displacement if ρ(x, F(x)) ≤ ε for all x ∈ Y . Theorem 16.12 (The theorem on ε-displacement) Let B be the unit ball of a finitedimensional Banach space X and let F : B → M be a continuous ε-displacement, 0  δ < 1 − ε. Then δ · B ⊂ F(B). Proof Assume to the contrary that there exists a point x0 ∈ (δ · B)\F(B). Since F(B) is a closed set, some neighbourhood of x0 does not contain points from F(B). Using this fact, we construct a fixed-point-free continuous mapping ξ of the ball B into the boundary ∂B. To this end, we draw through a point x ∈ B a ray emanating at the point F(x) and passing through x0 ; let ξ(x) be a point on this ray and on the boundary ∂B that is separated by x0 from the point F(x) on this ray. This mapping ξ from B into ∂B is continuous and has no fixed points:  x − ξ(x) ≥ F(x) − ξ(x) −  x − F(x) >  x0 − ξ(x) − ε ≥ ξ(x) − ε −  x0  ≥ 1 − δ − ε > 0, 

which contradicts Brouwer’s theorem. Let us now proceed with evaluation of widths of the sets n  p     xi  p Bn (r) := x ∈ Rn    ≤ 1}, 1 ≤ p < ∞, r i=1 i  x      i Bn∞ (r) := x ∈ Rn  max   ≤ 1 , i=1,...,n ri q

r = (r1, . . . , rn ), r1 ≥ r2 ≥ · · · ≥ rn > 0, in the spaces n , 1 ≤ q ≤ ∞. It turns out that this problem is closely related to the following fundamental isoperimetric problem; (see [542, p. 123]):

16.4 Evaluation of Widths of  p -Ellipsoids n

383 p

|xi | q → sup,

(x1, . . . , xn ) ∈ Bn (r) .

(16.8)

i=1

We define αk (r, p, q) =

k+1 

pq

rip−q

 p−q pq

,

βk (r, p, q) =

n 

i=1

pq

rip−q

 p−q pq

,

(16.9)

i=k+1

1 ≤ p, q < ∞, and the same notation will be used for p or q = +∞. Theorem 16.13 (A. Pietch, M. I. Stesin, A. B. Khodulev) Let n ∈ N, k ∈ Z+ , k ≤ n, 1 ≤ p, q ≤ ∞. Then p

p

p

p

p

p

p

p

p

q

p

q

p

q

p

q

p

q

ak (Bn (r), n ) = bk (Bn (r), n ) = dk (Bn (r), n ) = λk (Bn (r), n ) p p = πk (Bn (r), n ) = rk+1, ak (Bn (r), n ) = dk (Bn (r), n ) = λk (Bn (r), n ) = πk (Bn (r), n ) = βk (r, p, q), p > q, ak (Bn (r), n ) = αk (r, p, q),

p < q.

Proof We first prove the inequalities: p q (1) πk (Bn (r), n ) ≤ βk (r, p, q), p > q; p q (2) ak (Bn (r), n ) ≤ αk (r, p, q), p < q. To prove (1), we construct a projection τ ∈ P(Rn, M), where L ∈ L n (Rn ) is the subspace generated by the first k basis vectors from Rn : τ(x) = (x1, . . . , xk , 0, . . . , 0),

x = (x1, . . . , xn ) ∈ Rn .

p

Now, for all x ∈ Bn (r), we have q

 x − τ(x)q = ≤

n

|xi | q =

i=k+1 n 

n  q

 xi  q   · ri r i=k+1 i

n  x p  q  q p  p−q p  i p q rip−q ≤ βk (r, p, q),   r i i=k+1 i=k+1

where we used Hölder’s inequality. Hence p

q

πk (Bn (r), n ) ≤ βk (r, p, q). p

To prove (2), we choose an arbitrary x ∈ Rn , set ξi = riq−p |xi |, and arrange the sequence (ξi ) in increasing order: ξi1 ≤ · · · ≤ ξin . We define τ(x) = (y1, . . . , yn ) ∈ Rn ,

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16 Width. Approximation by a Family of Sets

yi1 = · · · = yin−k−1 = 0,  p rin−k q−p yi j = xi j − |xin−k | sgn xi j , j = n − k, . . . , n . ri j 

Note that yin−k also vanishes; that is, the range of τ lies in the union of coordinate planes of dimension k. The mapping thus constructed is piecewise linear and continuous in the closure of every domain from Rn on which it is linear. Hence it is also continuous on the whole space Rn . To estimate the error of approximation of p x ∈ Bn (r) by τ(x), we change from x-variables to ξ-variables: q

 x − τ(x)q =

n−k−1

pq

|xi j | q + riq−p · |xin−k | q · n−k

j=1

=

n

pq

rip−q j

j=n−k

n−k−1

pq p−q

q

ξi j ri j

q

+ ξin−k ·

j=1

n

pq

rip−q . j

j=n−k p

With this change, the inclusion x ∈ Bn (r) becomes  n  n pq

 xi j  p p   = ξi j · rip−q ≤ 1.  ri  j j j=1 j=1 From this representation of the approximation error and since the sequence (ξi j ) is monotone, it follows that the maximum of the norm of the difference  x − τ(x)q is attained if ξin−k = · · · = ξin . Hence it suffices to estimate  x − τ(x)q only in the case that  n  n pq pq

 xi j  p n−k−1 p p p−q   = ξ · r + ξ rip−q ≤ 1.  ri  ij in−k ij j j=1

j

j=1

j=n−k

This leads to a problem (with fixed s > 1, ρ j > 0, j = 1, . . . , n − k) of the form n−k

σjs ρ j → max

j=1

n−k

σj ρ j ≤ 1,

0 ≤ σ1 ≤ · · · ≤ σn−k .

j=1

Its solution is attained with σj = 0, j = 1, . . . , n − k − 1, σn−k · ρn−k = 1, which gives  x − τ(x)q ≤ αk (r, p, q). Theorem 16.13 is proved.



Theorem 16.14 (A. B. Khodulev) Let X, Y , Z be normed linear spaces on the same linear space X = Y = Z = L, and let BX , BY be the unit balls in the spaces X and

16.4 Evaluation of Widths of  p -Ellipsoids

385

Y , k1, k2 ∈ Z+ . Then the Alexandrov width has the following multiplicity property: ak1 +k2 (BX , Z) ≤ ak1 (BX , Y ) · ak2 (BY , Z). Proof We choose Mi ∈ Cki (L), i = 1, 2, and mappings f ∈ C(X, M1 ), g ∈ C(Y, M2 ). Given x ∈ L, we set b1 = sup  x − f (x)Y , x ∈B X

b2 = sup  y − g(y)X , y ∈BY

y(x) =

1 (x − f (x)). b1

By definition, y(x) ∈ BY for any x ∈ BX . Let h(x) = f (x) + b1 · g(y(x)). Then  x − h(x)X =  x − f (x) − b1 · g(y(x))X = b1 ·  y(x) − g(y(x))X ≤ b1 · b2 . Furthermore, Im h = Im f + b1 · Im g ⊂ M1 + b1 · M2 ∈ Ck1 +k2 (L). So ak1 +k2 (BX , Z) =

inf

inf

sup  x − τ(x) Z

M ∈ Ck1 +k2 (L) τ ∈ C(Z, M) x ∈B X

≤ sup  x − h(x) Z ≤ b1 · b2 . x ∈B X

Remark 16.4 A similar argument for the Kolmogorov and linear widths gives that dk1 +k2 (BX , X) ≤ dk1 (BX , Y ) · dk2 (BY , X) , λk1 +k2 (BX , X) ≤ λk1 (BX , Y ) · λk2 (BY , X) . Lemma 16.2 Let X be a Banach space and let there exist a norm-one linear projection π : M → L onto a linear space L ∈ L n+1 (M) \ L n (M), n ∈ Z+ . Let C ⊂ X be such that (δ · BX ∩ L) ⊂ C for some δ > 0. Then an (C, X) ≥ δ. Proof Assume the contrary: an (C, M) < δ. Then we can choose a pair (τ, K), where τ ∈ C(X, K), K ∈ Cn (M), and a positive θ such that  x − τ(x) ≤ δ − θ for all x ∈ C. Consider the mapping ψ : C → L defined by ψ = (π ◦ τ). The range of ψ lies in the union of planes of dimension ≤ n, and moreover, ψ is a continuous (δ − θ)displacement of the ball (δ · BX ∩ L) from L, because for all x ∈ (δ · BX ∩ L) ⊂ C,  x − (π ◦ τ)(x) = π(x) − (π ◦ τ)(x) ≤  x − τ(x) ≤ δ − θ. By the ε-displacement theorem, the range of ψ also contains a ball of dimension (n + 1), which contradicts the fact that the range of ψ can be covered by a finite number of planes of dimension ≤ n.  Corollary 16.4 Let n ∈ N, 1 ≤ q ≤ ∞. Then q

q

an−1 (Bn , Bn ) = 1.

386

16 Width. Approximation by a Family of Sets

To complete the proof of Pietch–Stesin–Khodulev’s theorem, we derive the converse inequality to inequality 2) from the beginning of the proof: p

q

1 1 ≥ q p q p an−k−1 (Bn , n (r)) πn−k−1 (Bn , n (r)) 1 1 = p p = πn−k−1 (n ( r1 ), Bn ) βn−k−1 ( r1 , q, p)

ak (n (r), Bn ) ≥

= αk (r, p, q) , where q > p. Arguing as before, we show that p

1 1 = q p an−k−1 (Bn , n (r)) an−k−1 (np ( r1 ), Bnp ) 1 = = βk (r, p, q) , q < p, αn−k−1 ( r1 , q, p) q

ak (n (r), Bn ) ≥



proving Pietch–Stesin–Khodulev’s theorem.

Theorem 16.15 (A. N. Kolmogorov, A. A. Petrov, Yu. M. Smirnov, S. B. Stechkin) The following equality holds:  n−k 1 2 . dk (Bn, n ) = n k Proof Let us prove the lower estimate. Let L ∈ Lk (n2 ), dim L = k and let {bi }i=1 be an orthonormal system in L. Then n

bmi bmj = δi j ,

1 ≤ i, j ≤ k,

⇒

m=1

k n

b2mi = k.

i=1 m=1

k

For every point x ∈ n2 , i=1 (x, bi )bi is an element of best approximation to x by vectors of the subspace L, and the best approximation of x ∈ n2 is equal to ρ2 (x, L) = (x, x) −

k

(x, bi )2 . i=1

So to approximate the set Bn1 , it suffices to approximate the set of all its extreme points, which are the vertices {±ei } (the endpoints of the canonical basis for Rn ). Hence by averaging the approximations to vertices of the set Bn1 , we get a lower estimate for the approximation of the entire set of vertices, and hence a lower estimate for the best approximation of the set Bn1 :

16.5

Dranishnikov–Shchepin Widths and Their Relation to the CE-Problem

387

    n  k 

1 1 2 (ei, ei ) − dk (Bn, n )  (ei, bi )2 n i=1 i=1     n  k  n − k

1 2 1− . = bi = n i=1 n i=1 The upper estimate is obtained if we find an extreme subspace L 0 ∈ Lk (n2 ) on which the value of the width is attained. This subspace is invariant under a cyclic shift of coordinates of vectors from L 0 , and in the case k = 2m, it is made of invariant two-dimensional subspaces spanned by pairs of vectors  πl j x   πm j x    πjx  , . . . , cos , . . . , cos , j = 1, . . . , m; a j = cos m m m  πl j x   πm j x     πjx  , . . . , sin , . . . , sin , j = 1, . . . , m. b j = sin m m m In the case k = 2m + 1, these subspaces are augmented with the vector (1, . . . , 1). Note that all the vertices of Bn1 are obtained by a cyclic shift of two symmetric (with respect to the origin) vertices (±1, 0 . . . , 0). Hence the space L 0 is at the same  distance from all vertices of the ball Bn1 .

16.5 Dranishnikov–Shchepin Widths and Their Relation to the CE-Problem Long ago, a famous conjecture known as the CE-problem was formulated. It has three equivalent formulations (see [204]). (1) Can one assert that a compact set is finite-dimensional if all its subcompacta are acyclic in all dimensions starting from some n? (2) Is the image of a finite-dimensional compact set under a cell-like map finitedimensional? (3) Is every cell-like decomposition of Rn stably shrinkable, that is, does it become shrinkable on forming the product with the trivial decomposition of some Rm ? The CE-problem was answered in the negative in 1993 by Dranishnikov [203]. The first question was historically the first. It arose in connection with Alexandrov’s theorem on the homological characterization of dimension, which asserts that for a finite-dimensional compact set, its dimension is the same as the maximum of its homological dimensions, that is, the dimension over the group of rotations of the circle. The following problem of the width of a skeleton turned out to be closely related to the CE-problem. Alexandrov’s problem of the width of a skeleton. Let M be a compact rectilinear polyhedron (polytope) in Rn . Then is the k-dimensional Alexandrov width

388

16 Width. Approximation by a Family of Sets

ak (M, Rn ) of this polyhedron the same as the k-dimensional width of the (k + 1)skeleton of any triangulation of it? Dranishnikov showed that the answer to this question is affirmative for the onedimensional widths a1 (M, Rn ). Moreover, he further showed that an affirmative solution to the question in the general case implies an affirmative solution to the CE-problem (see [202]). So Alexandrov’s problem of the width of the skeleton of the k-dimensional Alexandrov widths ak (M, Rn ), k ≥ 2, has negative solution. V. M. Tikhomirov put forward in the Alexandrov seminar the following conjecture on widths of compacta, the affirmative solution of which would also imply an affirmative solution of the CE-problem. Tikhomirov’s question. Is it true that every finite-dimensional compact set M contains (k + 1)-dimensional compact sets N such that their k-dimensional Alexandrov widths ak (N, X) approximate arbitrarily closely the k-dimensional width ak (M, X) of the compactum itself? In this regard, Dranishnikov [202] was able to construct for every ε > 0 a fourdimensional polyhedron M = Mε4 for which a2 (M, X) = 1, while for every threedimensional subset N ⊂ M, the width a2 (N) is less than ε. Thus the question of the width of the skeleton answers Tikhomirov’s question in the negative. In [204], Dranishnikov and Shchepin introduced the two-index width, which they called the binary width. Recall that a polyhedron is a figure composed of ordinary ‘rectilinear’ simplices; a simplex is the convex hull of linearly independent points in a Euclidean subspace or a homeomorphic image thereof. We give a more rigorous definition. A simplicial complex K is defined as a family {τjk } of simplices in Rn satisfying the following conditions: 1) if τjk ∈ K, then every face of τjk also lies in K; 2) two simplices can intersect only in their common face. A simplicial complex is finite if it consists of a finite number of simplices. Consider the set-theoretic union K ⊂ Rn of all simplices from K and equip K with the strongest topology in which the embedding mapping of each simplex in K is continuous. In other words, a set A ⊂ K is closed in this topology if and only if A ∩ τjk is closed in τjk for each τjk ∈ K. The space K, and moreover, each topological space X homeomorphic to K, is called a polyhedron. The Dranishnikov–Shchepin width akn (M, X) of a polyhedron M ⊂ X is defined as the supremum of the Alexandrov widths ak (N, X) over its at most n-dimensional closed subsets N ⊂ M. For an arbitrary compact set M ⊂ X, the width akn (M, X) is defined as the infimum of the Dranishnikov–Shchepin (n, k)-widths of its neighbourhood. This definition of a width comes from a result of Henderson’s [289], according to which an infinite-dimensional compact set may fail to contain finite-dimensional subcompacta of positive dimension. It turns out that just as the equality ak (M, X) = 0 characterizes the condition dim M  k, so the homological dimension of a compact set is characterized by the Dranishnikov–Shchepin width: akk+1 (M, X) = 0 if and only if the homological dimension of M is ≤ k (see [204, Sect. 5]). We set ak∞ (M, X) = lim akn (M, X). Note that ak∞ (M, X)  ak (M, X). n→∞

16.6

Bernstein Widths in the Spaces L ∞ [0, 1]

389

Remark 16.5 The equality ak∞ (M, X) = ak (M, X) does not hold for any compact set M ⊂ X =  2 , because it would imply the positive solution of the CE-problem (see [204, Sect. 5]). The concept of the Dranishnikov–Shchepin width allows one to formulate the problem of the width of a skeleton in a form that, according to Dranishnikov and Shchepin, has greater chances for an affirmative solution (see [204]). The new skeleton width problem. Let P be a rectilinear polyhedron from Rm and let P k be the k-skeleton of its triangulation. Can we then assert that ank (P, Rm ) = ank (P k , Rm )

(n  k)?

Finally, we recall an old problem posed by Alexandrov closely related to this theme that is unsolved even in dimension 3. Problem of the width of the boundary. Let M = M n be a smooth compact n-dimensional submanifold of Rn (or a polyhedral body). For all k < n − 1, is it true that ak (∂ M, Rn ) = ak (M, Rn )?

16.6 Bernstein Widths in the Spaces L ∞ [0, 1] Let Q be an arbitrary linearly ordered set. In some cases, we shall consider continuously linearly ordered sets for topological spaces Q — by definition this means that x0  y0 for arbitrary sequences (xk ) and (yk ) that converge, respectively, to points x0 and y0 from Q and such that xk  yk (k ∈ N). By  ∞ (Q) we denote the set of bounded functions x : Q → R for which  x = supt ∈Q |x(t)|. Theorem 16.16 (I.G. Tsar’kov) Let Q be an arbitrary linearly ordered set, V ⊂  ∞ (Q) an arbitrary space of dimension n ∈ N. Then for all ε > 0, there exist n−1 from Q such that a function ϕ ∈ V, ϕ = 1, and a set of points {ti }i=0 t0 < t1 < · · · < tn−1 and ϕ(ti ) = (−1)i, i = 0, . . . , n − 1. Corollary 16.5 Let Q be an arbitrary continuously linearly ordered compact set, V ⊂  ∞ (Q) an arbitrary subspace of dimension n ∈ N. Then there exist a function n−1 from Q such that ϕ ∈ V, ϕ = 1, and a set of points {ti }i=0 t0 < t1 < · · · < tn−1 and ϕ(ti ) = (−1)i, i = 0, . . . , n − 1. Corollary 16.6 Let Q be an arbitrary continuously linearly ordered set, V ⊂ C0 (Q) an arbitrary subspace of dimension n ∈ N. Then there exist a function ϕ ∈ V, n−1 from Q such that ϕ = 1, and a set of points {ti }i=0 t0 < t1 < · · · < tn−1 and ϕ(ti ) = (−1)i, i = 0, . . . , n − 1.

390

16 Width. Approximation by a Family of Sets

∞ be an arbitrary subspace of Corollary 16.7 Let m, n ∈ N, m  n, and let V ⊂ m n−1 dimension n. Then for every tuple of numbers {εi }i=0 of 1’s and −1’s of length n in which the number of 1’s and −1’s is  [n/2], there exist a function ϕ ∈ V, ϕ = 1, n−1 from Q such that and a set of numbers {ki }i=0

k0 < k1 < · · · < k n−1

and

ϕ(ki ) = εi, i = 0, . . . , n − 1.

Corollary 16.8 Let V ⊂ C[0, 1] be an arbitrary subspace of dimension n ∈ N. Then n−1 from [0, 1] such there exist a function ϕ ∈ V, ϕ = 1, and a set of points {ti }i=0 that t0 < t1 < · · · < tn−1 and ϕ(ti ) = (−1)i, i = 0, . . . , n − 1. Proof We give a simple proof of this fact proposed by V. M. Tikhomirov. Let Ln−1 ⊂ C[0, 1] be a Chebyshev subspace of dimension n − 1. By S we denote the unit sphere of the space V (the dimension of S is n − 1). Consider the odd continuous mapping f : S → Ln−1 that associates with each function x ∈ S its nearest function from Ln−1 . By Borsuk’s theorem (Theorem 16.3), there exists a function x ∈ S having the constantly zero function as a nearest point from Ln−1 . By Chebyshev’s alternation theorem (see Sect. 2.1), there exists a tuple of points 0  t0 < t1 < . . . < tn−1  1 such that x(ti ) = (−1)i ε, i = 0, . . . , n − 1, where ε = −1 ∨ 1. Among the functions x and −x there is an appropriate one. The corollary is proved.  Let W11 [0, 1] and V[0, 1] be, respectively, the Sobolev class of functions and the class of functions of bounded variation. The seminorms in these spaces are defined, ∫1 respectively, as : 0 |x(t)| dt and Var10 x( · ). Corollary 16.9 The Bernstein width satisfies the relation bn (B, L∞ [0, 1]) =

1 , 2n

where B is the unit ball of the space W11 [0, 1] or V[0, 1].

16.7 Widths of Function Classes The first results related to the widths dn (W pr , Lq ) were obtained by Kolmogorov in the 1930s. The exact values of these widths were calculated in the 1960s by Stechkin, Rudin, and Tikhomirov for some p and q, and Tikhomirov [541] put forward a duality relation between the Kolmogorov and Gelfand widths. In the case of univariable functions, asymptotic expressions for the Kolmogorov widths were calculated in the 1970s. The order of the width was found for p ≥ q in [541]–[418]. For 2 ≥ q > p, the order was obtained by Ismagilov [300], who also found that the equivalence dn (W pr , Lq )  n−r+( p − q )+ 1

1

16.7

Widths of Function Classes

391

is violated for p = 1, q = ∞. A method was developed in [300] and later in [416] for reducing the problem of evaluating the order of dn (W pr , Lq ) to that of finding order estimates for dn (Bpk , lqk ). This method was called the discretization of the problem of widths of function classes. Kashin [325] obtained estimates for dn (Bpk , lqk ) and used the discretization method to establish the order of decrease for dn (W pr , Lq ) in the cases p ≤ 2 ≤ q and 2 ≤ p < q. p q In the ‘finite-dimensional’ problem, in the problem of evaluation of dn (Bm, m ) with arbitrary m, n, the most significant achievements are due to Ismagilov, Kashin, and Gluskin for the ‘lower triangle’ and by Pietch and Stesin for the ‘upper triangle’ (see [541]). A final answer to this problem was given by Gluskin [273], except for the case q = ∞, 1 ≤ p < 2 (the solution in this case is unknown at present). We also note that the orders of decrease for the Alexandrov widths an (W pr , Lq )  n−r were calculated by Stesin in 1975. Among problems on the estimation of widths, the Alexandrov widths play the important role of a yardstick. Evaluation of widths for classes of periodic functions of several variables was  α, L˜ 2 ) was evaluated. The order of begun by Babenko in [52], where the width d N (W 2  pα, L˜ p ), where α is a vector with integer coordinates, 1 < p < ∞, decrease of d N (W  pα and classes of functions determined by hypoelliptic operators, was found for W  pα (where p is by Mityagin [436]. The order of best approximation for the class W a scalar) in the L˜ p -metric, 1 < p < ∞, was found by Nikol’skaya [449]. The order pα, L˜ q ) for 1 < q = p < ∞, 2 < q ≤ p < ∞ and α ∈ Rn was determined of d N (W  pα, L˜ q ) for 1 < p < q < ∞ (p is a scalar) by Galeev [248], [249]. The order of d N (W was established by Temlyakov [538]–[539]. Later, Galeev [250] found the order of best approximation in the case of vectorpα¯ , L˜ q ) valued parameters p and q, 1 < q ≤ p < ∞, and also the width d N (W αj ¯ α p (α j , p and q are vectors) in the p = ∩W for a finite intersection of balls W cases 1 < q ≤ p < ∞, 1 < p ≤ q ≤ 2, 2 ≤ p ≤ q < ∞, and 1 < p ≤ 2 ≤ q < ∞, where inequalities for vectors are understood in the coordinatewise sense. Below, following Skorikov [524], we unify the notions of width and cowidth into one notion of a two-parameter family of set-theoretic characteristics. Namely, combining the Kolmogorov and Gelfand widths, Skorikov put forward the definition of the information Kolmogorov width. In this definition, additional a priori information about functions is specified in an optimal way and then is used to find a best approximation subspace. According to Skorikov [524], for equal parameters, this characteristic is significantly better on Sobolev classes than the Kolmogorov and Gelfand widths for q ≥ 2 ≥ p and sufficiently high smoothness (r > 1); namely, the order of the newly defined width coincides with that of the Alexandrov width.

392

16 Width. Approximation by a Family of Sets

16.7.1 Definition of the Information Width We recall the basic types of widths (with slight modification of the above definitions). Let X be a normed space and let M be a nonempty subset of X. We shall say that M is centrally symmetric if it is centrally symmetric with respect to the origin. Let BX denote a closed unit ball in X. The inclusion Lm ⊂ X with subscript m will be understood to mean that Lm is a subspace of X of dimension m, m ∈ Z+ . The inclusion V n ⊂ X with superscript n will be understood to mean that V n is a subspace of X of codimension n, n ∈ Z+ . Definition 16.5 Let L be a subspace of X. The deviation of a set M from a subspace L is defined as E(M, L)X := sup inf  f − ϕX . f ∈M ϕ ∈L

The Kolmogorov width of a set M ⊂ X is defined as follows: dm (M, X) := inf E(M, Lm + a)X . L m ⊂X a ∈X

(If M is centrally symmetric, then this definition is equivalent to the following one: dm (M, X) := inf Lm ⊂X E(M, Lm )X .) The Kolmogorov width of the empty set is defined to be zero. We note that for m = 0, the above definition coincides with that of the Chebyshev radius rX (M) of M. The Gelfand width of a set M is defined as d n (M, X) := inf

sup

V n x ∈M∩V n

 xX .

We also give the definitions of some other widths that will be used in this chapter. We denote by L(U, V) the set of all bounded linear operators from a normed space U to a normed space V. The linear width of a set M of order n is defined by λn (M, X) :=

inf sup  f − A f − aX .

inf

L n ⊂X, A∈ L(X, L n ) a ∈X f ∈M

(If M is centrally symmetric, then this definition can be equivalently written as λn (M, X) :=

inf

sup  f − A f X .

L n ⊂X f ∈M A∈ L(X, L n )

We denote by P(U, V) the set of all bounded linear projections from a normed space U to a normed space V. The projection width of order n of a set M is defined as inf inf sup  f − P f − aX . πn (M, X) := L n ⊂X a ∈X f ∈M P ∈ P(X, L n )

16.7

Widths of Function Classes

393

For centrally symmetric sets M, this definition is equivalent to the following one: πn (M, X) :=

inf

sup  f − P f X .

L n ⊂X f ∈M P ∈ P(X, L n )

The Bernstein width of order n of a centrally symmetric set M is defined as bn (M, X) := sup{R | ∃Ln+1 ⊂ X : RBX ∩ Ln+1 ⊂ M }. Definition 16.6 The information Kolmogorov width (see [524]) is defined as   dmn (M, X) := inf sup dm M ∩ (V n + b), X , V n b ∈X

where infV n is taken over all subspaces of codimension n. The name of this width reflects the fact that in approximating a set, it is useful to take into account some a priori (the best-chosen in the case under consideration) information. Definition 16.7 The central information Kolmogorov width of a set M in a Banach space X is defined as (see [524]) d˚m (M, X) := infn dm (M ∩ V n, X). V

It follows from the above definitions that dmn (M, X) =

d˚m (M, X) ≤ dmn (M, X),   inf sup dm M ∩ (V n1 + b), X .

V n1 : codim V n1 ≤n b ∈X

p

p

By Bn we denote the unit ball in n . If X is a Banach space, x ∈ X and x ∗ ∈ X ∗ , then the action of x ∗ on x will be denoted by x, x ∗ . We denote by x+ (x− ) the positive (negative) part of a number x and by p the conjugate of p: p1 + p1 = 1.

16.7.2 Estimates for Information Kolmogorov Widths We need the following lemma. Lemma 16.3 Let Lm+n ⊂ X and V n ⊂ X be two subspaces of a normed space X. Then Lm+n ∩ V n is a subspace of dimension at least m. (This result is proved in linear algebra.) To estimate the Kolmogorov widths from below, V. M. Tikhomirov proposed the following theorem (see [542, p. 220]). Theorem 16.17 For an arbitrary convex centrally symmetric set M, bn (M, X) ≤ dn (M, X).

394

16 Width. Approximation by a Family of Sets

The next theorem is, in a sense, a generalization of Theorem 16.17. Theorem 16.18 For an arbitrary convex centrally symmetric set M, bn+m (M, X) ≤ d˚m (M, X). Proof Consider an arbitrary subspace V n ⊂ X. By definition of the Bernstein width, for an arbitrary δ > 0, there exists a subspace L˜ m+n+1 such that sup{R : RBX ∩ L˜ m+n+1 ⊂ M } > bn+m (M, X) − δ. By Lemma 16.3, the subspace L˜ m+n+1 ∩ V n has dimension at least m + 1. Hence bm (M ∩ V n, X) ≥ bn+m (M, X) − δ. By Theorem 16.17, dm (M ∩ V n, X) ≥ bm (M ∩ V n, X). Combining the above estimates, we have, since δ, V n are arbitrary, d˚m (M, X) ≥ bm+n (M, X).

(16.10) 

The theorem is proved. Theorem 16.19 (the ball width theorem) Let m, n ∈ N, m + n < dim X. Then dmn (BX , X) = d˚m (BX , X) = 1.

Proof The upper bound follows from Definition 16.7; the lower bound is a consequence of Theorem 16.18 and the fact that bn+m (B, X) = 1.  Theorem 16.20 Let M ⊂ X be an arbitrary nonempty set. Then dmn (M, X) ≤ λm+n (M, X). Proof Given an arbitrary number ε > 0, we construct a subspace Ln+m and a bounded linear operator A: X → Ln+m such that sup  f − A f  ≤ λm+n (M, X) + ε.

f ∈M

Since A is a bounded linear operator with finite-dimensional range, it can be represented in the form m+n

(x, yk )ϕk , Ax = k=1

X ∗,

n+m is a linearly independent k = 1, . . . , m + n, φk ∈ Ln+m , and {φk }k=1 where yk ∈  n system in Ln+m . We form the space V := k=1,...,n Ker yk . Then  f , y j  = b, y j  for arbitrary vectors b ∈ X and f ∈ V n + b and all j = 1, . . . , n. Therefore,

16.7

Widths of Function Classes

λm+n (M, X) + ε ≥ =

sup

sup

f ∈M∩(V n +b)

 f − Af 



n m+n





f − (b, yk )φk − ( f , yk )φk



f ∈M∩(V n +b)

≥ inf

395

sup

k=1

k=n+1



m+n



  n

f − a −

( f , y )φ k k ≥ dm M ∩ (V + b), X .

a ∈X f ∈M∩(V n +b)

k=n+1

This proves the theorem, since ε and b are arbitrary.



16.7.3 Some Exact Inequalities Between Widths. Projection Constants Throughout this section, the set M ⊂ X is assumed to be nonempty, convex, and centrally symmetric. It follows from the definitions of the widths dn , λn , and πn that dn (M, X) ≤ λn (M, X) ≤ πn (M, X). In [273], Gluskin posed the following question: what is the smallest constant C for which the inequality λn (M, X) ≤ Cdn (M, X) holds? In the Hilbert space setting, we have (see [542]) d n (M, H) ≤ dn (M, H). There arises the question whether this estimate is true in an arbitrary Banach space. Theorem 16.21 below answers these questions with an exact constant for the class of centrally symmetric convex sets. We require some definitions. Definition 16.8 The coprojection constant of order n of a space X is defined as α n (X) := sup

inf

L n PL n ∈ P(X, L n )

 Id −PLn ,

where the supremum is taken over all subspaces of codimension n, and the infimum, over all linear projections to this subspace. Definition 16.9 The projection constant of order n of a space X is defined as αn (X) := sup

inf

L n PL n ∈ P(X, L n )

PLn .

Note that αn (X) − 1 ≤ α n (X) ≤ αn (X) + 1.

(16.11)

Clearly, if X = H is a Hilbert space, then αn (H) = = 1. Kashin [325] (see also [348]) evaluated the asymptotic behavior of the projection constants of some classical spaces: α n (H)

396

16 Width. Approximation by a Family of Sets 1 1  p  αn [Cn]  n | 2 − p |,

C > 1,

1 1   αn L˜ p (T1 )  n | 2 − p | .

Now from (16.11) we have, for C > 1, 1 1  p   p  αn [Cn]  α n [Cn]  n | 2 − p |,

1 1     αn L˜ p (T1 )  α n L˜ p (T1 )  n | 2 − p | .

Definition 16.10 We define α n,m (X) := sup L n+m

inf

sup

inf  x − PLn x − lm X .

L n, L m, PL n x ∈B X lm ∈L m L n +L m =L n+m

Theorem 16.21 (E. M. Skorikov) The following equalities hold: λn (M, X) πn (M, X) = sup d (M, X) n M M dn (M, X) n d n (M, X) d (M, X) = sup = sup 0 = α n (X), M dn (M, X) M dn (M, X) d n (M, X) = α n,m (X) ≤ α n (X), sup m M dn+m (M, X) sup

where the supremum extends over all convex centrally symmetric sets M ⊂ X. To prove Theorem 16.21, we require the following lemma. Lemma 16.4 Let : 2X → R+ be a function of centrally symmetric sets that satisfies the following conditions: (1) M1 ⊂ M2 ⇒ 0 ≤ (M1 ) ≤ (M2 ), (2) (λM1 ) = λ(M1 ) for every λ, λ > 0. Then (M) = sup (Ln + BX ), sup M dn (M, X) Ln where BX is the unit ball in X. Proof Consider an arbitrary number ε > 0 and a centrally symmetric set M. There is a subspace Ln such that E(M, Ln ) ≤ dn (M, X)(1 + ε). Hence

1 (M) (M) ≤ . 1 + ε dn (M, X) E(M, Ln )

By the definition of E(M, Ln ), M ⊂ Ln + E(M, Ln )BX := N,

E(N, Ln ) = E(M, Ln ).

In view of this fact and the properties of the function , we obtain

16.7

Widths of Function Classes

397

  N (N) (M) ≤ = = (Ln + BX ). E(M, Ln ) E(N, Ln ) E(N, Ln ) Comparing the above estimates and taking into account the arbitrariness of ε and M, we obtain (M) . sup (Ln + BX ) ≥ sup d n (M, X) Ln M The reverse estimate readily follows if sets of the form Ln + BX are taken as M. The lemma is proved.  Proof (of Theorem 16.21) We note that all the widths mentioned in the theorem possess the properties required for the application of Lemma 16.4. (1) Let us prove that πn (M, X) = α n (X). sup M dn (M, X) By Lemma 16.4, the left-hand side can be written as sup M

πn (M, X) = sup πn (Ln + BX , X) dn (M, X) Ln = sup

inf

sup

L n {L n1 , P 1 } {x ∈B X , l ∈L n } L

 x + l − PLn1 x − PLn1 l .

n

It can easily be seen that if Ln1  Ln , then sup

{x ∈B X , l ∈L n }

 x + l − PLn1 x − PLn1 l  = ∞

for an arbitrary linear projection PLn1 : X → Ln1 . Hence sup

inf

sup

L n {L n1 , P 1 } {x ∈B X , l ∈L n } L

 x + l − Px − Pl 

n

= sup inf sup  x − Px := α n (X). L n PL n x ∈B X

(2) A similar analysis shows that sup M

λn (M, X) = sup λn (Ln + BX , X) dn (M, X) Ln = sup

inf

sup

L n {L n1 , A: X→L n1 } {x ∈B X , l ∈L n }

 x + l − Ax − Al .

It is easy to see that if Ln1  Ln , then sup

{x ∈B X , l ∈L n }

 x + l − Ax − Al  = ∞

for every linear operator A: X → Ln1 . Hence

398

16 Width. Approximation by a Family of Sets

sup λn (Ln + BX , X) = sup

inf

sup

L n A: X→L n {x ∈B X , l ∈L n }

Ln

 x + l − Ax − Al .

If A is not a projection, then there is a vector l ∗ ∈ Ln such that Al ∗  l ∗ . In this case, sup

{x ∈B X , l ∈L n }

 x + l − Ax − Al  ≥ sup γl ∗ − γ Al ∗  = ∞. γ>0

Hence A: X → Ln is a projection, and sup Ln λn (Ln + BX , X) = α n (X). (3) Let us prove the equality sup M

d0n (M, X) dn (M, X)

= α n (X).

Using Lemma 16.4, we get d0n (M, X)

= sup d0n (Ln + BX , X) dn (M, X) Ln   = sup inf sup rX (Ln + BX ) ∩ (V n + b) . sup M

L n V n b ∈X

Note that the subspace V n should be chosen in such a way that dim (Ln ∩V n ) = 0. To such a pair (Ln, V n ) there corresponds a unique linear projection PLn onto Ln along V n . We take a biorthogonalsystem ({ϕ1, . . . , ϕn }, {y1, . . . , yn }) such that Ln = span{ϕ1, . . . , ϕn } and V n = nj=1 Ker y j and consider the Chebyshev radius rX ( · ) of the set (Ln + BX ) ∩ (V n + b). This set is centrally symmetric relative to PLn b ∈ Ln , and hence   sup l + x − PLn b rX (Ln + BX ) ∩ (V n + b) := l ∈L n, x ∈B X : PL n (l+x)=PL n b

= sup  x − PLn x. x ∈B X

As a result, we get sup d0n (Ln + BX , X) = sup inf sup  x − PLn x = α n (X). L n PL n x ∈B X

Ln

Note that the set Ln + BX is invariant under a translation along Ln , and therefore, without loss of generality, PLn b can be replaced by 0. We have thus shown that sup M

d n (M, X) = α n (X). dn (M, X)

(4) Let us prove the last equality, sup M

dmn (M, X) = α n,m (X). dn+m (M, X)

16.7

Widths of Function Classes

399

Using Lemma 16.4, we find that sup M

dmn (M, X) = sup d n (BX + Ln+m, X) = A, dn+m (M, X) Ln+m m

where A = sup inf sup

inf

L n+m V n b ∈X L m,a ∈X

(16.12)

  E (BX + Ln+m ) ∩ (V n + b), Lm + a .

Note that by Lemma 16.3, we have dim(Ln+m ∩ V n ) ≥ m; moreover, dim(Ln+m ∩ V n ) = m, since otherwise, the deviation of the set Ln+m ∩ V n from an arbitrary m-dimensional subspace would be infinite. Moreover, the subspace Lm is specified uniquely, Lm = Ln+m ∩ V n,   since otherwise, we would have E (BX + Ln+m ) ∩ V n, Lm = ∞. Let Ln be the complement of Lm in Ln+m (Ln + Lm = Ln+m ). In the estimates below, we set for brevity sup = sup , sup = sup . sup = sup , x ∈B X

x

ln ∈L n

ln

lm

lm ∈L m

We have A = sup

inf

L n+m

= sup L n+m (i)

= sup L n+m

(ii)

= sup L n+m

= sup L n+m

= sup L n+m

sup inf

inf  x + ln + lm − l 2 − a

sup

b ∈X a ∈X x,ln ,lm V n, L n l 2 ∈L m x+ln +lm ∈V n +b L n +L m =L n+m

sup

inf  x + ln − l 2 − a

inf

sup inf

inf

sup inf

inf

sup

sup

inf  x + ln − l 2 − PLn b

inf

sup

sup

inf  x + ln − l 2 − PLn x − ln 

b ∈X a ∈X x,ln V n, L n l 2 ∈L m x+ln ∈V n +b L n +L m =L n+m

inf  x + ln − l 2 − a

sup

L n , PL n b ∈X a ∈X x,ln l 2 ∈L m PL n (x+ln )=PL n b L n +L m =L n+m L n , PL n b ∈X x,ln l 2 ∈L m PL n (x+ln )=PL n b L n +L m =L n+m

L n , PL n b ∈X x,ln l 2 ∈L m PL n (x+ln )=PL n b L n +L m =L n+m inf sup inf  x − PLn x − l 2  x l 2 ∈L m L n , PL n L n +L m =L n+m

= α n,m (X).

Here (i) follows from the fact that infV n can be replaced by inf PL n taken over all linear projections onto the subspace Ln ⊂ Ln+m with the condition PLn (x + ln ) = PLn b instead of x + ln ∈ V n + b. Equality (ii) is secured by the fact that the set (BX + Ln ) ∩ (V n + b), in which we now have V n = Ker PLn , is centrally symmetric with respect to PLn b. It remains to prove the inequality α n,m (X) ≤ α n (X). We have α n,m (X) := sup L n+m

inf

L n , PL n L n +L m =L n+m

sup inf  x − PLn x − l 2  x l 2 ∈L m

(16.13)

400

16 Width. Approximation by a Family of Sets

≤ sup L n+m

inf

L n , PL n L n +L m =L n+m

sup  x − PLn x

≤ sup inf sup  x − PLn x = α n (X). L n , L m PL n

(16.14)

x

(16.15)

x



The theorem is proved.

Note that in the spaces L˜ q = L˜ q (T1 ), q ≥ 2, the above estimate is attained with respect to the order on the Sobolev classes:  α, L˜ q ) d n (W 2  α n (Lq ). α  ˜ dn (W , Lq ) 2

The specific nature of the width means that generally speaking, some inequalities that hold for the Kolmogorov width are false for the information Kolmogorov width. To find an upper estimate for the Kolmogorov width of function classes in the spaces L˜ p (Tn ), the following obvious inequality is used: dn+m (B + C, X) ≤ dn (B, X) + dm (C, X). It turns out that a similar estimate does not hold for the information Kolmogorov width. Example 16.4 The following inequality holds: sup

M1 , M2

d˚0 (M1 + M2, X) ≥ α n (X), d n (M1, X) + d0 (M2, X)

where the supremum is taken over all pairs of centrally symmetric convex sets. Indeed, we set M2 = B X . As M1 we take all n-dimensional subspaces L n of X. Note that d0 (B X , X) = 1 (the ball width theorem) and d n (L n, X) = 0. Hence the quantity under study is estimated as d0n (M1 + M2, X) sup ≥ sup d0n (L n + B X , X). n M1 , M2 d (M1, X) + d0 (M2, X) Ln By Theorem 16.21, the above quantity is precisely α n (X).

However, if one requires that the sets be ‘mutually orthogonal’, then the preservation of the inequality under consideration can be guaranteed. Theorem 16.22 (E. M. Skorikov [524]) Let L N and V N be subspaces in X such that L N + V N = X and let dim X = ∞. If M1 ⊂ L N and M2 ⊂ V N , then n1 +n2 n1 n2 (M1 + M2, X) ≤ d˚m (M1, X) + d˚m (M2, X) d˚m 1 2 1 +m2

for all nonnegative n1, n2, m1, m2 ∈ Z+ , n1 < N. n1 (M1, X) + ε, d2 = Proof Consider an arbitrary number ε > 0 and write d1 = d˚m 1 n d˚m22 (M2, X) + ε. Next, we choose subspaces Lm1 , Lm2 , V n1 , V n2 such that

16.7

Widths of Function Classes

401

M1 ∩ V n1 ⊂ d1 BX + Lm1 ,

(16.16)

M2 ∩ V

(16.17)

n2

⊂ d2 BX + Lm2 .

Consider the subspaces n1 := V n1 ∩ L N + V N , V

n2 := V n2 ∩ V N + L N . V

Note that dim(V n1 ∩ L N ) ≥ N − n1 and codim(V n2 ∩ V N ) ≤ N + n2 . Hence n1 ) = codim(V N ) − dim(V n1 ∩ L N ) ≤ N − (N − n1 ) = n1, codim(V n2 ) = codim(V n2 ∩ V N ) − dim L N ≤ N + n2 − N = n2 . codim(V n1 , M2 ∩V n2 = M2 ∩V n2 . Now from (16.16), (16.17) Furthermore, M1 ∩V n1 = M1 ∩V we have n1 ⊂ d1 BX + Lm1 , M1 ∩ V n2 ⊂ d2 BX + Lm2 . M2 ∩ V

(16.18) (16.19)

n1 ∩ V n2 . By construction, Consider m1 ∈ M1 and m2 ∈ M2 such that m1 + m2 ∈ V N n n n  1, and therefore m2 ∈ V  1. Hence m1 ∈ V  2 , since M1 ⊂ L N ⊂ V n2 . M2 ⊂ V ⊂ V Consequently, n1 ∩ V n2 ⊂ M1 ∩ V n1 + M2 ∩ V n2 ⊂ (d1 + d2 )BX + Lm1 + Lm2 (M1 + M2 ) ∩ V 

in view of (16.18), (16.19). The theorem is proved.

Note that generally speaking, Theorem 16.22 and Example 16.4 imply that the information Kolmogorov width (as well as the Gelfand width) is discontinuous (as a set function) in the Hausdorff metric. We choose a space X isometric to X ⊕ X = X1 ⊕ X2 and consider the set RBX ∩ (BX1 + Ln ), Ln ⊂ X1 , where R ∈ R is sufficiently large that   1 d n RBX ∩ (BX1 + Ln ), X ≥ α n (X). 2 We then ‘slightly turn’ the subspace Ln in such a way that h(RBX ∩ Ln, RBX ∩ L˜ n ) < ε,

L˜ n ∩ X1 = {0}.

Hence

  h RBX ∩ (Ln + BX1 ), RBX ∩ ( L˜ n + BX1 ) < ε,   and now, using Theorem 16.22, we get d n RBX ∩ ( L˜ n + BX1 ), X = 1. Note that if M is a compact set, then lim

ε→+0

sup

N ⊂X, h(N, M)≤ε

dmn (N, X) = dmn (M, X).

402

16 Width. Approximation by a Family of Sets

16.7.4 Some Order Estimates and Duality of Information Width We begin this section by considering the following question: do the central information Kolmogorov and information Kolmogorov widths coincide (with respect to the order of the parameters) for convex centrally symmetric sets? The theorem below gives a positive answer to this question. Theorem 16.23 (E. M. Skorikov [524]) Let M be a convex centrally symmetric set. Then d˚m (M, X) ≤ dmn (M, X) ≤ 2d˚m (M, X). Proof The lower estimate for dmn (M, X) follows readily from the definition. Let us prove the upper estimate. Recall width: dmn (M, X) :=   the definition of the n n infV n supb ∈X dm M ∩ (V + b), X . We fix a subspace V and consider an arbitrary vector b ∈ X. Let Mb = M ∩ (V n + b). Then  1 1 (Mb + M−b ) = M ∩ (V n + b) + M ∩ (V n − b) 2 2 1 ⊂ (M + M) ∩ (V n + b + V n − b) = M0 . 2 As a corollary, dm (Mb, X) ≤ 2dm (M0, X), which proves the theorem.

(16.20) 

Let us consider the duality relation for the width d˚m (B, C). To this end, we introduce the necessary definitions. Let X be a Banach space and let X ∗ be its dual space. Definition 16.11 Let L be a subspace of X. The set L ⊥ := {x ∗ ∈ X ∗ | ∀ x ∈ L (x, x ∗ ) = 0} is called the annihilator of L. For L ∗ ⊂ X ∗ , the preannihilator is defined as (L ∗ )⊥ := {x ∈ X | ∀ x ∗ ∈ L ∗ (x, x ∗ ) = 0}. Duality relations between widths of different types can be useful in width evaluation. These relations are based on the following operation on sets. Definition 16.12 The polar of a set M ⊂ X is the set  M 0 := x ∗ ∈ X ∗ | ∀ x ∈ M (x, x ∗ ) ≤ 1 . The prepolar of a set M ∗ ⊂ X ∗ is defined as  (M ∗ )0 := x ∈ X | ∀ x ∗ ∈ M ∗ (x, x ∗ ) ≤ 1 . Let X be a Banach space and let M ⊂ X. The Minkowski functional of M (the gauge function) is defined as μ(x | M) := inf{λ ≥ 0 | x ∈ λM }.

16.7

Widths of Function Classes

403

Let M be a centrally symmetric bounded convex set and let L be the linear span of M. We denote by XM the completion of L in the norm μ( · | M). Tikhomirov [298] established a duality between the Kolmogorov and the Gelfand widths. Theorem 16.24 (See [542, p. 146]) Let B and C be centrally symmetric bounded closed convex sets in a Banach space X and let 0 ∈ int B. Then dn (C, XB ) = d n (B0, XC∗ 0 ). The next theorem establishes the self-duality of the central information Kolmogorov width. Theorem 16.25 (E. M. Skorikov [524]) Let B and C be centrally symmetric bounded closed convex sets in a Banach space X and let 0 ∈ int B and 0 ∈ int C 0 . Then n (C, XB ) = d˚nm (B0, XC∗ 0 ). d˚m

Proof Let L be a subspace of X and let C, 0 ∈ C, be a closed convex subset of X. The following relations were proved in [542, p. 146]: (C 0 )0 = C,

(C + L)0 = C 0 ∩ L ⊥,

(C ∩ L)0 = C 0 + L ⊥, 1 (αB)0 = B0 . α

B ⊂ C ⇒ C 0 ⊂ B0,

(16.21) (16.22)

n = (Similar relations hold for the preannihilator and the prepolar.) We write d˚m n (C, X ). For an arbitrary ε > 0, we choose subspaces V n and L such that d˚m B m   n C ∩ V n ⊂ d˚m + ε (B + Lm ).

Now, using (16.21) and (16.22), we have 1 ⊥ ) ⊂ C 0 + (V n )⊥ (B0 ∩ Lm n +ε d˚m   n   n   ⊥ ⇒ B 0 ∩ Lm ⊂ d˚m + ε C 0 + (V n )⊥ ⊂ d˚m + 2ε C 0 + (V n )⊥ , where the rightmost inclusion is secured by the condition 0 ∈ int C 0 . Since ε is arbitrary, we have n (C, XB ). d˚nm (B0, X ∗ 0 ) ≤ d˚m C

The converse estimation is carried out in exactly the same way (it is only necessary to interchange B and C 0 , C and B0 , X and X ∗ , m and n, the annihilator and the preannihilator, and the polar and the prepolar). As a result, we get the estimate  n 0 (C )0, X(B0 )0 ≤ d˚nm (B0, XC∗ 0 ). d˚m Now the theorem follows from (16.21).



404

16 Width. Approximation by a Family of Sets

Corollary 16.10 The following duality relations hold: n (Bpk , lqk ) = d˚nm (Bqk , lpk ). d˚m

(16.23)

We set dwv = dwv (C, XB ), d˚wv = d˚wv (C, XB ). Theorem 16.26 Let C ∈ XB be a centrally symmetric convex set, let 0 ∈ int B, and let 0 ∈ int C 0 . Then  1 1 n d n+k+l, k ∗ dm+k+l max 2αl (XB ) m 2α (XC 0 )  n+k n ≤ min 2α k (XB )dm+k+l , 2αl (XC∗ 0 )dmn+k+l , ≤ dm+l  1 1 n+k+l max l , k ∗ d˚m+k+l d˚m α (XB ) α (XC 0 )  k n+k n+k+l . ≤ d˚m+l ≤ min α (XB )d˚m+k+l, αl (XC∗ 0 )d˚m Proof Consider an arbitrary subspace V n . By Theorem 16.21, k k (C ∩ V n, XB ) ≤ dm (C ∩ V n, XB ) ≤ α k (X)dm+k (C ∩ V n, XB ). d˚m

Since V n is arbitrary, we have k+n k (C, XB ) = infn d˚m (C ∩ V n, XB ) d˚m V

≤ α k (X) infn dm+k (C ∩ V n, XB ) = α k (X)d˚m+k (C, XB ). V

Using Theorem 16.23, we get n dmk+n (C, XB ) ≤ 2α k (X)dm+k (C, XB ).

Now the right-hand estimate of the theorem is secured by Theorem 16.25. The lefthand estimate then follows from the right-hand estimate. The theorem is proved.  In particular, Theorem 16.26 implies that if the values of the parameters k and l are not large (of the order of a constant), then it is of no importance in the calculation of the order of the width whether they are written as superscripts or subscripts. To calculate the orders of widths for Sobolev classes, we shall need the following generalization of a certain multiplicative inequality. Pietsch [473] was the first to establish such an inequality for Kolmogorov widths, and Khodulev [335] then established it for linear and Alexandrov widths. Theorem 16.27 (See [473], [335]) Let B, C, D ⊂ X be centrally symmetric convex sets. Then dn+m (B, XC ) ≤ dn (B, XD ) dm (D, XC ), an+m (B, XC ) ≤ an (B, XD ) am (D, XC ), λn+m (B, XC ) ≤ λn (B, XD ) λm (D, XC ), for all nonnegative integers n and m.

16.7

Widths of Function Classes

405

Theorem 16.28 (E. M. Skorikov [524]) Let B, C, D ⊂ X be centrally symmetric convex sets. Then (n2 −m1 )+ n1 +n2 n1 (B, XC ) ≤ d˚m (B, XD ) d˚m (D, XC ) d˚m 2 1 1 +m2

for all nonnegative integers n1 , n2 , m1 , and m2 . n1 (B, XD ), for an arbitrary ε > 0, there is a pair Proof By definition of the width d˚m 1 n 1 of subspaces V , Lm1 such that  n1  B ∩ V n1 ⊂ d˚m (B, XD ) + ε (D + Lm1 ). 1

Therefore, n1 +n2 n2 d˚m (B, XC ) ≤ d˚m (B ∩ V n1 , XC ) 1 +m2 1 +m2  n2  n1 (B, XD ) + ε d˚m (D + Lm1 , XC ). ≤ d˚m 1 1 +m2

(16.24)

Since D + Lm1 is centrally symmetric, it follows that n2 (D+Lm1 , XC ) := inf d˚m 1 +m2

sup

inf

{V n2 , L m1 +m2 } ( f ,l 1 )∈(D, L m ) l ∈L m1 +m2 1 ( f +l 1 )∈V n2

≤

inf

 f + l1 − l 

inf  f − l 2 .

sup

(16.25)

{V n2 , L m2 } ( f ,l 1 )∈(D, L m ) l 2 ∈L m2 1 ( f +l 1 )∈V n2

In the case that n2 ≤ m1 , we obtain inf

{V n2 , L

m2 }

inf  f − l 2 

sup ( f ,l 1 )∈(D, L

m1

( f +l 1 )∈V n2

) l 2 ∈L

m2

≤ inf sup inf  f − l 2  := dm2 (D, XC ). L m2 f ∈D l 2 ∈L m

(16.26)

2

If n2 > m1 , then let the = Ker P n2 hold for some linear bounded projecn 2 tion along V onto a subspace complementing V n2. Then the set { f : f ∈ D, ∃l 1 ∈ Lm1 , ( f + l 1 ) ∈ V n2 } coincides with the set { f ∈ D : P n2 f ∈ P n2 (Lm1 )}. We set L˜ = P n2 (Lm1 ). Note that dim L˜ ≤ m1 . Let F be the complement of L˜ in Im P n2 , that is, dim F ≥ n2 − m1 . (16.27) Im P n2 = L˜ + F, relation V n2

˜ Then We denote by P1 the projection from Im P n2 onto F along L. { f ∈ D | P n2 f ∈ P n2 (Lm1 )} = D ∩ Ker P1 ◦ P n2 . Since Im P1 ⊂ Im P n2 , the operator P1 ◦ P n2 is a projection onto F. Consequently, in view of (16.27), we have codim Ker P1 ◦ P n2 ≥ n2 − m1 . We write V 1 = V 1 (V n2 , Lm1 ) = Ker P1 ◦ P n2 and extend the inequality in (16.25):

406

16 Width. Approximation by a Family of Sets

inf

inf  f − l 2 

sup

{V n2 , L m2 } f ∈D∩V 1 (V n2 , L m1 ) l 2 ∈L m2

≤

inf

{V n2 −m1 , L

m2 }

sup

f

inf  f − l 2 .

∈D∩V n2 −m1 l 2 ∈L

(16.28)

m2

From (16.24), (16.28), (16.26) and since ε is arbitrary, we have (n2 −m1 )+ n1 +n2 n1 (B, XC ) ≤ d˚m (B, XD )d˚m (D, XC ). d˚m 2 1 1 +m2



The theorem is proved.

Corollary 16.11 Let B, C, D ⊂ X be arbitrary sets. Then for all nonnegative integer numbers n, m, n d˚m (B, XC ) ≤ d n (B, XD )dm (D, XC ). This is a consequence of Theorem 16.28 with m1 = 0, n2 = 0.

16.7.5 Some Order Estimates for Information Kolmogorov Widths of Finite-Dimensional Balls To estimate the orders of widths for  Sobolev  classes, we shall need to know the Cn . orders of decrease of the widths d˚nn BCn p , lq Theorem 16.29 (A. Pietch [473], M. I. Stesin [531]) Let 1 ≤ q ≤ p ≤ ∞ and n < m. Then 1 1  p q dn Bm, m = (m − n) q − p . Theorem 16.30 (E. D. Gluskin [273]) Let 2 ≤ p < q ≤ ∞ and n < m. Then 1  p q  1  ( 1 − 1 )/( 1 − 1 ) dn Bm, m  min{1, m q n− 2 } p q 2 q .

Theorem 16.31 (E. M. Skorikov [524]) For all p and q, 1 ≤ p ≤ q ≤ ∞, the asymptotic relation 1 1  p q dnn B4n, 4n  n q − p holds with a constant depending on p and q. Proof By the duality relations (16.23), it suffices to prove the theorem for p ≤ q, 1 1 p + q ≤ 1. We prove the lower estimate. If p < q, then  x p ≤ (4n) p − q  xl q , 4n 4n 1  p q q  T16.19 q1 − p1 q1 − p1 − p1 ˚  q ˚ q n . dn B4n, 4n = 4 dn B4n, 4n ≥ (4n) 1

1

(Here and in subsequent relations, we indicate the underlying theorems.)

16.7

Widths of Function Classes

407

For q > p ≥ 2, the upper estimate is secured by Theorem 16.30 and since  p q  p q d˚n B4n, 4n ≤ dn B4n, 4n . Let us prove the upper estimate for q ≥ 2 ≥ p. By Corollary 16.11,  p q  2 q p 2 )dn B4n , 4n d˚n B4n, 4n ≤ d n (B4n, 4n T16.30



T16.24

=

 2 q p 2 dn (B4n , 4n )dn B4n , 4n

n2− p nq −2 = nq − p . 1

1

1

1

1

1



The theorem is proved. Theorem 16.32 Let 1 ≤ q ≤ p ≤ ∞. Then 1 1  p q  dnn B10n, 10n  n q − p .

Proof We prove the lower estimate. By Theorem 16.28,  p  q p  q  p  3n  p B10n, 10n ≤ d˚n B10n, 10n d˚2n B10n, 10n . d˚3n  q  q p  p  From Theorem 16.19 and since d˚2n B10n, 10n ≤ d˚n B10n, 10n , it follows that  p  q q  p  d˚n B10n, 10n d˚n B10n, 10n ≥ 1. 1 1  q p  Further, an appeal to Theorem 16.31 shows that dnn B10n, 10n  n p − q , and hence 1 1  p q  d˚n B10n, 10n  n q − p .  p q  The upper estimate follows from Theorem 16.29 and the inequality dnn B10n, 10n ≤  p q  dn B10n, 10n . The theorem is proved.

16.7.6 Order Estimates for Information Kolmogorov Widths of Function Classes Order estimates for information Kolmogorov widths of function classes in the space L˜ p (Tn ) will be found using the standard discretization machinery for this problem. This technique was described in [300]–[273] and then developed by many authors. Our analysis will depend on results from [250], [200] on this subject. In what follows, relations and inequalities for vectors will be understood in the coordinatewise sense. The n-dimensional vectors consisting only of 1’s and ∞’s will be denoted by 1 and ∞, respectively. Let Tn = [−π, π]n be the standard n-dimensional torus. We let L˜ p = L˜ p (Tn ),

p = (p1, . . . , pn ),

1 ≤ p j < ∞,

j = 1, . . . , n,

denote the space of functions x(t) = x(t1, . . . , tn ) that are 2π-periodic with respect to every argument and possess finite mixed norms

408

16 Width. Approximation by a Family of Sets



1  x p = 2π

∫

π

−π



1 ... 2π

∫

π

−π

  x(t) p1 dt1

 pp2 1

...

 pp n

n−1

1 pn

.

Given a function x ∈ L˜ p having zero mean with respect to all arguments,

xk ei(k,t), where Z˚ n = {k ∈ Zn | k1 · · · k n  0} x(t) = ˚n k ∈Z

(the convergence of the series is understood in the sense of the L˜ p -convergence), and a vector α ∈ Rn , we formally introduce the operation of fractional differentiation defined by  n

! π (ik j )α j xk ei(k,t), (ik j )α j = |k j | α j exp i α j sign k j . x (α) = 2 ˚ n j=1 k ∈Z

Definition 16.13 For vectors α, p ∈ Rn , 1 ≤ p j < ∞, j = 1, . . . , n, the Sobolev class is defined as the set of functions having zero mean with respect to all arguments:  pα := x( · ) |  x (α) p ≤ 1 . W For a finite m-tuple p¯ = {p1, . . . , pm }, 1 ≤ pr < ∞, r = 1, . . . , m, and α¯ = {α 1, . . . , α m } ⊂ Rn , consider the class ¯ pα W ¯ =

m "

pαrr . W

r=1

 α¯ will be denoted by W pα¯ . If pr = p for all r = 1, . . . , m, then W p¯ Definition 16.14 With a vector s ∈ R+n we associate the set ˚ n | 2s j −1 ≤ |k j | < 2s j , j = 1, . . . , n}. s = {k ∈ Z Then for absolutely convergent series, we have

x(t) = xk ei(k,t) = δs x(t), ˚n k ∈Z

where δs x(t) =



k ∈ s

s ∈N n

xk ei(k,t) .

Lemma 16.5 If x ∈ L˜ q , then

 xp ≤  xq

for 1 ≤ p ≤ q ≤ ∞. As before, the cardinality of the set S ⊂ Zn will be denoted by |S| or card S. Let S ⊂ Nn , S := card{k ∈ s : s ∈ S}. We shall require the following result.

16.7

Widths of Function Classes

409

 Theorem 16.33 (Dinh Dung [200]) Let αr ∈ R+n , r = 1, . . . , m, S = s ∈ R+n | (αr , s) ≤ 1, r = 1, . . . , m . Then

card s  2μM μl s ∈μS∩N n

(the constants involved do not depend on μ), where M is the value of this problem and l is the dimension of the solution set of the problem (s, 1) → sup, s ∈ S. Given M ⊂ Rn , we set ZC (M) = {k ∈ Zn | ∃m ∈ M : k − ml∞n ≤ C}. m ⊂ Rn , let S = {s ∈ Rn : (α r , s) ≤ 1, Theorem 16.34 (Dinh Dung [200]) Let {αr }r=1 + + r = 1, . . . , m}, let F be the solution set of the problem (s, 1) → sup, s ∈ S, let M be the value of this problem, and let l be the dimension of F. Then

2(s,1)  2μM μl, |Z1 (μN)|  μl s ∈Z1 (μ N )

(the constants involved do not depend on μ). Theorem 16.35 (J. E. Littlewood – R. C. Paley) Let α ∈ Rn and let 1 < p < ∞. Then

  12



(α) (α,s) 2

|2 δs x|

 x p 

s ∈N n

p

(the constants involved depend on α and p). Theorem 16.36 (See [250]) Let S ⊂ Nn , x = |S|

( 12 − p1 )−



2

s ∈S

 |S|

( 12 − p1 )+

(α,s)





s ∈S

p δs x p

2

 p1

(α,s)

δs x, α ∈ Rn , 1 < p < ∞. Then   x (α)  p

p δs x p

 p1

s ∈S

(the constants involved depend on α, p). Theorem 16.37 (E. M. Galeev [250]) For 1 < p < ∞ and s ∈ Nn , the space of 2(s,1) are isomortrigonometric polynomials x(t) = k ∈s xk ei(k,t)  and the space R j phic. Under this isomorphism, the vector θ(x) j=1,...,2(s,1) with coordinates

410

16 Width. Approximation by a Family of Sets (s,1)

{xm (τr )} ∈ R2

,

xm (t) =

xk ei(k,t),

sign k l =sign m l

l=1,...,n

m =(±1, . . . , ±1) ∈ R ,

τ = (π2

n

r

ri = 1, . . . , 2si −1,

2−s1

r1, . . . , π22−sn rn ),

(16.29)

i = 1, . . . , n,

is associated with a function x( · ), and the order relation 

−(s,1)

 x p  2

(s,1) 2

|θ(x) |

j p

 p1

j=1

holds. Theorem 16.38 (E. M. Skorikov) Let q, pr , αr ∈ Rn , 1 < q, pr < ∞, r = 1, . . . , m, and let A be the convex hull of {α1, . . . , α m }. Then 1 ¯ ˜ l pα M d NN (W ¯ , Lq )  (N log N)

for

˚ +n  , (A − 1) ∩ R

where M is the value and l the dimension of the solution set of the problem (s, 1) → sup, (s, αr ) ≤ 1, r = 1, . . . , m. Proof From Lemma 16.5 it follows that ¯ ˜ N  α¯ ˜ pα d NN (W ¯  , Lq ) ≤ d N (Wp¯ , Lq ),

¯ ˜ N  α¯ ˜ pα d NN (W ¯ , Lq ) ≤ d N (Wp¯ , Lq )

for (p)r ≥ pr , r = 1, . . . , m, q ≥ q. Therefore, if the inequalities 1  pα¯¯ , L˜ q ) ≤ d N (W  pα¯¯ , L˜ q )  (N logl N) M1 (N logl N) M  d NN (W N

(16.30)

are satisfied, where  ( p¯ )rj ≡ max 2, max{prj˜˜ | r˜ = 1, . . . , m, j˜ = 1, . . . , n} ,  (q ) j ≡ min 2, min{q j˜ | j˜ = 1, . . . , n} ,  ( p¯ )rj ≡ min 2, min{prj˜˜ | r˜ = 1, . . . , m, j˜ = 1, . . . , n} ,  (q ) j ≡ max 2, max{q j˜ | j˜ = 1, . . . , n} , for r = 1, . . . , m, j = 1, . . . , n, then the assertion of the theorem will follow. (1) Let us prove the lower estimate in (16.30). Throughout this subsection, it will be assumed that q = q  and p¯ = p¯ . We note that 1 < q ≤ 2 ≤ p < ∞. Let S = {s ∈ R+n : (αr , s) ≤ 1, r = 1, . . . , m} and let F be the solution set of the  problem (s, 1) → sup, s ∈ S. We set Sμ = ZC (μF) and Γ = span{ei(k,t) : k ∈ s, s ∈ Sμ }, where μ is determined by the equation N = 2μM μl and C > 0 is a constant such that dim Γ ≥ 10N. By Theorem 16.34, these μ and C exist, and we have |Sμ |  μl .

16.7

Widths of Function Classes

411

By the Littlewood–Paley theorem,  pα¯ , L˜ q ) ≥ d N (W  pα¯ ∩ Γ, L˜ q ) T16.35  pα¯ ∩ Γ, L˜ q ∩ Γ). d NN (W  d NN (W N If x( · ) ∈ Γ, then x( · ) = s ∈Sμ δs x( · ), and so we have T16.36

 x (α)  p  |Sμ | 2 − p 1

1



p

2(α,s) δs x p

(16.31)

 p1

s ∈Sμ T16.37

 2

μ− μpM

μ

l l 2− p

(s,1)  2

|θ(δs x) |

j p

 p1

.

s ∈Sμ j=1

Hence  pα¯ x( · ) ∈ W

(s,1)  2

for

|θ(δs x) j | p

 p1

 2−μ+

μM p

l

l

μ p−2 .

(16.32)

s ∈Sμ j=1

On the other hand, if y ∈ Γ, then







 yq =

δs y

s ∈Sμ

T16.37



− μqM

2

T16.36

 |Sμ |

1 1 2−q



 q1

s ∈Sμ

q

μ

q δs yq

(s,1)  2

l l 2−q

|θ(δs y) j | q

 q1

(16.33) .

s ∈Sμ j=1

By Theorem 16.37, there is a natural isomorphism between the space Γ of trigonometric polynomials and the finite-dimensional space R Sμ  . Hence from (16.32), (16.33), and (16.31), we have μM μM l l  q   pα¯ , L˜ q )  2−μ+ p − q μ p − q d N B10N d NN (W p , 10N . N Now an appeal to Theorem 16.32 shows that  pα¯ , L˜ q )  2−μ+ d NN (W  2−μ+

μM p

μM p

− μqM

− μqM

l

l

μp−q N q−p l

1

1

l

μ p − q (2μM μl ) q − p = 2−μ  (N −1 logl N) M . 1

1

1

(2) Now let us prove the upper estimate in (16.30). Recall that in this case, it suffices to assume that 1 < p ≤ 2 ≤ q ≤ ∞. For ν, k ∈ N, n ≤ k ≤ νM, we set  Sνk := s ∈ Nn | ν − 1 ≤ max (α j , s) < ν, (s, 1) = k . j=1,...,m

Hence Sνk  = 2k |Sνk |. Next, we set # Sνk , Nνk = μM+μγM−2νγM+γk |Sνk |, 2

if if

n ≤ k ≤ νM, n ≤ k ≤ νM,

ν ≤ μ, ν > μ,

412

16 Width. Approximation by a Family of Sets

where μ satisfies the relation 2μM μl = N and the constant γ, 1 > γ > 0, obeys the condition (γ + 1)M < 1 (this can be guaranteed because by the hypotheses of the ˚ +n  , and so M < 1). Therefore, theorem, we have (A − 1) ∩ R νM

Nνk =

ν ≥1 k=n

μ νM

Sνk  +

=

2μM+μγM−2νγM+γk |Sνk |

ν>μ k=n

ν=1 k=n

νM

+

(s,1)

2

ν>μ

(α j ,s)≤μ

2μM+μγM−2νγM+γ(s,1) .

(α j ,s)≤ν

j=1,...,m

j=1,...,m

Further, by Theorem 16.33, we have νM

T16.33



Nνk

2μM μl +

2μM+μγM−νγM ν l  2μM μl = N,

ν>μ

ν ≥1 k=n



since ν>μ 2−νγM ν l  2−μγM μl .  pα¯ , then If x = s ∈Sν k δs x ∈ W m≥

m

j

p T16.36

 x (α )  p  |Sνk | ( 2 − p )p 1

1

m

2(α

j ,s)

δs x p

j=1 s ∈Sν k

j=1

 2 |Sνk | pν

( 12 − p1 )p

δs x

p T16.37 pν−k

 2

|Sνk |

( 12 − p1 )p

s ∈Sν k

and hence if x =



s ∈Sν k

(s,1)

2

|θ(δs x) j | p,

s ∈Sν k j=1

 pα¯ , then δs x( · ) ∈ W

(s,1)  2

|θ(δs x) j | p

 p1

k

 2−ν+ p |Sνk | p − 2 . 1

1

(16.34)

s ∈Sν k j=1

On the other hand,







δs x



s ∈Sν k

T16.36

≤

|Sνk |

1 1 2−q



δs x

q

 q1

s ∈Sν k

q T16.37



− qk

2

|Sνk |

1 1 2−q

(s,1)  2

|θ(δs x) |

j q

 q1

(16.35) .

s ∈Sν k j=1

Carrying out discretization (see Theorem 16.28) and using Theorem 16.22), we get    T16.22 N ¯ ˜ ¯ α N νk α  ˜ d Nν k W p ∩ x = δs x , L q d N (W p , Lq ) ≤ s ∈Sν k

ν,k

T16.35



ν,k

     pα¯ ∩ x = δs x , L˜ q ∩ x = δs x d NNννkk W s ∈Sν k

s ∈Sν k

16.7

Widths of Function Classes (16.34), (16.35)



νM

ν>μ k=n

T16.28



νM

ν>μ k=n

413

k k 1 1  q 2−ν+ p − q |Sνk | p − q d NNννkk BpSν k ,  S

νk

 

k k 1 1    q 2−ν+ p − q |Sνk | p − q d Nν k BpSν k ,  2Sν k  d Nν k B2Sν k ,  S

νk

 

.

By Theorem 16.30, νM

 pα¯ , L˜ q )  d NN (W =

νM

ν>μ k=n

=

νM

ν>μ k=n

k

k

1

1

2 2 2−ν+ p − q |Sνk | p − q Sνk  1− p Nνk Sνk  q Nνk 1

1

1

1

−1 2−ν+k |Sνk |Nνk

2−ν+k |Sνk | |Sνk | −1 2−μM−μγM+2νγM−γk

ν>μ k=n

=

νM

2−ν+k−μM−μγM+2νγM−γk .

ν>μ k=n

We finally get  pα¯ , L˜ q )  d NN (W

2−ν−μM−μγM+2νγM

ν>μ



νM

2k−γk

k=n −ν−μM−μγM+2νγM ν M−γν M

2

2

ν>μ

=

2−ν−μM−μγM+νγM+ν M

ν>μ

= 2−μM−μγM

2−ν+νγM+ν M  2−μM−μγM+μM+μγM−μ

ν>μ −μ

=2 The theorem is proved.

 (N

−1

1

logl N) M . 

N (W  α¯ , L˜ q ), where N and M We note that the order of decrease of the quantity d M p¯ are of the same order, is estimated in a similar way.

414

16 Width. Approximation by a Family of Sets

16.8 Relation Between the Jung Constant and Widths of Sets In this section, we note one relation between the Jung constant and widths of sets. Assume that some system of closed sets of diameter ≤ ε covers a set M so that every point x ∈ M is contained in at most k sets from the cover and that there exists at least one point from M that is covered by precisely k sets. In this case, one says that there is an ε-covering of the set M of multiplicity k. The infimum of ε for which there exists an ε-covering of M of multiplicity n + 1 is called the Urysohn width of the set M (denoted by un (M)). The quantity un (M) (which was first called a width) was introduced by Urysohn (see [542]). Let M be a compact subset of a Banach space X. Then (see [542, § 4.1]) a N (M, X) ≤ u N (M) ≤ 2a N (M, X) J(X)

(16.36)

(a N (M, X) is the Alexandrov width of the set M). This inequality implies that u N −1 (B 2 ) = N

1 . J(2N )

The spaces X for which J(X) = 1/2 are called centrable [347], [542] (see Definition 15.25). From (16.36), it follows that in a centrable space X, u N (M) = 2a N (M) for every compact set   M ⊂ X (see also [542, Proposition 3, p. 222; Proposition 5, p. 276]).

16.9 Sequence of Best Approximations Consider general properties of the sequence of best approximations ρ(x, Ln ) by an infinite system of expanding subspaces (Ln ) of a Banach space. Bernstein posed and solved in the affirmative the following inverse best approximation problem (see, for example, [102], [545, § 2.5], [262, § 2.15]): is it possible from a given arbitrary number sequence (dn ) such that d0 ≥ d1 ≥ d2 ≥ . . . > 0,

dn → 0,

to find a continuous function f ∈ C[a, b] satisfying En ( f ) = dn,

n = 0, 1, 2, . . . ,

where En ( f ) is the best approximation to f by algebraic polynomials of degree at most n?

16.9

Sequence of Best Approximations

415

Bernstein’s proof was carried over by Timan [545, § 2.5] to the case of an arbitrary system L1 ⊂ L2 ⊂ . . . of strictly embedded finite-dimensional subspaces of an arbitrary Banach space X. Namely, for every sequence of finite-dimensional subspaces  Ln = X, {0} = L0 ⊂ L1 ⊂ . . . ⊂ Ln ⊂ . . . , n

dimLn = n, of a Banach space X and a number sequence (dn ) of nonnegative numbers that monotonically tends to zero, there exists an element x0 ∈ X such that ρ(x0, Ln ) = dn , n = 0, 1, 2, . . . . Indeed, augmenting the given system of subspaces by subspaces of the required dimensions, we can assume without loss of generality that the sequence 0 = L0 ⊂ L1 ⊂ . . . ⊂ Ln ⊂ . . .  is such that dim Ln = n (n ∈ N) and n Ln = X. Consider the intersection of all closed neighbourhoods " An (An := Ln + B(0, dn )). A := n

It is easily shown that A is a compact set in X. For each k ∈ N, we translate the subspace Lk by a vector xk ∈ Lk+1 so as to have  xk  = dk and ρ(0, Lk + xk ) =  xk  = dk . In the plane Lk , there exists a vector xk−1 such that ρ(0, Lk−1 +xk +xk−1 ) =  xk + xk−1  = dk−1 , and so on. So for each m < k, we can translate Lm by the vector xk + xk−1 + . . . + xm (xm ∈ Lm+1 ) so as to have ρ(0, Lm + xk + xk−1 + . . . + xm ) =  xk + xk−1 + . . . + xm  = dm . For the vector yk = xk + xk−1 + . . . + x0 , we have ρ(yk , Lm ) = dm , m = 0, . . . , k. It is easily checked that the sequence (yk ) ⊂ A has a cluster point x ∈ A and that ρ(x, Ln ) = dn (n ∈ N). The following problem is still open. Problem 16.1 Let L1 ⊂ L2 ⊂ . . . be a system of strictly embedded (closed) linear subspaces (not necessarily finite-dimensional) of an infinite-dimensional Banach space X, let this system be complete in X, and let d1 ≥ d2 ≥ . . . , dn → 0, be a sequence of nonnegative numbers. Is it true that there exists an element x ∈ X such that the best approximations ρ(x, Ln ) to x by the subspaces Ln are equal to dn : ρ(x, Ln ) = dn,

n = 1, 2, . . .?

(16.37)

If such an element exists for all subspaces Ln and numbers dn , then X is called a Bernstein space. In 1963, V. N. Nikol’skii noted that if X is a Bernstein space, then X is reflexive. Tyuremskikh [572] showed that every Hilbert space is a Bernstein space. He indicated some infinite-dimensional subspaces of C[a, b], L p [a, b], c0 that are Bernstein spaces. Among Hilbert spaces, at present no other example of a Bernstein space

416

16 Width. Approximation by a Family of Sets

is known. From results of Yu. A. Brudnyi, it follows that for every nonincreasing convex sequence dn → 0 (dn ≤ (dn−1 + dn+1 /2)) and every sequence (Ln ) of strictly nested subspaces of X, there exists x ∈ X such that ρ(x, Ln ) ≥ dn for all n and ρ(x, Ln ) ≤ Cdn for infinitely many n and some constant C. Tyuremskikh [573] investigated the existence of an element with given best approximations for a contracting system of strictly nested subspaces Y1 ⊃ Y2 ⊃ . . . . He proved the following result. Theorem 16.39 Let X be an infinite-dimensional Banach space. A necessary and sufficient that for every system of strictly nested subspaces Y1 ⊃ Y2 ⊃ . . . of X and every sequence d1 ≤ d2 ≤ . . . of nonnegative numbers, dn → d > 0, there exist an x ∈ X such that ρ(x, Yn ) = dn , n = 1, 2, . . . , is that X be a reflexive space. The answer in the case of rapidly decreasing best approximations was given by P. A. Borodin. Theorem 16.40 Let X be an infinite-dimensional Banach space, L1 ⊂ L2 ⊂ . . . an arbitrary countable system of strictly nested subspaces of X, and let a number sequence (dn ) be such that dn > ∞ k=n+1 dn for all n ≥ n0 for which dn > 0. Then there exists x ∈ X such that ρ(x, Ln ) = dn , n = 1, 2, . . . .

Chapter 17

Approximative Properties of Arbitrary Sets in Normed Linear Spaces. Almost Chebyshev Sets and Sets of Almost Uniqueness

In approximation of functions, one frequently encounters approximation problems involving nonlinear or even nonconvex sets. This is the case, for example, of approximation by rational fractions, splines with free knots, and exponential sums (see Sect. 4.3.2 and Chap. 11). For such sets M, approximative properties of existence, uniqueness, and approximative compactness are of special importance. This leads to questions of when and in what spaces X, the sets E(M) := {x ∈ X | PM x  }, U(M) := {x ∈ X | |PM x| ≤ x}, T(M) := {x ∈ X | |PM x| = 1}. have certain properties for every closed M ⊂ X (or sometimes for every M  ). These properties can be of various kinds. For example, one can consider ‘size’ properties (density, category, dimensional), connectedness, and descriptive properties. The principal definitions are given in Sect. 17.1. Approximation properties of sets in strictly convex spaces are discussed in Sect. 17.2. In Sect. 17.3, we consider constructive characteristics of spaces, that is, characteristics of spaces in terms of approximative properties of their subsets. Sets in locally uniformly convex spaces are discussed in Sect. 17.4, and sets in uniformly convex spaces in Sect. 17.5. Examples illustrating the above theorems and the concepts involved are given in Sect. 17.6. Density and category properties of the sets E(M), AC(M), and T(M) will be considered in Sect. 17.7. Category properties of U(M) are studied in Sect. 17.8. Other size characteristics of approximatively defined sets will be examined in Sect. 17.9. The farthest-point problem is briefly discussed in Sect. 17.10. Zajíček classes of smallness (Zk ) will be considered in Sects. 17.11, 17.12, and 17.13. Zajíček-smallness of the classes of sets Rk (M) in Euclidean spaces is discussed in Sect. 17.14. Almost Chebyshevsets are studied in Sect. 17.15. Almost Chebyshev subspaces of finite

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2_17

417

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dimension in the space C(Q) of continuous functions defined on a metrizable compact set Q with Chebyshev norm are considered in Sect. 17.16. In particular, we give Garkavi’s criterion for a subspace to be almost Chebyshev in the space C(Q).

17.1 Approximative Properties of Arbitrary Sets The important direction in geometric approximation theory of approximative properties of arbitrary sets, which deals with problems of this sort, stems from a paper of Stechkin [529], who was the first to examine density and category properties of points of uniqueness, existence, and approximative compactness for sets of fairly arbitrary structure. In [529], he posed a number of problems. Later, his studies were continued by many researches both in Russia and abroad (see the surveys [123], [173], [358], [359]). Some results mentioned in this chapter were used in the theory of optimal control of distributed systems (Fursikov [243]–[245], Yashina [607], [608], and so on). With a set M ⊂ X we associate the following approximatively defined sets: – the set of points of existence E(M) := {x ∈ X | |PM x| ≥ 1}, where | · | (or card ( · )) is the cardinality of a set; – the set of points of uniqueness U(M) := {x ∈ X | |PM x| ≤ 1}; – the set of points of existence and uniqueness T(M) := {x ∈ X | |PM x| = 1} (if T(M) = X, then M is a Chebyshev set); – the set of points of nonuniqueness R(M) := {x ∈ X | |PM x| ≥ 2}; – the set AC(M) of points of approximative compactness (see Sect. 4.3) — these are x ∈ X such that for every sequence yn ∈ M, n = 1, 2, . . . , the condition  x − yn  → ρ(x, M), n → ∞, implies the existence of a subsequence of (yn ) converging to some element y ∈ M (such a point always lies in PM x, and therefore AC(M) ⊂ E(M)). We also consider the set of points of approximative uniqueness TAC(M) = T(M) ∩ AC(M).

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Approximative Properties of Arbitrary Sets

419

If a set M is closed and X is a Banach space, then in this case, an equivalent definition of TAC(M) is as follows:   δ TAC(M) = x ∈ X | lim diam PM x=0 , (17.1) δ→0+

δ x := {y ∈ M |  x − y ≤ ρ(x, M) + δ} (see [358]). where PM Recall that a set M ⊂ X is nowhere dense if its closure does not contain interior points (in X). A set of first category (or a meager set) is a union of countably many nowhere dense sets. In what follows:

(I) is the class of subsets of X of first category; (II) is the class of complements of sets of first category in X. Sets of first category should be studied in complete spaces X, because in such spaces, the sets A ∈ (II) are characterized by the fact that every such an A contains a dense subset in X that is an intersection of a countable number of open sets. It is well known that a subset of second category of a complete metric space is dense in it. In this chapter, we shall be mostly concerned with density and category properties of the sets E(M), U(M), T(M), and AC(M). S. B. Stechkin characterized the Banach spaces X in which, for an arbitrary closed set M ⊂ X, the set of its points of uniqueness U(M) is dense in this space. These spaces are precisely the strictly convex spaces. Stechkin also showed that in a locally uniformly convex1 Banach space X, the set U(M) is a set of second category (that is, U(M) ∈ (II)) for every M ⊂ X. Throughout this chapter it is assumed that M is a nonempty set. Stechkin’s results from [529] can be briefly summarized as follows. Theorem 17.1 The following conditions are equivalent: (1) U(M) is dense in X for every M ⊂ X; (2) X is strictly convex. In connection with Theorem 17.1, we note that if X is not strictly convex, then there exists a hyperplane M ⊂ X such that U(M) = M. Theorem 17.2 If X is a locally uniformly convex Banach space, then U(M) ∈ (II) for all M ⊂ X. Theorem 17.3 (1) If X is a uniformly convex Banach space, then TAC(M) ∈ (II) for every closed nonempty M ⊂ X. (2) If X is a uniformly convex Banach space and M ⊂ X is a closed nonempty subset of X, then T(M) ∈ (II).

1 Recall that a space X is locally uniformly convex (X ∈ (LUR);) see Definition 4.13) if x  = x n  = 1 and lim x + x n  = 2 imply that x n → x. An equivalent definition is as follows: for all ε > 0 and x ∈ S, there exists δ = δ(ε, x) > 0 such that y + x  ≤ 2(1 − δ) whenever y ∈ S, y − x  ≥ ε.

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Theorem 17.4 If X is a strictly convex Banach space and M ⊂ X is boundedly compact (that is, the intersection of M with every closed ball is compact), then U(M) ∈ (II). Remark 17.1 The conclusion of Theorem 17.4 also holds in a slightly more general setting for precompact sets (see, for example, [173]). Theorem 17.5 The space X = R2 contains a set M such that the complement of U(M) is dense in some nondegenerate disc. Theorems 17.1–17.3 show that for a sufficiently wide class of spaces X, quite interesting results about the sets E(M), U(M), AC(M) can be obtained (to the effect, for example, that such sets are quite large) under fairly minimal assumptions on the set M ⊂ X (recall that a set M is assumed to be closed) or in general even for an arbitrary M. Before the appearance of Theorems 17.1–17.3, such results were known only in a very few particular cases. The above theorems gave rise to a large number of studies on approximative properties of arbitrary sets in normed spaces (after the appearance of [529], Stechkin himself never published any results on this topic). So the paper [529] initiated a new fruitful branch of geometric approximation theory. For more results in this direction, see the surveys by Konyagin [358], Cobzaş [173], and Zajíček [619]. Below, we shall prove Theorems 17.1–17.4 and give a brief survey of the extensions of Stechkin’s results. As before, by F (X) we denote the class of nonempty closed subsets of a space X.

17.2 Sets in Strictly Convex Spaces   Recall that a normed linear space X is strictly convex if its unit sphere S =  x = 1 does not contain nondegenerate intervals. To prove Theorem 17.1, we require the following simple property of strictly convex spaces (see, for example, Proposition 1.4). Let X be a strictly convex normed linear space, M ⊂ X, x0 ∈ X, x0  M, and let y0 ∈ PM x0 . Then PM x = {y0 }

for any x from(x0, y0 ].

(17.2)

We shall first prove the implication 2) ⇒ 1) in Theorem 17.1: if X is a strictly convex normed linear space, then for every set M ⊂ X, U(M) = X,

(17.3)

that is, U(M) dense in X. Indeed, let x be an arbitrary point in X. We need to show that every neighbourhood of this point contains a point x ∈ U(M). If x ∈ U(M), then there is nothing to prove. If x  U(M), then x  M and PM x contains at least two points, y1, y2 ∈ M. Hence by (17.2), for every point x ∈ (x, y1 ), we have x ∈ U(M), because PM x = {y1 }.

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In particular, every neighbourhood of x  U(M) contains a point x ∈ U(M). This proves (17.3). Proof (of Theorem 17.4) Since U(M) = X \ R(M) and since R(M) ⊂ R(M), in order to prove the theorem it suffices to verify that  if M is a boundedly compact set, then R(M) can be written in the form R(M) = ∞ n=1 Φn , where each Φn is closed and is nowhere dense. Let d(x) = diam PM x be the diameter of PM x. We define   Φn := x ∈ X | d(x) ≥ 1/n , n ∈ N. (17.4)  Now R(M) = ∞ n=1 Φn , and it suffices to show that each Φn , n = 1, 2, . . . , is closed and is nowhere dense. We claim that Φn is closed. Let (xk ) be a sequence from Φn converging to some element x ∈ X. Since M is boundedly compact, the set PM xk = M ∩ B(xk , ρ(xk , M)) is compact and nonempty for all k ∈ N. Hence there exist zk , zk ∈ PM x such that zk − zk  = d(xk ) ≥

1 . n

That the sequences (zk ) and (zk ) are bounded follows from the inequality zk − x ≤ zk − xk  +  xk − x, k ∈ N (the inequality for zk is similar). So we can assume without loss of generality that there exist z , z ∈ M such that zk → z ,

zk → z

as k → ∞

and

zk − zk  ≥

1 . n

But in this case, z − x = lim zk − xk  = lim ρ(xk , M) = ρ(x, M), k→∞

z .

k→∞

z , z

So ∈ PM x, diam PM x ≥ z − z  ≥ 1/n. This shows and similarly for that x ∈ Φn , and so the set Φn is closed. Let us now show that Φn is nowhere dense. Since by the above, Φn is closed, it suffices to show that Φn does not contain interior points. But if x ∈ Φn , then there exist z , z ∈ PM x, z − z  ≥ 1/n. But then by (17.2), the interval (x, y) lies in U(M) and hence is disjoint from Φn . This shows that Φn has no interior points. Theorem 17.4 is proved.  Problem 17.1 Characterize the Banach spaces in which U(M) = X for any compact set M ⊂ X (that is, M is a weakly almost Chebyshev set) or U(M) ∈ (II) (that is, M is an almost Chebyshev set). Is it true that a space with this property is strictly convex?

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17.3 Constructive Characteristics of Spaces By constructive characteristics of a space one means its characteristics in terms of approximative properties of its subsets. As examples of constructive characteristics of the strictly convex spaces we mention the following well-known results, in which by a uniqueness set one means a set M ⊂ X satisfying U(M) = X. (A) A Banach space X is strictly convex if and only if every subspace L of X is a uniqueness set. (B) A Banach space X is strictly convex if and only if every convex subset V of X is a uniqueness set. For the sake of completeness, we give the proofs. Since each subspace L is a convex set, it suffices to prove the necessity in (B) and the sufficiency in (A). Necessity. Let us prove that if X is a strictly convex normed space, then each convex subset of X is a uniqueness set. Indeed, if for some point x0 ∈ X, the set PV x0 contains at least two distinct points y0 and y1 , then by the convexity of V, the point y2 = (y0 + y1 )/2 also lies in V, and since X is strictly convex,  x0 − y2  <  x0 − y1 , which contradicts the definition of PV x0 . Sufficiency. Suppose that every subspace L of X is a uniqueness set. Assume to the contrary that X is not strictly convex. Namely, let x0, x1 ∈ S and (x0 + x1 )/2 = 1. Then the line  passing through x0 and x1 is at distance 1 from 0. We construct a subspace L consisting of the points of the form x − x0 , where x ∈ . In this case, ρ(−x0, L) = 1 and 0, x1 − x0 ∈ PL (−x0 ); so L is not a uniqueness set. Below, we shall find a characteristic of the strictly convex spaces depending on properties of arbitrary subsets M of X. Definition 17.1 We say that M is a set of weak almost uniqueness if U(M) is dense in X; M is a weakly almost Chebyshev set if T(M) is dense in X (cf. Sect. 17.15 below). The next theorem is a restatement of Theorem 17.1. Theorem 17.6 A Banach space X is strictly convex if and only if every subset M of X is a set of weak almost uniqueness (that is, U(M) = X). The necessity in this theorem is contained in the above implication 2) ⇒ 1) (see (17.3)). Let us prove the sufficiency. To this end, we shall show that if a space X is not strictly convex, then it contains a hyperplane L for which U(L) = L. Let x1, x2 ∈ S, x1  x2 , and let (x1 + x2 )/2 = 1. We set x := (x1 + x2 )/2 and construct a functional f (x) that attains its norm on x ; that is, f (x ) =  f  > 0. Further, setting L := {y | f (y) =  f }, we shall show that this is a required subspace. It is clear that x1, x2, x ∈ L. Further,

17.4

Sets in Locally Uniformly Convex Spaces

423

ρ(0, L) = 1, and if x ∈ X \ L is an arbitrary point, then f (x) = α  f , where α  1, and hence the (distinct) points y1 = x + (1 − α)x1,

y2 = x + (1 − α)x2,

y = x + (1 − α)x ,

lie in PM x, which proves Theorem 17.6.

17.4 Sets in Locally Uniformly Convex Spaces Recall that locally uniformly convex spaces were defined in Definition 4.13.) Let us give an equivalent definition. A normed linear space X is said to be locally uniformly convex if for every point x1 ∈ S and ε > 0, there exists δ = δ(ε, x1 ) such that the inequality (17.5) (x1 + x2 )/2 ≤ 1 − δ is satisfied for every point x2 ∈ S with  x1 − x2  ≥ ε. It is easily shown that each locally uniformly convex space is strictly convex (but generally the converse assertion is not true). In this section, we shall prove Theorem 17.2, which was formulated in Sect. 17.3. We need the following result on the structure of locally uniformly convex spaces (see [173], [183], [529]). In [183], Lemma 17.1 is called Stechkin’s lens lemma (because in it, the set Mε has a lenticular shape). Lemma 17.1 Let X ∈ (LUR) (X is a locally uniformly convex space) and let x0 ∈ S, 0 < α < 1. Consider the set ˚ 1)} ∩ B(αx0, 1 − α + ε). Mε := {X \ B(0, Then diam Mε → 0

as ε → +0.

Proof (of Lemma 17.1) To begin with, we note that for fixed x0 and ε, the sets Mε = Mε (α) do not increase with increasing α. So it suffices to verify our claim for 0 < α ≤ 1/2. Given x0 ∈ S and α0 ∈ (0, 1/2), define ˚ 1)} ∩ B(α0 x0, 1 − α0 + ε). Mε0 = {X \ B(0, Assume that diam Mε0 > 2d0 > 0 for all ε > 0 and some x0 ∈ S and α0 ∈ (0, 1/2). We claim that for sufficiently small ε, this assumption leads to a contradiction. For convenience, we assume that the point α0 x0 is the origin. We have

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˚ Mε0 = {X \ B(−α 0 x0, 1)} ∩ B(0, 1 − α0 + ε), and now the point x0 becomes the point x0 = (1 − α0 )x0 . Since x0 ∈ Mε0 and since diam Mε0 > 2d0 , there exists y1 ∈ Mε0 such that  x0 − y1  > d0 . It is clear that 1 − α0 ≤  y1  ≤ 1 − α0 + ε. Setting y0 =

x0  y1 

y1 , we find that  y0  =  x0  = 1 − α0 ,  y1 − y0  =

  y  1 − 1  x0  =  y1  −  x0  ≤ ε  x0 

(17.6)

and  x0 − y0  ≥  x0 − y1  −  y1 − y0  > d0 − ε >

d0 2

for ε < d0 /2.

Let us estimate  y1 + α0 x0  from above. We have α0 x y1 + α0 x0 = y1 − y0 + y0 + 1 − α0 0  α0  α0 y0 + = y1 − y0 + 1 − (y0 + x0 ). 1 − α0 1 − α0 By the definition of a locally uniformly convex space (see (17.5)), there exists δ > 0 depending on x0 , α0 , and d0 /(1 − α0 ) such that  y0 + x0  2

< (1 − δ)(1 − α0 ).

(17.7)

Using (17.6), the equality  y0  = 1 − α0 , and (17.7), we have  α0  α0  y0  +  y0 + x0   y1 + α0 x0  ≤  y1 − y0  + 1 − 1 − α0 1 − α0  α0  2α0 (1 − α0 ) + ≤ ε+ 1− (1 − α0 )(1 − δ) 1 − α0 1 − α0 = 1 + ε − 2α0 δ. So  y1 +α0 x0  < 1 for ε < ε0 , which contradicts the condition y1 ∈ Mε0 . Lemma 17.1 is proved.  Definition 17.2 Let X be an arbitrary Banach space, x ∈ X, M ⊂ X. Given ε > 0, we set   ε PM (x) := B x, ρ(x, M) + ε ∩ M, ε dε (x) := diam PM (x), d0 (x) := lim dε (x). ε→+0

By Fα = Fα (M) (α > 0) we denote the set of all points x ∈ X at which d0 (x) ≥ 1/α.

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Sets in Locally Uniformly Convex Spaces

425

Clearly, d0 (x) ≥ d(x) for all x ∈ X, and hence Fα ⊇ Φα (α > 0). (If a set M is boundedly compact, then Fα = Φα , α > 0). The sets Fα have better topological behavior than the sets Φα . For example, the following result holds (see [173, p. 271]). Lemma 17.2 Let X be an arbitrary Banach space. Then for all M ⊂ X and α > 0, the set Fα (M) is closed. Proof (of Lemma 17.2) Let xk ∈ Fα (M), k = 1, 2, . . . , and xk → x0 as k → ∞. We set ρ(x0, M) =: γ0 . ρ(xk , M) =: γk , For a given ε > 0, let k0 (ε) be such that  xk − x0  ≤ ε/3 for all k ≥ k0 (ε). Hence for all k ≥ k0 , we have γk ≤ γ0 +  xk − x0  ≤ γ0 +

ε , 3

which gives B(xk , γk + ε/3) ⊂ B(xk , γ0 + 2ε/3) ⊂ B(x0, γ0 + ε), ε/3 ε PM (xk ) ⊂ PM (x0 ),

and hence

1 ≤ dε/3 (xk ) ≤ dε (x0 ), α that is, x0 ∈ Fα (M). Lemma 17.2 is proved.



Note that the set Gα = Gα (M) = X \ Fα (M) consists of all points x at which d0 (x) < 1/α; by the above, Gα is open. Now we can prove Theorem 17.2, which is the main result of this section. Proof (of Theorem 17.2) We claim that each set Φn , n = 1, 2, . . . , is nowhere dense; that is, every nonempty open ball O contains a ball O disjoint from Φn . Note that the set Φn need not be closed. We fix an arbitrary ball O. If O ∩ Φn = , then the proof is complete. So we can assume that x ∈ O ∩ Φn . Hence x  M. Let y0 ∈ PM x ε (z) → 0 as ε → +0; that and z ∈ (x, y0 ). From Lemma 17.1, it follows that diam PM is, z ∈ G n . But by Lemma 17.2, the set G n is open, and hence there exists an entire ball O = O (z) ⊂ O lying in G n . For every point z in this ball, we have z ∈ G n , z  Fn , z  Φn ; that is, O ∩ Φn = . So each Φn is nowhere dense, and hence  R(M) = ∞ n=1 Φn is a set of first category. Recalling that U(M) = X \ R(M), we conclude the proof of Theorem 17.2. 

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17.5 Sets in Uniformly Convex Spaces A normed linear space X is called uniformly convex (see Sect. 9.2) if for every ε > 0, there exists δ = δ(ε) > 0 such that x + x 1 2 ≤ 1−δ 2 for all x1, x2 ∈ S such that  x2 − x1  ≥ ε, that is, if the condition of local uniform convexity holds uniformly with respect to x1 ∈ S. For uniformly convex spaces, Lemma 17.1 can be refined as follows (see also [183, Lemma 2.1], [173, Propositions 2.10 and 2.11]). Lemma 17.3 Let X be a uniformly convex normed linear space, δ > 0, 0 < α < 1,  x0  = α, and let ˚ 1)} ∩ B(x0, 1 − α + ε). Mδ = Mδ (x0 ) = {X \ B(0, Then for every ε > 0, there exists δ0 = δ0 (ε, α) such that for all δ ≤ δ0 and all x0 ,  x0  = α, we have diam Mδ (x0 ) < ε. The proof is similar to that of Lemma 17.1 and hence omitted. As in Sect. 17.4, given x ∈ X, M ⊂ X, we define   ε PM (x) =B x, ρ(x, M) + ε ∩ M ε > 0, ε d0 (x) = lim diam PM (x). ε→+0

Theorem 17.7 (S. B. Stechkin [529], [173]) Let X be a uniformly convex Banach space and let   M ⊂ X. Then the set K(M) := {x ∈ X | d0 (x) = 0} is a set of second category of type G δ . Assertion 1) of Theorem 17.3 follows from Theorem 17.7 in view of (17.1). Proof (of Theorem 17.7) We have K(M) = X \ H(M),

where

H(M) :=

∞

Fn .

(17.8)

n=1

Hence it suffices to show that each set Fn is closed and is nowhere dense. The closedness of the sets Fn is secured by Lemma 17.2 (in which X is an arbitrary Banach space). We claim that if a space X is uniformly convex, then the sets Fn , n = 1, 2, . . . , are nowhere dense, or (which is the same, since Fn are closed), that every neighbourhood O(x) of a point x ∈ Fn contains a point x1  Fn .

17.5

Sets in Uniformly Convex Spaces

427

Translating the origin if necessary and scaling, we can assume without loss of generality that x = 0 and ρ(0, M) = 1. We fix an arbitrary n ∈ N and a neighbourhood O := O(0), and let a number α > 0 be such that B(0, α) ⊂ O. Let ε ∈ (0, 1/n). By Lemma 17.3, we can find δ0 > 0 such that for every point x0 ∈ X,  x = α, and δ ∈ (0, δ0 ), we have diam Mδ (x0 ) < ε
0

ε

ε

ε (x) is closed, it follows that P 1 (x) ⊂ P 2 (x) for Since in our setting, the set PM M M 0 < ε1 < ε2 , and since ε dε (x) = diam PM (x) → 0,

ε → +0,

because x ∈ K(M), the set PM x consists of a unique point y = y(x) ∈ M, because X is complete. From Theorem 17.7 we have the following corollary, which is assertion 2) of Theorem 17.3. Corollary 17.1 Let X be a uniformly convex Banach space and M a closed nonempty subset of X. Then T(M) ∈ (II) and TAC(M) ∈ (II). If a space X is nonreflexive, then it contains an antiproximinal hyperplane M (see Proposition 1.7), for which, of course, T(M) is nowhere dense in X.

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17.6 Examples In this section, we shall give some examples illustrating the above theorems and the concepts involved. (1) Let X = L p [0, 1], p > 1, and let M = Mn be the set of all rational fractions of the form a0 + a1 t + . . . + an t n Rn (t) = b0 + b1 t + . . . + bn t n that are finite on [0, 1]. It can be easily checked that M is closed in L p . The space L p , p > 1, is uniformly convex by Clarkson’s theorem (Theorem 9.3), and hence the corollary to Theorem 17.7 implies that in L p there exists a set Q1 of second category such that each function x ∈ Q1 has precisely one best rational L p -approximant Rn (t). It should also be pointed out that by Efimov–Stechkin’s theorem, the set Mn is not a set of uniqueness. (2) In an infinite-dimensional Hilbert space H, there exists a bounded closed set M for which the set Φn (see (17.4)) is not closed for some fixed natural number n. Let (ek ), k = 0, 1, 2 . . . , be an orthonormal basis for H. We set 1 y2k−1 = (1 + εn )ek + e0, 2

1 y2k = (1 + εk )ek − e0 2

(k = 1, 2, . . . ),

where εk > 0, εk → 0 (k → ∞), and define M := {yk }

(k = 1, 2, . . . ).

Consider the points xk = εk ek . For these points, we have

√ 5 1  xk − y2k−1  =  xk − y2k  = ek ± e0 = , 2 2 1  xk − y2l−1  =  xk − y2l  = εk ek + (1 + εl )el ± e0 2 √ 5 1 2 2 = εk + (1 + εl ) + > (k  l). 4 2

Hence

√

5 , P(xk ) = {y2k−1, y2k }, 2 d(xk ) = diam{P(xk )} =  y2k−1 − y2k  = e0  = 1. ρ(xk , M) =

So, xk ∈ Φ1 (k = 1, 2, . . . ). √ Further, we note that xk → 0, ρ(0, M) = 5/2, and PM 0 = , which gives 0  Φ1 . Hence the set Φ1 is nonclosed. The above example can be slightly reworked to produce nonclosed Φn for every natural number n.

17.7

Density and Category Properties of the Sets E(M), AC(M), and T(M)

429

(3) The set H(M) (see the definition in (17.8)) need not be closed. This remark is trivial: as X it suffices to consider the Euclidean plane R2 , and as M, a closed circular arc with angle exceeding π. (4) Let E be the unit disc in the Euclidean plane R2 . Then there exists a closed set M ⊂ R2 for which H(M) (see (17.8)) is dense in E. We fix an arbitrary dense sequence xn ∈ E, n = 1, 2 . . . , in E and construct a closed set M ⊂ R2 such that for every n = 1, 2, . . . , the set PM xn has the cardinality of the continuum. It is clear that xn ∈ H(M), n = 1, 2 . . . . Augmenting if necessary the sequence (xn ) with the point 0 and relabeling, it can be assumed that x1 = 0. The set M can be constructed by excluding from the plane R2 a countable set of open discs On with centres at the points xn and of radius rn < 2 −  xn . Let r1 = 1. Assume that the numbers r2, . . . , rn are already defined. We set Mn := X \

n

Oν .

ν=1

n Oν lies inside the disc E2 of radius 2 with By the constraints on rn , the set ν=1 centre at 0. Moreover, the boundary of Mn consists of a finite number of circular arcs. As On+1 we take a disc whose interior contains a part of the boundary of Mn of cardinality of the continuum and such that for every circular arc composing the boundary of Mn , the interior of On+1 contains at most 1/3n of its arc. Clearly, these conditions are compatible with the condition rn < 2 −  xn . Finally, we get  O M := X \ ∞ n=1 n . So the boundary of M contains a set of the cardinality of the continuum, and this set lies on the boundary of On for every n, and hence PM xn , n = 1, 2 . . . , has the cardinality of the continuum. Note that in this example, the set PM xn has positive linear measure for all n ∈ N.

17.7 Density and Category Properties of the Sets E(M), AC(M), and T(M) Throughout this section, we assume that M ∈ F (X), where F (X) is the class of nonempty closed sets in a space X. The principal problem considered here is as follows. Given M ∈ F (X), how large are the sets E(M), AC(M), and T(M), and when they are nonempty? By James’s theorem (Theorem 1.5), every nonreflexive Banach space X contains a hyperplane M such that E(M) = M. Hence in order that E(M) be dense in X for all M, it is necessary that X be reflexive. However, Edelstein [218], [220] showed that this condition is insufficient (see Remark 4.7). The paper [388] provides less restrictive (than in [529]) conditions on X under which E(M) ∈ (II) for all M. The spaces X with this property were characterized by Lau [389] and Konyagin [350] (see Theorem 17.8 below). Various structural properties of the set E(M) versus the

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geometry of a space X and the geometry of a set M ∈ F (X) were studied in the papers [121], [122], [183], [210], [218], [220], [350], [389], [388], [492], [493], [529], [615] (the list is not complete). For asymmetrically seminormed spaces, the sets E(M) were studied by G. E. Ivanov [304]. The next theorem (see, for example, [352, Theorem 2.1.2], [359, Theorem 3], [389]), which characterizes the Efimov–Stechkin spaces,2 partially strengthens Theorem 17.3. Below, (EC) is the class of Efimov–Stechkin spaces. Theorem 17.8 Let X be a Banach space. Then the following conditions are equivalent: (a) X is a Efimov–Stechkin space; (b) AC(M) ∈ (II) for all M ∈ F (X); (c) E(M) ∈ (II) for all M ∈ F (X); (d) AC(M) is dense in X for all M ∈ F (X); (e) E(M) is dense in X for all M ∈ F (X); (f) AC(M) weakly dense in X for all M ∈ F (X); (g) E(M) is dense in X for all M ∈ F (X); (h) for each M ∈ F (X), there exists a set U ∈ (II) such that U ⊂ E(M) and the mapping PM |U is continuous; (i) for each M ∈ F (X), there exists a set V ∈ (II) admitting a continuous selection of the metric projection operator; (j) AC(M) is a set of category II in L for all M ∈ F (X) and any subspace L of finite codimension; (k) E(M) is a set of category II in L for all M ∈ F (X) and any subspace L of finite codimension. Corollary 17.2 Let X be a Banach space. The following conditions are equivalent: (a) X ∈ (EC) ∩ (R) (that is, X is a strictly convex Efimov–Stechkin space;3) (b) TAC(M) ∈ (II) for every closed nonempty set M ⊂ X; (c) for every functional f ∈ S ∗ , diam Nε ( f ) → 0,

ε → +0,

where Nε ( f ) := { f (x) ≥ 1 − ε} ∩ B(0, 1). We also note that Borwein [121] proved that E(M) ∈ (II) 2 Efimov–Stechkin spaces were defined in Chap. 9. Note that the class of Efimov–Stechkin spaces coincides with the class of reflexive Kadec–Klee spaces (the class of Kadec–Klee spaces is defined by the condition that the weak convergence of a sequence of points from the unit sphere S of such a space to a point from S implies the norm convergence; see Sect. 9.1). 3 Such spaces are also called strongly convex spaces; see also [173, pp. 274–275], [122, Sect. 6].

17.7

Density and Category Properties of the Sets E(M), AC(M), and T(M)

431

in a Kadec–Klee space4 whose dual can space be renormed to be strictly convex, provided that M is a boundedly weakly compact set (which means by definition that the intersection of M with an arbitrary closed ball is weakly compact; see Sect. 4.1). Density properties of the sets E(M) and AC(M) are important in the study of the connectedness of these sets. The following result is obtained using Theorem 17.8. Theorem 17.9 The following conditions are equivalent: 1) E(M) is connected for all M ∈ F (X); 2) AC(M) is connected for all M ∈ F (X); 3) X is a Efimov–Stechkin space. The equivalence 1) ⇔ 3) was proved by Konyagin [351]; 2) ⇔ 3), by Tsar’kov [550]. Density properties of the set E(M) are also used in the study of conical points of sets [221] and in the subdifferential calculus [297]. We note the following open question [357]. Problem 17.2 (S. V. Konyagin) Is it true that for every reflexive space X and nonempty closed set M  X, there exists at least one point x ∈ E(M) \ M? For the Efimov–Stechkin spaces (which are a fortiori reflexive), Problem 17.2 has a positive solution, because in these spaces, E(M) = X by Theorem 17.8. At the same time, if X is not a Efimov–Stechkin space, then the complement of E(M) can be quite large. In this regard, we mention the following result of Konyagin [351]. Theorem 17.10 If X is not a Efimov–Stechkin space, then X contains a closed set M and a functional x ∗ ∈ X ∗ such that   E(M) ∩ x | |x ∗ (x)| < 1 = . Konyagin (see [350], [122]) also showed that if a Banach space X is not a Efimov– Stechkin space, then X contains a closed bounded subset M and an open nonempty set U ⊂ X \ M such that U ∩ E(M) =  and the distance function ρ( · , M) is affine on U (in particular, it is Fréchet differentiable on U). We also mention some cases in which Problem 17.2 has an affirmative solution. If X is reflexive, then every weakly closed (and in particular, closed convex) set is sequentially weakly compact and hence E(M) = X. On the other hand, by James’s theorem (Theorem 1.5), every nonreflexive Banach space X contains a hyperplane M such that E(M) = M. This remark shows that Problem 17.2 has negative answer in the class of all reflexive spaces. The next result shows that Problem 17.2 has positive solution for bounded subsets of reflexive spaces [351], [357].

4 Kadec–Klee spaces are also called spaces with the A-property [591]. Examples of such spaces include  p , 1 ≤ p < ∞. For a Banach space, the condition X ∈ (EC) is equivalent to saying that X is a reflexive Kadec–Klee space.

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Theorem 17.11 (S. V. Konyagin) If a space X is reflexive, M ⊂ X nonempty and bounded, then there exist a point y ∈ M and a point z ∈ X such that the ray  = {y + tz | t > 0} is contained in E(M) \ M. Proof Let N := conv M be the closure of the convex hull of M. A point y ∈ N is called strongly exposed if there exists x ∗ ∈ X ∗ , x ∗  0, such that x ∗ (y) = supu ∈N x ∗ (u) and for every sequence (yn ) ⊂ N, if x ∗ (yn ) → x ∗ (y), then  yn − y → 0. A strongly exposed point is necessarily an exposed point; that is, x ∗ (y) > x ∗ (u) for all u ∈ N \ {y}. Troyanski [548] showed that a weakly compact convex subset of a Banach space is the closure of the convex hull of its strongly exposed points. Therefore, N contains a strongly exposed point. Let us show that y ∈ M. Indeed, let x ∗ ∈ X ∗ be the corresponding functional for the point y. We have sup x ∗ (u) = sup x ∗ (x) = x ∗ (y),

u ∈M

x ∈N

and hence there exists a sequence (yn ) ⊂ M ⊂ N such that x ∗ (yn ) → x ∗ (y). Now  y − yn  → 0, and hence y ∈ M. In our case, the unit ball B ⊂ X is w ∗ -weakly compact, and hence the functional x ∗ attains its norm on B at some point z ∈ S; that is, x ∗ (z) =  x ∗ . For each t > 0, consider x = y + tz and take an arbitrary point u ∈ M \ {y}. Hence  x − y ·  x ∗  ≤ t  x ∗  = x ∗ (x) − x ∗ (y) < x ∗ (x) − x ∗ (u) ≤  x − u ·  x ∗ , which shows that y ∈ PM x and x ∈ E(M) \ M, the result required.



We note two more results (see [352, Theorem 2.1.3], [220]). Theorem 17.12 (S. V. Konyagin) Every infinite-dimensional space X contains a closed set M such that int E(M) = . Of course, if dim X < ∞, then clearly, E(M) = X for every closed M (M ∈ F (X)). Theorem 17.13 (M. Edelstein) If X is a Banach space with the Radon–Nikodým property and if M is a nonempty convex closed bounded subset of X, then E(M) \ M

w

⊃ X \ M.

17.8 Category Properties of the Set U(M) The following problem, posed by S. B. Stechkin, is one of the most intriguing problems in this field. Problem 17.3 (S. B. Stechkin) Is it true that U(M) ∈ (II)

17.8

Category Properties of the Set U(M)

433

for every strictly convex space X and closed M ⊂ X? In other words, is it true that the set U(M) is the complement of a set of first category in X? An equivalent reformulation is as follows: is it true that U(M) contains a dense G δ -subset of X and hence is ‘large’ from the categorical point of view? Stechkin himself gave a positive answer to this question for boundedly compact subsets of strictly convex spaces (Theorem 17.4). Advances in this problem require finding less-restrictive conditions on a strictly convex space X that guarantee the above property. Broader than the class of locally uniformly convex spaces is the class with the following property: for all x, y, x1, x2, . . . , the conditions  x =  y =  x1  =  x2  = . . . and lim  x + xn  = lim  y + xn  = 2 imply x = y. Such spaces were called T2 -convex by S. V. Konyagin. The following results (see [349], [613]) extend Theorem 17.4. Recall that in this chapter, it is assumed that M ∈ F (X); that is, M is a nonempty closed set. Theorem 17.14 If X is a T2 -convex space and M ⊂ X, then U(M) ∈ (II) (that is, R(M) ∈ (I)). Theorem 17.15 If X is a separable strictly convex space and M ⊂ X, then U(M) ∈ (II) (that is, R(M) ∈ (I)). It is still unknown whether there exist a strictly convex space X and a set M ⊂ X such that R(M) is not a set of first category. Another condition implying the conclusion of Theorem 17.4 is the existence on X of a Fréchet-differentiable function with nonempty bounded support [350]. More general results were obtained by Zhivkov [624], [625]. By (Λ) he denotes the class of spaces X satisfying the condition that for every locally Lipschitz function f : X → R there exists a set A ∈ (II) such that ∀x ∈ A ∃x ∗ ∈ X ∗ ∀h ∈ X x ∗, h ≤ lim sup( f (x + th) − f (x))/t. t→0+

Zhivkov proved that if X ∈ (Λ) is strictly convex and M ⊂ X, then U(M) ∈ (II). The class (Λ) involves all weakly compactly generated spaces X (such a space coincides with the closure of the span of some weakly compact subset of this space), and in particular, all reflexive spaces [625]. Kenderov [331]–[333] showed that if X is strictly convex, then U(M) ∈ (II) for a wider class of sets than that given in Theorem 17.4 (in particular, for approximatively compact sets M, for which AC(M) = X). In [350] it was established that in a strictly convex X, AC(M) \ U(M) ∈ (I)

∀M ∈ F (X).

(17.10)

Note that condition (17.10) does not characterize the strictly convex spaces: Kamuntavichius [321] showed that this property holds, in particular, for the space L(T, σ, μ) of integrable functions, where (T, σ, μ) is a space with measure. Every

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such a space of dimension > 1 is not strictly convex. In [321], property (17.10) was also investigated for metric spaces. For further advances, see [173], [231], [232], [623], [492], [123]. At present, Problem 17.3 is open for strictly convex spaces.

17.9 Other Characteristics for the Size of Approximatively Defined Sets The above results show that under certain assumptions about a space X, the sets X \ E(M) and X \ U(M) are small in the categorical sense. However, it is desirable to have more meaningful information about the size of such sets. For example, a subset of first category (a meager set) of a d-dimensional space X, 1 ≤ d < ∞, can be of full d-dimensional measure. The following question arises naturally in relation to Theorem 17.2: given a d-dimensional strictly convex space X, is it true that the set R(M) = X \ U(M) has zero d-dimensional measure? It turns out that this is so, and moreover, the naturally defined (for example, the Hausdorff) dimension of R(M) is at most d − 1. In parallel with the set R1 (M) = R(M), we consider the set R2 (M) = {x ∈ X : |PM x| > 2}. The study of the dimensional properties of the sets R1 (M) and R2 (M) partially stems from Stechkin’s paper [529]. However, the first results in this direction for finite-dimensional Euclidean spaces X were obtained earlier by Paúc [463] and Erdős [228]. Let X = Rd , 2 ≤ d < ∞, k = 1 or k = 2. In [463] for d = 2 and in [228] for d > 2, it was proved that Rk (M) can be covered by countably many (d − k)-dimensional Lipschitz surfaces. For d = k = 2, the last condition means that R2 (M) is at most countable. These results were extended in [349] to the case of strictly convex spaces X of finite dimension d ≥ 2; however, for k = 2 < d it was required in addition that the space X be smooth (by definition, in a smooth space, a support hyperplane through a given point of the unit sphere is unique). We do not know whether the condition that X be smooth is essential. In particular, it is not known whether for every three-dimensional strictly convex space X and M ⊂ X, the set R2 (M) can be covered by countably many rectifiable curves. To carry over dimensional results to the infinite-dimensional setting, it was required to give a definition of a surface in an infinite-dimensional space. This was done by L. Zajíček. A Lipschitz hypersurface A in X associated with a nonzero vector v ∈ X is defined by {z + f (z)v | z ∈ Z }, where Z ⊂ X is a hyperplane, v  Z, f : Z → R is a Lipschitz function. If instead of the Lipschitz continuity of f one supposes that f is the difference of two continuous convex functions on Z, then the hypersurface is said to be δ-convex. The following results are due to L. Zajíček.

17.9

Other Characteristics for the Size of Approximatively Defined Sets

435

Theorem 17.16 If X is a separable strictly convex space, then R(M) can be covered by countably many Lipschitz hypersurfaces. Theorem 17.17 If X is a separable strictly convex smooth space and (vn )∞ n=1 is a complete sequence in X, then R(M) can be covered by Lipschitz hypersurfaces An associated with vn . Theorem 17.18 If X is a separable Hilbert space, then R(M) can be covered by countably many δ-convex hypersurfaces. Theorems 17.16 and 17.17 were proved in [613], and Theorem 17.18, in [615]. In can be shown that the separability condition in these theorems is essential. Clearly, every space X of nonzero dimension contains a subset M with disconnected U(M): it suffuses to take as M a two-point set. At the same time, from Theorem 17.16, it can be derived that for a separable strictly convex space X and M ⊂ X, there exists a countable set A ⊂ X such that U(M) ∪ A is path-connected [351]. Both the strict convexity condition and the separability of X cannot be relaxed. Similar dimensional considerations were used in [351] to show that in this setting, the set X \ R2 (M) is path-connected. We do not know whether the same result holds for nonseparable spaces. In relation to Theorem 17.18, L. Zajíček proposed the following problem: characterize the σ-ideal Γ generated by sets of the form R(M) in a separable Hilbert space X (that is, to characterize the subsets of all possible countable unions of sets of the form R(M)). The question was answered by Konyagin [354], who showed the sharpness of Theorem 17.18: Γ is the class of subsets of X that can be covered by countably many δ-convex hypersurfaces. Definition 17.3 A set A ⊂ X is said to be globally very porous [618] if there exists c > 0 such that every ball in X of radius r > 0 contains a ball of radius cr that has common points with A. A set is called σ-globally very porous if can be covered by countably many globally very porous sets. Let α > 0. A set A ⊂ X is called α-angle porous [617] if for every x ∈ A and ε > 0, there exist z ∈ X and g ∈ X ∗ \ {0} such that   z − x < ε, M ∩ y ∈ X : g, y − z > αg ·  y − z = . A set is said to be σ-angle small if for every α > 0, it can be covered by countably many α-angle porous sets. The concept of porosity was introduced by A. Denjoy in 1920. It is easily seen that for 0 < α < 1, each α-angle porous set is globally very porous, and hence each σ-angle small set is σ-globally very porous. Using Theorem 17.2 in [617], one can show that if X is a separable uniformly convex and uniformly smooth space (Sect. 9.5; the latter condition means that ( x + th − 1)/t as t → 0 uniformly over x ∈ S, h ∈ S), then for every M ⊂ X, the set X \ TAC(M) is σangle small, and hence is σ-globally very porous. De Blasi, Myjak, and Papini [183] showed that X \TAC(M) is a σ-globally very porous set in every uniformly convex X. Let us see how Theorem 17.5 was developed. Unlike Theorems 17.2–17.4, Theorem 17.5 is a negative result: it shows that already in the Euclidean plane, the set

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R(M) (that is, the complement of U(M)) can be quite thick. This phenomenon has general character. Every space X of dimension > 1 contains a closed set M such that R(M) is dense in X [349], and moreover, every infinite-dimensional X contains an M such that X \ E(M) is dense in X (see [353]). Vaněček [580] used Stechkin’s construction to show that for every space X of dimension > 1 and countable set A ⊂ X, there exists M such that A ⊂ R(M). Various refinements of Theorem 17.5 have certain value even for X = R2 . Zajíček [617] constructed a set M ⊂ X in the plane such that the set R(M) ∩ U has positive Hausdorff one-dimensional measure for every open set U ⊂ {x :  x ≤ 1}. Karlov [324] investigated what degree of smoothness of M guarantees that R(M) is nowhere dense. He provided an example of a closed curve M ⊂ R2 for which the tangent vector is Lipschitz-dependent on the natural parameter and such that the set R(M) is dense in some ball. At the same time, if X is a Hilbert space and M is a finite union of compact C 2 -smooth manifolds (possibly with C 2 -smooth boundary), then R(M) is nowhere dense (see [324]). To conclude this section, we mention other developments in the paper [529]. In addition to approximative compactness, one can also consider different properties related to one or another type of stability of the metric projection operator. This leads to new approximatively defined sets. Density and category properties of some such were considered in [623], [353], [171], [184].

17.10 The Farthest-Point Problem In this section, in place of the metric projection operator PM , we consider the maxprojection F M to an arbitrary closed bounded subset M, for which some analogues of the results are established (see [45], [46], [171], [183], [217], [623]). Given a set M ⊂ X, we define r(x, M) = sup{ x − z | z ∈ M }, F M x :={z ∈ M |  x − z = r(x, M)}, F E(M) = {x ∈ X | F M x  }, = {x ∈ X | |F M x| ≤ 1}, F T(M) = {x ∈ X | |F M x| = 1};

F U(M)

here F M x is the max-projection of a point x to a set M (the set of farthest points in M for a given point x), r(x, M) is the max-distance (or the deviation) from x to M. A set M is remotal if F M x   for all x ∈ X. Edelstein [217] showed that in a uniformly convex space X, the set F E(M) is dense in X for every nonempty closed set M ⊂ X. For reflexive locally uniformly convex Banach spaces, a similar result was established by Asplund [45]; for relatively compact subsets of arbitrary Banach spaces, see Lau [386] (Theorem 17.19). For a survey on remotal sets, see Cobzaş [173].

17.10

The Farthest-Point Problem

437

Theorem 17.19 (K.-S. Lau) Let M   be a weakly compact subset of a Banach space. Then X \ F E(M) ∈ (I). Deville and Zizler [194] proved a sort of a converse result to Theorem 17.19. Theorem 17.20 Let X be a Banach space, M   a closed convex subset of X. If M is remotal for every equivalent norm on X, then M is weakly compact. Edelstein and Lewis [219] constructed an example of a closed convex bounded set M ⊂  2 for which a farthest point fails to exist from any point from a dense linear manifold 02 ⊂  2 consisting of finitely supported sequences. Here F M∩ 2 x =  for 0

all x ∈ 02 . Deville and Zizler [194, Proposition 3] (see also [437, Proposition 21]) showed that if X is a Banach space with Radon–Nikodým property (see [128]), and M   is a w ∗ -compact subset of X ∗ , then F E(M) contains a dense G δ -subset of X ∗ . It worth pointing out an interesting relation of the spaces with Mazur intersection property to remotal sets. By definition, a space X satisfies the Mazur intersection property (X ∈ (MIP)) if each convex bounded closed nonempty subset of X can be represented as an intersection of closed balls. For a finite-dimensional space Xn , Phelps [469] showed that ext S ∗ = S ∗ if and only if Xn ∈ (MIP) (for the infinitedimensional setting, see [269]). Phelps also showed that the Banach spaces with Fréchet-differentiable norms lie in the class (MIP). Definition 17.4 By Far M we denote the set of all points z ∈ M such that z ∈ F M x for some x ∈ X (see also Sect. 15.8). The following result holds (see [386], [173], [74]). Theorem 17.21 (1) If X ∈ (MIP), then conv Far M = M for every weakly compact set   M ⊂ X. (2) If X is reflexive, then X ∈ (MIP) if and only if conv Far M = M for every closed convex set   M ⊂ X. For further advances, see [74], in which an example of a space X with Fréchetdifferentiable norm (as a corollary, X ∈ (MIP)) and a closed bounded nonremotal subset of X were constructed (see also [75, example 2.3]). We mention the following results. Theorem 17.22 (See [75]) Let X be a strictly convex space and let C be one of the following classes of sets: K = {all compact subsets of the space X }; F = {all compact sets of finite affine dimension in X }; W = {all weakly compact subsets of X }. Then each   M ∈ C is an intersection of balls if and only if conv Far M = M for all M ∈ C .

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Theorem 17.23 (See [437]) Let M   be a weakly compact subset of a Banach space X and let x ∈ X. Then F M (x)   ⇐⇒ Fconv M (x)   (that is, x has a farthest point in M if and only if x has a farthest point in conv M). In Theorem 17.23, the requirement that a set be weakly compact is essential: in [368], [369], Kraus constructed an example of a weakly closed bounded subset M of c0 such that F M (0) = , but F convM (0)  . An analogue of Theorem 17.23 for reflexive spaces was established in [503]: if a space X is reflexive and M ⊂ X is weakly closed,5 then Fconv (x)   ∀x ∈ X

⇐⇒ F M (x) =  ∀x ∈ X.

The following results can be found in [503]: (1) in a reflexive space X, every closed nonempty bounded convex set is remotal if and only if X is finite-dimensional; (2) in a normed space X, every closed nonempty bounded set is remotal if and only if X is finite-dimensional. However, the proofs of these two results given in [503] have raised objections (see [369], [489], [428, example 5]). Later, Rao [489] established that in every Banach space that is not a Schur space,6 there exists a closed nonempty bounded convex nonremotal set (in particular, such a set exists in every infinite-dimensional reflexive space). Martín and Rao [428] showed that every infinite-dimensional Banach space X contains a closed bounded convex nonempty nonremotal set M (that is, F M (x) =  for some point x ∈ X). In [428] it was also shown that if X is an infinite-dimensional Banach space, then X ∗ contains a w ∗ -compact convex nonempty nonremotal set. As in the case of the classical metric projection operator, the strict convexity of a space guarantees the density of elements with unique farthest point in a given subset of the space. The following result holds [194]. Theorem 17.24 Let X be a strictly convex space, M ⊂ X a closed bounded set with F E(M) = X. Then F T(M) = X. Balashov and Ivanov [68] established that the condition ‘for each point of a space lying sufficiently far from the set there exists a unique farthest point in this set’ characterizes the strong convexity of the set in the finite-dimensional normed spaces in which the unit ball is a strictly convex generating set. Ivanov [303] showed that for every point x lying sufficiently far from a convex closed bounded subset M of a uniformly convex Banach space X with Fréchet-differentiable norm, a remotal point for x exists, is unique, and depends continuously on x if and only if the Minkowski 5 Theorem 2.6 of [503] contains a typo: a set should be weakly closed, not closed. 6 A space is a Schur space if in it, the norm convergence of a sequence is equivalent to its weak convergence.

17.11

Classes of Small Sets (Z k )

439

sum7 of the set M with some set gives a ball. Moreover, the farthest point in M for x depends continuously on the set M in the Hausdorff metric (see also [67], [66]). Balashov and Ivanov [71] established the duality between farthest points of a set A and the nearest points of the set C dual to A. The duality of convex sets A and C is understood in the sense that the Minkowski sum of these two sets gives the ball B(0, r). In this case, the farthest point in A for x and the nearest point in (−C) for x lie on the same line. With the help of this duality one can describe all farthest points in terms of the nearest points, and vice versa.

17.11 Classes of Small Sets (Z k ) Definition 17.5 Given a Banach space X and its subspaces Z and V, we write X = Z ⊕ V if every element x ∈ X can be uniquely represented in the form z + v (z ∈ Z, v ∈ V). Let k ∈ N, k  dim X. A set A ⊂ X is called a Lipschitz surface of codimension k in X associated with a subspace V of dimension k if there exist a subspace Z and a C-Lipschitz mapping f : Z → V such that X = Z ⊕V

and

A = {z + f (z) | z ∈ Z }.

Note that for k = dim X, a Lipschitz surface of codimension k degenerates to a point. Definition 17.6 A subset A of a Banach space X is said to be Zk -small (written A ∈ (Zk )) if in the case k  dim X (in the case k > dim X, respectively), the set A is contained in a union of a countable number of Lipschitz surfaces of codimension k (the set A is empty). These classes were introduced by Zajíček [614], [616], who showed that all points of nonuniqueness in a separable strictly convex Banach space form a Zk -small set. We give a more general result obtained by S. V. Konyagin in his PhD thesis [352]. Definition 17.7 A subset A of a Banach space X is associated with a subspace V ⊂ X if (17.11)  x1 − x0 − v  εv for some ε > 0 and all x1, x0 ∈ A and v ∈ V. Theorem 17.25 A set A ⊂ X can be covered by a Lipschitz surface of codimension k if and only if there exists a k-dimensional subspace V ⊂ X associated with the set A. Proof Necessity. Let A be a surface of codimension k associated with a subspace V ⊂ X, dim V = k, and let Z and f be the corresponding subspace and the mapping (see Definition 17.5). Next, let numbers L, C > 0 be such that

7 The Minkowski sum (or simply the sum) of two subsets A and C of a linear space X is the set A + C = {a + c | a ∈ A, c ∈ C }.

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 f (z1 ) − f (z0 ) ≤ Lz1 − z0 

(17.12)

z + v  Cz

(17.13)

for all z, z0, z1 ∈ Z and v ∈ V. Then for every x0 = z0 + f (z0 ), x1 = z1 + f (z1 ), and v ∈ V, if v  2 x1 − x0 , then  x1 − x0 − v = (z1 − z0 ) + ( f (z1 ) − f (z0 ) − v) C C ≥ Cz1 − z0    x1 − x0   v, L+1 2(L + 1) and if v > 2 x1 − x0 , then  x1 − x0 − v  v −  x1 − x0  > v2  . Sufficiency. Let Z be the subspace complementing V (that is, Z ⊕ V = X). For each x ∈ A, there corresponds z = ψ(x) ∈ Z such that x − z ∈ V. By (17.11), the mapping ψ is a bijection, and moreover, z1 − z0  =  x1 − x0 − v  εv for all 0 x0, x1 ∈ A, v = x1 − z1 − x0 + z0 . Hence (x1 − z1 ) − (x0 − z0 )  z1 −z . ε Setting f (z) = x − z for x ∈ A and z = ψ(x), we see that the mapping f : A → V is Lipschitz continuous. It can be extended with the preservation of Lipschitz continuity to the whole space Z (see [522]). The theorem is proved. 

17.12 Contingent One frequently encounters functions that are neither smooth nor convex — for example, the distance to a nonconvex set. One method of studying the smoothness of nonsmooth mappings is the approach based on the contingent of an arbitrary set M at a point x0 . The concept of the contingent (the contingent cone) was introduced by G. Bouligand in 1930 in [127]. For more details, see [506, Sect. IX.2], [417], [253]. Definition 17.8 Let A ⊂ X, x ∈ A. By contg A x we denote the contingent of a set A at a point x, which is defined as the set of unit vectors v for which there exists a sequence {xn } ⊂ A \ {x} such that xn → x

(n → ∞)

and

xn − x →v  xn − x

(n → ∞).

Thus the contingent of M at x0 is the union of all one-sided tangents to the set M emanating from the point x0 . The following result is due to S. V. Konyagin (a stronger variant of this result in the finite-dimensional setting is known as Roger’s theorem (see [506], Sect. IX.13)). Theorem 17.26 Let X be a Banach space, k ∈ N, V a k-dimensional subspace of X, and let A be a subset of X such that contg A x ∩ V =  for all x ∈ A. Then the set A can be covered by countably many Lipschitz surfaces of codimension k associated with V. Therefore, A is a Zk -small set.

17.13

Zajíček-Smallness of the Classes of Sets R(M) and R∗ (M)

441

Proof We take arbitrary points x ∈ A and s0 ∈ V ∩ S. There exists n ∈ N such that 1 x − x − s > (17.14)  x − x n for arbitrary points x ∈ O1/n (x) ∩ A, x  x, and s ∈ O1/n (s0 ) ∩ V ∩ S. Since V ∩ S is compact, there exists n such that inequality (17.14) holds for all x ∈ O1/n (x) ∩ A such that x  x, and all s ∈ V ∩ S. Denoting, for each n ∈ N, by An the set of all x ∈ A satisfying this condition, we get

An . (17.15) A= n∈N

Let Z be the subspace of X that complements V. We write V as a countable union of sets Vm of diameter smaller then 1/2n and define An,m = {x ∈ An | ∃z ∈ Z, ∃v ∈ Vm : x = z + v}. We claim that the inequality x − x − s > ε  x − x

(17.16)

holds for some ε > 0 for all x , x ∈ An,m such that x  x and s ∈ V ∩ S. If  x − x < n1 , this claim follows from the definition of the sets An . Let  x − x  n1 . Consider a number C > 0 such that for all z ∈ Z and v ∈ V, inequality (17.13) holds: z + v  Cz. Let x = z + v, x = z + v , and z, z ∈ Z, v, v ∈ Vm . Hence we have z − z  x − x − v − v v − v 1/(2n) 1  = 1 − > 1− = ;  x − x  x − x  x − x 1//n 2 and moreover, for every vector s ∈ V ∩ S, z − z x − x z − z   v − v − s  − − + s  C > C/2.  x − x  x − x  x − x  x − x So ε = min{1/n, C/2}. This shows that the set An,m is associated with V. Now  An,m .  the conclusion of the theorem follows from the equality A = n,m∈N

17.13 Zajíček-Smallness of the Classes of Sets R(M) and R∗ (M) We recall the following definitions (for an arbitrary set M in a space X). As before, U(M) is the set of uniqueness for M, R(M) := X \ U(M) is the set of nonuniqueness (see Sect. 17.1). Further, we define Rk (M) (k ∈ N) as the set of all points x ∈ X for which there exist y1, . . . , yk+1 ∈ PM x not lying in any (k − 1)-dimensional plane. It is easily

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checked that R(M) = R1 (M) = X \ U(M); R∗k (M) (k ∈ N) is the set of all points x ∈ X for which there exist y1, . . . , yk+1 ∈ PM x not lying in any (k − 1)-dimensional plane and such that (yi + y j )/2 − x < (x, M) for all 1  i < j  k + 1. We define R∗1 (M) := R∗ (M). If X is a strictly convex space, then R∗k (M) = Rk (M), and R2 (M) is the set of all points x ∈ X for which PM x contains at least three elements. We have already noted that according to Zajíček’s theorem, we have R(M) ∈ (Z1 ) for an arbitrary set M in a separable strictly convex Banach space X. It follows that for a nondegenerate Gaussian measure on X, the set R(M) has zero measure, and hence the set of uniqueness U(M) is a set of full measure. This provides a theoretical basis for constructing algorithms based on a random choice of points x for which a nearest point exists almost surely. The next result extends Zajíček’s theorem. Theorem 17.27 (S. V. Konyagin) Let M be a separable subset of a Banach space X. Then R∗ (M) ∈ (Z1 ). Proof Let Y be the subspace spanned by the set M and let (hm ) be a sequence dense y−x in Y ∩ S. We set Sx = SM x = { y−x  | y ∈ PM x}, x  M, and put

1 ϑn,m = x ∈ X | ∃s1, s2 ∈ SM x, |s1 + s2   2 − , n s −s 1 1 2 1 (x, M)  , − hm  . n s2 − s1  4n  Since R∗ (M) = n,m Θn,m , it suffices to show that all sets Θn,m lie in (Z1 ). By Lemma 17.26, to verify this claim it suffices to check that for every point x ∈ Θn,m , hn  contgΘn, m x.

(17.17)

Consider an arbitrary point x ∈ Θn,m and the corresponding elements s1, s2 ∈ 1 SM x and set y j = x + (x, M)s j , j = 1, 2, and s = ss22 −s −s1  . Next, consider the

1 function f (α) =  y2 − (x + αs). We set α0 = s2 −s (x, M). Since x ∈ Θn,m , we 2 1 1 have s2 + s1   2 − n and (x, M)  n , which gives

α0 =

2 − s1 + s2  1 2s2 − (s1 + s2 ) (x, M)  (x, M)  2 . 2 2 2n

Since f is a convex function and since (x, M) y + y 1 (x, M)  1 2 − x = s1 + s2   2− , f (α0 ) = 2 2 2 n we have

(17.18)

17.14

Zajíček-Smallness of the Classes of Sets R k (M) in Euclidean Spaces

f (α)  for 0  α 

443

α α0 − α α α/n  (x, M) − f (0) + f (α0 )  (x, M) − α0 α0 s − 2 − s1  2n 1 2n2

 α0 . But (x + αs, M)  f (α), and so (x + αs, M)  (x, M) −

|α| 2n

(17.19)

for 0  α  2n1 2 . If we consider the function g(α) =  y1 − (x + αs), then a similar analysis proves equality (17.19) for − 2n1 2  α  0. Now let h be an arbitrary element from Y ∩ S ∩ O1/4n (hm ). For 0 < |α| < 2n1 2 , we have (x + αh, M)  (x + αs, M) + αs − αh  (x + αs, M) |α| |α| |α| + |α|s − hm  + |α|hm − h < (x, M) − + + = (x, M). 2n 4n 4n (17.20) Assume that (17.17) does not hold. Then there exists an element x ∈ Θn,m such that 0 <  x − x < 2n1 2 and h = xx −x −x  ∈ O1/4n (hm ). By (17.19), (x , M) < (x, M). Swapping x and x and proceeding as before, we find that (x, M) < (x , M). This contradiction completes the proof of (17.17), and therefore of the theorem.  Theorem 17.28 (S. V. Konyagin) Let X be a separable Banach space that is either two-dimensional or smooth and let M ⊂ X. Then R∗2 (M) ∈ (Z2 ). S. V. Konyagin [352] constructed an example of a non-strictly convex space X such that R(M) ∈ (Z1 ) for every separable M ⊂ X. In a uniformly convex space it can be shown that the points of approximative compactness form a large subset in the set of points of existence. Theorem 17.29 (I. G. Tsar’kov) Let X be a separable uniformly convex Banach space. Then E(M) \ TAC(M) ∈ (Z1 ) for every set M ⊂ X. In particular, for a set of existence M ⊂ X (a proximinal set M), we have X \ TAC(M) ∈ (Z1 ), that is, the set of all points of nonuniqueness together with the points of nonapproximative compactness is Zajíček-small.

17.14 Zajíček-Smallness of the Classes of Sets R k (M) in Euclidean Spaces According to the previous section, for k = 1, 2, the sets Rk (M) are Zk -small for a sufficiently broad class (or possibly all) of separable strictly convex Banach spaces. The situation changes if k  3.

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The results of Sect. 17.14 belong to S. V. Konyagin. For an arbitrary function ρ : X → R and arbitrary x, h ∈ X, we set ρ+h (x) = lim

t→0+

ρ(x + th) − ρ(x) . t

Lemma 17.4 Let X be a separable Banach space, let a function ρ : X → R satisfy the condition (17.21) | ρ(x1 ) − ρ(x2 )|   x1 − x2  (x1, x2 ∈ X); let H be a compact subset of the unit sphere S ⊂ X, and A = {x ∈ X | ρ+h (x) +  An so that for all ρ+−h (x) < 0}. Then the set A can be represented in the form n∈N

n ∈ N,

contg An x ∩ H = 

∀x ∈ An .

(17.22)

Proof Given x ∈ A, consider the following sequence of real functions on the set H: gn (h) = sup

0 0. Now consider a dense set of vectors {hl } in X and the sequence of all subspaces Vm of dimension k each of which is spanned by k vectors of the system {hl }. There exists a subspace Vm such that ∀h ∈ Vm ∩ S

∃h ∈ V ∩ S :

h − h   ε.

Further, by (17.30) and (17.31), we have, for all h ∈ Vm ∩ S,

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ρ+h (x) + ρ+−h (x)  −ε < 0. Let Am bethe set of all x ∈ Uk (ρ) for which (17.29) holds for all h ∈ Vm . Then Am . By Lemma 17.4, every Am can be written as a union of the sets Uk (ρ) = m∈N

Am,n (m, n ∈ N) such that for x ∈ Am,n , contg Am, n x ∩ Vm = . Lemma 17.26 shows that in this case, Am,n ∈ (Zk ). Hence

Uk (ρ) = Am,n ∈ (Zk ). m,n∈N

The lemma is proved.



Theorem 17.30 Let k  3, let X be a separable reflexive space, dim X  k. Then the following conditions are equivalent: (a) X is Euclidean; (b) Rk (M) ∈ (Zk ) for every set M ⊂ X; (c) for no set M ⊂ X does the set Rk (M) contain a subspace X of codimension k − 1. Proof (a) ⇒ (b). In view of Lemma 17.5, it suffices to show that Rk (M) ⊂ Uk (ρ). Let x ∈ Rk (M) and let y1, y2, . . . , yk+1 be points from PM x in general position. As V we consider the subspace spanned by y2 − y1, . . . , yk+1 − y1 . For each h ∈ V, h  0, there exist yi and y j such that (h, yi − y j ) < 0. As the function ρ(x) we take the distance function (x, M) (by this we also define the corresponding functions ρ+h (x) and ρ−h (x)). We have  y j − x − th −  y j − x (h, x − y j ) = , t→0+ t (x, M)  yi − x + th −  yi − x (−h, x − yi ) ρ+−h (x)  lim = , t→0+ t (x, M) (h, yi − y j ) < 0, ρ+h (x) + ρ+−h (x)  (x, M) ρ+h (x)  lim

which gives us x ∈ Uk (ρ). (b) ⇒ (c). We need to verify thata Zk -small set cannot contain a subspace Y ⊂ X of codimension k − 1. Let A = n∈N An , where each of the sets An is associated with a subspace Vn of dimension k. Then Vn ∩ Y  {0}, and hence An ∩ Y (which is associated with Vn ∩ Y ) is a Z1 -small subset of Y . Therefore, A ∩ Y =  n∈N (An ∩ Y )  Y . (c) ⇒ (a). Let X be a reflexive space such that Y ∩Q k (M)   for every subspace Y of codimension k − 1 and set M ⊂ X. In particular, this also holds for the set M = {x ∈ X | (x, Y )}. There exists y ∈ Y such that the plane spanned by PM y has dimension < k. But the plane L spanned by PM 0 has the same dimension, because

17.14

Zajíček-Smallness of the Classes of Sets R k (M) in Euclidean Spaces

447

PM 0 = −y + PM y. The point 0 lies in L, because if x ∈ PM 0, then PM 0 also contains −x. Hence L is a subspace. Let G be the set formed by the points x ∈ X for which 0 ∈ PY x. For each x ∈ G, x  0, we have 0 ∈ PY (x/ x), which shows that x/ x ∈ L and x ∈ L. Therefore, G ⊂ L. Every point x ∈ X can be written as g + y, where g ∈ G, y ∈ Y . Indeed, since X is reflexive, we have E(Y ) = X; if y ∈ PY x and g = x − y, then PY g = −y + PY x  0, but since codim Y = k − 1  dim L, we have X = L ⊕ A and G = L. Let us show that the space X is strictly convex. If this were not so, then there would exist a nondegenerate interval Δ lying on the unit sphere. We draw the plane of codimension k − 1 that supports the sphere and contains this interval. Let V be the corresponding subspace. Then Δ ⊂ L. Therefore, L has a common line with V. But this is impossible. In a strictly convex reflexive space X, for every subspace V of codimension k − 1 and x ∈ X, the set PV x is a singleton; we denote it by v. We have x = v + (x − v), v ∈ V, x − v ∈ L. Therefore, for every subspace V of codimension k − 1, the operator  PV is linear. Now the space X is Euclidean by Singer’s theorem [519]. Given a set M ⊂ X and l ∈ Z+ , we denote by Rl (M) the set of all x ∈ X for which the subspace spanned by PM x has codimension  l. It is easily checked that if dim X = n < ∞ and l < n, then Rl (M) = Rn−l (M). Theorem 17.31 Let l ∈ Z+ and let X be a normed linear space, dim X  l + 3. Then the following conditions are equivalent: (a) X is Euclidean; (b) for every M ⊂ X and subspace Y ⊂ X of dimension m > l, the set Rl (M) ∩ Y is Zm−l -small in Y ; (c) for no M ⊂ X does the set Rl (M) contain an (l + 1)-dimensional linear subspace. Proof (a) ⇒ (b). As the function ρ(x) we take the distance function (x, M). In the proof of the above theorem it was shown that if x ∈ X and V is the space spanned by the elements of the form y1 − y2 , y1, y2 ∈ PM x, then ρ+h (x) + ρ+−h (x) < 0 for all h ∈ V \ {0}. Since codim V  l for x ∈ Rl (M), we have dim Z  m − l for Z = V ∩Y . So for all x ∈ Rl (M) ∩ Y , there exists a subspace Z ⊂ Y of dimension m − l such that ρ+h (x) + ρ+−h (x) < 0 for every vector h ∈ Z \ {0}. Hence by Lemma 17.5, the set Rl (M) ∩ Y is a Zm−l -small subset of Y . The implication (b) ⇒ (c) is clear. The implication (c) ⇒ (a) can be proved as the corresponding implication in Theorem 17.30 with the only difference that instead of Singer’s theorem, one should employ the following Rudin and Smith’s theorem from [502]: if X be a normed linear space, dim X  3, m ∈ N, m  dim X − 2, then the space X is Euclidean if and only if X is strictly convex and for every subspace  Y ⊂ X of dimension m, the metric projection operator PY : X → Y is linear. Corollary 17.3 In a Euclidean space X, for every set M ⊂ X, the intersection R0 (M) with a finite-dimensional subspace is at most countable. However, as was shown by S. V. Konyagin, the set R0 (M) itself can have the cardinality of the continuum for some set M ⊂ 2 .

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Approximative Properties of Arbitrary Sets . . .

17.15 Almost Chebyshev Sets If X is a strictly convex space, then every finite-dimensional subspace of X is a Chebyshev subspace. If X is not only strictly convex but is also reflexive, then every subspace of X is a Chebyshev subspace. In particular, in such a space, every subspace of finite codimension is a Chebyshev subspace. We also note that if the space X ∗ is the dual space of a separable space X, then it always contains a Chebyshev subspace of codimension 1. A good account of results on the existence of Chebyshev subspaces of finite dimension or codimension can be found in [259], [262, Sect. 11]. It is well known that C(Q) does not always contain nontrivial Chebyshev subspaces (of dimension > 1). For example, in the space of continuous functions on the (solid) square, there are no Chebyshev subspaces of finite dimension > 1; in L 1 [0, 1], no nontrivial finite-dimensional subspace in L 1 [0, 1] is a Chebyshev subspace; however, such a subspace is an almost Chebyshev subspace. This is a big difference between L 1 (Q, μ) and C(Q), because in C(Q), not every finite-dimensional subspace is almost Chebyshev. In this section, we shall consider more general almost Chebyshev sets and subspaces. Definition 17.9 A nonempty subset M of a normed linear space X is called an almost Chebyshev set (respectively, an almost uniqueness set) if T(M) ∈ (II) (U(M) ∈ (II)); that is, the set X \ T(M) (X \ U(M)) is a set of first category (a meager set) in X (by the definition, such a set is an at most countable union of nowhere dense sets). Recall that the sets T(M) and U(M) were defined above in Sect. 17.1. The concept of an almost Chebyshev set was introduced by Stechkin [529], who showed that a closed nonempty subset of a uniformly convex Banach space is an almost Chebyshev set. Later, various problems related to almost Chebyshev sets and almost uniqueness sets were studied by Garkavi [259], [261], Lau [387], [389], Deutsch and Kenderov [192], Bartelt and Schmidt [80], Borwein and Fitzpatrick [122], Ustinov [579], Li and Wang [400], Cobzaş [172]. This list is by no means complete. In a strictly convex Efimov–Stechkin space, each nonempty set is a set of almost uniqueness (cf. Theorem 17.1 and Corollary 17.2). Garkavi [259] showed that every reflexive subspace L0 of a separable space X contains an almost Chebyshev subspace L ⊂ X that is isomorphic to L0 . So every separable space contains almost Chebyshev subspaces of every finite dimension. Lau [387] showed that if X is a separable Banach space that is either locally uniformly convex or that satisfies the Radon–Nikodým property, then ‘almost all’ (closed) subspaces in X are almost Chebyshev subspaces. Garkavi [261] characterized the finite-dimensional almost Chebyshev subspaces in C(Q), where Q is a compact Hausdorff space (see Theorem 17.37 below). A similar characterization for the space L 1 (μ) was given by Rozema [499], which relates the almost Chebyshev property of a subspace in L 1 (μ) with the lack of a continuous selection of the metric projection operator onto that subspace.

17.15

Almost Chebyshev Sets

449

Definition 17.10 A Banach space X is called a U-space (Lau [389]) if for every ε > 0, there exists δ > 0 such that if  x =  y = 1, (x + y)/2 > 1 − δ, then (x ∗ + y ∗ )/2 > 1 − ε for all functionals x ∗, y ∗ ∈ S ∗ that support the sphere S at the points x and y, respectively. The class of Banach U-spaces contains the Banach uniformly convex spaces, the spaces with uniformly Fréchet-differentiable norm, and is contained in the class of uniformly nonsquare Banach spaces [389]. A uniformly nonsquare space is not necessarily a U-space — it suffices to consider as a unit ball a hexagon in the plane. Since a Banach uniformly nonsquare space is reflexive, so is every Banach U-space (see [389]). A Banach space X is a U-space if and only if so is X ∗ . The next theorem8 strengthens some results of S. B. Stechkin (Theorems 17.3 and 17.4). Theorem 17.32 (K.-S. Lau [389]) In a locally uniformly convex U-space, every closed nonempty subset is an almost Chebyshev set. Theorem 17.32 is a particular case of Corollary 17.2. In the following theorem, Corollary 17.2 is extended to the case of nonreflexive spaces. Theorem 17.33 (J. M. Borwein, S. Fitzpatrick [122]) In a strictly convex Kadec– Klee space, a nonempty boundedly weakly compact closed set is an almost Chebyshev set. From Theorem 17.33, it follows that a nonempty closed subset of a strictly convex Efimov–Stechkin space is an almost Chebyshev set [122, Corollary 6.5]. We note in passing the following characterization of strictly convex Efimov– Stechkin spaces (in other terminology, strongly convex spaces) in terms of weakly almost Chebyshev sets (weakly almost Chebyshev sets were defined in Sect. 17.3). Theorem 17.34 (See [352], [122]) (1) The following conditions on a Banach space X are equivalent: (a) X is a strictly convex Efimov–Stechkin space9; (b) the norm of the space X ∗ is Fréchet-differentiable; (c) every nonempty closed subset of X is an almost Chebyshev set; (d) every nonempty closed set M ⊂ X is a weakly almost Chebyshev set; that is, T(M) = X in M. (2) In a strictly convex Kadec–Klee space, a nonempty boundedly weakly compact closed set is an almost Chebyshev set. Theorem 17.35 Let M ⊂ X be an approximatively compact subset of a Banach space X. If a set M is a weakly almost Chebyshev set (a weakly almost uniqueness set), then it is an almost Chebyshev set. 8 In fact, the class of spaces indicated in this theorem is contained in the class of strongly convex spaces (see Corollary 17.2) above. 9 Such spaces are also called strongly convex spaces.

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Approximative Properties of Arbitrary Sets . . .

Proof (of Theorem 17.35) Since M is approximatively compact, for each point x ∈ T(M) = U(M) and an arbitrary number ε >  0,there exists δ = δ(x) > 0 such  Oδ (x) | x ∈ T(M) = U(M) is that diam PM (Oδ (x)) < ε. Hence each set Aε := open and dense in X. By construction, the intersection  A1 A := n∈N

n

consists of points of existence and uniqueness, and its complement is a nowhere dense set. Hence the theorem follows.  We note another simple result (Rozema [498]). Theorem 17.36 Let M be a proximinal set. If the metric projection PM is lower semicontinuous and M is an almost Chebyshev set, then M is a Chebyshev set. Using Theorem 17.36 and a result of A. L. Brown, Rozema [498] showed that in the spaces c0 and n1 , every finite-dimensional almost Chebyshev subspace is a Chebyshev subspace. Rozema [499] also proved that in L 1 (μ), where μ is an atomless measure, every convex finite-dimensional subset is an almost Chebyshev set. In [499], he also constructed an example of a non-Chebyshev almost Chebyshev subspace of the space  1 admitting a continuous selection of the metric projection operator.

17.16 Almost Chebyshev Systems of Continuous Functions In this section, following [259] and [261], we study almost Chebyshev subspaces of finite dimension in the space C(Q) of continuous functions defined on a metrizable compact set Q with the Chebyshev norm  f  = maxu ∈Q | f (u)|. In particular, we give Garkavi’s criterion for a subspace to be almost Chebyshev in the space C(Q) (see [261]). Let Sn = {ϕ1 (u), . . . , ϕn (u)} be a basis for a subspace L ⊂ C(Q). As usual, an element p from L is called a polynomial in the system Sn . Bases of Chebyshev (almost Chebyshev) subspaces are called Chebyshev (almost Chebyshev) systems of functions. An infinite system {ϕ1, ϕ2, . . .} is called a Markov (almost Markov) system if for every n < ∞, the system {ϕ1, . . . , ϕn } is a Chebyshev (almost Chebyshev) system. The property of a subspace L being an almost Chebyshev subspace means that if near the unit sphere one describes the ‘cylinder’ whose axis is the subspace L, then ‘almost all’10 ‘generators’ of the cylinder have a unique common point with the sphere. In some problems, for example in interpolation problems on compact sets, almost Chebyshev systems are capable, to a certain extent, of compensating for 10 Here, ‘almost all means’ all except a meager set in the space of cylinder generators equipped with the distance between them.

17.16

Almost Chebyshev Systems of Continuous Functions

451

the lack of Chebyshev systems, which are commonly used for interpolation on an interval. By Haar’s theorem, a system Sn is a Chebyshev system if and only if each nontrivial polynomial in this system vanishes at at most n − 1 points of a (metrizable) compact set Q. According to Mairhuber (see Sect. 2.5), this condition can be satisfied only if Q is homeomorphic to some closed subset of a circle. This shows that the study of almost Chebyshev systems in C(Q) is reasonable, because almost Chebyshev systems exist on every compact set. In this section, we shall characterize almost Chebyshev systems and give a constructive proof of the existence of almost Chebyshev and almost Markov systems on an arbitrary compact set. We also establish some properties of almost Chebyshev systems that are close to those of Chebyshev systems. We also indicate the possibility of using almost Chebyshev systems in interpolation problems on compact sets. In this section, Q is a metrizable compact set. We begin with some notation and definitions. (1) Ln = L(Q, Sn ) is the subspace of polynomials in a system Sn on a compact set Q. (2) E = E(Q, Sn, f ) = ρ( f , L) is the best approximation (the distance) from a function f to the set of polynomials in a system Sn on a compact set Q. (3) P(Q, Sn, f ) is the set polynomials p ∈ L(Q, Sn ) of best approximation to a function f on Q. (4) T(Q, Sn ) is the class of functions f ∈ C(Q) for which a polynomial of best approximation in the subspace L(Q, Sn ) exists and is unique; that is, x ∈ T(L(Q, Sn ))) (this class will be called the class of uniqueness). ˚ 0, r) is an open ball in a metric space Q with centre at u0 and radius r; that (5) B(u ˚ 0, r) := {u | (u0, u) < r }. is, B(u (6) R = R(Q, Sn, f , p) is the set of points of maximal deviation of a polynomial p from a function f ; that is, R(Q, Sn, f , p) := {u | | f (u) − p(u)| =  f − p}.  (7) M(Q, Sn, f ) = {R(Q, Sn, f , p) | p ∈ P(Q, Sn, f )}, where the intersection is taken over all polynomials of best approximation p ∈ P(Q, Sn, f ) to f . (8) With each set G ⊂ Q we associate the number Nn (G) defined as the cardinality of G if it is smaller than n and equal to n otherwise. In what follows, it is always assumed that the number of points of a compact set Q is not smaller than n. We need some simple lemmas. Lemma 17.6 The minimal set M = M(Q, Sn, f ) has the following properties: (a) on M, all polynomials from P(Q, Sn, f ) are equal; (b) there exists a polynomial p∗ ∈ P(Q, Sn, f ) such that the set R(Q, Sn, f , p∗ ) coincides with the set M(Q, Sn, f ); (c) if f (u) is not contained in the class of uniqueness T(Q, Sn ), then at least n − Nn (M) + 1 linearly independent polynomials from L(Q, Sn ) vanish identically on M. Lemma 17.7 Let f ∈ C(Q). For each ε > 0, there exists δ > 0 such that for every function  f ∈ C(Q) satisfying the condition  f −  f  < δ and for each polynomial f ), there exists a polynomial p ∈ P(Q, Sn, f ) such that p − p < ε. p(u) ∈ P(Q, Sn, 

452

17

Approximative Properties of Arbitrary Sets . . .

The proofs of these results are carried out by standard machinery of the theory of best uniform approximation and are hence omitted. We only note that the proof of assertion (c) of Lemma 17.6 depends on assertions (a) and (b) of this lemma. Lemma 17.8 If for points u1, . . . , un of a compact set Q, the determinant    ϕ1 (u1 ) . . . ϕn (u1 )    ∂(ϕ1, . . . , ϕn ) :=  . . . . . . . . .  ∂(u1, . . . , un ) ϕ1 (un ) . . . ϕn (un ) is nonzero, then for every ε > 0, there exists δ > 0 such that |p(u)| < ε (u ∈ Q) for each polynomial p(u) in {ϕ1, . . . , ϕn } satisfying the conditions |p(uk )| < δ (k = 1, . . . , n). For a Chebyshev system on an interval, a proof of this lemma can be found, for example, in [361]. The proof is the same for an arbitrary system. The next theorem characterizes almost Chebyshev systems. Theorem 17.37 A system Sn = {ϕ1, . . . , ϕn } is an almost Chebyshev system on a compact set Q if and only if at most n − Nn (G) linearly independent polynomials in the system Sn vanish identically on each open subset G ⊂ Q. Proof Sufficiency. Let Fm be the class of functions f from C(Q) such that from the set M(Q, Sn, f ) one can take m points u1, . . . , um for which the rank of the matrix 

 ϕ (u ) . . . ϕ1 (um ) ϕ1, . . . , ϕn  1 1  :=  . . . . . . . . .  u1, . . . , um  ϕn (u1 ) . . . ϕn (um ) 

(17.32)

is m. From assertion c) of Lemma 17.6 it follows that Fn ⊂ T(Q, Sn ). If f is not contained in the class of uniqueness T(Q, Sn ), then f ∈ Fm for 0  m  n − 1, and f  Fn . The sufficiency will be proved once we show that for every f ∈ Fm (0  m  n − 1) and an arbitrary ε > 0, there exists a function  f ∈ Fm+1 such that f −  f  < ε. Indeed, in this case, the class Fn , and hence T(Q, Sn ), is everywhere dense in C(Q), and hence since C(Q) is separable, L(Q, Sn ) is an almost Chebyshev subspace. Let f ∈ Fm and f  Fn . We can assume without loss of generality that the set P = P(Q, Sn, f ) contains the identically zero function. In this case, for u ∈ M = M(Q, Sn, f ) we have | f (u)| = E = E(Q, Sn, f ), and so p = 0 (p ∈ P). Under the conditions of the theorem, the set M (for f ) contains at least one limit point u0 of the compact set Q (otherwise, there would exist n − m linearly independent polynomials that vanished on the open subset consisting of m+1 isolated points). For definiteness, we assume that f (u0 ) = +E. Consider the function Φ(u) = sup p ∈ Pp(u). This function is continuous, nonnegative, and Φ(u0 ) = 0. Furthermore, for each point u ∈ Q, there exists a polynomial pu ∈ P such that pu (u) = Φ(u). Let u1, . . . , um be the points at which the rank of the matrix (17.32) is m. Assuming that ε < E, we ˚ 0, δ) does not contain these points (except choose δ > 0 so small that the ball B(u ˚ 0, δ), possibly u0 ) and such that for u ∈ B(u

17.16

Almost Chebyshev Systems of Continuous Functions

0  Φ(u) 

ε , 2

0  E − f (u) 

453

ε 2

(ε < E).

(17.33)

  ˚ 0, δ ) contains a point y such that the rank of the matrix ϕ1,...,ϕn is The set B(u 2 u1,...,um,y  ϕ ,...,ϕ  n n m + 1. Indeed, if u11,...,umm is the principal minor of the matrix (17.32), then the  ϕ ,...,ϕ ,ϕ  n1 n m n m+1 nontrivial polynomial p = det does not vanish identically on the u1,...,um,y δ ˚ 0, ), because that set is open and infinite. Let py (y) = Φ(y). We construct set B(u 2 a continuous function ϕ so as to have 0  ϕ(u)  1,  ˚ 0, δ ) 1, u ∈ B(u 2 ϕ(u) = ˚ 0, δ). 0, u  B(u ˚ 0, δ) on which (E + py (u))ϕ(u) > f (u). We denote by R0 the part of the set B(u Consider the function   E + py (u) ϕ(u), u ∈ R0,  (17.34) f (u) = f (u), u  R0 . Using (17.33), it can be easily shown that  f (u) is continuous and

We claim that

 f (u0 ) = f (u0 ) = E,  f (y) = E + py (y),  f− f  < ε.

(17.36)

E(Q, Sn,  f )  E.

(17.37)

(17.35)

For a point u  R0 , we have |  f (u) − py (u)| = | f (u) − py (u)|  E. Now let u ∈ R0 . Then  f (u) − py (u) = (E + py (u))ϕ(u) − py (u). If this difference is positive, then it is majorized by E + py (u) − py (u) = E. If (E + py (u))ϕ(u)  py (u), then again in view of (17.33) and (17.37), we get |  f (u) − py (u)|  |E + py (u)|ϕ(u) + |py (u)|  2|py (u)|  2 ε2 = ε < E. Consider an arbitrary polynomial pfrom P(Q, Sn,  f ). We have  f (u) = f (u) outside R0 . Hence for u  R0 , we have | f (u) − p(u)|  E(Q, Sn,  f )  E.

(17.38)

If u ∈ R0 and p(u)  p(u), then by (17.33), we have  f (u)  f (u)  p(u)  p(u), and hence for such points u, | f (u) − p(u)|  |  f (u) − p(u)|  E.

(17.39)

If u ∈ R0 and p(u)  p(u), then for sufficiently small λ > 0, we have py (u)  p(u)  f (u). Therefore, setting r(u) = (1 − λ)py (u) + λ p(u), we get (1 − λ)py (u) + λ

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Approximative Properties of Arbitrary Sets . . .

at these points | f (u) − r(u)|  E.

(17.40)

But by (17.38), (17.39), inequality (17.40) holds also for all remaining u ∈ Q, and hence r(u) ∈ P(Q, Sn, f ), which implies that r(uk ) ≡ λ p(uk ) + (1 − λ)py (uk ) = 0

(k = 1, . . . , m),

r(y) ≡ λ p(y) + (1 − λ)py (y)  Φ(y) = py (y).

(17.41)

From (17.41), we have p(uk ) = 0

(k = 1, . . . , m),

p(y)  py (y).

(17.42) (17.43)

Using (17.36), we have  f (y) = E + py (y). Hence p(y) > py (y), for otherwise, f ). Taking into account (17.36) and we would have |  f (y) − p(y)| > E  E(Q, Sn,  (17.43), we get that p(y)  py (y)

and

| f (y) − p(y)| = E.

(17.44)

An appeal to (17.37) shows that E(Q, Sn,  f ) = E. Next, since uk  R0 (uk  u0 ) f (u0 ), we have f (uk ) =  f (uk ) = ±E (k = 1, . . . , m). Now from and since f (u0 ) =  (17.42) and (17.44) it follows that u1, . . . , um, y are points of maximal deviation of the f ), these polynomial p from f . Since p(u) is an arbitrary polynomial from P(Q, Sn,  f ). Moreover, the rank of the matrix (17.32) points lie in the minimal set M(Q, Sn,  is m + 1, and hence  f ∈ Fm+1 . Necessity. Assume the contrary. Let an open subset G1 ⊂ Q be such that the number of linearly independent polynomials from L(Q, Sn ) that vanish identically on G1 is n − m1  n − Nn (G1 ) + 1. This is equivalent to saying that the number of functions from Sn that are linearly independent on G1 is m1  Nn (G1 ) − 1 < n. Consider all open subsets G1 of G1 of cardinality at least m1 . There are two cases to consider: (a) on each G1 , the number of linearly independent functions is m1 ; (b) there exists a set G2 from {G1 } on which the number of linearly independent functions is m2  m1 − 1. In the latter case, we again consider all open sets {G2 } from G2 of cardinality ≥ m2 points. As before, there are two possibilities. In the case similar to b), we again choose an open set G3 on which the number of linearly independent functions is m3  m2 − 1. Since n > m1 > m2 > . . ., we arrive after finitely many steps at one of the following cases. (A) On some open set G of cardinality ≥ m+1, the number of linearly independent functions is m > 0. These functions remain linearly independent on each open subset G of G of cardinality ≥ m.

17.16

Almost Chebyshev Systems of Continuous Functions

455

(B) On some open subset G of cardinality m  1, there is no linearly independent function from the system Sn (that is, all functions vanish identically on G). The case (B) is trivial and hence omitted. Consider Case (A). For convenience, we assume that on G, the functions ϕ1, . . . , ϕm are linearly independent and ϕm+1, . . . , ϕn vanish identically. We claim that in Case A), in the set G, one can choose m + 1 points u1, . . . , um+1 such that all the determinants   ϕ1, . . . , ϕm Dk ≡ det , k = 1, . . . , m + 1 u1, . . . , uk−1, uk+1, . . . , um+1 will be nonzero. If G contains at least m + 1 isolated points of Q, then as {uk } one can take all m + 1 points among these points. Moreover, the determinants Dk are nonzero, because every set of m isolated points forms an open subset on which, by condition (A), the system Sm = {ϕ1, . . . , ϕm } is linearly independent. Assume now that the number of isolated points in G is ≤ m and that u0 ∈ G is a cluster point of ˚ 0, δ0 ) lies wholly the compact set Q. We choose δ0 > 0 to be so small that the ball B(u ˚ 0, δ0 ) is an open infinite set, in G and does not contain isolated points of Q. Since B(u then by condition (A), the system Sm is linearly independent on this set. This also ˚ δ) for every δ > 0 and point implies that Sm is linearly independent on each ball B(u, ˚ 0, δ0 ). In other words, no nontrivial polynomial in the system Sm vanishes u ∈ B(u ˚ δ) (u ∈ B(u ˚ 0, δ0 )). We choose m points u1, . . . , um from identically on the balls B(u, ˚ 0, δ0 )) such that the determinant B(u   ϕ1, . . . , ϕm Dm+1 ≡ det . (17.45) u1, . . . , um is zero and consider m nontrivial polynomials   ϕ1, . . . , ϕm Dk (u) = det , u1, . . . , uk−1, u, uk+1, . . . , um

k = 1, . . . , m.

˚ δ) (u ∈ By the above, since Dk (u) is continuous, it follows that for every ball B(u, ˚ δ1 ) ⊂ B(u, ˚ δ) ˚ 0, δ0 )) and for each polynomial Dk (u), one can find a ball B(u, B(u such that this polynomial is nonzero on this ball. Therefore, there exists a point ˚ 0, δ0 )) at which all polynomials Dk (u) are nonzero. Hence taking into um+1 ∈ B(u account (17.45), we get   ϕ1, . . . , ϕm Dk ≡ det  0, k = 1, . . . , m + 1, (17.46) u1, . . . , uk−1, uk+1, . . . , um+1 proving the claim. Consider now a compact set Q0 consisting of the above m + 1 points u1, . . . , um+1 . On these m + 1 points, consider a function f (u) that on Q0 does not identically equal any polynomial in Sm . Conditions (17.46) mean that Sm is a Chebyshev system on Q0 . Hence for f , there exists a unique polynomial p0 ∈ P(Q0, Sm, f ). We can assume without loss of generality that p0 (u) = 0 for u ∈ Q0 , so that maxu ∈Q0 | f (u)| = E(Q0, Sm, f ) = E. We construct a function F(u) on the compact

456

17

Approximative Properties of Arbitrary Sets . . .

set Q such that  F(u) =

F   E, f (u), 0,

u ∈ Q0, u ∈ Q \ G.

Since ϕm+1 (u) = . . . = ϕn (u) = 0 if u ∈ Q0 and since F   E, the set P(Q, Sn, F) contains the identically zero function and E(Q, Sn, F) = E(Q0, Sm, f ). Furthermore, all polynomials p(u) ∈ P(Q, Sn, F) vanish at the points of the compact set Q0 . Now  be an arbitrary function from C(Q) satisfying the inequality F − F  < δ , let F   E/2. where δ > 0 is not greater than E/4 and is sufficiently small that E(Q, Sn, F) For a given ε > 0, we can pick δ such that for the polynomial  p = a1 ϕ1 (u) + . . . + am ϕm (u) + . . . + an ϕn (u) ∈ P(Q, Sn, F), there exists by Lemma 17.7 a polynomial p ∈ P(Q, Sn, F) such that p − p < ε.

(17.47)

From (17.47), it follows that if u ∈ Q0 , then p(u)| = |a1 ϕ1 (u) + . . . + am ϕm (u)| < ε, and hence if ε is sufficiently small, then by Lemma 17.8, we have |a1 ϕ1 (u) + . . . + am ϕm (u)|
1). For example, it can be easily checked that if the cardinality of the set of isolated points of a topological set S is greater than that of the continuum, then the space C(S) does not have any almost Chebyshev system consisting of more than one function.

Appendix A

Chebyshev Systems of Functions in the Spaces C, C n , and L p

In this appendix, we consider classical abstract and concrete Chebyshev systems in in the spaces C, C n , and L p . We formulate extreme problems, some of which are still open. Structural formulas for extended polynomials are discussed. Conjectures in classical extreme problems are formulated. We first give fundamental definitions of Chebyshev systems, which date back to [322], [371], [185], [186]; our presentation here is based on [188], [187]. A subspace Ln+1 (n ≥ 0) of the space C[a, b] of dimension (n + 1) is called a Chebyshev space (or a T-space) of nth order if every nontrivial function from Ln+1 has on [a, b] at most n distinct zeros. Every basis in such a T-space is called a T-system of order n on [a, b]. An (n + 1)-dimensional subspace Ln+1 of the space C n [a, b] is called an extended Chebyshev space (or an ET-space) of order n if every nontrivial function in this space has on [a, b] at most n zeros, counting their multiplicity. n for a T-space L A basis {ek ( · )}k=0 n+1 is called a complete Chebyshev system (or m (0 ≤ m ≤ n) a CT-system) of order n on [a, b] if all subspaces Lm = span{ek ( · )}k=0 are Chebyshev spaces. A subspace Ln+1 with this basis is called a complete Chebyshev space (or a CT-space) of order n. n for an ET-space Ln+1 is called an extended complete ChebyA basis {ek ( · )}k=0 m for all shev system (or an ECT-system) of order n on [a, b] if Lm = span{ek ( · )}k=0 0 ≤ m ≤ n is an extended Chebyshev system. A subspace Ln+1 with such basis is called an extended complete Chebyshev space (or an ECT-space) of order n. Elements of such spaces will be called extended polynomials. The classical and important example of an ECT-space of order n in C[a, b] is the subspace Pn of classical algebraic polynomials of degree ≤ n. A detailed account of ECT-spaces can be found in Demidovich and Kochurov [435]. Among other examples of ECT-systems of order n on [a, b] that generate the corresponding ECT-spaces of order n in C[a, b], we mention exponential functions: {1, eα1 t , . . . , eαn t },

0 < α1 < . . . < αn ;

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. TsarŠkov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2

459

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460

power functions (a > 0): {1, t α1 , . . . , t αn },

0 < α1 < . . . < αn ;

rational fractions (a > 0):  1 1  1, , , . . ., t + α1 t + αn

0 < α1 < . . . < αn ;

logarithmic functions (a > 0): {1, ln(t + α1 ), . . . , ln(t + αn )},

0 < α1 < . . . < αn ;

Gaussian functions (a > 0): {1, e−

(t +α1 )2 β

, . . . , e−

(t +α n )2 β

},

0 < α1 < . . . < αn, β > 0;

exponential-power functions (a > 0): {1, . . . , t β0 −1, eα1 t , . . . , t β1 −1 eα1 t , . . . , eα j t , . . . , t β j −1 eα j t }, 0 < α1 < . . . < α j , βi ∈ N,

0 ≤ i ≤ j,

j 

βi = n + 1.

i=0

Approximation theory deals with some extremal problems associated with the subspace Pn of the space C[a, b] (the space of classical algebraic polynomials of degree ≤ n; see, for example, [542]). Among such problems, we mention the problem of construction in Pn of a polynomial of least deviation from zero in the space C[a, b] with the Chebyshev norm and also in L 1 - and L 2 -spaces. In all such cases, explicit formulas for such polynomials are available. We give them for the canonical interval [−1, 1] with the help of the classical G. G. Lorentz’s representation for polynomials (see [435], Chap. 1, § 1.2.5). On the canonical interval [−1, 1], the monic polynomial Tn ( · ) of least deviation from zero in the uniform norm can be written as Tn (t) =

n 

ϑk · (t + 1)n−k (t − 1)k ,

k=0

(A.1)

(2n) ! . ϑk = 2n−1 2 (2k)! (2n − 2k)! P. L. Chebyshev was the first to obtain such a representation, and so the polynomial Tn ( · ) is now called the Chebyshev polynomial of the first kind after him. The representation (A.1) for Tn ( · ) is a direct consequence of Paszkowski’s formula (see [462], Chap. I, § 1, formula (26)). In turn, on the canonical interval [−1, 1], the monic polynomial Un ( · ) of least deviation from zero in the L 1 -norm reads as

A Chebyshev Systems of Functions in the Spaces C, C n , and L p

Un (t) =

n 

461

υk · (t + 1)n−k (t − 1)k ,

k=0

(2n + 2)! . υk = 2n+1 2 (2k + 1)!(2n − 2k + 1)!

(A.2)

A. N. Korkin and E. I. Zolotarev (former students of Chebyshev) were the first to derive an explicit formula for such a polynomial. After them, the polynomials Un ( · ) are called Chebyshev polynomials of the second kind. The representation (A.2) for Un ( · ) is also a direct consequence of another Paszkowski formula (see [462], Chap. I, § 1, formula (27)). Finally, on the canonical interval [−1, 1], the monic polynomial G n ( · ) of least deviation from zero in the L 2 -norm reads as G n (t) =

n 

γk · (t + 1)n−k (t − 1)k ,

k=0

(n!)4 γk = . (2n)![k!(n − k)!]2

(A.3)

This representation can be obtained from the well-known Rodrigues’s formula (see, for example, [535], Chap. IV, § 1, formula (4)), according to which n! [(t + 1)n (t − 1)n ](n) . (A.4) (2n)! n Cnk · u(k) (t) v (n−k) , we get in Indeed, applying Leibniz’s rule [u(t) · v(t)](n) = k=0 succession G n (t) =

[(t + 1)n (t − 1)n ](n) n  = {Cnk · [n(n − 1) · · · (n − k + 1)] (t + 1)n−k k=0

× [n(n − 1) · · · (k + 1)] (t − 1)k } = n  n(n − 1) · · · (n − k + 1)(n − k)! (t + 1)n−k {Cnk · = (n − k)! k=0

n(n − 1) · · · (k + 1)k! (t − 1)k } k! n  n! n! = (t + 1)n−k · (t − 1)k Cnk · (n − k)! k! k=0 ×

=

n  k=0

n! n! n! · · · (t + 1)n−k (t − 1)k k!(n − k)! (n − k)! k!

= (n!)3 ·

n  k=0

1 · (t + 1)n−k (t − 1)k . [k!(n − k)!]2

(A.5)

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462

Substituting into (A.4) the expression for [(t + 1)n (t − 1)n ](n) from (A.5), we get (A.3). Legendre was the first to find such a representation, and hence G n ( · ) are called Legendre polynomials. Such polynomials are frequently used in the theory of orthogonal polynomials (see, in particular, [462]). It would be interesting to obtain explicit formulas for extended polynomials of least deviation from zero that extend (A.1)–(A.3) from the classical setting of the space of algebraic polynomials Pn to the case of an arbitrary ECT-space of corresponding order.

A.1 Statement of the Problem Let an (n + 1)-dimensional subspace Ln+1 of the space C n [a, b] be an ECT-space of order n, and let n ∈ C n ([−1, 1]) (A.6) {ek ( · )}k=0 be a basis for Ln+1 that is an ECT-system of order n on the interval [−1, 1]. We denote by Φ the class of extended monic polynomials of order n (from the ECT-space Ln+1 n−1 , under consideration) with arbitrary coefficients (xk )k=0 n−1    Φ = ϕ = ϕ(t) : ϕ(t) = en ( · ) + xk ek ( · ) .

(A.7)

k=0

Considering the function f (t, ϕ) := |ϕ(t)| = |en (t) +

n−1 

xk ek (t)|,

(A.8)

k=0

we formalize the corresponding generalizations of problems (A.1)–(A.3). 1. Problem of extended polynomials of order n of least deviation from zero in the Chebyshev norm. If we set f1 (ϕ) := max f (t, ϕ), t ∈[−1,1]

(A.9)

then this problem can be formally written as f1 (ϕ) → min,

ϕ ∈ Φ.

(P1 )

2. Problem of extended polynomials of order n of least L 1 -deviation from zero. If we set ∫ 1 f (t, ϕ) dt, (A.10) f2 (ϕ) := −1

A Chebyshev Systems of Functions in the Spaces C, C n , and L p

463

then this problem can be formally written as f2 (ϕ) → min,

ϕ ∈ Φ.

(P2 )

3. Problem of extended polynomials of order n of least L 2 -deviation from zero. If we set ∫ 1 f 2 (t, ϕ) dt, (A.11) f3 (ϕ) := −1

then this problem can be formally written as f3 (ϕ) → min,

ϕ ∈ Φ.

(P3 )

We first note that all three problems are convex finite-dimensional extremal unconstrained problems. All such extremal problems can be reformulated in the following general form: n−1 f (x) → min, x ∈ Rn, x := (xk )k=0 . (P)

A.2 Basic Facts About Structural Formulas for Extended Polynomials of Least Deviation from Zero Following Demidovich and Kochurov [187], we state some conjectures about structural formulas for polynomials that solve problems (P1 )–(P3 ). In problem (P1 ), the structural formula for Tn ( · ) (the extended monic polynomial of order n of least C-deviation from zero on the canonical interval [−1, 1]) can be written in the following form:      e0 (−1) · · · en (−1)   e0 (−1) · · · en (−1)        e0 (−1) · · · en (−1)   e  (−1) · · · en (−1)     0  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  (n−2)    (n−2) e e(n−2) (−1) · · · e(n−2) (−1) (−1) n n  0(n−1) (−1) · · · e(n−1)  0 e   e0 (+1) · · · en (+1)   0 (−1) · · · en (−1)    e (t) · · · e (t)   e0 (t) · · · en (t)  0 n Tn (t) = Θ0   +...  + Θ1   e0 (−1) · · · en−1 (−1)   e0 (−1) · · · en−1 (−1)        (−1)   e  (−1) · · · e  (−1)   e0 (−1) · · · en−1 n−1   0   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    (n−2)  (n−2) e(n−2) (−1) · · · e(n−2) (−1) e    0 n−1 (−1) n−1  0(n−1) (−1) · · · e(n−1)  e0 (+1) · · · en−1 (+1)  e (−1) · · · en−1 (−1) 0

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   e0 (+1) · · · en (+1)      e0 (+1) · · · en (+1)    . . . . . . . . . . . . . . . . . . . . . .  (n−2)  (n−2) e (+1) n  0(n−1) (+1) · · · e(n−1) e   0 (+1) · · · en (+1)  e (t) · · · e (t)  0 n + . . . + Θn  ,  e0 (+1) · · · en−1 (+1)     (+1)   e0 (+1) · · · en−1   . . . . . . . . . . . . . . . . . . . . . .  (n−2)  (n−2) e  n−1 (+1)  0(n−1) (+1) · · · e(n−1) e (+1) · · · en−1 (+1) 0 n are some constants. where {Θk }k=0 It is unclear at present how to find such constants. In turn, in problem (P2 ), the structural formula for Un ( · ) (which is the extended monic polynomial of order n of L 1 -least deviation from zero on the canonical interval [−1, 1] can be written in the following form:

     e0 (−1) · · · en (−1)   e0 (−1) · · · en (−1)          e0 (−1) · · · en (−1)   e (−1) · · · en (−1)     0  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  (n−2)   (n−2)  (n−2) (n−2) e e  (−1) n  0(n−1) (−1) · · · e(n−1)  0 (−1) · · · en (−1) e     0 (−1) · · · en (−1)  e0 (+1) · · · en (+1)   e (t) · · · e (t)   e0 (t) · · · en (t)  0 n Un (t) = Υ0   +...  + Υ1   e0 (−1) · · · en−1 (−1)   e0 (−1) · · · en−1 (−1)        (−1)   e  (−1) · · · e  (−1)   e0 (−1) · · · en−1 n−1   0   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    (n−2)  (n−2) e(n−2) (−1) · · · e(n−2) (−1) e    0 n−1 (−1) n−1  0(n−1) (−1) · · · e(n−1)  e0 (+1) · · · en−1 (+1)  e (−1) · · · en−1 (−1) 0    e0 (+1) · · · en (+1)       e0 (+1) · · · en (+1)    . . . . . . . . . . . . . . . . . . . . . .  (n−2)  (n−2) e (+1) n  0(n−1) (+1) · · · e(n−1) e   0 (+1) · · · en (+1)  e (t) · · · e (t)  0 n + . . . + Υn  ,  e0 (+1) · · · en−1 (+1)     (+1)   e0 (+1) · · · en−1   . . . . . . . . . . . . . . . . . . . . . .  (n−2)  (n−2) e  n−1 (+1)  0(n−1) (+1) · · · e(n−1) e (+1) · · · en−1 (+1) 0

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n where {Υk }k=0 are some constants. It is also unclear how to evaluate the constants n {Υk }k=0 . Finally, in problem (P3 ), the structural formula for G n ( · ) (the extended monic polynomial of order n of least L 2 -deviation from zero) on the canonical interval [−1, 1] can be written as follows:      e0 (−1) · · · e n (−1)   e0 (−1) · · · e n (−1)       e  (−1) · · · e  (−1)   e  (−1) · · · e  (−1)  n n   0   0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       (n−2)   (n−2) (n−2)   e0  (−1) · · · e (n−2) (−1) (−1) · · · e n (−1) e n   (n−1)   0 (n−1)  e   (−1) · · · e (−1) (+1) · · · e (+1) e n n   0   0  e (t )  e (t ) ··· e n (t )  ··· e n (t )  0 0 G n (t ) = Γ0  + Γ  + Γ2  1   e0 (−1) · · · e n−1 (−1)   e0 (−1) · · · e n−1 (−1)       e  (−1) · · · e  (−1)   e  (−1) · · · e  (−1)    0   0 n−1 n−1     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   (n−2)   (n−2) (n−2) (n−2)  e  e (−1) · · · e (−1) (−1) · · · e (−1) n−1 0 n−1     0(n−1) (n−1)  e0 (+1) · · · e n−1 (+1)   e (−1) · · · e (−1) 0 n−1

   e0 (−1) · · · e n (−1)    e  (−1) · · · e  (−1) n   0 . . . . . . . . . . . . . .      e0 (+1) · · · e n (+1)     e (+1) · · · e n (+1)   0  e (t ) · · · e (t )  0

n

   e0 (−1) · · · e n−1 (−1)    e  (−1) · · · e  (−1)   0 n−1   . . . . . . . . . . . . . . . .    e0 (+1) · · · e n−1 (+1)    e  (+1) · · · e  (+1) 0

n−1

   e0 (+1) · · · e n (+1)     e  (+1) · · · e  (+1)  n   0 . . . . . . . . . . . . . . . . . . .     (n−2)   e0 (+1) · · · e (n−2) (+1) n   (n−1) (n−1)  e (+1) · · · e (+1) n   0  e (t ) ··· e n (t )  0 + · · · + Γn  ,  e0 (+1) · · · e n−1 (+1)     e  (+1) · · · e  (+1)    0 n−1   . . . . . . . . . . . . . . . . . . .   (n−2) (n−2)  e (+1) · · · e (+1) n−1   0(n−1) e (+1) · · · e (n−1) (+1) 0

n−1

n are some constants. where {Γk }k=0 n for the corresponding extended monic polynomial of The constants {Γk }k=0 order n of least L2 -deviation from zero on the canonical interval [−1, 1] in an arbitrary Chebyshev space (different from Pn ) can be evaluated as follows. It is known that every Chebyshev system is linearly independent. Moreover, for every linearly independent system on the interval [−1, 1], every monic polynomial G n ( · ) of the form (A.2) can be written in the Gram form    ∫1  ∫1 ∫1 ∫1  2 e0 (t) dt e0 (t)e1 (t) dt · · · e0 (t)en−1 (t) dt e0 (t)en (t) dt    −1  −1 −1 −1  1  ∫1 ∫1 ∫1  ∫  2 (t) dt  e1 (t)e0 (t) dt  e · · · e (t)e (t) dt e (t)e (t) dt 1 n−1 1 n   1  −1  −1 −1 −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    ∫1  ∫1 ∫1 ∫1   2  en−1 (t)e0 (t) dt en−1 (t)e1 (t) dt · · · en−1 (t) dt en−1 (t)en (t) dt    −1 −1 −1 −1    e1 (t) ··· en−1 (t) en (t) e0 (t)   ;  ∫1  ∫1 ∫1   2 e (t) dt e0 (t)e1 (t) dt · · · e0 (t)en−1 (t) dt    −1 0  −1 −1  1  ∫1 ∫1  ∫  2 (t) dt  e1 (t)e0 (t) dt  e · · · e (t)e (t) dt 1 n−1   1  −1  −1 −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    ∫1  ∫1 ∫1   2  en−1 (t)e0 (t) dt en−1 (t)e1 (t) dt · · · en−1 (t) dt    −1

−1

−1

this representation is a simple consequence of the corresponding orthogonal decomposition.

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Using this formula and the last expression for G n ( · ) and equating the coefficients multiplying ek ( · ), k = 0, 1, . . . , n, we get a nondegenerate system of linear equations n . with respect to the unknowns {Γk }k=0

A.3 Further Results and Inequalities for Derivatives Given an (n + 1)-dimensional subspace Ln+1 of C n [a, b] that is an ECT-space of order n on [a, b] and given a basis for Ln+1 n {ek ( · )}k=0 ∈ C n [a, b]

(A.12)

that is an ECT-system of order n on [a, b], one can introduce the operation of generalized differentiation of order k, 0 ≤ k ≤ n, as follows: Dk : C n [a, b] → C n−k [a, b].

(A.13)

This operation extends the classical kth derivative operation — here the kth generalized derivative of a function x( · ) ∈ C n [a, b] is defined by Dk x( · ) =

W(e1 ( · ), . . . , ek ( · ), x( · )) , W(e1 ( · ), . . . , ek ( · )) D0 x( · ) = x( · ),

1 ≤ k ≤ n,

(A.14)

where W( · ) is the corresponding Wronskian. In [188], the problem of maximization on an ECT-space of the generalized derivative at a point τ with constraint on a function in the C-metric on a given interval was considered. This problem can be formally written as Dk x(τ) → max, τ ∈ [a, b],  x( · )C([a,b ]) ≤ 1, [a , b] ⊂ [a, b], x( · ) ∈ Ln+1 . In the case τ ∈ [a, b]\[a , b], [a , b] ⊂ [a, b],



(P 1 )

(A.15)

it was shown in [188] that as a solution of this extrapolation extremal problem one can take the value of the kth generalized derivative at the point τ of the extended polynomial of order n of least C-deviation from zero, as constructed for the interior interval [a , b] and taken with the corresponding normalized factor. Remark A.1 In the space of polynomials Pn , the corresponding estimate in the case [a , b] = [a, b] reads as (n2 )(n2 − 12 )(n2 − 22 ) · · · (n2 − (k − 1)2 ) 2k (2k − 1) · (2k − 3) · · · 5 · 3 · 1 (b − a)k ∀pn ( · ) ∈ Pn . (A.16) × pn ( · ) C[a,b]

p(k) n ( · ) C[a,b] ≤

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This inequality, obtained by the brothers A. A. and V. A. Markov, is sharp and becomes an equality for Chebyshev polynomials. Similarly, one can pose the problem of maximization on an ECT-space of the value of the generalized derivative at a point τ subject to the condition that the function satisfy an L p -constraint on a given interval. This problem can be formally written as 

Dk x(τ) → max,

(P 2 )

τ ∈ [a, b], [a , b] ⊆ [a, b],  x( · ) L p [a,b ] ≤ 1, x( · ) ∈ Ln+1 .

A.4 Some Facts from Convex Analysis The above problems can be treated using the machinery of convex analysis. We recall some fundamental facts from this theory (for more details, see [544], [189]). Assume that X and Y are dual with respect to the bilinear form · , ·  : X ×Y → R. x ∈ X, and let f ( x ) be finite. The set (possibly empty) Let f : X → R, ∂ f ( x ) = {y ∈ Y | f (x) − f ( x ) ≥ x − x, y

∀ x ∈ X}

is called the subdifferential of f at x . The elements of ∂ f ( x ) are called subgradients of the function f at the point x . If a function f is in addition differentiable at x , then ∂ f ( x ) is a singleton, which is f ( x ) (the derivative of f at this point). Theorem A.1 (Fermat’s theorem for convex functions, subdifferential form) Let f : Rn → R be a convex function. A point x ∈ Rn is a point of minimum of f if and only if 0 ∈ ∂ f ( x ). (A.17) The next result is concerned with the subdifferential of a maximum for convex functions. Theorem A.2 (Dubovitskii–Milyutin) Let convex functions fi : Rn → R, 1 ≤ i ≤ m, be continuous at a point x and let f1 ( x ) = . . . = fm ( x ). Then ∂ max( f1, . . . , fm )( x ) = conv

m

∂ fi ( x ),

i=1

where as before, conv M is the convex hull of a set M. We next recall the decomposition lemma (Lemma 6.5). Let a function F : [a, b] × Rn → R be such that (a) the function x → F(t, x) is convex on Rn for all t ∈ [a, b]; (b) the function t → F(t, x) is continuous on [a, b] for all x ∈ Rn ;

(A.18)

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(c) M = inf x ∈Rn maxt ∈[a,b] F(t, x) > −∞. Then there exist a natural number m ≤ n + 1 and points τi ∈ [a, b], 1 ≤ i ≤ m, such that M = infn max F(τi, x). x ∈R 1≤i ≤m

Theorem A.3 (Tikhomirov–Magaril-Il’yaev) Let a function F : [a, b] × Rn → R satisfy conditions a), b) of the decomposition lemma. Then x ∈ Rn is a point of minimum for the function x → f (x) = max F(t, x) t ∈[a,b]

(A.19)

if and only if there exist a natural number m ≤ n+ 1, points τi ∈ [a, b], vectors m x ), and numbers αi > 0, i = 1, . . . , m, i=1 αi = 1, such that yi ∈ ∂Fx (τi, m 

αi yi = 0,

(A.20)

i=1

and moreover, f ( x ) = F(τi, x ),

i = 1, . . . , m.

(A.21)

This result will be proved later, in § B.2.

A.5 Tests for Least Deviation from Zero for Extended Polynomials from Chebyshev Spaces n Let Ln+1 be an ECT-space of order n in C n [a, b] and let {ek ( · )}k=0 ∈ C n [a, b] be a basis for Ln+1 that is an ECT-system of order n on [a, b]. Recall that we introduced the family of extended monic polynomials of order n in Ln+1 : n−1    Φ = ϕn ( · ) = en ( · ) + xk ek ( · ) , k=0 n−1 (xk )k=0

where the coefficients are arbitrary. Using (A.9)–(A.11), we formalized the problems on extended nth-order polynomials of least deviation from zero in the metrics C, L 1 , and L 2 as convex extremal problems (P1 )–(P3 ), which are well known to possess a unique solution, provided that the original space is smooth and the ECT-space is finite-dimensional. Recall the following tests of least deviation from zero for extended polynomials in the metrics C, L 1 , and L 2 . Theorem A.4 (On zeros of extended polynomials of least deviation from zero) For each of the metrics C, L 1 , L 2 , all zeros of the corresponding extended monic polynomial of order n of least deviation from zero on the interval [a, b] are real, distinct, and lie on the interval (a, b).

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With the help of this theorem and the above facts from convex analysis, for each of the extremal problems (P1 )–(P3 ), one can formulate conditions under which ϕˆ n ( · ) = en ( · ) +

n−1 

xk ek ( · ) ∈ Φ ⊂ L N +1

(A.22)

k=0

is the corresponding extended polynomial of order n of least deviation from zero. Namely, the following result holds. Theorem A.5 Test (for a solution of problem (P1 )) A necessary and sufficient condition that an extended polynomial ϕˆ n ( · ) = en ( · ) +

n−1 

xk ek ( · ) ∈ Φ

(A.23)

k=0

be a solution of problem (P1 ) is that there exist points τ1 < · · · < τn < τn+1 ∈ [a, b] and numbers n+1 {αi }i=1 > 0,

such that

n+1  

n+1 

(A.24)

αi = 1

(A.25)

j = 0, 1, · · · , n − 1,

(A.26)

i=1

αi sgn ϕˆ n (τi ) e j (τi ) = 0,

i=1

and moreover, | ϕˆn (τi )| =  ϕˆn ( · ) C ,

j = 0, 1, · · · , n − 1.

(A.27)

Remark A.2 The system of equalities (A.26) is equivalent to the system n+1  

αi sgn ϕˆ n (τi ) ϕ(τi ) = 0

∀ ϕ( · ) =

i=1

n−1 

xk ek ( · ) ∈ Ln−1,

(A.28)

k=0

which is commonly known as the principal identity for problem (P1 ). Taking this into account, Theorem A.5 can be reformulated as the following generalized Chebyshev criterion: among all extended monic polynomials of order n, an extended polynomial (A.24) is a polynomial of least C-deviation on the interval [a, b] if and only if the maximum of its absolute value is attained at n + 1 different points at which the sign alternates (such points are commonly known as points of a Chebyshev alternant). For problem (P2 ), the following result holds. Theorem A.6 (Test for a solution of problem (P2 )) A necessary and sufficient condition that an extended polynomial

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ϕˆ n ( · ) = en ( · ) +

n−1 

xk ek ( · ) ∈ Φ

(A.29)

k=0

be a solution of problem (P2 ) is that ∫ b sgn ϕˆ n (t)e j (t) dt = 0,

j = 0, 1, · · · , n − 1.

a

(A.30)

Remark A.3 According to Theorem A.6, all zeros of the extended polynomial (A.29) are real, distinct, and lie in the interval (a, b). If theses zeros {ξ j } nj=1 are ordered a < ξ1 < ξ2 < · · · < ξn < b,

(A.31)

and if we set a =: ξ0,

b =: ξn+1,

then (A.30) can be written as equalities ∫ ξi+1 n  (−1)n−i e j (t) dt = 0,

j = 0, · · · , n − 1.

ξi

i=0

(A.32)

(A.33)

The system of equalities (A.33) is equivalent to the system n 

(−1)n−i

i=0

∫

ξi+1

ξi

ϕ(t) dt = 0

∀ϕ( · ) =

n−1 

xk ek ( · ) ∈ Ln−1,

(A.34)

k=0

which is known as the principal identity for problem (P2 ). For problem (P3 ), the following result holds. Theorem A.7 (Test for a solution of problem (P3 )) A necessary and sufficient condition that an extended polynomial ϕˆ n ( · ) = en ( · ) +

n−1 

xk ek ( · ) ∈ Φ

(A.35)

j = 0, 1, · · · , n − 1.

(A.36)

k=0

be a solution of problem (P3 ) is that ∫ b ϕˆ n (t)e j (t) dt = 0, a

Remark A.4 The system of equalities (A.36) is equivalent to the system ∫ a

b

ϕˆ n (t)ϕ(t) dt = 0

∀ϕ( · ) =

n−1 

xk ek ( · ) ∈ Ln−1,

k=0

which is known as the principal identity for problem (P3 ).

(A.37)

Appendix B

Radon, Helly, and Carathéodory Theorems. Decomposition Theorem

The results given here are auxiliary and can be found in the literature. Nevertheless, we list the ideas and facts that will be used widely throughout the book in order to facilitate its use. We formulate and prove the classical Radon, Helly, and Carathéodory theorems, which are fundamental for convex analysis and have numerous applications in geometry, analysis, and approximation theory.

B.1 Radon, Helly, and Carathéodory Theorems We first note an important combinatorial property discovered by J. Radon in 1921. Theorem B.1 (J. Radon) Every subset of Rn consisting of at least n + 2 points can be subdivided into two nonempty disjoint sets whose convex hulls have at least one common point. m ⊂ Rn (m  n+2) Proof Consider an arbitrary set of points {xi = (xi(1), . . . , xi(n) )}i=1 and consider the homogeneous system of (n + 1) equations m 

ti = 0,

i=1

m  i=1

(j)

xi ti = 0,

j = 1, . . . , n.

This system  has a nontrivial solution (t1, . . . , tm ). Let A = {i | ti  0}, B = {i | ti < 0}, τ = i ∈A ti . Then    ti = −τ and αi xi = (−αi )xi, where αi = ti /τ, i = 1, . . . , m. i ∈B

i ∈A

i ∈B

Theorem B.1 is proved.



From the definition of convexity, it is clear that the intersection of a family of convex sets is a convex set (possibly empty). Helly’s theorem provides a condition for which this intersection is nonempty. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. TsarŠkov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2

471

472 B Basic Facts About Structural Formulas for Extended Polynomials of Least Deviation . . .

Theorem B.2 (E. Helly) Let K be a family consisting of at least (n + 1) convex sets in Rn . Assume that this family either is finite or is a family of compact sets. Then if each (n + 1) sets of this family have nonempty intersection, then all sets from this family have a common point. Proof Consider the case of a finite family K . We argue by induction on the number m of elements in the family. 1◦ . For m = n + 1, the result is clear. 2◦ . Assume that the conclusion of the theorem holds for m  n + 1. Consider the family of convex sets K of cardinality m + 1  n + 2. By the induction step, for each set A ∈ K , there exists a point x A lying in the intersection of all sets of the family K \ { A}, and by Radon’s theorem, there exists a partition of the family K into two nonintersecting families M and N such that the convex hulls of, respectively, the sets {x A } A∈M and {x A } A∈N have nonempty intersection. Further, the convex hull of the first set lies in the intersection of all elements of N , and the convex hull of the second set lies in the intersection of all elements of M . Hence the intersection of all sets in K is nonempty. To complete the proof of Helly’s theorem, we need two well-known auxiliary results. Recall that a system of sets {Mα | α ∈ A} is said to be centred if the intersection of every finite collection of elements of the system is nonempty. The first simple result gives an equivalent (dual) definition of a compact set. Proposition B.1 A set X is compact if and only if every family of its closed subsets with empty intersection contains a finite subfamily with empty intersection.  Proof Necessity. Let X be compact and let {Fα | α ∈ A} be an arbitrary family of closed sets in X such that ∩Fα = . Then by De Morgan’s laws,1 the family Uα := X \ Fα forms an open cover of X. Since X is compact, there exists a finite family nX. Applying again De Morgan’s

n laws, we have

nUα1 , . . . , Uαnn that also covers Fαk = k=1 (X \ Fαk ) = k=1 Uαk = X, which gives k=1 Fαk = . X \ k=1 Sufficiency. Let {Uα | α ∈ A} be an arbitrary open cover of X. It is clear that the family {Fα | α ∈ A}, where Fα := X \ Uα , is a family of subsets closed in X with empty intersection, which by the condition contains a finite subset {Fα1 , . . . , Fαn }

n Fαk = . It is clear that Uα1 , . . . , Uαn forms a cover of X.  such that k=1 Proposition B.2 A set X is compact if and only if every centred system of closed subsets of X have nonempty intersection.  Proof (of Proposition B.2) Necessity. Let S = {Fα | α ∈ A} be an arbitrary centred system of sets that are closed in X. Then the intersection α∈ A Fα =  is 1 De Morgan’s laws in the algebra of sets read as 



 A\ Bβ = (A \ B β ), A \ Bβ = (A \ Bβ ). β

β

β

β

B Basic Facts About Structural Formulas for Extended Polynomials of Least Deviation . . . 473

nonempty, for otherwise, since X is compact and from Proposition B.1, there would exist a finite subsystem with empty intersection, but this is impossible, because S is a centred system. Sufficiency. Let {Fα | α ∈ A} be an arbitrary family of closed sets in X with empty intersection. Then this family contains a finite subsystem with empty intersection, because otherwise, the family {Fα } would be a centred family, which by the assumption would have nonempty intersection. So every family of closed sets in X with empty intersection contains a finite subfamily with empty intersection. By Proposition B.1, X is compact.  Let us complete the proof of Helly’s theorem. 3◦ . We proved the conclusion of the theorem for finite systems. As a result, the infinite family of closed convex sets from the hypothesis of the theorem is centred. A closed set in Rn is compact. Now the conclusion of Helly’s theorem follows from Proposition B.2. 

B.2 Decomposition Theorem By decomposition theorems one means necessary conditions that allow one to narrow the domain of optimization of a given functional without impairing its optimal value. As a simplest example of such a result we mention Fermat’s necessary condition for an extremum: in searching for an extremum, one should not go through all values of the variables, but rather consider only those at which the derivative is zero. As decomposition theorems one can also mention Chebyshev’s and de la Vallée Poussin’s alternation theorems. The proof of the decomposition theorem is based on Helly’s theorem. In the variant of the decomposition theorem that follows, we are concerned with minimization of convex functions defined as a pointwise maximum of convex functions. Theorem B.3 (Decomposition theorem; see [544]) Let T be a Hausdorff compact set, X a finite-dimensional space, card T ≥ dim X+1, and let a function F : T ×X → R be such that: (a) the mapping F(t, · ) : X → R is convex for each t ∈ T; (b) the mapping F( · , x) : T → R is upper semicontinuous for each x ∈ X; (c) M := inf max F(t, x) > −∞. x ∈X t ∈T

Then there exist r ∈ N, r ≤ dim X + 1, and points τ1, . . . , τr ∈ T, such that M = inf max F(τi, x). x ∈X 1≤i ≤r

Remark B.1 Note that every function on a finite set is continuous. Hence if a set T of finite, then the function F(t, x) is always continuous with respect to t. In this case, the decomposition theorem is stated as follows: from a finite set of convex functions

474 B Basic Facts About Structural Formulas for Extended Polynomials of Least Deviation . . .

Fk (x) , k = 1, . . . , N, one can take at most d + 1 (d = dim X) functions Fk1 , . . . , Fkn such that min max Fk (x) = min max Fki (x). x ∈R d k=1,..., N

x ∈R d i=1,...,n

Remark B.2 The decomposition theorem has great value in approximation theory. It says that for approximation of a function on an interval by linear combinations of several functions (for example, polynomials), it suffices to approximate it at only a finite number of points from that interval. Proof (of Theorem B.3) It is clear that M < +∞. With every τ = (τ1, . . . , τn−1 ) ∈ T n+1 we associate the number m(τ) := inf

max F(τi, x)

x ∈X 1≤i ≤n+1

and define m := supτ ∈T n+1 m(τ). It is clear that m ≤ M. Let us prove the converse inequality. Let ε > 0. Given t ∈ T, consider the set A(t) := {x ∈ X | F(t, x) ≤ m+ε}. For t, the set A(t) is nonempty, closed, and convex. x = x(ξ) ∈ X Next, let ξ = (ξ1, . . . , ξn ) ∈ T n . It is clear that there exists an element  such that F(ξi, x(ξ)) < m + ε, i = 1, . . . , n. We set V(ξ) := ξ  = (ξ1, . . . , ξn ) ∈  T n | F(ξi, x(ξ)) < m + ε, i = 1, . . . , n . Since the function F( · , x(ξ)) is upper semicontinuous, V(ξ) is an open set (containing ξ). From the open cover {V(ξ)}ξ ∈T n of the compact set T n , we choose a finite subcover and let it correspond to points := conv{x(ξ 1 ), . . . , x(ξ s )}. It is clear that A is a convex ξ 1, . . . , ξ s . Consider the set A compact set. has We claim that every collection of n + 1 sets from the family { A(t)}t ∈T ∪ { A} then the result nonempty intersection. Indeed, if among these sets there is no set A, where η = (η1, . . . , ηn ) ∈ T n . is clear. Now consider the sets A(η1 ), . . . , A(ηn ), A, i Since η ∈ V(ξ ) for some 1 ≤ i ≤ s, this means that F(η j , x(ξ i )) < m + ε,

j = 1, . . . , n,

i.e.

x(ξ i ) ∈

n 

A(η j ).

j=1

and hence the family A(η1 ), . . . , A(ηn ), A has nonempty intersection. But x(ξ i ) ∈ A, As a result, by Helly’s theorem, A ∩t ∈T A(t)  . It follows that M ≤ m + ε, and therefore M = m. The function τ = (τ1, . . . , τn+1 ) → max1≤i ≤n+1 F(τi, x) is upper semicontinuous on T n+1 for each x ∈ X. Hence so is the function τ → m(τ) := inf

max F(τi, x)

x ∈X 1≤i ≤n+1

on T n+1 . Consequently, it attains its maximum point at some point τ = ( τ1, . . . , τn+1 ). Retaining the unequal coordinates, we get the result of the theorem.  The convex hull of a set consists of all possible convex combinations of elements of the set. It turns out that in the finite-dimensional space Rn , this result can be substantially improved.

B Basic Facts About Structural Formulas for Extended Polynomials of Least Deviation . . . 475

Theorem B.4 (C. Carathéodory) If A is a nonempty subset of Rn , then every point from the convex hull conv A of A can be represented as a convex combination of at most n + 1 points from A. In other words, conv A =

 n

 n+1    λi xi  λ ≥ 0, λi = 1, xi ∈ A, 1 ≤ i ≤ n + 1 .

i=1

i=1

Proof Let A be a nonempty subset of Rn and let y be a point from its convex hull. m There exist points (xi ),im= 1, . . . , m, and positive numbers (αi ), i = 1, . . . , m, i=1 αi = 1, such that y = i=1 αi xi . We assume that m is the smallest of all such possible numbers. Assume that m  n + 2. Then, as in the proof of Radon’s theorem, m βi = 0 thereexists a nontrivial family of numbers (βi ), i = 1, . . . , m, such that i=1 m βi xi = 0. In the index set A = {i | βi < 0} there exists an index j ∈ A and i=1 α α such that β jj = maxi ∈A αβii , and hence αi − βi β jj  0 for all i ∈ A. Then m αj  βi xi = 0 β j i=1

and

y=

m   i=1

αi − βi

 αj xi, βj

where all coefficients in the second expression are nonnegative and their sum is 1, while the jth coefficient is zero; that is, y lies in the convex hull of (m − 1) points, which contradicts the minimality of m. Hence m  n + 1. The theorem is proved. 

Appendix C

Some Open Problems

In this section, we recall some open problems in geometric approximation theory. The most important open problem is the following: ◦ (Efimov–Stechkin–Klee problem) Prove or disprove that in an infinite-dimensional Hilbert space, every Chebyshev set is convex. (See Chap. 6). A closely related problem to the Efimov–Stechkin–Klee problem is the unique farthest point problem (UFPP): ◦ Must a uniquely remotal set (a max-Chebyshev set) in a Banach space be a singleton? In the Hilbert setting, UFPP is equivalent to the Efimov–Stechkin–Klee problem. A related open problem is as follows: Is every Chebyshev set in a strictly convex reflexive Banach space convex? (J. M. Borwein) The problems given below are still open, and some of them are nearly as old as the Efimov–Stechkin–Klee problem. 1. Characterize the n-dimensional Banach spaces (n ≥ 5) in which every Chebyshev set is convex. 2. Characterize the n-dimensional Banach spaces (n ≥ 4) in which every bounded strict sun (sun, α-sun) is convex. (For strict suns, the answer for n = 2 and 3 is known.) 3. (S. B. Stechkin’s problem) Does there exist an infinite-dimensional space in which every Chebyshev set is convex? 4. Is it true that every Chebyshev set with continuous metric projection is convex in a smooth space? 5. (L. P. Vlasov’s problem) In what spaces is every convex Chebyshev set compact? 6. (L. P. Vlasov’s problem) In what spaces has each convex Chebyshev set a continuous metric projection? 7. (P. A. Borodin’s problem) Does an approximately compact but not boundedly compact set exist in every infinite-dimensional Banach space? 8. (D. Kölzow’s problem) Does there exist a nonconvex Chebyshev set in some infinite-dimensional incomplete pre-Hilbert space? (See Chap. 6). © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. TsarŠkov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2

477

478

C Some Open Problems

9. Is it true that every sun in a three-dimensional normed space is B-connected (or equivalently, P-connected). Characterize the finite-dimensional spaces in which every sun is B-connected. (See §7.2.) 10. Is a B-connected sun in a finite-dimensional space P-cell-like? ˚ 11. Is it true that every strict sun is connected (B-connected, B-connected)? Moreover, the answer here is unknown even for Chebyshev suns! (The proof of this claim proposed in [140] has raised objections.) 12. (I. G. Tsar’kov’s problem) Is it true that a boundedly weakly compact strict sun in nonatomic L 1 (μ) is convex? (For related results, see §8.3.) 13. Suppose that in a space X, every moon is a strict protosun. Is it true that X is strongly nonlunar? (See §7.9.) 14. Characterize spaces in which every Chebyshev set with continuous metric projection is a sun. In particular, in which spaces is every approximatively compact Chebyshev set a sun? The answer is unknown even in Efimov–Stechkin spaces. (See §§10.3, 9.1.) 15. Is it true that a weakly compact Chebyshev set in a Banach space is a sun? (See §10.3.) 16. Characterize the n-dimensional Banach spaces (n ≥ 4) in which every Chebyshev set is monotone path-connected. (The answer for n = 2 and 3 is known.) 17. Is it true that a weakly compact (a boundedly weakly compact) monotone pathconnected set in C(Q) is a sun? (For related results, see §7.7.) 18. Characterize the Banach spaces, even finite-dimensional ones, in which every unimodal set is a strict sun. 19. Does there exist a separable Banach space without nontrivial Chebyshev subspaces? (See §8.2.) 20. When is the set of generalized rational fractions RV,W approximatively compact in L p -spaces? When is this set an existence set in C(Q)? in L p ? In cases in which RV,W is an existence set, it is a strict sun in C(Q), since every RV,W is always a strict protosun in C(Q) (See Chap. 11.) 21. Evaluate the Jung constant of the space n1 for all n (see §14.5). p 22. Find the relative Jung constant Js (pn ) of the space n for all n (see §14.5). 23. Does every weakly compact convex subset C of a Banach space have the fixedpoint property for nonexpansive mappings? This is the same as asking whether the condition in Theorem 14.9 that C have normal structure can be dropped (see §14.3). 24. (L. Veselý’s problem) It is unknown whether there are Banach spaces in which all finite sets, but not all compact sets, admit Chebyshev centers. (See Chap. 15.) 25. (A. L. Garkavi’s problem) Characterize the Chebyshev nets in C(Q) (or equivalently, find an analogue of Theorem 15.4). (See Chap. 15.) 26. Let L1 ⊂ L2 ⊂ . . . be a system of strictly embedded (closed) linear subspaces (not necessarily finite-dimensional) of an infinite-dimensional reflexive Banach space X, let this system be complete in X, and let d1 ≥ d2 ≥ . . . , dn → 0, be a sequence of nonnegative numbers. Is it true that there exists an element x ∈ X such that the best approximations ρ(x, Ln ) to x by the subspaces Ln are equal to dn (i.e., ρ(x, Ln ) = dn , n = 1, 2, . . . )? The answer to the same problem is

C Some Open Problems

27.

28.

29.

30.

479

unknown on the class of Banach spaces (not necessarily reflexive) if the above inequalities are strict. (See Chap. 16). (S. V. Konyagin’s problem) Is is true that for every reflexive space X and nonempty closed set M  X, there exists at least one point x ∈ E(M) \ M? (E(M) := {x ∈ X | PM x  }). (See Chap. 16.) (S. B. Stechkin’s problem) Is it true that U(M) ∈ (II) for every nonseparable strictly convex space X and closed M ⊂ X? In other words, is it true that the set U(M) is the complement of a set of first category in X? An equivalent reformation is as follows: is it true that U(M) contains a dense G δ -subset of X and hence is ‘large’ from the categorical point of view? (See Chap. 16.) Characterize the Banach spaces in which U(M) = X (U(M) is the set of points of uniqueness of the set M) for every compact set M ⊂ X (that is, M is a weakly almost Chebyshev set) or U(M) ∈ (II) (that is, M is an almost Chebyshev set). Is it true that a space with this property is strictly convex? (For a discussion, see Chap. 17). Let R2 (M) = {x ∈ X : |PM x| > 2}. It is not known whether the set R2 (M) can be covered by countably many rectifiable curves for every three-dimensional strictly convex space X and M ⊂ X. (See §17.9).

Most of the above classical open problems can also be formulated for asymmetrically normed spaces. For a background in asymmetric spaces and some advances, see Cobzaş [174].

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Index

Symbols α-sun, 85, 96, 189 β-sun, 189 δ-sun, 106, 189 γ-sun, 190 ε-displacement, 382 H-set, 168 R-strongly convex interval, 131 b-extension of a set, 193 A Acyclic space, 201 Admissible sequence of sets, 293 Alternant, 20, 219 Annihilator, 402 Asymmetric norm, 150, 151 Asymptotic radius (of a net), 328 B Ball hull, 130 Banach–Mazur hull, 130 Best n-net, 300 Best approximant, 1 Best approximation operator, 1 Body, 261 C Cantor property, 266 Cavern, 110 Chebyshev n-net, 300 Chebyshev centre, 264 relative, 297 self, 298 Chebyshev point, 300 Chebyshev radius, 264 relative, 297 self, 298

Chebyshev rank, 257 Chebyshev set, 98 Chebyshev system, 24, 229, 459 Chebyshev-centre map Z( · ), 296 Class of sets (E ), 116 ˚ 116 ( B), (AC), 116 (B), 116 (P), 116 (P0 ), 116 Class of spaces (ES), 167 (Ex-w ∗ s), 133 (KK), 168 (LUR, 419, 423 (LUR), 72 (MIP), 437 (MS), 144, 145 (MeI), 132 (P) (Brown), 140 (QUC), 318 (RED-Y), 319 (URED), 321 (URED-Y), 322 (KK1 ), 170 Complete Chebyshev system, 459 Constant co-projection, 395 projection, 395 Constant error curve, 239 Contingent, 440 Continuous selection, 380 Covering radius, 300 Critical value of a space, 293 CT-system, 459

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. R. Alimov and I. G. TsarŠkov, Geometric Approximation Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-90951-2

505

506 D Defect (of a fraction), 224 Deutsch convergence, 74 Diameter, 296 Distance to a set, 1 E ECT-space, 459 ECT-system, 459 Eikonal equation, 92 ET-space, 459 Existence set, 116 F Face, 10 Family of admissible subspaces, 330 Farthest point, 102, 319 Farthest-point mapping, 102 H Haar cone, 229 proper, 229 Haar system, 24, 229 Hadamard matrix, 276 Hadamard set, 276 Hausdorff distance, 109, 125 one-sided, 125 I Interpolating subspace, 320 Inversion in the unit sphere, 103 IRL-continuity, 127 J Jung constant, 263 K Klee cavern, 110 Kolmogorov set, 82 L LG-set, 81, 143, 144 Luminosity point, 2, 95 M Map δ-condensing, 272 η-condensing, 272 with property (C), 272 Mapping ε-displacement, 382 Fréchet differentiable, 181, 182 Gâteaux differentiable, 90

Index Hausdorff lower semicontinuous, 125, 204 Hausdorff upper semicontinuous, 125, 204 lower semicontinuous, 117 upper semicontinuous, 117, 342 Max-Chebyshev set, 102 Max-projection (set of farthest points), 436 Max-projection operator, 102 Measure of nonconvexity, 265 with the Cantor property, 266 Meta-sun, 190 Metasun, 109 Metric projection, 1 IRL continuous, 127 norm-τ-upper semicontinuous, 69 ORL continuous, 139 strongly unique, 178, 179 Minimizing sequence, 60 Modulus of convexity, 172, 173 of smoothness, 180, 181 Monotone curve, 134 Moon, 143, 144 N Net of cardinality n, 300 Chebyshev (best), 300 Nonnodal zero, 236 Norm Brown associated, 131, 133, 134 Normal structure coefficient, 271 Number CV-critical, 294 O ORL-continuity, 139 ORU-continuity, 125 P Point diametral, 273 exposed, 348 farthest, 102, 319, 351 luminosity, 2, 95 lunar, 143, 144 nearest, 1 of approximative compactness, 60 quasipolyhedral, 147, 148 regular, 94, 240 singular, 94, 240 smooth (smooth point), 348 solar, 2, 95 strict protosolar, 2

Index strict solar, 2 strongly nonlunar, 145, 146 Point of existence, 60 Pointwise convergence of sequences almost everywhere, 76 Polar, 402 Preannihilator, 402 Prepolar, 402 Problem1of best simultaneous approximation, 296 Property (P1 ), 343 (R1 ), 335 (P2 ), 345 Protosun, 96 R Radius asymptotic, 328 Regular τ-convergence, 67 Relative Jung constant, 285 Root number (of a Haar cone), 229 S Segment, 129, 131, 302 generalized, 130, 303 relative, 130 relative interior, 131 Selection continuous, 380 Lipschitz, 348 Self Chebyshev radius, 267, 285 Semi-sun, 190 Sequence maximizing, 355 minimizing, 60, 355 Set B-connected, 116 P-connected, 116 P0 -connected, 116 ˚ B-connected, 116 m-connected, 130 σ-globally very porous, 435 τ-closed, 69 τ-compact, 69 τ-open, 69 ε-compact, 204 a-convex, 195 a.e.-approximatively compact, 78 acyclic, 201 almost Chebyshev, 448 almost convex, 190 almost uniqueness, 448 angle porous, 435

507 antiproximinal, 14, 65 approximatively τ-compact, 68 approximatively compact, 60 approximatively weakly compact, 60 boundedly τ-compact, 69 boundedly compact, 59 boundedly sequentially τ-compact, 69 boundedly weakly compact, 59 Brosowski–Wegmann-connected, 138 CCF-set, 351 cell-like, 201 centrable, 354 Chebyshev, 2 contractible, 152, 153 diametral, 270 extremum characterizable, 86, 140 globally very porous, 435 interpolational regular, 234 Kolmogorov set, 82 LG-set, 81 locally solvent, 234 Menger connected, 130 monotone path-connected, 134 NCCF-set, 351 nontrivial, 294, 351 of asymptotic centres, 328 of existence, 1, 116 of exponential sums, 75 of points of approximative compactness AC(M), 60, 418 of points of approximative uniqueness TAC(M), 418 of points of existence E(M), 418 of points of existence and uniqueness T(M), 418 of points of nonuniqueness R(M), 418 of relative Chebyshev δ-centres, 343 of Steiner points, 360 of uniqueness, 1 of weak almost uniqueness, 422 points of uniqueness U(M), 418 regular, 138 sequentially weakly closed, 63 strictly monotone path-connected, 134 unimodal, 81, 143, 144 uniquely remotal, 102 varisolvent, 235 weakly almost Chebyshev, 422 with normal structure, 273 with Property Z, 235 with the density property, 235 with unique farthest-point property, 102 Zk -small, 439 Signature, 36

508 extremal, 37 Sobolev class, 408 Solar point, 2, 95 Space E-monotone, 250 U-space, 449 r-strictly convex, 257 (CCF)-space, 351 Bernstein, 415 centrable, 414 Efimov–Stechkin, 167 Kadec, 168 Kadec–Klee, 168, 431 locally uniformly convex, 72, 423 of type p, 359 quasi-uniformly convex, 318 quasipolyhedral, 147, 148 rotund, 12 Schur space, 438 smooth, 95 strictly E-monotone, 250 strictly convex, 12 strictly convex in every direction, 319 strongly convex, 430 strongly nonlunar, 145, 146 T2 -convex, 433 uniformly convex, 64, 172, 173 uniformly convex in every direction, 321 uniformly convex in every direction from Y, 322 uniformly convex in every direction in a subspace Y, 324 uniformly rotund in every direction, 301 uniformly smooth, 181, 182 weakly compact generated, 135 weakly uniformly convex, 321 with Mazur intersection property, 437 with normal structure, 270 with the Cantor property, 266 with the property (Aτ ), 72 Span, 130–132, 303 Steiner centre, 360 Strict protosolar point, 2 Strict protosun, 2, 96, 189 Strict solar point, 2, 96

Index Strict sun, 2, 96, 189 Strictly interpolating subspace, 325 Strong uniqueness constant, 35 Strong uniqueness of the metric projection, 35, 178, 179 Strongly unique element of best approximation, 35 Subspace Haar, 219 interpolating, 320 strictly interpolating, 325 Sun, 2, 189 Support cone, 83 System H-system, 257 almost Markov, 450 centred, 472 Chebyshev, 24, 229 E-Chebyshev system, 251 Haar, 24, 229 interpolation, 25 Markov, 450 with the uniqueness property, 256 T T-space, 459 T-system, 459 W Width Alexandrov, 370 Bernstein, 372, 393 binary, 388 central information Kolmogorov, 393 Dranishnikov–Shchepin, 388 entropy, 369 Fourier, 370 Gelfand, 392 information Kolmogorov width, 391 Kolmogorov, 369, 392 Kolmogorov width, 393 linear, 369, 392 projection, 369, 392 Urysohn, 414