Curvature measures of singular sets 9783030181826, 9783030181833

Citation preview

Springer Monographs in Mathematics

Jan Rataj Martina Zähle

Curvature Measures of Singular Sets

Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK Series Editors Sheldon Axler, San Francisco, USA Mark Braverman, Princeton, USA Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C. Güntürk, New York, USA Claude Le Bris, Marne la Vallée, France Pascal Massart, Orsay, France Alberto A. Pinto, Porto, Portugal Gabriella Pinzari, Padova, Italy Ken Ribet, Berkeley, USA René Schilling, Dresden, Germany Panagiotis Souganidis, Chicago, USA Endre Süli, Oxford, UK Shmuel Weinberger, Chicago, USA Boris Zilber, Oxford, UK

This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

More information about this series at http://www.springer.com/series/3733

Jan Rataj • Martina Zähle

Curvature Measures of Singular Sets

123

Jan Rataj Mathematical Institute Charles University Prague, Czech Republic

Martina Zähle Mathematisches Institut Friedrich-Schiller-Universität Jena Jena, Germany

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-030-18182-6 ISBN 978-3-030-18183-3 (eBook) https://doi.org/10.1007/978-3-030-18183-3 Mathematics Subject Classification (2010): Primary: 53C65, 28A75; Secondary: 28A80, 49Q15, 58A25, 53A17 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved 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, express 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

Preface

The curvature notions studied in this book have some forerunners in the nineteenth century. Carl Friedrich Gauss (1777–1855) and Bernhard Riemann (1826–1866) essentially stimulated the intrinsic versions in differential geometry. In particular, the Gauss curvature and, more generally, the Riemannian curvature tensor, nowadays, play an important role in many applications. Jacob Steiner (1796– 1863) and Hermann Minkowski (1864–1909) started with a parallel development in convex geometry with different tools. There is a long, related history up to now in both fields. The famous milestones are, e.g., the Gauss-Bonnet theorem, the Minkowski-Steiner formula for convex bodies and Weyl’s tube formula for compact smooth submanifolds, the Crofton formula, Blaschke, Santalo, and Chern’s principal kinematic formula, and Hadwiger’s characterization theorem. This concerns the Minkowski quermassintegrals in convex geometry, known as intrinsic volumes, or the integrals of higher-order mean curvatures as differential geometric counterparts, including the scalar curvature. Some background on these topics is summarized in Chaps. 2 and 3. One particular aim of the present book is to demonstrate that these functionals and their measure-geometric extensions fit into the program of Felix Klein (1849– 1925) to study geometric properties that are invariant under certain transformation groups of the underlying space. It turns out that these curvature functionals and measures form a complete system of Euclidean invariants, which are additive, like volume, and continuous in an appropriate sense. In modern algebraic and differential geometry, but also in mathematical physics, the extensions, sometimes, are called Lipschitz-Killing curvatures. We will keep this notation. Geometric measure theory has been developed since the late 1950s at the beginning mainly stimulated by the higher-dimensional Plateau problem and calculus of variations. (A survey on this topic can be found in [HP16].) Until now, Federer’s book [Fed69] is a standard reference for a huge variety of basic notions and relationships. In Chap. 1, we select those which are relevant for our purposes. There, we also refer to the textbook of Krantz and Parks [KP08]. The use of currents with additional properties makes it possible to treat geometric properties of first-order rectifiable sets with powerful analytical and algebraic tools. v

vi

Preface

Subsets of Euclidean spaces, which do not fit into the context of convex or differential geometry, are not only of interest in calculus of variations. Mathematical models for natural phenomena often lead to singular sets, from piecewise smooth submanifolds up to fractals. The first fundamental paper on curvatures, i.e., secondorder properties, of sets with possible singularities was [Fed59]. Therein, Federer unified and extended the abovementioned results from differential geometry and convex geometry for the more general sets with positive reach by methods of geometric measure theory. The curvature measures were introduced implicitly as coefficients of a generalized Steiner-Weyl polynomial. This approach was not included in [Fed69] and was not followed up until the 1980s. We understand the present book as a continuation of [Fed59] and a collection and extension of related investigations from the last 30 years. As a main tool, we use the explicit representation of the generalized Lipschitz-Killing curvatures by integrating certain differential forms over the associated unit normal bundle endowed with a topological index function. This allows us to treat the second-order analysis of the underlying geometric sets with tools of the first-order current theory. Except for the mentioned approaches from convex and differential geometry, there are also relationships and possible applications to algebraic geometry, geometric analysis, and mathematical physics. Furthermore, methods from this book have been applied to models in stochastic geometry, in particular to random tessellations, which describe real phenomena. The random sets under consideration can be singular in the sense of Chaps. 4, 5, and 9. We introduce and study the main notions, in particular those of normal cycle and curvature measures, and their relationships for the special cases of sets with positive reach (Chap. 4) and their locally finite unions (Chap. 5). For these classes, the basic definitions and mathematical tools are more accessible and prepare the reader for the study of more general singular sets. (Some extensions and related approaches are discussed in Chap. 9 and the bibliographical notes therein.) The corresponding integral geometric formulas are derived in Chap. 6. One of the main results is the translative integral formula, which leads also to a direct proof of the principal kinematic formula for such sets. Approximations of the curvature measures of singular sets by those of more regular sets in the present book are mainly used for the purposes of Chap. 8. But they are of independent interest for many theoretical and numerical applications. In Chap. 7, we show approximations of the normal cycles, and hence of the curvature measures, by those of parallel sets of small distances as well as of appropriate polytopes. Chapter 8 provides the corresponding characterization theorems for the curvature functionals and measures. For the case of sets with positive reach, this question was already asked by Federer. The problem was an appropriate notion of continuity. An answer for unions of sets with positive reach is given in terms of flat convergence of associated normal cycles. By means of the above approximations, we can reduce the characterization to the convex cases considered by Hadwiger [Had57] and Schneider [Sch78]. It is shown that the Lipschitz-Killing curvature functionals (or

Preface

vii

measures) form a basis for all Euclidean motion invariant continuous (measurevalued) valuations. This gives the interpretation in the spirit of Felix Klein’s ideas. Extensions of normal cycles and curvature measures to more general classes of singular sets are presented in Chap. 9. Moreover, a principal kinematic formula is proved, which includes that for sets with positive reach. Note that Chaps. 4, 5, 6, 7, 8 and 9 contain the full proofs of the statements, except for some basic tools from the literature. In Chap. 10, we leave the context of geometric measure theory and present some applications to fractal geometry, i.e., to much more irregular sets. Due to a result of Fu [Fu85], for arbitrary compact sets in Euclidean spaces of dimensions less than four, the closure of the complement to parallel sets of almost all distances has positive reach. In higher dimensions, this property is called tube regularity. By a reflection principle, it allows to introduce the curvature measures for parallel sets with this property, in particular, for those in certain classes of fractals. In the recent literature, appropriate convergences of the rescaled curvature measures for their parallel sets of small distances have been shown. The limits are interpreted as fractal curvature measures. This leads to new parameters for describing the geometry of such fractals. Some of these results with corresponding references are presented in this chapter, partially in a form of a survey. The Minkowski content is included as a special case. The latter has been studied by different authors with various methods. We are extremely grateful to Andreas Bernig, Ulrich Menne, Werner Nagel, Dušan Pokorný, Thomas Wannerer, Steffen Winter, and Ludˇek Zajíˇcek, who read parts of earlier versions of the manuscript and made valuable comments. We also thank Christopher Schneider for contributing the computer graphics in Chap. 10. Finally, we acknowledge the support of the Czech and the German Science ˇ and DFG, by some research grants (in particular, GACR ˇ 18– Foundations, GACR 11058S and DFG/ZA 242/5) on related topics during the work on this book. Prague, Czech Republic Jena, Germany

Jan Rataj Martina Zähle

Contents

1

Background from Geometric Measure Theory. . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Area, Coarea and Rectifiability .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Hausdorff Measure . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.2 Densities of Sets . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.3 Lipschitz Mappings.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.4 Tangent Cones . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.5 Approximate Differential . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.6 Rectifiable Sets . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.7 Purely Unrectifiable Sets . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Multilinear Algebra and Differential Forms . . . .. . . . . . . . . . . . . . . . . . . . 1.2.1 Multilinear Algebra.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.2 The Grassmannian .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.3 Multivector Fields and Differential Forms . . . . . . . . . . . . . . . . 1.3 Currents .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.2 Normal and Rectifiable Currents. . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.3 Product of Currents .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.4 Push-Forward . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.5 Constancy Theorems . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.6 Integral Currents and Compactness .. . .. . . . . . . . . . . . . . . . . . . . 1.3.7 Slicing of Currents.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.8 Indecomposable Integral Currents . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Computations on Grassmannians .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 1 3 4 7 7 8 10 11 11 15 17 19 19 21 22 23 25 28 29 30 31

2

Background from Convex Geometry .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Intrinsic Volumes .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Curvature and Area Measures . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Integral-Geometric Formulas . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Polar Cones .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Simplicial Complexes .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

37 37 39 41 42 42

ix

x

Contents

3

Background from Differential Geometry and Topology . . . . . . . . . . . . . . . 3.1 Curvature Measures of Smooth Submanifolds ... . . . . . . . . . . . . . . . . . . . 3.2 Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Integral Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Morse Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

47 47 50 51 51

4

Sets with Positive Reach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Geometric Properties .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Reach of Intersection .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Unit Normal Bundle.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Principal Curvatures and Directions .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Curvature Measures and Steiner Formula .. . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Normal Cycle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7 Basic Properties of Curvature Measures . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.8 Gauss-Bonnet Formula .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.9 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

55 55 63 66 68 71 77 80 83 85

5

Unions of Sets with Positive Reach . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 87 5.1 Topological Index Functions and Additive Extension of Normal Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 87 5.2 Curvature Measures, Generalized Principal Curvatures and Steiner Formula .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 93 5.3 Regular UPR -Representations and the Reflection Principle for Normal Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 97 5.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 102

6

Integral Geometric Formulas . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Translative and Kinematic Integral Formulas for Curvature Measures .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Local Representation of Mixed Curvature Measures .. . . . . . . . . . . . . . 6.3 Absolute Curvature Measures . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

106 118 128 136

7

Approximation of Curvatures .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Approximation by Parallel Sets. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Polytopal Approximation . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

139 139 148 157

8

Characterization Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Characterization of Lipschitz-Killing Curvatures.. . . . . . . . . . . . . . . . . . 8.2 Characterization of Associated Measures .. . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

159 159 164 169

9

Extensions of Curvature Measures to Larger Set Classes . . . . . . . . . . . . . 9.1 Legendrian Cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Normal Cycles .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Lipschitz Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

171 172 179 185 206

105

Contents

10 Fractal Versions of Curvatures.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Self-Similar Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3 (Average) Minkowski Content and Localized Versions . . . . . . . . . . . . 10.3.1 Characterization in Terms of Surface Area .. . . . . . . . . . . . . . . 10.3.2 Densities and Measures via Ergodic Theory . . . . . . . . . . . . . . 10.3.3 Total Values, Minkowski Measures and the Renewal Theorem . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.4 Relationships to Fractal Zeta Functions . . . . . . . . . . . . . . . . . . . 10.4 Extension to Fractal Curvature Measures . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4.1 Application of Renewal Theory .. . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4.2 Methods from Ergodic Theory: Curvature Densities and Measures . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

xi

209 209 217 223 223 224 226 234 235 235 239

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 245 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 251 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 255

Chapter 1

Background from Geometric Measure Theory

The role of the first chapter is to collect the notions and results from geometric measure theory which will be needed in the sequel. Most definitions and results from this chapter can be found in the Federer’s book [Fed69] and/or in the book of Krantz and Parks [KP08]. Other sources will be cited when needed. Most of these results are presented without proofs. The basic setting is the d-dimensional Euclidean space, Rd , with standard scalar √ ˚ r)) we denote product x · y and norm |x| := x · x, x, y ∈ Rd . By B(x, r) (B(x, d the closed (resp. open) ball with centre x ∈ R and radius r ≥ 0. Given an affine subspace L of Rd , pL denotes the orthogonal projection onto L and L⊥ stands for the orthogonal complement (as a linear subspace) to L. The Lebesgue measure in Rd is denoted by Ld . All frequently used notations can be found in the List of symbols. We assume that the reader is familiar with the basic linear algebra, analysis and measure theory.

1.1 Area, Coarea and Rectifiability 1.1.1 Hausdorff Measure The Hausdorff measures are defined as outer measures on Rd . An outer measure on a nonempty set X is a set function μ defined on the family P(X) of all subsets of X with values in [0, ∞] with the properties: (i) μ(∅) = 0, (ii) A ⊂ μ(A) ≤ μ(B),  B implies  ∞ (iii) μ( ∞ i=1 Ai ) ≤ i=1 μ(Ai ).

© Springer Nature Switzerland AG 2019 J. Rataj, M. Zähle, Curvature Measures of Singular Sets, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-18183-3_1

1

2

1 Background from Geometric Measure Theory

To an outer measure μ, the system of μ-measurable sets is assigned: Aμ := {A ⊂ X : μ(E) = μ(E ∩ A) + μ(E \ A) whenever E ⊂ X}. Theorem 1.1 ([KP08, §1.2]) Aμ is a σ -algebra and the restriction of μ to Aμ is a (σ -additive) measure. There is a plenty of outer measures whose σ -algebra is very poor or even trivial. There exists, however, a simple criterion assuring that the σ -algebra is rich enough. An outer measure μ on a metric space (X, ρ) is called metric if μ(A ∪B) = μ(A)+ μ(B) whenever dist (A, B) := inf{ρ(a, b) : a ∈ A, b ∈ B} > 0. Theorem 1.2 (Carathéodory’s criterion, [KP08, Theorem 1.2.13]) If μ is a metric outer measure on a metric space (X, ρ) then Aμ contains all Borel sets. A metric outer measure μ on X is said to be Borel regular if to any A ⊂ X there exists a Borel set B ⊃ A such that μ(A) = μ(B). Let s ≥ 0. Denote ωs =

π s/2 Γ ( 2s + 1)

(note that if s is an integer then ωs is the volume of the unit ball in Rs ). We define the s-dimensional Hausdorff measure in Rd as Hs (A) = lim

inf 



δ→0+ A⊂ i Gi diam Gi ≤δ i

 ωs

diam Gi 2

s .

(The infimum above is taken over all finite or countable coverings of A with (arbitrary) subsets G1 , G2 , . . . of Rd of diameters at most δ.) Proposition 1.3 ([Fed69, Section 2.10]) For any s ≥ 0, 1. 2. 3. 4. 5. 6.

Hs is a metric outer measure on Rd ; Hs is Borel regular; Hs is translation and rotation invariant; H0 is the counting measure; Hd is the Lebesgue measure (Hd = Ld ); Hs = 0 if s > d.

Note that the definition of Hs would not change if we would consider only coverings by open, or closed, or even compact convex, sets. Covering by balls would, however, produce another measure called spherical measure (though its values on “nice” sets would be the same).

1.1 Area, Coarea and Rectifiability

3

Let A be a subset of Rd . The Hausdorff dimension of A is defined as dimH A := inf{s ≥ 0 : Hs (A) < ∞}. The Hausdorff dimension has the following meaning. Proposition 1.4 ([KP08, Proposition 2.1.2]) If s < dimH A then Hs (A) = ∞. If s > dimH A then Hs (A) = 0. Examples Any nonempty open set in Rd has Hausdorff dimension d. A nonempty m-dimensional C 1 -submanifold has Hausdorff dimension m. Any countable set has Hausdorff dimension 0. 2 The middle-third Cantor set in R1 has Hausdorff dimension log log 3 (see, e.g., [Mat95, §4.10]; cf. Chap. 10). • The trajectory of Brownian motion in Rd has almost surely Hausdorff dimension 2 (nevertheless, its two-dimensional Hausdorff measure vanishes; see, e.g., [MP10, Ch. 4]).

• • • •

Definition 1.5 A set A ⊂ Rd is called Ahlfors s-regular (0 ≤ s ≤ d) if there exist constants c, C > 0 such that cr s ≤ Hs (A ∩ B(a, r)) ≤ Cr s ,

a ∈ A, 0 < r < diam A.

(1.1)

A is said to be upper (lower) Ahlfors s-regular if the upper (lower, respectively) inequality holds.

1.1.2 Densities of Sets Let A be a subset of Rd and a ∈ Rd a point. Let s > 0. Define Θ ∗s (A, a) = lim sup r→0+

Θ∗s (A, a) = lim inf r→0+

Hs (A ∩ B(a, r)) , ωs r s Hs (A ∩ B(a, r)) , ωs r s

the upper and lower s-dimensional density of A at a. If both the upper and lower densities agree we call the common value s-dimensional density of A at a and denote it by Θ s (A, a).

4

1 Background from Geometric Measure Theory

Theorem 1.6 ([Fed69, §2.10.19]) 1. If A ⊂ Rd is Lebesgue measurable then Θ d (A, ·) equals 1 Ld -almost everywhere on A and equals 0 Ld -almost everywhere on the complement of A (Lebesgue density theorem). 2. If Hs (A) < ∞ then Θ ∗s (A, ·) ≤ 1 Hs -almost everywhere on A.

1.1.3 Lipschitz Mappings A mapping f : A → Rn defined on a set A ⊂ Rd is Lipschitz if there exists a number L ≥ 0 such that |f (y) − f (x)| ≤ L|y − x|

for all x, y ∈ A.

The infimum of all constants L with the above property as called the Lipschitz constant of f and denoted Lip f . The following result says that we can mostly work with Lipschitz mappings defined on the whole space. Theorem 1.7 (Kirszbraun, [Fed69, §2.10.43]) Any Lipschitz mapping from a subset of Rd to Rn can be extended to a Lipschitz mapping defined on the whole Rd , with the same Lipschitz constant. Lipschitz mappings are often used in geometric measure theory instead of C 1 mappings from the classical calculus. The following two results make this possible. Theorem 1.8 (Rademacher, [Fed69, §3.1.6]) A Lipschitz mapping f : Rd → Rn is differentiable Ld -almost everywhere. Theorem 1.9 (Whitney, [Fed69, §3.1.16]) Let f : Rd → Rn be Lipschitz and let ε > 0. Then there exists a C 1 mapping g : Rd → Rn such that Ld {x ∈ Rd : f (x) = g(x)} < ε. The behaviour of Hausdorff measure under Lipschitz mappings is given in the following simple proposition. Proposition 1.10 ([KP08, Proposition 2.4.7]) If A ⊂ Rd and f : A → Rn is Lipschitz then Hs (f (A)) ≤ (Lip f )s Hs (A), s ≥ 0. As a consequence we obtain the Ahlfors regularity of compact Lipschitz submanifolds. Definition 1.11 A set M ⊂ Rd is an m-dimensional Lipschitz submanifold (1 ≤ m ≤ d − 1) if for any a ∈ M there exist an open neighbourhood U of a in Rd and a bi-Lipschitz homeomorphism f : M ∩ U → im f ⊂ Rm (called Lipschitz chart of M). A collection (fi ) of charts with domains covering M is called an atlas of M.

1.1 Area, Coarea and Rectifiability

5

Proposition 1.12 Any compact m-dimensional Lipschitz submanifold M ⊂ Rd is Ahlfors m-regular (1 ≤ m ≤ d − 1). Proof Let a ∈ M be given, let f : M ∩ U → im f ⊂ Rm be a Lipschitz chart, denote L := max{Lip f, Lip f −1 }, and let ε > 0 be such that B(a, ε) ⊂ U . Then, for any 0 < r ≤ ε, f −1 (B(f (a), L−1 r)) ⊂ M ∩ B(a, r) ⊂ f −1 (B(f (a), Lr)). Applying Proposition 1.10, we obtain L−k ωk (L−1 r)k ≤ Hk (M ∩ B(a, r)) ≤ Lk ωk (Lr)k , which shows that the set A∩B(a, ε) is Ahlfors k-regular with constants c = ωk L−2k and C = ωk L2k in (1.1) independent of a ∈ M and ε > 0. Since M is compact, it has a finite atlas of Lipschitz charts, and it is not difficult to verify that M fulfills (1.1) as well.   Corollary 1.13 For any bounded set F ⊂ Rd and r > 0, the boundary ∂Fr is upper Ahlfors (d − 1)-regular. Proof If r > 0 is a regular value of the distance function x → dist (x, F ) then ∂Fr is a (d − 1)-dimensional Lipschitz manifold by the inverse function theorem for Lipschitz mappings (see [Cla76]). This appears, in particular, if r > diam F (cf. [Fu85]). In such a case, ∂Fr is Ahlfors (d − 1)-regular by Proposition 1.12. Let now r > 0 be arbitrary and partition F = F 1 ∪ · · · ∪ F n into finitely many pieces with diam F i < r, i = 1, . . . , n. Then, each ∂(F i )r is upper Ahlfors (d − 1)regular by the above observation, and, since clearly ∂Fr ⊂ ∂(F1 )r ∪ · · · ∪ ∂(Fn )r , ∂Fr is upper Ahlfors (d − 1)-regular.   Let a mapping f with values in Rn be differentiable at a point a ∈ Rd (with differential Df (a)) and let 0 ≤ k ≤ d be an integer. The k-dimensional Jacobian of f at a is defined as Jk f (a) := sup{Hk (Df (a)(C)) : C is a k-dimensional unit cube in Rd }.

(1.2)

(Note that Df (a) maps the cube C always to a parallelepiped and we are considering its maximal k-volume.) Another description using the multilinear notation from  Sect. 1.2.1 is Jk f (a) =  k Df (a). Particular cases (cf. [KP08, pp. 127–8]) 1. If k = d = n then Jd f (a) = | det Df (a)|. 2. If k = d < n then 

Jd f (a) = det Df (a)T Df (a) .

6

1 Background from Geometric Measure Theory

Moreover, Jd f (a) = Hd (Df (a)(C)) for any unit cube C ⊂ Rd , or Jd f (a) = Hd (Df (a)(A))/Ld (A) for any measurable subset A ⊂ Rd of positive finite Lebesgue measure. 3. If k = n < d then 

Jn f (a) = det Df (a) Df (a)T . If the rank of Df (a) is less than n then Jn f (a) = 0. If the rank of Df (a) equals n then Jn f (a) = Hn (Df (a)(C)) for any unit n-cube in (ker Df (a))⊥ , the orthogonal complement of the kernel of Df (a) (ker Df (a) = {u ∈ Rd : Df (a)u = 0}), or Jn f (a) = Hn (Df (a)(A))/Hn (A) for any measurable subset A ⊂ (ker Df (a))⊥ of positive finite n-dimensional Hausdorff measure. Two basic formulas for change of variables in the setting for Lipschitz functions follow. Notice that we use the notion “integrable function” in the usual sense for a measurable function with finite integral. Theorem 1.14 (Area formula, [Fed69, §3.2.3]) Let f : Rd → Rn be Lipschitz, d ≤ n, and let A ⊂ Rd be Lebesgue measurable. Then 

 Jd f dLd =

A

Rn

card(A ∩ f −1 {z}) Hd (dz).

If, moreover, h is an Ld -integrable (or nonnegative Ld -measurable) function on A then    h(x)Jd f (x) Ld (dx) = h(x) Hd (dz). Rn

A

x∈A∩f −1 {z}

Theorem 1.15 (Coarea formula, [Fed69, §3.2.11]) Let f : Rd → Rn be Lipschitz, d ≥ n, and let A ⊂ Rd be Lebesgue measurable. Then 

 Jn f dLd = A

Rn

Hd−n (A ∩ f −1 {z}) Hn (dz).

If, moreover, h is an Ld -integrable (or nonnegative Ld -measurable) function on A then    h(x)Jn f (x) Ld (dx) = h(x) Hd−n (dx)Hn (dz). A

Rn

A∩f −1 {z}

1.1 Area, Coarea and Rectifiability

7

1.1.4 Tangent Cones If A ⊂ Rd and a ∈ Rd , a vector u ∈ Rd is said to be tangent to A at a if u = 0 or if there exist sequences (ai ) ⊂ A \ {a} and (ri ) ⊂ (0, ∞) such that ai → a and ri (ai − a) → u, i → ∞. Another (equivalent) description is that a nonzero vector u is tangent to A at a if and only if there exist points ai = a from A such that ai −a u |ai −a| → |u| . We denote by Tan(A, a) the set of all tangent vectors to A at a. It is always a closed cone with vertex at the origin. Given an integer 0 ≤ k ≤ d, the cone of (Hk , k)-approximate tangent vectors of A at a is defined as Tank (A, a) = {Tan(E, a) : E ⊂ A, Θ k (A \ E, a) = 0}. Clearly, Tank (A, a) is a closed subcone of Tan(A, a).

1.1.5 Approximate Differential A function f : A → Rn (A ⊂ Rd ) is said to be (Hk , k)-approximatively differentiable at a ∈ A if there exists a mapping g : Rd → Rn differentiable at a and such that Θ k ({x ∈ A : f (x) = g(x)}, a) = 0. The mapping (Hk , k)ap Df (a) := Dg(a) | Tank (A, a) (restriction of Dg(a) to Tank (A, a)) is called the (Hk , k)-approximate differential of f at a. We often write only ap Df (a), or even only Df (a), for brevity. (Note that the approximate differential coincides with the classical one whenever the latter exists, hence, there is no risk of confusion.) It can be shown that ap Df (a) does not depend on the choice of the function g. Assume that f is (Hk , k)-approximatively differentiable at a ∈ A and that Tank (A, a) is a k-dimensional subspace of Rd . For an integer 0 ≤ m ≤ k, we define the m-dimensional approximate Jacobian of f at a as ap Jm f (a) = sup{Hm (ap Df (a)(C)) : C is a unit m-cube in Tank (A, a)}. We will also use the short notation Jm f (a) for the approximate Jacobian of f , i.e., Jm f (a) := ap Jm f (a).

8

1 Background from Geometric Measure Theory

1.1.6 Rectifiable Sets Definition 1.16 Let k ∈ [0, d] be an integer. A set A ⊂ Rd is called • k-rectifiable if A is a Lipschitz image of a bounded subset of Rk ; • countably k-rectifiable if A is a countable union of k-rectifiable sets; • locally k-rectifiable if for any a ∈ A there exists r > 0 such that A ∩ B(a, r) is k-rectifiable; • countably Hk -rectifiable if A = A0 ∪ A1 ∪ A2 ∪ . . . with Hk (A0 ) = 0 and Ai k-rectifiable, i ≥ 1; • locally Hk -rectifiable (in an open set U ⊃ A) if it is countably Hk -rectifiable and Hk (A ∩ K) < ∞

for any compact K ⊂ Rd

(K ⊂ U, resp.).

Remark 1.17 1. The reader should be warned that the terminology for rectifiable sets is not unified in the literature. Federer [Fed69] used the term “(Hk , k)-rectifiable set” for a set that we call countably Hk -rectifiable with the additional condition of finite Hk measure. Krantz and Parks [KP08] use “countably k-rectifiable” for what we call “countably Hk -rectifiable”. We adopt here the terminology of Ambrosio et al. [AFP00]. 2. The local Hk -rectifiability does not imply Hk -measurability in general. (Consider a non-measurable subset B ⊂ Rk and take A = B × {0} ⊂ Rd .) We will mostly use the rectifiability together with the measurability assumption. Examples 1. A k-dimensional Lipschitz submanifold of Rd is locally k-rectifiable. 2. The graph of a Lipschitz function of d − 1 variables is locally (d − 1)-rectifiable. 3. If A ⊂ Rd is countably Hk -rectifiable and f : A → Rn Lipschitz then f (A) is countably Hk -rectifiable as well. 4. If A ⊂ Rd is locally Hk -rectifiable and f : A → Rn Lipschitz and proper (i.e., preimages of compact sets are compact) then f (A) is locally Hk -rectifiable as well. Theorem 1.18 ([Fed69, §3.2.18]) Let A ⊂ Rd be Hk -measurable and locally Hk rectifiable and γ > 1. Then there exist bi-Lipschitz mappings gi : Rk → im gi ⊂ Rd with Lip gi ≤ γ and Lip gi−1 ≤ γ and compact subsets Ki ⊂ Rk , i = 1, 2, . . . , such that gi (Ki ) ∩ gj (Kj ) = ∅ for i = j and

H

k

A\

 i

 gi (Ki ) = 0.

1.1 Area, Coarea and Rectifiability

9

Proposition 1.19 ([Fed69, §3.2.19]) Let A ⊂ Rd be Hk -measurable and locally Hk -rectifiable and f : A → Rn be Lipschitz. Then for Hk -almost all a ∈ A, Θ k (A, a) = 1, Tank (A, a) is a k-dimensional subspace, and f is (Hk , k)approximately differentiable at a. Theorem 1.20 ([Fed69, §3.2.29]) A set A ⊂ Rd is countably Hk -rectifiable if and only if there exist k-dimensional C 1 -submanifolds M1 , M2 , . . . of Rd such that k H (A \ i Mi ) = 0. We present now a common extension of Theorems 1.14 and 1.15 for rectifiable sets. It is an easy extension of [Fed69, §3.2.22] (where finite global Hk -measure is assumed), see also [KP08, Theorem 5.4.9]. Theorem 1.21 (General Area-coarea theorem) Let A ⊂ Rd be Hk -measurable and locally Hk -rectifiable and Z ⊂ Rn be Hn -measurable and locally Hm rectifiable, k ≥ m, and let f : A → Z be Lipschitz. Then 1. for Hk -almost all x ∈ A, either ap Jm f (x) = 0 or im ap Df (x) = Tanm (Z, f (x)) is an m-dimensional subspace, 2. f −1 {z} is locally Hk−m -rectifiable and Hk−m -measurable for Hm -almost all z ∈ Z, 3.   k ap Jm f dH = Hk−m (f −1 {z}) Hm (dz), A

Z

4. for any Hk -integrable (or nonegative Hk -measurable) function h on A,    k ap Jm f (x)h(x) H (dx) = h(x) Hk−m (dx) Hm (dz). A

Z

f −1 {z}

Example 1.22 Given two subspaces Lp , Lq of Rd , of dimension p, q, respectively, we define J (Lp , Lq ) = Jr (pLq |Lp ), the r-dimensional Jacobian of the orthogonal projection to Lq restricted to Lp , where r = min{p, q}. Let A ⊂ Rd be Hk -measurable and locally Hk -rectifiable, let L be a j dimensional subspace of Rd and set r = min{j, k}. If f = pL |A : A → L is the orthogonal projection from A to L then Jr f (a) = J (Tank (A, a), L)

10

1 Background from Geometric Measure Theory

for Hk -almost all a ∈ A. The area-coarea theorem thus yields 

 J (Tank (A, a), L) Hk (da) = A

L

−1 Hk−j (A ∩ pL {z}) Hj (dz)

if k ≥ j and   k k J (Tan (A, a), L) H (da) = A

pL (A)

−1 Hj −k (A ∩ pL {z}) Hk (dz)

if k ≤ j . Assume now that k = j ; integrating the last formula with respect to the invariant probability measure νkd over the Grassmannian G(d, k) of all k-subspaces (for exact definition see Sect. 1.2.2) and using Fubini theorem (provided that the measurability is shown), we get the Crofton formula (consequence of [Fed69, §2.10.15, §3.2.6]): Theorem 1.23 (Crofton formula for Hausdorff measures) If A ⊂ Rd is Hk measurable and locally Hk -rectifiable and 0 ≤ k ≤ d then 



H (A) = c(d, k) k

G(d,k) L

−1 card(A ∩ pL {z}) Hk (dz)νkd (dL)

with  c(d, k) =

Γ

k+1 2



Γ



 Γ 

d+1 2

d−k+1 2

 

Γ

1 2

 .

The following theorem is about the rectifiability of level sets of Lipschitz maps. It as a consequence of [Fed69, §3.2.31]. Theorem 1.24 If W ⊂ Rd is Hk -measurable and locally Hk -rectifiable, f : W → Rn Lipschitz and 0 ≤ m ≤ k, then the set {y ∈ Rn : Hk−m (f −1 {y}) > 0} is countably Hm -rectifiable.

1.1.7 Purely Unrectifiable Sets A set E ⊂ Rd is called purely k-unrectifiable if it contains no k-rectifiable subset of positive Hk -measure.

1.2 Multilinear Algebra and Differential Forms

11

Proposition 1.25 Any Hk -measurable set W ⊂ Rd with Hk (W ) < ∞ can be written as disjoint union W = A ∪ E of a countably k-rectifiable set A and purely k-unrectifiable set E. Theorem 1.26 (Structure theorem, [Fed69, §3.3.13]) Assume that Hk (E) < ∞. If E is purely k-unrectifiable then Hk (pL (E)) = 0 for almost all k-subspaces E of Rd . Consequently, the Crofton formula fails for purely unrectifiable sets with positive Hk -measure.

1.2 Multilinear Algebra and Differential Forms 1.2.1 Multilinear Algebra Let V be a finite (d-)dimensional vector space over R (usually V = Rd ). If k ≥ 0 is k an integer, we denote by V the linear space of all k-linear functions α : V k → k  R. Elements of V are called covariant k-tensors and dim k V = d k . Let Σ(k)be the set of all permutations of {1, . . . , k} . A covariant k-tensor α is called antisymmetric if for any σ ∈ Σ(k) and for all vectors v1 , . . . , vk ∈ V ,

α vσ (1) , . . . , vσ (k) = (sgn σ )α (v1 , . . . , vk ) .  The set of all antisymmetric covariant k-tensors will be denoted by k V . It is k k a linear subspace of V . Elements of V are called k-covectors or multicovectors . It follows from the antisymmetry that α(v1 , . . . , v k ) = 0 whenever the vectors v1 , . . . , vk are linearly dependent. Thus, if k > d then k V is trivial. Given   two multi-covectors α ∈ k V , β ∈ m V , their exterior (or wedge) product is defined by the “shuffle formula” (α ∧ β)(v1 , . . . , vk+m )  (sgn σ )α(vσ (1) , . . . , vσ (k) )β(vσ (k+1) , . . . , vσ (k+m) ), =

(1.3)

σ ∈Σ(k,m)

where Σ(k, m) = {σ ∈ Σ(k + m) : σ (1) < · · · < σ (k), σ (k + 1) < · · · < σ (k + m)}.  It is easy to verify that α ∧ β ∈ k+m V . k-covectors of the form α1 ∧ · · · ∧ αk ,  where α1 , . . . , αk ∈ V ∗ = 1 V , are called simple. Any k-covector can be written as a linear combination of simple k-covectors.

12

1 Background from Geometric Measure Theory

The direct sum ∗

V =

∞  k

V

k=0

is called exterior algebra of V ; it is obviously an algebra with respect to addition and exterior multiplication. For a basis b1 , . . . , bn in an n-dimensional real vector space the dual basis is denoted by b1∗ , . . . , bn∗ , i.e., bi∗ (bj ) = δij . If {e1 , . . . , ed } is a basis of V then {ei∗1 ∧ · · · ∧ ei∗k : 1 ≤ i1 < · · · < ik ≤ d}  is a basis of k V . The space of k-vectors or multivectors is introduced as 

kV

:=

k

V∗,

 where V ∗ = 1 V is the dual space to V . Then the exterior product of multivectors is defined as well. Using the identification V ∗∗ = V simple k-vectors are given by v1 ∧ · · · ∧ vk for v1 , . . . , vk ∈ V and for the above basis in V the k-vectors {ei1 ∧ · · · ∧ eik : 1 ≤ i1 < · · · < ik ≤ d} form a basis of



kV .

Example 1.27 1. If u = u1 e1 + · · · + ud ed and v = v 1 e1 + · · · + v d ed then u∧v =

 (ui v j − uj v i )(ei ∧ ej ). i 0 if and only if the vectors (v1 , . . . , vk ) and (w1 , . . . , wk ) have the same orientation. A linear mapping L : V → W between two vector spaces induces mappings on the spaces of alternating as follows:  forms  and multivectors  The linear mapping k L : k W → k V with (

k

L)α(v1 , . . . , vk ) := α(Lv1 , . . . , Lvk ), α ∈



kW

(1.5)

,

is called pullback of the alternating k-linear forms under the mapping L. It commutes with exterior multiplication: k+l

L(α ∧ β) =

k

L(α) ∧

l

L(β), α ∈

k

W, β ∈

l

W.

The counterpart called pushforward of multivectors is the linear mapping   V → W determined by k k (



k L)(v1

∧ · · · ∧ vk ) := Lv1 ∧ · · · ∧ Lvk .

(1.6) 

kL

:

(1.7)

14

1 Background from Geometric Measure Theory

(Using multilinearity and Proposition 1.28 one can see that this definition does not depend on the choice of the representing vectors v1 , . . . , vk for the k-vector v1 ∧ . . . ∧ vk .) Then one obtains the duality ξ, (

k

L)α = (



k L)ξ, α , ξ

∈



kV ,

α∈

k

(1.8)

W.

The operations of interior multiplication are introduced as follows. Let ξ ∈   and α ∈ m V . If k ≤ m we define ξ α ∈ m−k V by η, ξ and if k ≥ m we define ξ ξ

α = η ∧ ξ, α, α∈



k−m V

η∈





kV

(1.9)

m−k V ,

by

α, β = ξ, α ∧ β,

β∈

k−m

(1.10)

V.

Let now V = Rd and {e1 , . . . , ed } be its canonical orthonormal  basis. u · v denotes the basic scalar product. We define a scalar product on k Rd by means of the values on the induced basis as follows. Assume that i1 < · · · < ik and j1 < · · · < jk and set  (ei1 ∧ · · · ∧ eik ) • (ej1 ∧ · · · ∧ ejk ) =

1 if i1 = j1 , . . . , ik = jk , 0 otherwise.

 The corresponding norm on k Rd will be denoted by  · . Similarly the dual space k d R gets a Euclidean structure. Here besides the dual norm ·, we define another norm (called comass) α = sup{|ξ, α| : ξ ∈



kR

d

simple , |ξ | = 1},

α∈

k

Rd .

Direct calculations lead to the following formula for the scalar product of simple k-vectors:

k (u1 ∧ · · · ∧ uk ) • (v1 ∧ · · · ∧ vk ) = det ui · vj i,j =1 .

(1.11)

From this one can derive that for v1 ∧ · · · ∧ vk = 0 the norm |v1 ∧ · · · ∧ vk | is equal to the k-volume of the convex hull of the vectors v1 , . . . , vk , i.e., of the spanned parallelepiped. Because of this and Proposition 1.28 a nonzero simple k vector in k Rd has the geometric interpretation of an equivalence class of oriented parallelepipeds with the same k-dimensional spanned vector space in Rd and the same k-volume. This plays a role in applications to geometric integration theory.

1.2 Multilinear Algebra and Differential Forms

15

Recall that in the general Euclidean setting the dual element v ∗ of an arbitrary vector v is defined by the relation v ∗ (u) = u · v for all vectors u, where · means the corresponding scalar product. Definition 1.29   For any 0 ≤ k ≤ d, the Hodge star operator is the linear mapping  : k Rd → d−k Rd given by (e1∗ ∧ · · · ∧ ed∗ ))∗ .

 : ξ → (ξ

Note that ξ is simple whenever ξ is, and that ξ ∧ ξ, e1∗ ∧ · · · ∧ ed∗  = |ξ |2 .

1.2.2 The Grassmannian Let G(d, k) denote the set of all k-dimensional linear subspaces of Rd . Due to Proposition 1.28, via the mapping 0 = ξ = v1 ∧· · ·∧vk → L(ξ ) and the orientation of (v1 , . . . , vk ) we can identify G0 (d, k) := {ξ ∈



kR

d

simple, |ξ | = 1}

d with the set of oriented linear k-subspaces  dof R . In [Fed69, §3.2.8] it is shown ∞ that G0 (d,k) is a C -submanifold of k R . The Grassmannian G(d, k) is then obtained as a factorspace, by identifying k-vectors from G0 (d, k) that differ by sign only:

G(d, k) = G0 (d, k)/sgn , and the mapping  : ξ → L(ξ ) is the natural projection from G0 (d, k) to G(d, k). We have dim G0 (d, k) = dim G(d, k) = k(d − k) (see also Sect. 1.4). (These representations are equivalent to those from classical differential geometry and Lie groups, where G0 (d, k) is identified with the factor group SO(d)/(SO(k) × SO(d − k)) for the special orthogonal groups and G(d, k) with O(d)/(O(k) × O(d − k)) for the orthogonal groups.) The Hausdorff measure of G0 (d, k) is equal to 2cd,k := Hk(d−k)(G0 (d, k)) = 2π

k(d−k) 2

k  j =1

Γ (j/2) Γ ((d − j + 1)/2)

(1.12)

16

1 Background from Geometric Measure Theory

(see [Fed69, §3.2.28]). Since the Hausdorff measures in Euclidean spaces are motion invariant and −1 (L) has two elements for any L ∈ G(d, k), it follows that   −1 k(d−k) νkd (·) := cd,k H |G0 (d,k) −1 (·) (1.13) is the (unique) O(d)-invariant probability measure on G(d, k).  Here we identify an orthogonal mapping g ∈ O(d) with the orthogonal mapping k g on G(d, k) in the above interpretation. Then νkd corresponds to the Haar measure on O(d)/(O(k) × O(d − k)). Lemma 1.30 Given any ξ ∈ G0 (d, k) and u, v, w ∈ Rd , we have: u∗ = (u · v) ξ

(v ∧ ξ ) ξ L(ξ

v∗ = ξ

if u ⊥ L(ξ ),

(1.14)

(pL(ξ ) v)∗ ,

(1.15)

v ∗ ) = L(ξ ) ∩ v ⊥ if v ∈ L(ξ )⊥ ,

(1.16)

and (ξ

v ∗ ) • (ξ

w∗ ) = (pL(ξ )v) · (pL(ξ ) w).

Proof Denote for brevity L := L(ξ ) (recall (1.4)). If ψ ∈ then (v ∧ ξ )

k

(1.17)

Rd and 0 = u ⊥ L

u∗ , ψ = v ∧ ξ, u∗ ∧ ψ = u, v ∗ ξ, ψ = (u · v)ξ, ψ

(we have used (1.3) in the second equality). This proves (1.14). Further, denote a := pL v. If a = 0 then it follows from the definition that a ξ v ∗ = 0 and, hence, (1.15) holds. If a = 0 we can write ξ in the form |a| ∧ζ a·v ∗ with some ζ ∈ G0 (d, k − 1), a, v ⊥ L, and we get ξ v = |a| ζ = |pL v|ζ by (1.14), and both (1.15), (1.16) follow. Finally, note that both sides of (1.17) depend linearly on both vectors v, w, and they wanish whenever one of these vectors is perpendicular to L. Thus, it is sufficient to verify (1.17) in the case when v, w are elements of a given orthonormal basis of L, which follows easily using (1.11).   Definition 1.31 Let L, M be a k, l-dimensional subspace of Rd , respectively, and let {a1 , . . . , ad }, {b1 , . . . , bd } be two orthonormal bases of Rd such that L = Lin {a1 , . . . , ak } and M = Lin {b1 , . . . , bl }. We define     d d     [L, M] :=  ai ∧ bj  , i=k+1 j =l+1 

1.2 Multilinear Algebra and Differential Forms

17

where the norm on the right hand side is understood in the space of (2d − k − l)vectors. Note that [L, M] vanishes unless L + M span the whole Rd . If this is the case, [L, M] equals J (L, M ⊥ ) from Example 1.22. Geometrically, [L, M] is the volume of the orthogonal projection to M ⊥ of a unit (d − l)-cube in L ∩ (L ∩ M)⊥ . The following integral equality will be useful later. For any 0 ≤ k ≤ d,  G(d,k)

[L, M]2 νkd (dL) =

 −1 d , k

M ∈ G(d, d − k).

(1.18)

To see this, note first that the integral clearly does not depend on M ∈ G(d, d − k) since νkd is rotation invariant. Further, note that [L(ξ ), L(ζ )⊥ ]2 = (ξ • ζ )2 whenever  ξ, ζ ∈ G0 (d, k), and that the Parseval’s identity in k Rd gives 

(ξ • (ei1 ∧ · · · ∧ eik ))2 = 1,

ξ ∈ G0 (d, k), |ξ | = 1.

1≤i1 n, 3. (F) limi→∞ Ti , f, y = T , f, y for Ln -almost all y ∈ Rn .

1.3.8 Indecomposable Integral Currents Definition 1.67 A current T ∈ Ik (Rd ) is called indecomposable if there exists no S ∈ Ik (U ) with S = 0 = T − S and N(T ) = N(S) + N(T − S). Proposition 1.68 ([Fed69, §4.2.25]) For any T ∈ Ik (Rd ) there exists a sequence of indecomposable currents Ti ∈ Ik (Rd ) such that T =

∞  i=1

Ti and N(T ) =

∞ 

N(Ti ).

i=1

The following result can be found in a paper by Hardt [Har77], see a reformulation of Theorem 1 on page 1 therein. In fact, the result is formulated for real functions, but an extension to vector-valued mappings is obvious. Theorem 1.69 (Hardt [Har77]) Let T = (Hk W ) ∧ ξ ∈ Ik (Rd ) be d n 1 indecomposable and let h : R → R be a C -mapping such that ξ Dh = 0 Hk -almost everywhere on W . Then h|spt T is constant.

1.4 Computations on Grassmannians

31

1.4 Computations on Grassmannians In this section we derive several formulas which will be useful later. For computations of differentials and Jacobians of functions on the Grassmannian, we will need to work with tangent vectors to G0 (d, k). These can be described as follows. First, consider a unit simple multivector ξ ∈ G0 (d, k) be given, let L := L(ξ ) be the associated k-subspace, and let v ∈ L and w ∈ L⊥ be unit vectors. Then, (ξ

v ∗ ) ∧ w ∈ Tan(G0 (d, k), ξ )

(recall (1.10) for the definition of interior multiplication). To see this, consider the multivectors    ξi := (ξ v ∗ ) ∧ i −1 w + 1 − i −2 v ∈ G0 (d, k), i = 1, 2, . . . , and note that ξ = (ξ i(ξi − ξ ) = (ξ

v ∗ ) ∧ v and

v ∗ ) ∧ w + (ξ

v∗ ) ∧ i



 1 − i −2 − 1 v → (ξ

v∗ ) ∧ w

as i → ∞. Consequently, we can obtain on orthonormal basis of Tan(G0 (d, k), ξ ) in the following way. Choose an orthonormal basis {u1 , . . . , ud } of Rd such that ξ = u1 ∧ · · · ∧ uk and denote ηi,j = u1 ∧ · · · ∧ ui−1 ∧ uj ∧ ui+1 ∧ · · · ∧ uk ∈



kR

d

,

1 ≤ i ≤ k, k + 1 ≤ j ≤ d.

Then, {ηi,j , 1 ≤ i ≤ k, k + 1 ≤ j ≤ d} is an orthonormal basis of Tan(G0 (d, k), ξ ). Note that the given basis vectors of Tan(G0 (d, k), ξ ) are unit simple k-vectors (but, of course, not all tangent vectors are simple). Lemma 1.70 Let v ∈ S d−1 be a fixed unit vector and consider the mapping g : ξ → pL(ξ ) v,

ξ ∈ G0 (d, k).

Fix a ξ ∈ G0 (d, k) such that v ∈ L := L(ξ ) and v ∈ L⊥ . Then, the differential of g at ξ is given by Dg(ξ )((ξ

u∗ ) ∧ w) = (w · pL⊥ v)u + (u · pL v)w,

32

1 Background from Geometric Measure Theory

v ∈ L, w ∈ L⊥ . The Jacobian of g is Jd−1 g(ξ ) = sink−1  (v, L) cosd−1−k  (v, L). Proof Note that we can describe g equivalently by g(ξ ) = ξ

(v

ξ ∗ ),

hence, ξ is clearly differentiable. Let ui , ηi,j be as above and assume, moreover, that u1 = pL v/|pL v| and ud = pL⊥ v/|pL⊥ v|. The differential of g fulfills Dg(ξ )(η1,j ) Dg(ξ )(ηi,d ) Dg(ξ )(η1,d ) Dg(ξ )(ηi,j )

= = = =

|pL v|uj , k < j < d, |pL⊥ v|ui , 1 < i ≤ k, |pL⊥ v|u1 + |pL v|ud , 0, 1 < i ≤ k, k < j < d.

Hence, the form for the differential follows by linearity and the Jacobian is  k  d−1      Jd−1 g(ξ ) =  |pL⊥ v|ui ∧ |pL v|ui ∧ (|pL⊥ v|u1 + |pL v|ud )   i=2

i=k+1

= |pL⊥ v| = sin

k−1 

k−1

|pL v|

d−1−k

(v, L) cosd−1−k  (v, L).  

In the sequel the normalized orthogonal projection onto a linear subspace L of Rd will be denoted by πL , i.e., πL w :=

pL w , w ∈ Rd . |pL w|

Lemma 1.71 Let the mapping h be given as h(ξ ) := πL(ξ )v,

ξ ∈ G0 (d, k), L(ξ ) ⊥ v

(as above, v is a fixed unit vector). Then its Jacobian equals Jk−1 h(ξ ) = tank−1  (v, L(ξ )). Proof We will use the same basis of the tangent space as in the proof of Lemma 1.70. Now we have   Dg(ξ )η . Dh(ξ )η = pg(ξ )⊥ |pL v|

1.4 Computations on Grassmannians

33

Hence, Dg(ξ )(η1,j ) Dg(ξ )(ηi,d ) Dg(ξ )(η1,d ) Dg(ξ )(ηi,j )

= = = =

uj , k < j < d, (tan  (v, L))ui , 1 < i ≤ k, ud , 0, 1 < i ≤ k, k < j < d,  

and the assertion follows. Lemma 1.72 Given v ∈ S d−1 fixed, consider the mapping q : ξ →

ξ |ξ

v∗ , v∗ |

ξ ∈ G0 (d, k), L(ξ ) ⊥ v

(Note that L(q(ξ )) = L(ξ ) ∩ v ⊥ , see (1.15).) Then, J(k−1)(d−k)q(ξ ) = cos  (v, L(ξ ))−(k−1) . Proof We have Dq(ξ )η = pq(ξ )⊥

η |ξ

v∗ . v∗ |

Using the same basis vectors as in the previous proofs, and the fact that |ξ |pL v|, we obtain

v∗ | =

Dq(ξ )(η1,j ) = 0, k < j ≤ d, Dg(ξ )(ηi,j ) = ±(u2 ∧ · · · ∧ ui−1 ∧ uj ∧ ui+1 ∧ · · · ∧ uk ), 1 < i ≤ k < j < d, 1 < i ≤ k, Dg(ξ )(ηi,d ) = ±|pL v|−1 (v  ∧ u2 ∧ · · · ∧ uk ), where v  is a unit vector in Lin {u1 , ud } perpendicular to v, and the assertion follows.   In the following considerations, we will consider the special orthogonal group and we SO(d) in Rd , which is known to be a smooth manifold of dimension d(d−1) 2 d d consider it to be embedded into the Euclidean space L(R , R ) of linear mappings from Rd to Rd (see [Fed69, §3.2.28]). We construct a particular mapping from the sphere to SO(d) which will be useful later. Fix a unit vector e ∈ Rd and consider the mapping Fe : S d−1 \ {−e} → SO(d) defined as follows. Given a unit vector u = ±e, Fe (u) is the orthogonal mapping acting as the rotation in the plane Lin (e, u) sending e to u, and leaving the

34

1 Background from Geometric Measure Theory

orthogonal complement to this plane unchanged. Formally, we can write Fe (u)z =

−(z · u) + (e · u)(z · e) (e + u) + (z · e)(u − e) + z, 1 + (e · u)

z ∈ Rd .

(1.35)

Clearly, Fe is differentiable, and its differential can be computed as follows. For z ∈ Rd fixed and direction v ⊥ u, we have D[Fe (u)z](v) = +

−(z · v)(1 + e · u) + (v · e)(z · (e + u)) (e + u) (1 + e · u)2 −(z · u) + (z · e)(1 + 2e · u) v. 1+e·u

(1.36)

Lemma 1.73 Let W ⊂ (Rd × S d−1 )2 satisfy Hp (W ) = 0. Then Hq (A) = 0, where A = {(u1 , v1 , u2 , v2 , ρ) ∈ W × SO(d) : v1 + ρv2 = 0} and q = p +

(d−1)(d−2) . 2

Proof We will use the locally Lipschitz mapping Fe from (1.35) and the following two easy facts: 1. The mapping ρ → ρ −1 is an isometry on SO(d). 2. Given a fixed rotation σ ∈ SO(d), the mappings ρ → ρ ◦ σ and ρ → σ ◦ ρ are isometries on SO(d). Given e ∈ S d−1 , we fix some isometry ψ : e⊥ → Rd−1 . and, to any σ ∈ SO(d − 1) we attach σ e ∈ SO(d) which leaves e fixed and acts as σ ◦ ψ on e⊥ . From the above statements, we see that   (u1 , v1 , u2 , v2 , σ ) → u1 , v1 , u2 , v2 , Fe (−v1 ) ◦ σ e ◦ (Fe (v2 ))−1 is a locally Lipschitz mapping from (W ∩ Qe ) × SO(d − 1) onto A ∩ (Qe × SO(d)), where Qe = {(u1 , v1 , u2 , v2 ) : v1 = −e, v2 = −e}. Since Hp (W ) = 0 and SO(d − 1) is a smooth manifold of dimension we have (see [Fed69, §2.10.45]

(d−1)(d−2) , 2

Hq (W × SO(d − 1)) = 0. Hence, also Hq (A ∩ (Qe × SO(d))) = 0. Choosing three different unit vectors e, we cover the whole set A and get Hq (A) = 0.  

1.4 Computations on Grassmannians

35

In the following result, we associate with a locally rectifiable “unit normal bundle” W ⊂ Rd × S d−1 its “m-flag space” by adding all m-subspaces orthogonal to the normal vectors, show its rectifiability and describe the approximate tangent vectors. This will be needed later for some integral-geometric formulas. Lemma 1.74 Let W be a locally Hk -rectifiable Borel subset of Rd × S d−1 , 0 ≤ m ≤ d − 1 − k, and consider the set ⊥

X = {(x, n, ζ ) : (x, n) ∈ W, ζ ∈ Gn0 (d − 1, m)} ⊥

(Gn0 (d−1, m) denotes the space of unit simple m-vectors in the (d−1)-dimensional subspace n⊥ ). Then, X is locally Hp -rectifiable with p = k + m(d − 1 − m) and for any (x, n, ζ ) ∈ X, the approximate tangent space Tanp (X, (x, n, ζ ) is spanned by the multivectors (0, 0, η),

⊥

η ∈ Tan(Gn0 (d − 1, m)),

and

u, v, n ∧ (ζ

v∗ ) ,

(u, v) ∈ Tank (W, (x, n)).

Proof Let Fe be the mapping from (1.35) (e ∈ S d−1 ) and consider the mapping  

 Φe : (x, n, ξ ) → x, n, F (n) ξ e k ⊥

defined on (W ∩ (Rd × (S d−1 \ {−e})) × Ge0 (d − 1, m). It is clear that, since Fe is locally Lipschitz, Φe is locally Lipschitz as well. Further, since W is locally Hk ⊥ rectifiable and Ge0 (d − 1, m) is a smooth manifold of dimension m(d − 1 − m), the ⊥ product (W ∩ (Rd × (S d−1 \ {−e})) × Ge0 (d − 1, m) is locally p-rectifiable (see [Fed69, §3.2.23]) and its locally Lipschitz image im Φe = {(x, n, ζ ) ∈ X : n = e} retains the same property. Repeating the same construction for another unit vector e, we can cover the whole set X. Let now (x, n, ζ ) ∈ X be given and let V be the m-subspace associated with ζ . We will find a basis of Tanp (X, (x, n, ζ )). Note that for given (x, n, ξ ) ∈ W × ⊥ ⊥ Ge0 (d − 1, m), (u, v) ∈ Tank (W, (x, n)) and η ∈ Tan(Ge0 (d − 1, m), ξ ) we have   



DΦe (x, n, ξ )(u, v, η) = u, v, D k Fe (n) v ξ + Fe (n)η .

36

1 Background from Geometric Measure Theory ⊥

Clearly, the vectors (0, 0, η) with η ∈ Tan(Gn (d − 1, m) are tangent to X. They form a subspace of dimension m(d − 1 − m). The remaining k basis tangent vectors can be obtained as follows. Assume that e = n and let ξ = ζ = w1 ∧ · · · ∧ wk be a representation such that the vectors w1 , . . . , wk are orthonormal and w1 = pV v/|pV v| if |pV v| > 0 (otherwise, we choose w1 arbitrarily). Then we have (see (1.36)) D[Fn (n)wi ](v) = |v|(w1 · n)n =

 |pV v|n,

i = 1, i = 2, . . . , k.

0,

Thus we get k k  



 D k Fn (n) v wi = DF (n)v w1 ∧ F (n)wi i=1



i=2

= |pV v| n ∧ ζ (F (n)w1 )∗

(pV v)∗ = n∧ ζ

= n∧ ζ v∗ (see (1.15)), which implies the assertion.



 

Chapter 2

Background from Convex Geometry

2.1 Intrinsic Volumes Let K be a convex body (i.e., a nonempty, convex and compact set) in Rd . We use the notation for the r-parallel body (r ≥ 0) Kr := {z ∈ Rd : dist (z, K) ≤ r}.

(2.1)

(Equivalently, we can write using Minkowski summation Kr = K + rB d , where B d is the unit ball centred in the origin in Rd .) The Steiner formula expresses the volume of Kr as a polynomial: Ld (Kr ) =

d 

ωk r k Vd−k (K),

r ≥ 0.

(2.2)

k=0

The coefficient Vk (K) is called the k-th intrinsic volume of K (k = 0, 1, . . . , d). Another representation of intrinsic volumes is Vk (K) =

   d ωd Lk (πL K) νkd (dL). k ωk ωd−k G(d,k)

This means that Vk (K) is, up to a constant factor, the mean volume of the projections of K into the k-dimensional subspaces of Rd . From classical convex geometry, the re-indexed and re-normed versions ωk Wkd (K) = d Vd−k k

© Springer Nature Switzerland AG 2019 J. Rataj, M. Zähle, Curvature Measures of Singular Sets, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-18183-3_2

37

38

2 Background from Convex Geometry

are known as the quermassintegrals or Minkowski functionals of K. One advantage of the intrinsic volumes is their independence of the dimension of the embedding space. In particular, if K ⊂ Rd is k-dimensional then Vk (K) = Hk (K). For any convex body K we have V0 (K) = 1 and Vd (K) = Ld (K). If K has d−1 inner points then also Vd−1 (K) = 12 Hd−1 (∂K). The number 2ω dωd V1 (K) is known as the mean width of K. It is easy to show that the intrinsic volumes are valuations on the space of convex bodies, i.e., they are additive in the sense that for two convex bodies K, L such that K ∪ L is convex as well, Vk (K ∪ L) + Vk (K ∩ L) = Vk (K) + Vk (L). Further, Vk is invariant with respect to all Euclidean motions, and it is continuous with respect to the Hausdorff metric. The latter is defined for nonempty closed sets A and B in Rd by

dH (A, B) := min inf dist (a, B), inf dist (b, A) . a∈A

b∈B

(2.3)

It follows from the general theory of valuations that Vk can be extended to an additive and motion invariant functional on the convex ring (the space of finite unions of convex bodies) denoted again by Vk . (Here one defines additionally Vk (∅) = 0, k = 0, 1, . . . , d.) Vd still coincides with the Lebesgue measure and Vd−1 (R) = 12 Hd−1 (∂R) whenever R = K1 ∪ · · · ∪ KN is a set from the convex ring with full-dimensional convex bodies K1 , . . . , KN . The zeroth intrinsic volume, V0 , is the Euler-Poincaré characteristic. (The Vk ’s are, however, no more continuous on the convex ring with respect to the Hausdorff distance; as a counterexample, consider two approaching parallel segments.) First proofs were given in Hadwiger [Had57], Groemer [Gro78], Schneider [Sch80], see also the survey McMullen and Schneider [MS83]. The following famous Hadwiger’s characterization theorem [Had57] says that the intrinsic volumes form a basis of all additive, continuous and motion invariant functionals on the family of convex bodies. Simpler proofs were given in Klain [Kla95], Klain and Rota [KR97], and Chen [Che04]. Here and in the sequel we will use the notation C  for the space of convex bodies (i.e., noempty convex compact sets in Rd ), and P  for the subspace of nonempty convex polytopes (i.e., convex hulls of nonempty finite sets), both being equipped with the Hausdorff distance. Note that the exclusion of the empty set is necessary in order that the Hausdorff distance is well defined. Theorem 2.1 (Hadwiger [Had57]) If ψ is an additive, continuous and motion invariant functional on C  then there exist real constants c0 , . . . , cd such that ψ=

d  k=0

ck V k .

2.2 Curvature and Area Measures

39

It is an open problem, whether this characterization remains valid if C  is replaced by P  .

2.2 Curvature and Area Measures Let ΠK denote the metric projection onto the convex body K, i.e., for z ∈ Rd , ΠK (z) is determined by the property |z − ΠK (z)| = dist (z, K). The following local Steiner formula is due to Federer [Fed59] who proved it even for more general sets, see Chap. 4. If B is a Borel subset of Rd then −1 Ld (Kr ∩ ΠK (B)) =

d 

ωk r k Cd−k (K, B),

r ≥ 0,

(2.4)

k=0

with nonnegative coefficients Cd−k (K, B) which depend σ -additively on B and, hence Ck (K, ·) is a finite Borel measure on Rd called kth curvature measure of K. Clearly, we have Ck (K, Rd ) = Vk (K). Regarding additionally the directions of the metric projection onto K, the Steiner formula has the following generalization. Consider the mapping   z − ΠK (z) , Π˜ K : z → ΠK (z), |z − ΠK (z)|

z ∈ Rd \ K,

and let E be a Borel subset of Rd × S d−1 . Then d

 −1 d−k (K, E), Ld (Kr \ K) ∩ Π˜ K (E) = ωk r k C

r ≥ 0,

(2.5)

k=1

d−k (K, E) which again depend σ -additively on E with nonnegative coefficients C  and, hence Ck (K, ·) is a finite Borel measure on Rd × S d−1 ; it is called sometimes k (K, ·) kth support measure of K in convex geometry. Of course, the image of C under the first projection mapping π0 : (x, n) → x is Ck (K, ·). The image under the second projection π1 : (x, n) → n is a finite Borel measure on the sphere, k (K, Rd × ·), and called kth area measure of K. Curvature denoted Sk (K, ·) = C and area measures can again be extended additively to the convex ring. Considering also the dependence on the convex body K, we can consider the curvature measures as functionals Ck : C  ×B(Rd ) → [0, ∞), where B(Rd ) denotes the family of Borel subsets of Rd . The curvature measures possess the following properties. (Again we define Ck (∅, ·) = 0, k = 0, . . . , d.)

40

2 Background from Convex Geometry

Additivity: Ck (K ∪ L, ·) + Ck (K ∩ L, ·) = Ck (K, ·) + Ck (L, ·) whenever K, L and K ∪ L are convex bodies. Motion covariance: Ck (gK, gA) = Ck (K, A) for any convex body K, Borel subset A of Rd and Euclidean motion g : Rd → Rd . Weak continuity: If a sequence of convex bodies Kn converges to K as n → ∞ in the Hausdorff distance then Ck (Kn , ·) → Ck (K, ·) weakly. Local determinancy: If K ∩ G = L ∩ G for convex bodies K, L and an open set G ⊂ Rd then the restrictions of Ck (K, ·) and Ck (L, ·) to G coincide. The following generalization of Hadwiger’s representation theorem was shown by Schneider [Sch78, Theorem 6.1]. An analysis of the proof shows that here the corresponding statement remains valid for the space of convex polytopes. Theorem 2.2 Let Φ : C  × B(Rd ) → [0, ∞) (or Φ : P  × B(Rd ) → [0, ∞)) be a functional such that Φ(K, ·) is a finite Borel measure for any K ∈ C  (for any K ∈ P  , respectively). Assume further that Φ is additive, motion covariant, weakly continuous and locally determined, in the sense given above. Then there exist nonnegative constants c0 , . . . , cd such that Φ(K, ·) =

d 

ck Ck (K, ·),

K ∈ C

(K ∈ P  ).

k=0

An analogous result holds for the area measures Sk (K, ·), K ∈ C  . Note that these measures are motion covariant in a slightly different sense: we have Sk (gK, g0 B) = Sk (K, B) whenever K is a convex body, B a Borel subset of S d−1 and g a Euclidean motion in Rd with linear component g0 . Also, a different locality condition applies. Given a convex body K, we use the notation for the support function of K h(K, u) := sup{x · u : x ∈ K},

u ∈ Rd ,

and for the support set of K and u ∈ S d−1 K(u) := {x ∈ K : x · u = h(K, u)}. The following representation result was shown by Schneider [Sch75, Satz 2]. Note that here the functional considered need not be nonnegative. Theorem 2.3 Let Ψ : C  × B(S d−1) → R be a functional such that Ψ (K, ·) is a finite signed Borel measure for any K ∈ C  . Assume further that Ψ is additive, motion covariant and weakly continuous in the sense given above, and that it satisfies the following locality condition: If

 u∈B

K1 (u) =

 u∈B

K2 (u) then Ψ (K1 , B) = Ψ (K2 , B),

B ∈ B(S d−1 ),

2.3 Integral-Geometric Formulas

41

K1 , K2 ∈ C  . Then there exist real constants c0 , . . . , cd such that Ψ (K, ·) =

d 

K ∈ C .

ck Sk (K, ·),

k=0

2.3 Integral-Geometric Formulas The above representation theorems make it possible to find simple proofs of main integral-geometric results as Principal kinematic theorem and Crofton formula (see [Sch78]). Let Gd be the group of all rigid motions (i.e., isometries preserving orientation) in Rd . Any g ∈ Gd is an affine mapping of the form g(·) = b + ρ(·) with a shift b ∈ Rd and linear mapping ρ from the special orthogonal group SO(d). (We will often call ρ a “rotation”, though it need not be a rotation in the usual narrower sense.) Let ϑd denote the normalized Haar measure on SO(d). Then, the measure Θd on Gd given by 

 Gd

f (g) dΘd (g) =

 SO(d) Rd

f (b + ρ) Ld (db)ϑd (dρ)

for any nonnegative measurable function f on Gd , is a (both left and right) Haar measure on Gd . We will often write only dg instead of Θd (dg) in the integrals in order to save space. Theorem 2.4 (Principal kinematic formula) If K, L are convex bodies, 0 ≤ k ≤ d and A, B Borel subsets of Rd then 





Ck K ∩ gL, A ∩ gB dg =

γ (d, r, s)Cr (K, A)Cs (L, B),

1≤r,s≤d r+s=d+k

Gd

where  Γ 

γ (d, r, s) = Γ

r+1 2



 Γ 

r+s−d+1 2

s+1 2



Γ



d+1 2

.

Let further A(d, j ) be the space of all affine j -subspaces of Rd , 0 ≤ j ≤ d. Any affine subspace E ∈ A(d, j ) can be represented uniquely as E = V + z with

42

2 Background from Convex Geometry

V ∈ G(d, j ) and z ∈ V ⊥ . The motion invariant measure μdj on A(d, k) is defined by μdj (dE) = Ld−j (dz)νjd (dV ), using this correspondence. Theorem 2.5 (Crofton formula) Let K be a convex body, 0 ≤ k ≤ j ≤ d, and A ⊂ Rd a Borel set. Then  Ck (K ∩ E, A ∩ E) μdj (dE) = γ (d, j, k)Cd+k−j (K, A). A(d,j )

2.4 Polar Cones A cone (with centre at the origin) is a set C with the property that if x ∈ C then also tx ∈ C for any t > 0. Definition 2.6 The polar cone to a set C ⊂ Rd is given by C o := {v ∈ Rd : u · v ≤ 0 for all u ∈ C}. It is clear from the definition that C o is always a closed convex cone. If C is already a cone then its second polar satisfies C oo = conv C.

(2.6)

If C, D ⊂ Rd are closed convex cones then (C ∩ D)o = C o + D o ,

(C + D)o = C o ∩ D o

(2.7)

(see [Roc70, Sect. 14, Corollary 16.4.2]).

2.5 Simplicial Complexes A k-simplex is the convex hull σ := conv {x0 , . . . , xk } of k + 1 affinely independent points x0 , . . . , xk ∈ Rd . A j -face of σ (j = 0, 1, . . . , d) is any j -simplex given as the convex hull of j + 1 points (vertices) of {x0 , . . . , xk }. The (d − 1)-faces of d-simplices are called facets, the 1-faces are edges and the 0-faces are vertices of σ . If σ is a d-simplex and F is a facet, we denote by ν(F ) ∈ S d−1 its unit outer normal vector. A simplicial k-complex Σ in Rd is a family of k-simplices with the following property: If σ, σ  belong to Σ then σ ∩ σ  is either empty or a face of both σ and σ  . A simplicial k-complex Σ is closed (or face-to-face) if for any σ ∈ Σ, any (k − 1)-face μ of σ is a face of another k-simplex σ  ∈ Σ.

2.5 Simplicial Complexes

43

Finally, a simplicial d-polytope is the union of a locally finite simplicial d complex, P = Σ. If a j -face (facet, edge, vertex) of a simplex σ ∈ Σ lies in ∂P , it as called a j -face (facet, edge, vertex) of P . We denote by Fk (P ) the set of all kfaces of P , k = 0, . . . , d −1. Of course, the notion of a face depends not only on the union P , but on the d-complex Σ. Therefore, whenever speaking about a simplicial d-polytope, we consider the union P together with the simplicial decomposition Σ. When speaking about a d-polytope, we consider only the union P . Fatness is the crucial property of “good” triangulations (we use the term triangulation for simplicial decomposition in any dimension). Given a k-simplex σ = conv {x0 , . . . , xk } in Rd , its size is ρ(σ ) := max |xi − xj |, 0≤i,j ≤k

its altitude h(σ ) := min dist (xi , aff(σi )), 0≤i≤k

where σi := conv {xj : 0 ≤ j ≤ k, j = i}, and and its fatness Θ(σ ) :=

h(σ ) . kρ(σ )

Remark 2.7 The following properties can be easily verified. 1. If μ is a face of a simplex σ then h(μ) ≥ h(σ ), ρ(μ) ≤ ρ(σ ) and Θ(μ) ≥ Θ(σ ). 2. If all edges of a d-simplex σ have lengths at least ε and μ is a k-face of σ then Hk (μ) ≥ Θ(σ )k−1 εk . In order to construct polytopal approximations (Lemma 1.19) we will use Delaunay triangulations and its weighted variant. Let ε > 0, C > 0 and S ⊂ Rd be given and let M be a locally finite point set in S such that any two different points of M have distance at least ε > 0, and dist (x, M) ≤ Cε for all x ∈ S. Let us call such a set M an (ε, C)-net in S. Note that a nonempty set S ⊂ Rd always contains an (ε, 1)-net (consider a maximal subset of points having any two points at least ε apart). Let now M be an (ε, C)-net in Rd . If the points of M lie in general mutual position, meaning that no d + 2 points of M lie on a common (d − 1)-sphere, we define the Delaunay d-complex of M as the simplicial d-complex ΣM consisting of all d-simplices σ with vertices in M and such that (int Bσ ) ∩ M = ∅,

(2.8)

44

2 Background from Convex Geometry

where Bσ denotes the circumscribed ball to σ ; its radius is called circumradius of σ and will be denoted by Rσ . Note that Rσ ≤ Cε,

σ ∈ ΣM ,

since M is an (ε, C)-net. If the mutual position assumption is violated a Delaunay dcomplex of M (i.e., a d-complex with vertices in M whose each d-simplex satisfies (2.8)) still exists but need not be uniquely determined. We refer to [BCY18] and to the references given therein. If d ≤ 2, it is not difficult to show that such a Delaunay complex ΣM consists of uniformly fat triangles (i.e., infσ ∈ΣM Θ(σ ) > 0). This is, however, not true in higher dimension, where “bad” simplices may occur: Consider a 3-simplex whose four vertices lie on a 2ε-sphere, close to a common great sphere, and have mutual distances greater that ε. Such a simplex can have fatness arbitrarily close to 0; these simplices are called “slivers” in the literature. There exist, however, methods modifying slightly the Delaunay construction and removing such slivers. In particular, we can use a weighted Delaunay triangulation, see e.g. [BCY18, Chapter 5], known also as Laguerre Delaunay tessellation or power Delaunay tessellation. Assume that to any point x ∈ M a weight w(x) ∈ [0, ε/2) is assigned. Given a k-simplex σ = conv {x0 , . . . , xk } with vertices xi ∈ M, its orthocentre cσ and orthoradius Rˆ σ are defined by the conditions |xi − cσ |2 = Rˆ σ2 + w(xi )2 ,

i = 0, . . . , k,

and by the minimality of Rˆ σ . The ball B(cσ , Rˆ σ ) is called orthoball of σ ; this name comes from the fact that its boundary sphere intersects the boundary spheres of B(xi , w(xi )) orthogonally. The weighted Delaunay d-complex is a d-complex Σˆ M fulfilling |x − cσ |2 ≥ Rˆ σ2 + w(x)2 ,

x ∈ M,

σ ∈ Σˆ M

(note that this is condition (2.8) in the case w(x) = 0, x ∈ M). Again, the weighted Delaunay triangulation is not unique, unless we assume that no orthoball contains more than d + 1 points on its boundary. It can be easily seen that the orthoradius Rˆ σ of any simplex σ ∈ Σˆ M cannot be larger than Cε, thus we get 1 Rσ ≤ (C + )ε, 2

σ ∈ Σˆ M .

(2.9)

Boissonnat et al. [BCY18, Theorem 5.25] guarantees that there exist weights w(x), x ∈ M, such that the resulting weighted Delaunay d-simplex Σˆ has uniformly fat simplexes. It is formulated for Zd -periodic point patterns (or, equivalently, for point patterns in the torus Td = Rd /Zd ) (Fig. 2.1).

2.5 Simplicial Complexes

45

Fig. 2.1 The orthoball of σ = conv {x0 , x1 , x2 }

B(x2 , w(x2 ))

B(x1 , w(x1 )) cσ ˆσ R B(x0 , w(x0 ))

Theorem 2.8 ([BCY18, Theorem 5.25]) Let M be an (ε, C)-net in Rd , ε, C > 0, and assume that M is Zd -periodic. Then there exist a constant θ = θ (d, C) > 0 (independent of ε) and weights w(x) ∈ [0, ε/2), x ∈ M, such that the weighted Delauynay d-complex Σˆ M fulfills the fatness condition Θ(σ ) ≥ θ,

σ ∈ Σˆ M .

Chapter 3

Background from Differential Geometry and Topology

3.1 Curvature Measures of Smooth Submanifolds There exist close analogues to the curvature measures for convex bodies in classical differential geometry of smooth submanifolds. Basic notions have been developed nearly at the same time starting from the 1930th, mainly within the so-called integral geometry. Assume that a convex body K has a C 2 -boundary ∂K and νK (x) denotes the outer unit normal at x ∈ ∂K; the mapping νK : ∂K → S d−1 is called the Gauss map of K. The differential DνK (x) exists everywhere on ∂K and it is a self-adjoint linear map from Tan(K, x) ∼ = νK (x)⊥ to Tan(S d−1 , νK (x)) ∼ = ⊥ νK (x) . The (real) eigenvalues κ1 (x), . . . , κd−1 (x) ≥ 0 of Dν(x) are called principal curvatures of K at x; we assume that the corresponding orthonormal eigenvectors b1 (x), . . . , bd−1 (x) have the property that (b1 (x), . . . , bd−1 (x), ν(x)) is a positively oriented basis of Rd . For 0 ≤ k ≤ d − 1, we denote the elementary symmetric function of the principal curvatures of order k by sk (K, x) =



k 

κij (x),

x ∈ ∂K,

1≤i1 0, where the kth total curvature Ck (Mm ) of Mm can be expressed in modern notation as follows: Ck (Mm ) =

1 (m − k)ωm−k

 sm−k (Mm , x) Hm (dx)

(3.3)

Mm

if m − k is even. Here sm−k (Mm ; x) agrees with a constant multiple of the trace of the (m − k)/2-th power of the Riemannian curvature tensor R of the manifold at (m−k)/2 (x) is considered as an element point x, which is determined m−k m−k intrinsically. (R of (T M )⊗ (Tx Mm ), where trace(a ⊗b) := a •b for the inner product m−k x m in (Tx Mm ). For details see also Federer [Fed59, Remark 5.21].) If m − k is odd or k > m one obtains Ck (Mm ) = 0. When restricting the integrals of the mean curvature functions to Borel subsets B of Rd one obtains the corresponding curvature measures 1 Ck (Mm , B) := (m − k)ωm−k

 sm−k (Mm ; x) Hm (dx) . Mm ∩B

One of the basic results connecting the local differential geometry of a compact C 2 -submanifold Mm , m ≤ d, with its global topology is the Gauss-Bonnet theorem telling that the total Gauss curvature agrees with the Euler-Poincaré characteristic: C0 (Mm ) = χ(Mm ) .

(3.4)

This can be shown by means of Morse theory in the more general context of Riemannian geometry. Let us turn back to the case of a compact C 2 -domain. Then ∂A is a C 2 submanifold of dimension d − 1 and, calculating the contribution to the Steiner formula of ∂A, we have to consider both normal vectors at each x ∈ ∂A, νA (x) and −νA (x). The principal curvatures will change the sign, which means that the elementary symmetric function associated with −νA (x) equals (−1)k times the elementary symmetric function associated with νA (x). Consequently, the curvature measures of ∂A satisfy Ck (∂A, ·) = (1 + (−1)d−1−k )Ck (A, ·).

50

3 Background from Differential Geometry and Topology

This means, in particular, that for even d − 1 − k the curvature measures Ck (A, ·) are objects of the intrinsic geometry of ∂A. In general, the latter does not hold true for the mean curvatures Hd−1−k (A, x) of the domain A, and thus for their curvature integrals Ck (A, · ), if d − 1 − k is odd. More generally, in classical differential geometry and Riemannian geometry the above notions and partial results have been extended to arbitrary compact C 2 -manifolds with boundary.

3.2 Euler Characteristic Under the Euler-Poincaré characteristic χ(A) of a set A ⊂ Rd we will always mean the Euler characteristic with respect to the singular homology (see [Dol72, Ch. III]). (Nevertheless, we will mainly work with assumptions assuring the equivalence of all homology classes.) Definition 3.1 A set A ⊂ Rd is contractible to a point a ∈ A if there exists a continuous mapping h : A × [0, 1] → A such that h(x, 0) = x and h(x, 1) = a for any x ∈ A. The set A is locally contractible if for any a ∈ A and ε > 0 there exists a neighbourhood a ∈ V ⊂ U (a, ε) such that A ∩ V is contractible to a. Definition 3.2 A continuous mapping f : A → B ⊂ A is a retraction (onto B) if f (b) = b for any b ∈ B. A set A ⊂ Rd is a neighbourhood retract if there exists an open set U ⊃ A and a retraction f : U → A. Theorem 3.3 ([Hat02, Theorem A.7]) If A ⊂ Rd is compact and locally contractible then A is a neighbourhood retract. Theorem 3.4 ([Hat02, Corollary A.8]) If A ⊂ Rd is a compact neighbourhood retract then its homology groups and fundamental group are finitely generated. Hence, the Euler-Poincaré characteristic χ(A) is defined. Theorem 3.5 Assume that X, Y ⊂ Rd are compact and locally contractible, and that at least one of the Euler-Poincaré characteristics χ(X ∩ Y ), χ(X ∪ Y ) exists. Then, the other exists as well, and we have the additivity formula χ(X ∪ Y ) + χ(X ∩ Y ) = χ(X) + χ(Y ). Proof Since X, Y are compact, (Rd , X, Y ) is an excisive triad with respect to the ˇ Cech cohomology (see [Dol72, VIII.6.13,15]). Further, since both X and Y are compact and locally contractible, they are neighbourhood retracts by Theorem 3.3 and ˇ their Cech cohomologies coincide with the usual cohomologies [Dol72, VIII.6.12]. The existence and additivity of the Euler characteristic follows now by [Dol72, Propositions V.4.11 and V.5.8].  

3.4 Morse Theory

51

3.3 Integral Geometry Integral geometry as a branch of classical differential geometry studies kinematic relationships, as e.g. the famous Principal kinematic formula of Santaló, Blaschke and Chern for the integral of the Euler-Poincaré characteristic of the intersection of two bodies as above, one fixed and the other moving. H. Federer [Fed59] extended it to a more general situation including the total curvatures of all orders for compact C 2 -submanifolds and S.S. Chern [Che66] gave a direct differential geometric proof for the latter case (see also Sulanke and Wintgen [SW72] for a different approach to this version). In the above notations Chern proved the following Principal kinematic formula:  Ck (Mm ∩ gMn ) dg =



γ (d, r, s)Cr (Mm )Cs (Mn )

(3.5)

0≤r≤m, 0≤s≤n r+s=d+k

Gd

where 1 ≤ m, n ≤ d and the constants γ are as in the convex case. Moreover, the differential geometric analogue of the measure version of the Principal kinematic formula from the convex case (Theorem 2.4) remains valid:   Ck (Mm ∩ gMn , A ∩ gB) dg = γ (d, r, s)Cr (Mm , A)Cs (Mn , B) 0≤r≤m, 0≤s≤n r+s=d+k

Gd

for Borel sets A and B. The same holds true for the Crofton formula (Theorem 2.5). Note, that even in the convex case both measure versions go back to Federer [Fed59], although some of the global cases were developed much earlier.

3.4 Morse Theory Let A be a compact C 2 -domain in Rd . As above, we denote by Tan(∂A, x) the tangent space and by νA (x) ⊥ Tan(A, x) the unit outer normal vector to A at x ∈ ∂A. The second fundamental form of A at x ∈ ∂A is a symmetric bilinear form defined on Tan(∂A, x) as IIx (u, v) := u · (DνA (x)v),

u, v ∈ Tan(∂A, x).

Let further f be a C 2 -smooth function defined on an open neighbourhood of A without stationary points (i.e., its gradient is nonzero everywhere). We define the Hessian form of f relative to A as the symmetric bilinear form HA f (x) := D 2 f |Tan(∂A, x) − (∇f (x) · νA (x))IIx .

52

3 Background from Differential Geometry and Topology

Definition 3.6 We say that a point x ∈ ∂A is critical for f if −∇f (x) is a positive multiple of νA (x). The value f (x) is then called a critical value of f . Any real number c which is not a critical value is called a regular value of f . Further, we say that a critical point x ∈ ∂A of f is nondegenerate if HA f (x) has full rank (d −1). At a nondegenerate critical point x ∈ ∂A, the index of F relative to A, denoted indf,A (x), is defined as the number of negative eigenvalues of HA f (x). Finally, we say that a smooth function f without stationary points is a Morse function for A if 1. f has only nondegenerate critical points on ∂A, 2. for each c ∈ R there is at most one critical point x ∈ ∂A with f (x) = c. Note that the critical points of a Morse function must be isolated. Thus, any Morse function of a compact C 2 -domain has only finitely many critical points. The classical Morse theory says that given a Morse function for A, the domain A can be represented as a CW complex with each k-cell corresponding to a nondegenerate critical point of index k (see [MC69] or [Hir76]). This yields the following formula for the Euler-Poincaré characteristic of A. Theorem 3.7 If A is a compact C 2 -domain in Rd , f a Morse function for A and c ∈ R then  χ(A ∩ {f ≤ c}) = (−1)indf,A (x) x: f (x)≤c

(the summation is carried out over critical points of f ). In particular, χ(A) =

 (−1)indf,A (x). x

We will need a generalization of the above formula to a more general framework where ∂A is only C 1 smooth, but the unit outer normal νA is Lipschitz on ∂A. Definition 3.8 A closed C 1,1 -domain in Rd is a C 1 -domain (i.e., a d-dimensional C 1 -submanifold with boundary) such that the Gauss map x → νA (x) is Lipschitz. By the well-known Rademacher theorem, a Lipschitz mapping (in a finitedimensional space) is differentiable almost everywhere. Hence, if A is a closed C 1,1 -domain, its Gauss map νA is differentiable Hd−1 -almost everywhere on ∂A. We adopt Definition 3.6 for closed C 1,1 -domains (with the only difference that we understand the nondegeneracy condition so that it presumes the existence of DνA (x)). Fu [Fu89a] showed that the Morse formula for the Euler-Poincaré characteristic holds for compact C 1,1 -domains as well.

3.4 Morse Theory

53

Theorem 3.9 If A is a compact C 1,1 -domain in Rd , f a Morse function for A and c ∈ R then  (−1)indf,A (x) . χ(A ∩ {f ≤ c}) = x: f (x)≤c

In particular, χ(A) =

 (−1)indf,A (x). x

Fu also showed that given a compact C 1,1 domain in Rd , the “height function” hv : x → x · v,

x ∈ Rd ,

is Morse for A for Hd−1 -almost all v ∈ S d−1 [Fu89a, §5]. Note further that D 2 hv = 0 and ∇hv (x) = v, hence x is critical for hv |A if and only if v = −νA (x) and HA hv (x) = −IIx at critical points x ∈ ∂A. Thus, the index indhv ,A (x) agrees with λA (x) := the number of negative principal curvatures at x ∈ ∂A. Consequently, we get the following. Corollary 3.10 If A is a compact C 1,1 -domain in Rd then for Hd−1 -almost all v ∈ S d−1 and all t ∈ R,  χ(A ∩ {x : x · v ≤ t}) = (−1)λA (x) . −1 x∈νA {−v}: x·v≤t

In particular, χ(A) =



(−1)λA (x).

−1 x∈νA {−v}

Chapter 4

Sets with Positive Reach

4.1 Geometric Properties The metric projection to a set ∅ = X ⊂ Rd is not determined everywhere unless X is convex and closed. Sets with positive reach are sets with the property that the metric projection is defined on some open neighbourhood of X. Given a set ∅ = X ⊂ Rd , we define its distance function as dX : x → dist (x, X) = inf{|x − a| : a ∈ X}, and Unp X denotes the set of all x ∈ Rd for which there exists a unique point a ∈ X nearest to x. We write then a =: ΠX (x), and the mapping ΠX : Unp X → X is called the metric projection to X. Lemma 4.1 ΠX is continuous. Proof Assume for the contrary that ΠX is not continuous at a point x ∈ Unp X, i.e., there exist points xi ∈ Unp X, xi → x, infi |ΠX (xi ) − ΠX (x)| > 0. Turning to a subsequence, we may assume that ΠX (xi ) → y ∈ X. Clearly y = ΠX (x), but |y − x| = dX (x), which contradicts the assumption x ∈ Unp X.   The reach function of X is defined as ˚ r) ⊂ Unp X} reach (X, a) := sup{r ≥ 0 : B(a, and the reach of X is set as reach X := inf reach (X, a) a∈X

© Springer Nature Switzerland AG 2019 J. Rataj, M. Zähle, Curvature Measures of Singular Sets, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-18183-3_4

55

56

4 Sets with Positive Reach

x r

ΠX (x)

a

X

Unp X Fig. 4.1 A set X with positive reach, the metric projection ΠX to X and the reach r = reach (X, a). The dashed lines do not belong to Unp X

(i.e., reach X is the supremum of all r ≥ 0 such that any z ∈ Xr has its unique nearest neighbour in X). Note that reach ∅ = ∞ and that any set with positive reach is closed (Fig. 4.1). Examples of sets with positive reach: 1. If K is closed and convex then (and only then) reach K = ∞. 2. If F is finite nonempty then reach F = min{ 12 |x − y| : x, y ∈ F, x = y}. 3. If X is a compact C 2 -domain then reach X > 0. (Without compactness, the assertion is not true.) 4. The union of two disjoint compact sets with positive reach has positive reach. First we show a differentiability property of the distance function. Lemma 4.2 If the distance function dX is differentiable at some point x ∈ Unp X \ X then its gradient is ∇dX (x) =

x − ΠX (x) . |x − ΠX (x)|

Further, dX is continuously differentiable on int (Unp X \ X). Proof It follows from the triangle inequality that dX is 1-Lipschitz, i.e., |dX (y) − dX (x)| ≤ |y − x|,

x, y ∈ Rd .

4.1 Geometric Properties

57

Thus, |∇dX (x)| ≤ 1 whenever dX is differentiable at x. Note further that if x ∈ x−ΠX (x) Unp X \ X and u := |x−Π then X (x)| dX (x − tu) = dX (x) − t,

0 ≤ t ≤ dX (x).

Consequently, ∇dX (x) = u whenever x ∈ Unp X \ X and dX is differentiable at x. The second statement follows from Lemma 4.3 and the continuity of ΠX .   Lemma 4.3 Let f : U → R be Lipschitz, U ⊂ Rd open, g : U → R continuous and 1 ≤ i ≤ d. If ∂ f (x) = g(x) whenever f is differentiable at x ∈ U, ∂xi then ∂ f (x) = g(x) for all x ∈ U. ∂xi Proof Let x ∈ U and r > 0 be such that B(x, 2r) ⊂ U , and let ei denote the ith vector of the canonical basis of Rd . Since f is clearly absolutely continuous in the ith coordinate, we have 

t

f (y + tei ) − f (y) = 0

∂f (y + sei ) ds, ∂xi

y ∈ B(x, r), |t| < r.

We know by Rademacher’s theorem (Theorem 1.8) that f is differentiable almost everywhere on U . Hence, using the assumption and Fubini, 

t

f (y + tei ) − f (y) =

g(y + sei ) ds for almost all y ∈ B(x, r) and all |t| < r.

0

Since both f and g are continuous, we infer 

t

f (x + tei ) − f (x) =

g(x + sei ) ds,

|t| < r,

0

hence

∂f ∂xi (x)

= g(x).

 

Remark 4.4 Applying the implicit function theorem, we obtain that the level sets −1 {r} are (d − 1)-dimensional C 1 -manifolds, 0 < r < reach X. Note also that dX −1 dX {r} = ∂Xr if 0 < r < reach X. A stronger result will be obtained later (Corollary 4.22). Recall that the tangent cone of X ⊂ Rd at a point a ∈ X is defined as the set of all vectors u ∈ Rd such that either u = 0 or there exists a sequence of points ai ∈ X \ {a} with ai → a and ri > 0 such that ri (ai − a) → u, i → ∞. For general

58

4 Sets with Positive Reach

sets X, Tan(X, a) is always a closed cone. We define further the normal cone of X at a ∈ X as the polar cone to Tan(X, a) (see Definition 2.6), i.e., Nor(X, a) := Tan(X, a)o = {v : v · u ≤ 0 for any u ∈ Tan(X, a)}. Clearly, Nor(X, a) is always a closed convex cone. The following geometric property is crucial for sets with positive reach. Lemma 4.5 Let reach (X, a) =: r > 0, a ∈ ∂X and n ∈ S d−1 . The following statements are equivalent. (i) (ii) (iii) (iv)

ΠX (a + tn) = a for some t > 0, ΠX (a + tn) = a for all 0 < t < r, ˚ + rn, r) = ∅, X ∩ B(a n ∈ Nor(X, a).

Proof We start by showing (i) ⇒ (ii). Let (i) be true and denote t0 := sup{t > 0 : ΠX (a + tn) = a}. Note that t0 > 0 by assumption, and that clearly ΠX (a +tn) = a for any 0 < t < t0 . In order to show (ii), we have to verify that t0 ≥ r. Assume, for the contrary, that t0 < r. Then x0 := a + t0 n ∈ int Unp X (since reach X > t0 ). Consider the differential equation g  (s) = ∇dX ◦ g(s),

g(0) = x0 .

By the Peano existence theorem, there exists δ > 0 and a differentiable function g : (−δ, δ) → Rd such that g(0) = x0 and g  (s) = ∇dX (g(s)) for |s| < δ. Since ∇dX is always a unit vector (see Lemma 4.2), we have |g  (s)| = 1. Further, (dX ◦ g) (s) = ∇dX ◦ g(s) · g  (s) = g  (s) · g  (s) = 1, hence, for any −δ < s1 < s2 < δ,  s2 − s1 =

s2 s1



|g (s)| ds =



s2

(dX ◦ g) (s) ds

s1

= dX (g(s2 )) − dX (g(s1 )) ≤ |g(s2 ) − g(s1 )|. It follows that the image of g must be a straight segment of length 2δ, namely (x0 + (t0 − δ)n, x0 + (t0 + δ)n). By the definition of g, any point of this segment will have its metric projection onto X in a, which is a contradiction with the definition of t0 .

4.1 Geometric Properties

59

The implications (ii) ⇒ (iii) ⇒ (iv) are easy. We will show that (iv) ⇒ (i). Assume for the contrary that n ∈ Nor(X, a) ∩ S d−1 but (i) is not true, and denote x(t) := a + tn, t > 0. Clearly x(t) ∈ X for sufficiently small t > 0 (since otherwise, n would be a tangent vector to X at x, contradicting our assumption). Then, a(t) := ΠX (x(t)) = a for all sufficiently small t > 0 by our assumptions, nevertheless a(t) → a, t → 0, by the continuity of ΠX . We will show that the unit x(t )−a(t ) vectors n(t) := |x(t )−a(t )| converge to n as t → 0. We have lim sup(a(t) − a) · n ≤ 0 t →0

since n ∈ Nor(X, a). Thus (x(t) − a) · n + (a − a(t)) · n |x(t) − a(t)| t ≥ 1, ≥ lim inf t →0 |x(t) − a(t)|

lim inf n(t) · n = lim inf t →0

t →0

since |x(t) − a(t)| ≤ |x(t) − a| = t for all t. Hence n(t) → n, t → 0. By the already proven implication (i) ⇒ (iii) applied to a(t) and n(t), the open balls ˚ B(a(t) + rn(t), r) do not intersect X. But we have ˚ + rn, r) ⊂ B(a



˚ B(a(t) + rn(t), r),

t >0

˚ + rn, r) ∩ X = ∅. We have thus shown (iii) which clearly implies (i), a hence B(a contradiction.   Note that for any a, b ∈ Rd , n ∈ S d−1 and r > 0, ˚ + rn, r) if and only if (b − a) · n > b ∈ B(a

|b − a|2 . 2r

(4.1)

Reformulating Lemma 4.5 (iii), we thus obtain the following. Corollary 4.6 If reach X > 0, a, b ∈ X and v ∈ Nor(X, a) then (b − a) · v ≤

|b − a|2 |v| . 2reach X

Recall that Xr denotes the r-parallel set to X (see (2.1)). In the following we show that ΠX is Lipschitz on Xr if r < reach X. Lemma 4.7 If reach X > r and x, y ∈ Xr then |ΠX (y) − ΠX (x)| ≤

reach X |y − x|. reach X − r

60

4 Sets with Positive Reach

Fig. 4.2 X does not intersect the interior of B(a + rn, n) if r = reach X and n ∈ Nor(X, a) ∩ S d−1

B(a + rn, r)

a X

Proof Denote a := ΠX (x) and b := ΠX (y). Using Corollary 4.6, we get |y − x||b − a| ≥ (y − x) · (b − a) = ((y − b) + (b − a) + (a − x)) · (b − a) |b − a|2 |y − b| |b − a|2 |x − a| + |b − a|2 − 2reach X 2reach X   r ≥ |b − a|2 1 − , reach X

≥−

which implies the assertion (Fig. 4.2).

 

There is a connection of the reach of a set with a morphological property of closedness. A set X ⊂ Rd is said to be closed with respect to a set B ⊂ Rd if X = (X⊕(−B))!B (cf. [Mat75], where the Minkowski addition ⊕ and Minkowski subtraction ! are defined). Note that X is closed with respect to B if and only if the complement Rd \ X can be written as a union of translates of −B. ˚ r) Lemma 4.8 If reach X ≥ r then X is closed with respect to the open ball B(0, of radius r. Proof Assume that reach X ≥ r. We will show that Rd \ X is the union of open balls of radius r disjoint with X. Let a point x ∈ Rd \ X be given. If dX (x) ≥ r ˚ r) ∩ X = ∅. If 0 < dX (x) < r then x ∈ Unp X and, denoting then clearly B(x, x−a ˚ + rn, r) ∩ X = ∅ by Lemma 4.5 (iii), a := ΠX (x) and n := |x−a| , we have B(a and we are done.   Example 4.9 The following example shows that the reverse implication in Lemma 4.8 is not true in general. Let 0 < s < r, let x, y be two points in R2 ˚ r) ∪ B(y, ˚ r)). Then X is closed with respect with |x − y| = 2s and X := R2 \ (√ B(x, 2 2 ˚ to B(r), nevertheless, reach X = r − s < r. Definition 4.10 Given a set X ⊂ Rd with reach X > 0, we define the unit normal bundle of X as nor X := {(a, n) : a ∈ ∂X, n ∈ Nor(X, x) ∩ S d−1 }.

4.1 Geometric Properties

61

Lemma 4.11 nor X is a closed subset of R2d . ˚ i+ Proof Let (a, n) = limi→∞ (ai , ni ), (ai , ni ) ∈ nor X for all i. We have B(a ˚ rni , r) ∩ X = ∅ by Lemma 4.5 for r = reach X and all i, hence also B(a + rn, r) ∩ X = ∅. Thus (a, n) ∈ nor X, again by Lemma 4.5.   The closedness of nor X has important consequences. We present two of them. Corollary 4.12 Let reach X > 0 and a ∈ X. (i) If a ∈ ∂X then Nor(X, x) = {0}. (ii) If u ∈ S d−1 belongs to the topological interior of Tan(X, a) then the segment [a, a + εu] is included in X for some ε > 0. Proof (i). If a ∈ ∂X there exist points xi ∈ Rd \ X, xi → a. Then the metric i projections ai := ΠX (xi ) also converge to a and ni := |xxii −a −ai | ∈ Nor(X, ai ) by Lemma 4.5. Passing to a subsequence, we can achieve that ni → n ∈ S d−1 . Then (a, n) ∈ nor X since nor X is closed. (ii). Let u ∈ int Tan(X, a) ∩ S d−1 be given. Assume, for the contrary, that there exists a sequence εi → 0 such that xi := a + εi u ∈ X for all i. Then, as in i the proof of (i), ai := ΠX (a + εi u) → a, ni := |xxii −a −ai | ∈ Nor(X, ai ) and we can assume that ni → n ∈ S d−1 ∩ Nor(X, a). Corollary 4.6 implies that (a − ai ) · ni ≤ (2r)−1 |ai − a|2 with r := reach X, hence u · ni =

1 1 |ai − a|2 (xi − a) · ni ≥ (ai − a) · ni ≥ − . εi εi 2rεi

Letting i → ∞ we get u · n ≥ 0 (note that |ai − a| ≤ 2εi ) which, however, contradits the facts that n ∈ Nor(X, a) = Tan(X, a)o and u ∈ int Tan(X, a).   An important property of sets with positive reach is the convexity of the tangent cones. Recall that C o denotes the polar cone to a cone C (see Definition 2.6). Lemma 4.13 If reach X > 0 and a ∈ ∂X then Tan(X, a) = Nor(X, a)o , hence Tan(X, a) is convex. Proof First note that Nor(X, a)o = Tan(X, a)oo ⊃ Tan(X, a) by (2.6). Hence, it is enough to show that Nor(X, a)o ⊂ Tan(X, a).

(4.2)

62

4 Sets with Positive Reach

Let u be a unit vector which is not a tangent vector to X at a. Then, by the definition of tangent vectors, there exist ε, γ > 0 such that the cone V := {v : (v − a) · u > |v − a| cos γ } does not intersect X ∩ B(a, ε). We can assume that ε ≤ reach X. As in the proof x(t )−a(t ) of Lemma 4.5, denote x(t) := a + tu, a(t) := ΠX x(t), n(t) := |x(t )−a(t )| , 0 < t < ε. An easy geometric observation is that the open cone V contains the open ball ˚ B(x(t), t sin γ ). Clearly |x(t) − a(t)| ≤ t, and since a(t) ∈ X, it must lie outside V and we have thus t sin γ ≤ |x(t) − a(t)| ≤ t,

0 < t < ε.

Further, Corollary 4.6 implies (a − a(t)) · n(t) ≤

|a − a(t)|2 , 2reach X

0 < t < ε.

Using these two estimates, we obtain u · n(t) = t −1 ((x(t) − a(t)) − (a − a(t))) · n(t) = t −1 (|x(t) − a(t)| − (a − a(t)) · n(t)) ≥ sin γ −

|a(t) − a| . reach X

In view of the compactness of the unit sphere, we can find a sequence ti → 0+ such that n(ti ) → n ∈ S d−1 . Since also a(ti ) → a, we have (a, n) ∈ nor X by Lemma 4.11, hence, n ∈ Nor(X, x). As u · n ≥ lim inf u · n(t) > 0, t →0

we infer that u ∈ Nor(X, x)o , and the proof (4.2) is finished.

 

We conclude this subsection with an equivalent condition for reach X ≥ r. Proposition 4.14 Given a closed set X ⊂ Rd and r > 0, the following conditions are equivalent: (a) reach X ≥ r, (b) for any a, b ∈ X, dTan(X,a)(b − a) ≤

|b−a|2 2r .

Remark 4.15 Note that condition (b) can be reformulated as follows: If a, b ∈ X with 0 < |b − a| < 2r then there exists a tangent vector 0 = u to X at a with  (b − a, u) ≤ arcsin b−a . 2r

4.2 Reach of Intersection

63

Proof (a) ⇒ (b). Assume that reach X ≥ r and a, b ∈ X. If a = b or |b − a| ≥ 2r then condition (b) is clearly satisfied with 0 ∈ Tan(X, a). Assume thus that 0 < b−a |b − a| < 2r and denote u0 := |b−a| , γ := arcsin |b−a| 2r and C := {u : u · u0 ≥ |u| cos γ }. We have to show that Tan(X, a) has nontrivial intersection with C. Assume, for the contrary, that the intersection is trivial. Since both are closed convex cones, there must be a hyperplane strictly separating them, i.e., there exists a unit vector w such that u · w < 0 if u ∈ Tan(X, a) and v · w > 0 if v ∈ C. Note that clearly w ∈ Nor(X, a) = Tan(X, a)o . Consider the vector w0 := u0 − (sin γ )w. Clearly, the angle formed by u0 and w0 is less or equal to γ , hence, w0 ∈ C. Further, we have w·

b−a |b − a| = w · u0 = w · (u0 − w0 ) + w · w0 > w · (u0 − w0 ) = sin γ = . |b − a| 2r

But this contradicts Corollary 4.6. (b) ⇒ (a). Assume that (b) holds but reach X < r. Then there exists a point x with s := dX (x) < r which has at least two nearest neighbours in X, i.e., there ˚ s), any are two different point a, b ∈ X ∩ ∂B(x, s). Since X does not intersect B(x, tangent vector to X at a must form an angle with b − a of size at least arcsin |b−a| 2s , which contradicts property (b).  

4.2 Reach of Intersection Example 4.16 Let X be the x-axis and Y the graph of the function f (x) = x 4 sin x1 if x ∈ (−1, 0) ∪ (0, 1) and f (0) = 0. Then reach X, reach Y > 0, but reach (X ∩ Y, 0) = 0. Hence, the property to have positive reach is not preserved by intersections. Under additional assumptions, the intersection preserves positive reach, as the following proposition shows. Proposition 4.17 Let reach X, reach Y ≥ r > 0. Given a ∈ ∂(X ∩ Y ), denote 

 |u + v| η(a) = inf : u ∈ Nor(X, a), v ∈ Nor(Y, a), |u| + |v| > 0 . |u| + |v| Set η = infa∈∂(X∩Y ) η(a), and assume that η > 0. Then for all a ∈ ∂(X ∩ Y ), (i) Tan(X ∩ Y, a) = Tan(X, a) ∩ Tan(Y, a), (ii) Nor(X ∩ Y, a) = Nor(X, a) + Nor(Y, a), (iii) reach (X ∩ Y ) ≥ rη.

64

4 Sets with Positive Reach

Proof First, note that (i) and (ii) are equivalent by (2.7). We will show (ii). Assume that reach X, reach Y ≥ r and let a ∈ X ∩ Y be given. If a ∈ ∂X ∩ ∂Y then (ii) is obvious. If a ∈ ∂X ∩ ∂Y and m ∈ Nor(X, a), n ∈ Nor(Y, a) are unit ˚ + rm, r) ∩ X = B(a ˚ + rn, r) ∩ Y = ∅ by Lemma 4.5 (iii). Hence, vectors, then B(a ˚ + rm, r) ∪ B(a ˚ + rn, r)) ∩ (X ∩ Y ) = ∅. (B(a This implies that both m, n belong to Nor(X ∪ Y, a), and if m, n are linearly independent, then for any point z from the line segment [rm, rn], the open ball ˚ |z − a|) is contained in B(a ˚ + rm, r) ∪ B(a ˚ + rn, r), beeing thus disjoint B(z, with X ∩ Y , which implies that z − a ∈ Nor(X ∩ Y, a) (indeed, Tan(X ∩ Y, a) ⊂ ˚ |z−a|), a) = {u : u·(z−a) ≤ 0}). We have thus shown the inclusion Tan(Rd \ B(z, Nor(X, a) + Nor(Y, a) ⊂ Nor(X ∩ Y, a). For the opposite inclusion, take a vector u ∈ Nor(X, a) + Nor(Y, a), let H be a hyperplane separating u and the convex cone Nor(X, x) + Nor(Y, x), and let v be a vector perpendicular to H and forming an acute angle with u. Then u ∈ int Tan(X, a) ∩ int Tan(Y, a), hence, a + εv ∈ X ∩ Y for sufficiently small ε by Lemma 4.12. Hence v ∈ Tan(X ∩ Y, x) and u ∈ Nor(X ∩ Y, x). Thus (ii) is verified (Fig. 4.3). To show (iii), we will use Proposition 4.14. Let a, b ∈ X ∩ Y be given. We show that (b − a) · w ≤

|b − a|2|w| , 2ηr

w ∈ Nor(X ∩ Y, a).

(4.3)

If w = 0 the inequality is obvious. If w = 0 we can represent w = u + v with some u ∈ Nor(X, a), v ∈ Nor(Y, a) by (ii), at least one of u, v being nonzero. We have η(a)

Fig. 4.3 Two sets X, Y with positive reach and the value of η(a) at a point a ∈ ∂X ∩ ∂Y

Nor(X, a) Y a X Nor(Y, a)

4.2 Reach of Intersection

(b − a) · u ≤

|b−a|2 |u| 2r

65

and (b − a) · v ≤

(b − a) · (u + v) ≤

|b−a|2 |v| 2r

by Corollary 4.6, hence

|b − a|2 (|u| + |v|) |b − a|2|u + v| ≤ , 2r 2ηr

proving (4.3). We know already from (i) and (ii) that Tan(X ∩ Y, a) is a convex cone and Nor(X∩Y, a) = Tan(X∩Y, a)o . Denote u = ΠTan(X∩Y,a)(b−a) and w = b−a −u. Since Tan(X ∩ Y, a) is a cone we have u · w = 0 and, hence, |w| = |b − a| cos θ , where θ is the angle formed by w and b − a. Also w ∈ Nor(X ∩ Y, a), and we infer from (4.3) that cos θ ≤ |b − a|/(2ηr), hence |v| ≤ |b − a|2 /(2ηr). Thus we have shown that dTan(X∩Y,a)(b − a) ≤

|b − a|2 , 2ηr

which implies that reach (X ∩ Y ) ≥ ηr by Proposition 4.14.

 

Definition 4.18 We say that two sets X, Y with positive reach touch if there exist (a, n) ∈ nor X such that (a, −n) ∈ nor Y . Note that X, Y do not touch whenever the number η > 0 from Proposition 4.17 is positive. We will need later an extension of Proposition 4.17 for more than two sets: Theorem 4.19 Let X1 , . . . , Xk be sets with reach Xi ≥ r > 0, i = 1, . . . , k. Let η be the infimum of all numbers |v1 + · · · + vk | |v1 | + · · · + |vk |  such that vi ∈ Nor(Xi , a), i = 1, . . . , k, a ∈ ki=1 Xi and |v1 | + · · · + |vk | > 0. Assume that η > 0. Then for all a ∈ ∂(X1 ∩ · · · ∩ Xk ), (i) Tan(X1 ∩ · · · ∩ Xk , a) = Tan(X1 , a) ∩ · · · ∩ Tan(Xk , a), k 1 k (ii) Nor(X1 ∩ · · · ∩ X  , a) = Nor(X , a) + · · · + Nor(X , a), k i ≥ rηk−1 . (iii) reach i=1 X Proof For k = 2, this is just Proposition 4.17. For general k it follows by induction. Since the proof of (i) and (ii) is easy, we will show only (iii). Suppose that (iii) is valid for at most k − 1 sets and let X1 , . . . , Xk be sets with reach greater or equal to r fulfilling the assumptions. The sets X1 , . . . , Xk−1 also  i k−2 satisfy the assumptions with some η ≥ η and, hence, reach ( k−1 i=1 X ) ≥ rη k−1 i k by induction assumption. Further, let u ∈ Nor( i=1 X , a) and v ∈ Nor(X , a), |u| + |v| > 0. By (ii), we can write u = u1 + · · · + uk−1 for some ui ∈ Nor(Xi , a). Then we have using the assumption and the triangle inequality |u + v| = |u1 + · · · + uk−1 + v| ≥ η(|u1 | + · · · + |uk−1 | + |v|) ≥ η(|u| + |v|).

66

4 Sets with Positive Reach

 i k Hence, the two sets k−1 i=1 X and Xsatisfy the conditions of Proposition 4.17 with k−2 ≤ r and we infer reach ( ki=1 Xi ) ≥ r˜ η = rηk−1 .   r˜ = rη We close this section with a result about sections with small balls. Proposition 4.20 If reach X ≥ r > 0 then for all z ∈ Rd , X ∩ B(z, r) is either empty or reach (X ∩ B(z, r)) ≥ r. Proof Assume that X ∩ B(z, r) = ∅. We will prove that reach (X ∩ B(z, r)) ≥ r. Let a, b ∈ X ∩ B(z, r). Due to Proposition 4.14, we have to show that dTan(X∩B(z,r))(b − a) ≤

|b − a|2 =: ρ. 2r

Obviously, |b − a| ≤ 2r and B(z, r) contains the spindle-shaped set La,b given as the intersection of all closed balls of radius r containing a and b. One easily verifies that Tan(La,b , a) contains the ball B(b − a, ρ). Since reach X ≥ r, we have by Proposition 4.14 dTan(X,a)(b − a) ≤ ρ. Thus there is a vector v ∈ Tan(X, a) with |v − (b − a)| ≤ ρ.Then, also v ∈ B(b − a, ρ) ⊂ Tan(La,b , a) ⊂ Tan(B(z, r), a). Since X and B(z, r) do not touch, we infer v ∈ Tan(X, a) ∩ Tan(B(z, r), a) = Tan(X ∩ B(z, r), a)  

by Proposition 4.17(i), and we are done.

4.3 Unit Normal Bundle Recall that the unit normal bundle of a set X with positive reach is defined as nor X = {(a, n) : a ∈ ∂X, n ∈ S d−1 ∩ Nor(X, a)}. By means of the following mappings we parametrize the “collars” Xr \ X and boundaries ∂Xr for 0 < r < reach X. We define f : (a, n, t) → a + tn,

(a, n, t) ∈ nor X × (0, reach X),

and the sections f (t ) : (a, n) → f (a, n, t),

(a, n) ∈ nor X,

0 < t < reach X.

4.3 Unit Normal Bundle

67

Lemma 4.21 For each 0 < r < reach X, f (r) is a bi-Lipschitz homeomorphism from nor X onto ∂Xr and the restriction of f to nor X×(0, r] is a locally bi-Lipschitz homeomorphism from nor X × (0, r] onto Xr \ X. Proof The mapping f is clearly Lipschitz on nor X × (0, r]. Consider now the inverse mapping   z − ΠX (z) , dX (z) , f −1 : z → ΠX (z), |z − ΠX (z)|

z ∈ Xr \ X.

Its first and third component are Lipschitz on Xr \ X by Lemma 4.7. It is easy to verify that if g is a Lipschitz vector function on A ⊂ Rd and there exist c, C > 0 such that c ≤ |g(x)| ≤ C, x ∈ A, then the normalized vector function g/|g| is Lipschitz on A as well. Thus, the vector function z →

z − ΠX (z) |z − ΠX (z)|

is Lipschitz on Xr \Xs for any 0 < s < r < reach X. Consequently, f −1 is Lipschitz on nor X × (s, r] for any 0 < s < r, hence, locally Lipschitz on nor X × (0, r). The bi-Lipschitz property of f (r) follows immediately (Fig. 4.4).   Corollary 4.22 If 0 < r < reach X then Xr is a closed C 1,1 -domain (see Definition 3.8). Further, the unit normal bundle nor X is a (d − 1)-dimensional Lipschitz submanifold of R2d (in particular, it is locally (d − 1)-rectifiable). z−ΠX (z) Proof The Lipschitz property of the Gauss map z → |z−Π on ∂Xr follows from X (z)| 1 Lemma 4.21. Also, since ∂Xr is C -submanifold, nor X as its bi-Lipschitz image is a Lipschitz submanifold.  

Fig. 4.4 A set X with positive reach and its unit normal bundle

n a

X

68

4 Sets with Positive Reach

4.4 Principal Curvatures and Directions Let X be a set of positive reach in Rd and let (x, n) ∈ nor X be given. For any 0 < r < reach X, x + rn ∈ ∂Xr and the Gauss map νr on Xr (cf. Sect. 3.1) satisfies νr (x + rn) = n. Thus, if 0 < t < t + s < reach X then the mapping y → y + sνt (y) is a bi-Lipschitz homeomorphism between ∂Xt and ∂Xt +s (cf. Lemma 4.21), and its inverse is z → z − sνt +s (z). Consequently, νt is differentiable at y if and only if νt +s is differentiable at y + sn. Further, note that by Rademacher’s theorem, for each 0 < t < reach X, since νt is Lipschitz on the (C 1 -submanifold) ∂Xt , it is differentiable Hd−1 -almost everywhere on ∂Xt . It follows that for Hd−1 -almost all points (x, n) ∈ nor X, νt is differentiable at x + tn, for all 0 < t < reach X. We call points (x, n) ∈ nor X with this property regular. Proposition 4.23 If reach X > 0 and (x, n) ∈ nor X is regular then Tan(nor X, (x, n)) is a (d − 1)-dimensional subspace and there exist vectors b1 (x, n), . . . , bd−1 (x, n) in Rd and numbers κ1 (x, n), . . . , κd−1 (x, n)∈[−reach X, ∞] such that b1 (x, n), . . . , bd−1 (x, n), n form a positively oriented orthonormal basis of Rd and the vectors ⎛

⎞

⎝

1 1 + κi2 (x, n)

bi (x, n), 

κi (x, n) 1 + κi2 (x, n)

bi (x, n)⎠ ,

i = 1, . . . , d − 1,

form an orthonormal basis of Tan(nor X, (x, n)). (We set √ √∞

1+∞2

1 1+∞2

= 1.)

= 0 and

Proof Fix a regular point (x, n) ∈ nor X and 0 < r < reach X. Recall that νr (y) = n at y = x + rn, the Weingarten mapping −Dνr (y) exists and, consequently, there exist principal directions bir (y) ∈ Tan(∂Xr , y) and principal values (curvatures) r (y), n form a positively κir (y); we may assume that the vectors b1r (y), . . . , bd−1 oriented orthonormal basis of Rd . We will show that −

1 1 ≤ κir (y) ≤ , reach X − r r

i = 1, . . . , d − 1.

(4.4)

This follows from the fact that all directional derivatives of νr at y lie within the given bounds, which is a consequence of −

1 νr (z) − νr (y) z − y 1 ≤ · ≤ , reach X − r |z − y| |z − y| r

y, z ∈ ∂Xr .

4.4 Principal Curvatures and Directions

69

Here the left-hand side inequality follows from Corollary 4.6 since reach Xr ≥ reach X − r. For the right-hand side inequality, note that νr (z) − νr (y) z − y 1 ((z − ΠX z) − (y − ΠX y)) · (z − y) · = |z − y| |z − y| s |z − y|2   1 1 (ΠX z − ΠX y) · (z − y) = ≤ , 1− s |z − y|2 s since we have shown in the proof of Lemma 4.7 that (ΠX z − ΠX y) · (z − y) ≥ 0. Thus, (4.4) is proved. The vectors bir (y), i = 1, . . . , d − 1, form a basis of Tan(∂Xr , y). The inverse mapping to f (r) can be written in the form (f (r) )−1 (y) = (y − rνr (y), νr (y)), hence, it is differentiable as well, with differential D(f (r) )−1 (y) = (I − rDνr (y), Dνr (y)), mapping Tan(∂Xr , y) onto Tan(nor X, (x, n)). Thus, the vectors D(f (r))−1 (y)bir (y) = ((1 − κi(r)r)bir , κi(r) bir ),

i = 1, . . . , d − 1,

(4.5)

form a basis of Tan(nor X, (x, n)). Setting bi (x, n) := bir (y) and κi (x, n) :=

⎧ ⎨ ⎩

(r)

κi (y) (r)

1−κi (y)r

∞

if κi(r) r < 1, (r)

if κi

= 1,

i = 1, . . . , d − 1,

(recall that y = x + rn), we get the assertion.

 

Lemma 4.24 The values κi (x, n) from Proposition 4.23 are uniquely determined at any regular point (x, n) ∈ nor X, up to the order. Furthermore, for any 1 ≤ i ≤ d − 1, the subspace Lin {bj (x, n) : κj (x, n) = κi (x, n)} is uniquely determined. Proof Throughout the proof, we shall omit the argument (x, n) at κi and bi . Assume that ⎞ ⎛ κi ⎝ 1 bi ,  bi ⎠ , i = 1, . . . , d − 1, 2 1 + κi 1 + κi2

70

4 Sets with Positive Reach

and ⎛ ⎝

1 1 + (κi )2

bi , 

⎞

κi 1 + (κi )2

bi ⎠ ,

i = 1, . . . , d − 1,

are two orthonormal bases of Tan(nor X, (x, n)), where {bi }, {bi }, are two orthonormal bases of n⊥ . Then there exist coefficients cij such that 



1 1 + (κi )2 κi 1 + (κi )2

bi =



cij 

j

bi =



cij 

j

1 1 + κj2 κj 1 + κj2

bj ,

(4.6)

bj .

(4.7)

Fix some 1 ≤ i ≤ d − 1 and assume first that κi < ∞. Multiplying (4.6) with κi , we get  κ κi  bi = cij  i bj 1 + κj2 1 + (κi )2 j

(4.8)

and, comparing (4.7) and (4.8), we obtain that ⎛ cij ⎝ 

κj 1 + κj2

−

κi 1 + κj2

⎞ ⎠=0

for all j . Consequently, we have κj < ∞ and cij κi = cij κj for all j , hence, the alternative cij = 0 or κi = κj

(4.9)

holds for any j . Assume now that κi = ∞. Then we have zero on the left hand side of (4.6) which  implies that cij / 1 + κj2 = 0, hence cij = 0 or κj = ∞, for all j . Thus (4.9) holds again for all j . It follows from (4.9) that the sets of numbers {κi : 1 ≤ i ≤ d − 1} and {κi : 1 ≤ i ≤ d − 1} coincide and that any bi is a linear combination of those bj belonging to the same κi . From this property, the assertion follows.  

4.5 Curvature Measures and Steiner Formula

71

Definition 4.25 The numbers κ1 (x, n), . . . , κd−1 (x, n) ∈ [−reach X, ∞] are called (generalized) principal curvatures of X at (x, n), and the unit vectors b1 (x, n), . . . , bd−1 (x, n) are the associated principal directions. The function sk (X; x, n) =



%k

j =1 κij (x, n) %d−1  1 + κi2 (x, n) 1≤i1 s (indeed, if dX (y) = r + s and x = ΠX (y) then ΠXr (y) = sx+ry r+s ) and Xr+s = (Xr )s , we have also by the Steiner formula 

d−1 −1 (E) = k (Xr , E). Ld (Xr+s \ X) ∩ Π ωd−k s d−k C X k=0

Comparing the coefficients at sd−k in the above two expressions, we obtain the desired formula.   As a corollary we infer the vague convergence of curvature-direction measures of Xr to those of X. A much stronger continuity result will be shown in Chap. 7 (Theorem 7.3). Corollary 4.35 If reach X > 0 and k ∈ {0, 1, . . . , d − 1} then v k (Xr , ·) → k (X, ·), C C

r →0+.

v

Consequently, also Ck (Xr , ·) → Ck (X, ·), r → 0+. If X is bounded then also Ck (Xr ) → Ck (X), r → 0+. Let F be an affine j -subspace of Rd (0 < j < d) and let Y ⊂ F have positive reach. We can consider the curvature (curvature-direction) measures of Y to be defined in Rd as above, or to be defined relatively in F . (In fact, the curvature-direction measures are determined through the refined Steiner formula, cf. Theorem 4.30, using the j -volume in F .) We will use the upper index (F ) for the curvature(-direction) measures defined relatively in F . It turns out that the curvature

76

4 Sets with Positive Reach

measures are the same, whatever embedding space we choose, and the curvaturedirection measures (which live on different spaces) are simply related, see the following lemma. The symbol μvar denotes the variation of a (signed) measure μ. Lemma 4.36 Let Y ⊂ F ⊂ Rd have positive reach, where F ∈ Adj . Then, (F )

(i) Ck (Y, B) = Ck (Y, B), B ⊂ F Borel, k = 0, . . . , j , (F ) (Y, E) = C k (Y, ξ −1 (E)), E ⊂ F × S j −1 Borel, k = 0, . . . , j − 1, (ii) C L k (F ) )var (Y, E) = C var (Y, ξ −1 (E)), E ⊂ F × S j −1 Borel, k = 0, . . . , j − 1, (iii) (C k L k where L ∈ G(d, j ) is the j -subspace parallel to F and ξL : F × (S

d−1

⊥

\L )→F ×S

j −1

,

  pL n (x, n) → x, . |pL n|

Proof Let B ⊂ F be a bounded Borel set. Since Y ⊂ F , we can decompose the parallel set (in Rd ) as follows: Yr ∩ ΠY−1 (B) = {y + z : y ∈ F ∩ ΠY−1 (B), z ∈ L⊥ , dist (y, Y )2 + |z|2 ≤ r 2 }, and using subsequently the Fubini theorem and the Coarea theorem with z → |z|, we get Ld (Yr ∩ ΠY−1 (B)) =

 L⊥ ∩B(0,r)

Lj (Y√r 2 −|z|2 ∩ ΠY−1 (B)) Ld−j (dz) 

r

= (d − j )ωd−j

s d−j −1 Lj (Y√

0

r 2 −s 2

∩ ΠY−1 (B)) ds.

Applying the Steiner formula (Corollary 4.33) relatively in F , we obtain Lj (Y√

r 2 −s 2

∩ ΠY−1 (B)) =

j 

ωj −k (r 2 − s 2 )

j−k 2

Ck(F ) (Y, ΠY−1 (B)).

k=0

A routine calculation (use substitution s 2 = r 2 t) yields  (d − j )ωd−j ωj −k

r

s d−j −1 (r 2 − s 2 )

j−k 2

ds = ωd−k r d−k ,

0

hence, L

d

(Yr ∩ ΠY−1 (B))

=

j  k=0

ωd−k r d−k Ck (Y, ΠY−1 (B)). (F )

4.6 Normal Cycle

77

Comparing the polynomial coefficints with those of the Steiner formula (Corollary 4.33, now in Rd ), we arrive at (i). In order to show (ii), we use a similar decomposition with a bounded Borel set E ⊂ Y × S j −1 : −1 ((ξL )−1 (E)) (Yr \ Y ) ∩ Π Y −1 (E), z ∈ L⊥ , dist (y, Y )2 + |z|2 ≤ r 2 }, = {y + z : y ∈ F ∩ (Yr \ Y ) ∩ Π Y and we proceed exactly in the same way as when proving (i). Finally, in order to prove (iii), we use the description of the relative (in F ) unit normal bundle of Y   nor(F ) Y = ξL nor Y ∩ (Y × (S d−1 \ L⊥ )) . Let (y, u) ∈ nor(F ) Y be a regular point with principal curvatures κ1 (y, u), . . . , κj −1 (y, u). Any point (y, n) ∈ nor Y ∩ ξL−1 (y, u) can be written in the form n = (cos t)u + (sin t)v with some v ∈ L⊥ ∩ S d−1 and t ∈ [0, π2 ), and the principal curvatures at such a point (y, n) are κi (y, n) = (cos t)κi (y, u), i = 1, . . . , j − 1, and κi (y, n) = ∞, i = j, . . . , d − 1. Thus the symmetric functions of principal curvatures do not change signs on the fibres ξL−1 (y, v) and, hence, (iii) follows from (ii).  

4.6 Normal Cycle Proposition 4.23 provided an orthonormal basis of the tangent space to nor X at regular points (x, n) ∈ nor X. Thus, aX (x, n) :=

d−1 

⎞

⎛ 1

κi (x, n)

⎝ bi (x, n),  bi (x, n)⎠ 2 (x, n) 2 (x, n) 1 + κ 1 + κ i=1 i i

is a unit simple (d − 1)-vectorfield orienting the locally (d − 1)-rectifiable set nor X. Recall the definition of the bi-Lipschitz mapping f (r) : nor X → ∂Xr from Sect. 4.4 (0 < r < reach X), the notation for the Gauss mapping νr of ∂Xr , and the r (y) of X at a point y ∈ ∂X . principal directions b1r (y), . . . , bd−1 r r Lemma 4.37 We have for Hd−1 -almost all (x, n) ∈ nor X aX (x, n) =



d−1 D(f

(r) −1

)

 r (x + rn) (b1r (x + rn) ∧ · · · ∧ bd−1 (x + rn)).

Further, the mapping (x, n) → aX (x, n) is Hd−1 -measurable on nor X.

78

4 Sets with Positive Reach

Proof The formula was, in fact, shown in the proof of Proposition 4.23. The measurability of aX follows then from the Area formula for currents (Theorem 1.49) r ) and the mapping (f (r) )−1 .  applied with T = (Hd−1 ∂Xr ) ∧ (b1r ∧ · · · ∧ bd−1  Integration over oriented Hausdorff rectifiable sets can be treated by means of rectifiable currents (see Definition 1.43). Definition 4.38 If reach X > 0, we define the current NX ∈ Dd−1 (R2d ) by NX = (Hd−1

nor X) ∧ aX

and call it normal cycle associated with X. We state several important properties of the normal cycle. We will use the following notion. Definition 4.39 The contact 1-form is the differential form α ∈ D1 (R2d ) given by (u, v), α(x, n) = u · n,

u, v, x, n ∈ Rd .

A current T ∈ Dk (R2d ) (k ≥ 1) is called Legendrian if T

α = 0.

Proposition 4.40 The current NX has the following properties for any set X with positive reach. (i) (ii) (iii) (iv)

NX is a locally integral current. spt NX ⊂ ∂X × S d−1 . ∂NX = 0 (NX is a cycle). NX is Legendrian.

Proof The first two assertions follow immediately from the definition. To prove (iii), recall (Lemma 4.21) that there exists a bi-Lipschitz homeomorphism f (r) : nor X → ∂Xr whenever 0 < r < reach X. Then, the push-forward Tr := (f (r) )# NX is a (d − 1)-current given by integration over the oriented boundary ∂Xr which is a C1 -manifold without boundary (see Corollary 4.22), hence, ∂Tr = 0 by Theorem 1.41. Since f (r) is bi-Lipschitz, we have also NX = (f (r))−1 # Tr and −1 (r) ∂NX = (f )# ∂Tr = 0 due to (1.29). (iv) follows from the fact that  (NX α)(φ) = aX , α ∧ φ dHd−1 nor X

and aX , α ∧ φ = 0 almost everywhere on nor X since all the principal directions bi (x, n) are perpendicular to n.   In order to relate the normal cycle to the curvature measures, we need the following algebraic notion. Recall that π0 (x, y) = x, π1 (x, y) = y.

4.6 Normal Cycle

79

Definition 4.41 The Lipschitz-Killing curvature form of order k ∈ {0, . . . , d − 1} is the (d − 1)-form ϕk on R2d given by a1 ∧ · · · ∧ ad−1, ϕk (x, n)  1 = (d − k)ωd−k

πj1 (a1 ) ∧ · · · ∧ πjd−1 (ad−1 ) ∧ n, Ωd ,

j1 +···+jd−1 =d−k−1 ji ∈{0,1}

a1 , . . . , ad−1 ∈ R2d , (x, n) ∈ R2d . Since ϕk (x, n) does not depend on the first vector coordinate x, we write usually only ϕk (n). In particular, ϕ0 = (dωd )−1 (π1 )# (id

Ωd ).

Note that, applying ϕk to the vectorfield aX defined above, we get aX (x, n), ϕk (n) =

1 sd−1−k (X; x, n) (d − k)ωd−k

(4.14)

at any regular point (x, n) ∈ nor X. It follows from Definitions 4.28 and 4.38 that Theorem 4.42 If reach X > 0, 0 ≤ k ≤ d − 1 and E is a bounded Borel subset of R2d then k (X, E) = (NX C

1E )(ϕk ).

Remark 4.43 The orientation of aX is chosen so that, for Hd−1 -almost all (x, n) ∈ nor X, aX (x, n), ϕk(x,n)(n) > 0,

(4.15)

where k(x, n) = max{k ≤ d − 1 : aX (x, n), ϕk(x,n) (n) = 0}. (In other words, k(x, n) is the number of nonzero principal curvatures at (x, n).) Lemma 4.44 We have for Hd−1 -almost all (x, n) ∈ nor X,  (π0 + tπ1 )aX (x, n) d−1 = n, 0 < t < reach X | d−1 (π0 + tπ1 )aX (x, n)| (for the notion of the Hodge star operator , see Definition 1.29). Consequently, & 

d−1 (π0

'

+ tπ1 )aX (x, n) ∧ n, Ωd > 0,

0 < t < reach X.

80

4 Sets with Positive Reach

Proof In fact, both sides of the first equation are equal to b1 (x, n) ∧ · · · ∧ bd−1 (x, n) since (b1 (x, n), . . . , bd−1 (x, n), n) is supposed to be a positively oriented orthonormal basis. Note further that |



d−1 (π0

+ tπ1 )aX (x, n)| = Jd−1 f (t )(x, n)

with the mapping f (t ) from Lemma 4.21 which is bi-Lipschitz and, hence, its Jacobian is nonzero.   Corollary 4.45 Let X, Y ⊂ Rd be two sets with positive reach, assume that (x, n) is regular in both nor X and nor Y and that Tan(nor X, (x, n)) = Tan(nor Y, (x, n)). Then aX (x, n) = aY (x, n). Proof Let T denote the common tangent space of nor X and nor Y at (x, n) and denote Lt : n⊥ → T the inverse to the linear mapping π0 + tπ1 restricted to T . Then we have, using Lemma 4.44,  ( d−1 Lt )(n) aX (x, n) = aY (x, n) =  . |( d−1 Lt )(n)|

 

4.7 Basic Properties of Curvature Measures Given a Euclidean motion g ∈ Gd , we denote the associated mapping g(x, ˜ n) = (g(x), O(n)),

(x, n) ∈ R2d ,

where g(x) = O(x) + z is the decomposition of g into the shift by z ∈ Rd and orientation preserving orthogonal mapping O ∈ SO(d). Theorem 4.46 (Motion covariance) If X ⊂ Rd has positive reach and g ∈ Gd then NgX = g˜# NX .

(4.16)

4.7 Basic Properties of Curvature Measures

81

Consequently, for any 0 ≤ k ≤ d − 1, k (gX, gE) k (X, E) and Ck (gX, gB) = Ck (X, B) C ˜ =C

(4.17)

whenever E ⊂ R2d , B ⊂ Rd are bounded Borel sets. Proof From the definition of the unit normal bundle we find that nor gX = g(nor ˜ X). It is well-known in classical differential geometry that the principal curvatures are motion invariant and the principal directions are motion covariant. We infer from Corollary 4.26 that the same is true for sets with positive reach: κi (gX; g(x, ˜ n)) = κi (X; x, n),

bi (gX; g(x, ˜ n)) = Obi (X; x, n),

i = 1, . . . , d − 1 (we use here an extended notation for principal curvatures and directions refering to the set the are associated with). Consequently, since D g(x, ˜ n)(u, v) = (O(u), O(v)), we can write ˜ n)) = agX (g(x,



˜ n) d−1 D g(x,



aX (x, n).

Applying the Area formula for currents (Theorem 1.49), we obtain (4.16) (note that g˜ is a bijection). Formulas in (4.17) follows directly using Theorem 4.42.   The next result is the homogeneity of curvature measures. Note that reach tX > 0 whenever reach X > 0 and t > 0. Proposition 4.47 (Homogeneity) If reach X > 0, t > 0 and B ⊂ Rd is a bounded Borel set then Ck (tX, tB) = t k Ck (X, B),

i = 0, . . . , d.

Proof Applying the Steiner formula (Corollary 4.33) to tX and tB we get for 0 < tr < reach X d

 Ld (tX)t r ∩ Πt−1 ωd−k r d−k t d−k Ck (tX, tE). X (tE) = k=0

Since the Lebesgue measure is homogeneous of order d, the left-hand side equals t d Ld (Xr ∩ ΠX−1 (B)) = t d

d 

ωd−k r d−k Ck (X, B),

k=0

where we applied again the Steiner formula. Comparing the coefficients in the polynomials, we get the assertion.  

82

4 Sets with Positive Reach

Remark 4.48 Proposition 4.47 can be shown also directly from the integral representation of curvature measures in Definition 4.28, using the fact that the principal curvatures of tXat a point (tx, n) are those of X at (x, n) divided by t. An important property is that both NX and Ck (X, ·) depend additively on X. Theorem 4.49 (Additivity) Let X, Y ⊂ Rd be such that r0 := min{reach X, reach Y, reach (X ∪ Y )} > 0. Then we have: (i) reach (X ∩ Y ) ≥ r0 ; (ii) 1nor(X∪Y ) (x, n) = 1nor(X) (x, n) + 1nor(Y ) (x, n) − 1nor(X∩Y ) (x, n), (x, n) ∈ Rd × S d−1 ; (iii) NX∪Y = NX + NY − NX∩Y ;  ∪ Y, ·) = C(X,   ·) − C(X  ∩ Y, ·). (iv) C(X ·) + C(Y, Proof Let a point z satisfy dX (z) < r0 and dY (z) < r0 . If dX (z) = dY (z) then necessarily ΠX (z) = ΠY (z) (since otherwise, we would get a contradiction with the unique footpoint property for X ∪ Y ). If, on the other hand, say dX (z) < dY (z) then ΠX (z) = ΠX∪Y (z) and ΠY (z) = ΠX∩Y (z). The last equality can be seen as follows: by continuity of the distance, there must be a point w in the segment [z, ΠY (z)] having the same distance from X and Y , and by the same argument as above, ΠX (w) = ΠY (w), but this is only possible if w = ΠY (z) ∈ X. The opposite inequality dX (z) > dY (z) can be treated analogously. In all cases, it follows that z ∈ Unp (X ∩ Y ), hence, reach (X ∩ Y ) ≥ r0 . Further, if follows from the above z−ΠX (z) observation that (ii) holds whenever (x, n) = (ΠX (z), |z−Π ) for X = X or X (z)| X = Y and z ∈ X. If, on the other hand, (x, n) is not of this type then clearly it does not belong to any of the four unit normal bundles in appearing (ii). This verifies (ii). For (iii), we have to show that for Hd−1 -almost all (x, n), 1nor(X∪Y ) (x, n)aX∪Y (x, n) = 1nor(X) (x, n)aX (x, n) + 1nor(Y ) (x, n)aY (x, n) −1nor(X∩Y ) (x, n)aX∩Y (x, n). Due to Proposition 1.19, we can assume that Tan(nor X, (x, n)) is the same subspace whenever (x, n) ∈ nor X and X is X, Y, X ∪ Y or X ∩ Y . Then the last equality follows from (ii) and Corollary 4.45. (iv) follows from (iii) by definition.   The next property says that the curvature measures are locally determined. Proposition 4.50 (Locality of curvature measures) If X, Y are two sets of positive reach such that U ∩ X = U ∩ Y for some bounded open set U ⊂ Rd then NX

1U ×S d−1 = NY

1U ×S d−1 .

4.8 Gauss-Bonnet Formula

83

k (X, E) = C k (Y, E) for any Borel set E ⊂ U × S d−1 and k = In particular, C 0, . . . , d − 1. Proof Follows from the locality of principal curvatures (see Remark 4.29) and Definition 4.28.   In Chap. 8 we will show that the above properties characterize the curvature measures as some basic Euclidean invariants. Note that they extend the well-known relationships from convex geometry (cf. Chap. 2).

4.8 Gauss-Bonnet Formula The classical Gauss-Bonnet formula says that for a C 2 -smooth compact manifold M without boundary, C0 (M) = χ(M), where χ is the Euler-Poincaré characteristic (cf. (3.4)). We will show that the same is true for sets with positive reach. Assume first that X is a C 1,1 domain. Then, Corollary 3.10 says that for Hd−1 almost all v ∈ S d−1 ,  χ(X) = (−1)λX (x), (4.18) −1 x∈νX {v}

where νX is the Gauss map of X and λX (x) is the number of negative principal curvatures at x ∈ ∂X. We will show that this already implies the Gauss-Bonnet formula for C 1,1 -domains. Proposition 4.51 If X ⊂ Rd is a compact C 1,1 domain then C0 (X) = χ(X). Proof The total curvature of order zero is C0 (X) = (dωd )

−1



d−1 

κi (x) Hd−1 (dx)

∂X i=1

(cf. Sect. 3.1). The Jacobian of the Gauss map νX : x → νX (x) on ∂X is d−1      Jd−1 νX (x) =  κi (x)   i=1

84

4 Sets with Positive Reach

whenever νX is differentiable at x ∈ ∂X. Thus, using the Area formula (Theorem 1.21), we get C0 (X) = (dωd )−1 = (dωd )−1

 (−1)λX (x) Jd−1 νX (x) Hd−1(dx) 

∂X

 S d−1

(−1)λX (x) Hd−1 (dv),

−1 x∈νX {v}

and the last expression equals χ(X) by (4.18).

 

We further approximate a general set X of positive reach by its parallel neighbourhoods which are already C 1,1 domains. Lemma 4.52 If 0 < r < reach X then the set X and its parallel neighbourhood Xr are homotopy equivalent. Proof Consider the mapping h : Xr × [0, 1] → Xr ,

(z, t) → tΠX (z) + (1 − t)z.

It is an easy exercise to verify that h is a homotopy with h(z, 0) = z for all z and   h(Xr × {1}) = X. The last three statements, together with the continuity result (see Corollary 4.35) lim C0 (Xr ) = C0 (X)

r→0+

imply already the Gauss-Bonnet formula. Theorem 4.53 (Gauss-Bonnet formula for sets with positive reach) If X ⊂ Rd is compact and reach X > 0 then C0 (X) = χ(X). Finally, we show that (4.18) holds for compact sets with positive reach as well. Theorem 4.54 If X ⊂ Rd is compact and reach X > 0 then for Hd−1 -almost all n ∈ S d−1 ,  χ(X) = (−1)λX (x,n) , x: (x,n)∈nor X

where λX (x, n) is the number of negative principal curvatures of X at a regular point (x, n) ∈ nor X. Remark 4.55 Since regular points in nor X have full (d − 1)-dimensional measure, the right hand side in the above formula is well defined for Hd−1 -almost all n ∈ S d−1 .

4.9 Bibliographical Notes

85

Proof Let X be compact and 0 < r < reach X. Corollary 4.26 implies that the principal curvatures of X at a regular point (x, n) and of the parallel set Xr at x + rn have the same signs. Thus, λX (x, n) = λXr (x + rn) for Hd−1 -almost all (x, n) ∈ nor X and the assertion follows from (4.18) by approximating X with the compact C 1,1 -domain Xr .  

4.9 Bibliographical Notes 1. Sets with positive reach have been introduced in the fundamental paper by Federer in 1959 [Fed59] where also curvature measures, Steiner formula and Principal kinematic formula were proved. Representation of curvature measures as integrals of local curvatures, as well as the current representation by means of normal cycles, were proved in [Zäh86b]. 2. Sets with positive reach in Riemannian manifolds have been considered by Kleinjohann [Kle81] and Bangert [Ban82] who showed that the property reach (X, x) > 0 for all x ∈ X is independent of the Riemannian structure in a smooth connected manifold. In particular, a compact set has (locally) positive reach if and only if it is the sublevel set of a proper, bounded from below and semiconvex function at a weakly regular value. 3. Further characterizations and properties of sets with positive reach in Riemannian manifolds were obtained by Lytchak [Lyt04, Lyt05]. In particular, a compact set X has positive reach if and only if there exist positive constants r, K such that any two points in x, y ∈ X of distance dist (x, y) < r can be joined in X by a C 1,1 -path of length less than Kdist (x, y) and C 1,1 -norm less than K (see [Lyt05, Theorems 1.2, 1.3]). 4. Reach of graphs and subgraphs. Federer [Fed59, §4.20] stated that the graph of a Lipschitz function f : Rd → R has positive reach if and only if f has a Lipschitz derivative. Fu [Fu85, Theorem 2.3] showed that if f is locally Lipschitz then the subgraph of f has positive reach if and only if f is semiconcave (which means that f can be written in the form f (x) = C|x|2 − g(x) for some convex function g and constant C ≥ 0). 5. Let X ⊂ Rd have positive reach and topological dimension 0 < k ≤ d, and denote X(j ) := {x ∈ X : dim Nor(X, x) ≥ d − j }, 0 ≤ j ≤ d. Federer [Fed59] showed that X \ X(k−1) is open if k = d and is a C 1,1 k-dimensional submanifold of Rd if k < d. Further, he showed that X(k−1) is countably Hk−1 rectifiable. The last statement was strengthened in [RZ17] where it was shown that X(k−1) can be covered by locally finitely many DC surfaces of dimension k − 1. 6. Lower dimensional sets of positive reach. If X ⊂ Rd is a topological manifold of dimension 0 < k < d and reach X > 0 then X is already a k-dimensional C 1,1 submanifold of Rd (i.e., it can be locally represented as the graph of a C 1 function with Lipschitz differential). This was stated already by Federer [Fed59, Remark 4.20] and proved essentially by Lytchak [Lyt05, Proposition 1.4]. Note

86

4 Sets with Positive Reach

that, however, the mapping a → Tan(X, a) need not be globally Lipschitz on X (see [RZ17, Example 7.13]). 7. A full characterization of planar sets with positive reach and of one-dimensional sets of positive reach was obtained in [RZ17]. The topological structure of a general set in X ⊂ Rd of positive reach can be rather complicated; though, of course, X itself is locally contractible, its boundary need not be and its complement can have infinitely many components (consider the set {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ dK (x)2 } with an uncountable totally disconnected set K ⊂ [0, 1]). 8. The Steiner formula goes back to Jacob Steiner who proved it for polytopes and smooth convex bodies in dimensions 2 and 3. The related tube formula was obtained by Weyl [Wey39]. Its proof for sets of positive reach (in particular, for general convex bodies) is due to Federer [Fed59]. 9. Hug, Last and Weil [HLW04] proved the following Steiner-type formula valid for any closed set X ⊂ Rd and bounded measurable function f : Rd → R with compact support:  Rd \X

=

f (y) Ld (dy)

d−1  k=0

 Od−k

 N + (X) 0

δ(X,x,n)

t d−1−k f (x + tn) dt μk (X, d(x, n)),

where N + (X) =

   y − ΠX (y) : y ∈ Unp X \ X , ΠX (y), |y − ΠX (y)|

δ(X, x, n) = inf{δ > 0 : x + δn ∈ Unp X},

(x, n) ∈ N + (X),

and μk (X, ·) are certain signed Borel measures on N + (X) for which the above k (X, ·). integrals converge. If reach X > 0 then μ(X, ·) = C 10. The normal cycle has been considered by Sulanke and Wintgen [SW72] for smooth submanifolds. For sets with positive reach, the notion was introduced in [Zäh86b]. Closed integral currents with the Legendrian property have been studied in more general context by Fu, see e.g. [Fu89a, Fu94]. 11. Stochastic processes of sets with positive reach and their curvature measures were considered in [Zäh86a].

Chapter 5

Unions of Sets with Positive Reach

5.1 Topological Index Functions and Additive Extension of Normal Cycles Though the family PR of sets with positive reach covers both the classical geometric sets, i.e., convex sets and C 1,1 smooth submanifolds it does not include their natural singular extensions: the polyconvex sets and the piecewise smooth submanifolds, respectively. Therefore the natural question arises whether we can consider unions of sets with positive reach under the above aspects. To this aim we recall the additivity properties of the unit normal bundles, the normal cycles and the curvaturedirection measures of PR-sets from Theorem 4.49. If X, Y ⊂ Rd are such that r0 := min{reach X, reach Y, reach (X ∪ Y )} > 0. then we have: (i) reach (X ∩ Y ) ≥ r0 ; (ii) 1nor(X∪Y ) (x, n) = 1nor X (x, n) + 1nor Y (x, n) − 1nor(X∩Y ) (x, n), (x, n) ∈ Rd × S d−1 ; (iii) NX∪Y = NX + NY − NX∩Y ;  ∪ Y, ·) = C(X,   ·) − C(X  ∩ Y, ·). (iv) C(X ·) + C(Y, A first aim of this section is to extend the notion of normal cycle and of the curvature-direction measures in such a way that formulas (iii) and (iv) remain valid if we merely suppose that reach X > 0, reach Y > 0 and reach (X ∩ Y ) > 0. Definition 5.1 Let UPR denote the family of all subsets of Rd which can be represented as a locally finite union of sets with positive reach whose arbitrary finite intersections also have positive reach. If X ∈ UPR , we call X=

∞ 

Xi

(5.1)

i=1

© Springer Nature Switzerland AG 2019 J. Rataj, M. Zähle, Curvature Measures of Singular Sets, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-18183-3_5

87

88

5 Unions of Sets with Positive Reach

a UPR -representation of X, provided that 1. {i ∈ N : Xi ∩ K = ∅} is finite for any K ⊂ Rd compact, 2. reach i∈I Xi > 0 for any finite set I ⊂ N. Seeking for an additive extension of the normal cycle to UPR , we could take a UPR -representation (5.1) and apply the inclusion-exclusion principle: NX =



(−1)card(I )−1 N  Xi ,

I ∈P∗ (N)

(5.2)

i∈I

where P∗ (N) denotes the family of nonempty finite sets of natural numbers. This formula could be used as a definition of the normal cycle of the set X if the right hand side is independent of its UPR -representation. (Note that the sum is finite when applied to a differential form with compact support.) In order to make this rigorous we interpret formula (ii) in terms of a local topological index function via Euler numbers admitting the corresponding generalizations (Fig. 5.1). Definition 5.2 For any X ⊂ Rd , x ∈ Rd and n ∈ S d−1 , the index of X at x in direction n is defined by  iX (x, n) = 1X (x) 1 − lim



 lim χ X ∩ B(x + (ε + δ)n, ε)

(5.3)

ε→0+ δ→0+

provided that the right hand side is determined, where χ denotes the Euler-Poincaré characteristic in the sense of singular homology. In the proof of Theorem 5.6 below we will see that iX (x, n) = 1nor X (x, n) if reach X > 0 and, therefore Eq. (5.2) in this case corresponds to the additivity of the Euler characteristic. At the same time the latter will guarantee our additive extensions. In order to show the existence of the index function for UPR -sets we will need the following propositions which are also of independent interest.

n

Fig. 5.1 The index function iX (x, n) = 1 − 2 = −1 ε

x + (ε + δ)n

x X

5.1 Topological Index Functions and Additive Extension of Normal Cycles

89

Proposition 5.3 Any X ∈ UPR is locally contractible.  i i Proof Let X = ∞ i=1 X be aUPR -representation, x ∈ X, I (x) := {i : x ∈ X } i i and r > 0 be such that reach ( i∈J X ) > r for all J ⊂ I (x) and B(x, r) ∩ X = ∅ for any i ∈ / I (x). We will show that X ∩ B(x, r) is contractible. The main point is  the existence of a continuous mapping φ : B(x, r) → i∈I (x) Xi which preserves  points from i∈I (x) Xi . Then we can use the contraction given by g(y, t) := φ(tx + (1 − t)y),

y ∈ X ∩ B(x, r),

t ∈ [0, 1]

(cf. the proof of Lemma 1.49 where the metric projection in case X ∈ PR was used). We will define φ so that φ(z) ∈ Xi if and only if dXi (z) = dX (z). For given z ∈ B(x, r) denote di := dXi (z) and suppose that I (x) = {1, . . . , N} and d1 ≤ · · · ≤ dN . Let Π1,...,i be the metric projection onto X1 ∩. . .∩Xi . We introduce a sequence of points (zN , zN−1 , . . . , z1 ) by zN := Π1,...,N (z) and zi := Π1,...,i

   di di 1− Π1,...,i (z) + zi+1 , di+1 di+1

i = N − 1, . . . , 1

(setting 0/0 := 1). Then put φ(z) := z1 . The correctness of the definition of φ follows from the fact that if di+1 = di for some i then we obtain the same point zi for both the orderings of i, i + 1. Continuity of the metric projection implies that of φ.   Proposition 5.4 If X, Y, X ∩ Y ∈ UPR are compact, then the Euler-Poincaré characteristic of these sets and their union is defined and additive: χ(X ∪ Y ) + χ(X ∩ Y ) = χ(X) + χ(Y ). Proof Follows from Proposition 5.3 and Theorem 3.5.

 

The following auxiliary relationship leads, in particular, to the values of the index function iX for the special case of PR-sets X. Lemma 5.5 If 0 < r < reach X, then for any ball B(r) of radius r intersecting X we have χ(X ∩ B(r)) = 1. Proof By Proposition 4.20 for Z := X ∩ B(r) we obtain reach Z > r. Note that the convex hull conv Z of the intersection set Z lies in a distance not greater than r from Z. Therefore the metric projection from conv Z onto Z yields a homotopy between these sets. Consequently, χ(Z) = χ(conv Z) = 1.   Theorem 5.6 The index iX (x, n) is determined for any X ∈ UPR , x ∈ Rd and n ∈ S d−1 . For X ∈ PR we have the identity iX = 1nor X . Moreover, the index function is locally bounded and satisfies the additivity property iX + iY = iX∪Y + iX∩Y whenever X, Y, X ∩ Y, X ∪ Y ∈ UPR .

90

5 Unions of Sets with Positive Reach

Proof First we show that iX = 1nor X for X ∈ PR. If (x, n) ∈ nor X, then X ∩ B(x + (ε + δ)n, ε) = ∅ for sufficiently small ε and δ by the unique nearest point property of sets with positive reach, hence iX (x, n) = 1. Further, let (x, n) ∈ nor X and x ∈ X (the case x ∈ X is trivial). Then we have X ∩ B(x + (ε + δ)n, ε) = ∅ for all ε > 0 and sufficiently small

δ (cf. Lemma 4.5). Therefore Lemma 5.5 implies that χ X ∩ B(x + (ε + δ)n, ε) = 1 for ε < reach X and δ sufficiently small, hence iX (x, n) = 0. The existence and additivity of iX for general X ∈ UPR follows from this special case by means of Proposition 5.4, since the set X ∩ B(x + (ε + δ)n, ε) has a finite UPR -representation due to Proposition 4.20. The measurability of iX for X ∈ PR follows by the measurability of nor X, and for X ∈ UPR we use the additivity. The additivity and locality of the index function imply the following inclusion-exclusion formula valid for any (x, n): iX (x, n) =



(−1)card(I )−1 ii∈I Xi (x, n),

(5.4)

∅=I ⊂I (x)

where I (x) = {i : x ∈ Xi } and Xi are the PR-components from (5.1). Thus, |iX (x, n)| ≤ 2card I (x) and iX is locally bounded.   The identity iX = 1nor X for X ∈ PR suggests the following generalization. Definition 5.7 For X ∈ UPR , the unit normal bundle is defined by nor X := {(x, n) ∈ Rd × S d−1 : iX (x, n) = 0}. Note that, in general, the unit normal bundle nor X need not be closed. Proposition 5.8 For any X ∈ UPR , (i) nor X is a locally (d −1)-rectifiable and Hd−1 -measurable subset of Rd ×S d−1 , (ii) The index function iX is locally Hd−1 -integrable. Proof The measurability of nor X follows from the measurability of the index  i be a U function. Let X = ∞ X PR -representation of X (5.1). Note that by the i=1 definition of the unit normal bundle and by the additivity of the index function we have    (5.5) nor X ⊆ nor Xi I ∈P∗ (N)

i∈I

and the union on the right hand side is locally finite (i.e., the intersection of nor X with a bounded domain is covered by a finite union of unit normal bundles of sets of positive reach). The local rectifiability follows immediately from the same property of the unit normal bundles of sets with positive reach. In particular, nor X has locally finite Hd−1 -measure and since iX is locally finite, it is locally Hd−1 -integrable.  

5.1 Topological Index Functions and Additive Extension of Normal Cycles

91

Definition 5.9 For X ∈ UPR , the normal cycle NX ∈ Dd−1 (Rd × Rd ) is defined by NX = H d−1

nor X ∧ iX aX ,

where aX is the simple unit (d − 1)-vector field associated with Tand−1 (nor X, ·) with orientation (sign) given by (  ) (π0 + επ1 )aX (x, n) lim d−1 ∧ n, Ωd > 0 ε→0 | d−1 (π0 + επ1 )aX (x, n)| for sufficiently small ε > 0. Remark 5.10 1. In view of Lemma 4.44, the definition is consistent with that for PR-sets. 2. By construction, NX is locally (d − 1)-rectifiable with support spt NX ⊂ nor X ⊂ ∂X × S d−1 . 3. Due to Proposition 1.19, for Hd−1 -almost all (x, n) ∈ nor X, the approximate d−1 tangent space Tan (nor X, (x, n)) is a (d −1)-dimensional subspace and agrees d−1 with Tan (nor I Xi , (x, n)) for some ∅ = I ⊂ N finite. We will show that NX is the desired additive extension of the normal cycle and that the main geometric properties of normal cycles of PR-sets remain valid. (Recall Definition 4.39 for the contact 1-form α.) Theorem 5.11 If X, Y, X ∩ Y, X ∪ Y ∈ UPR , then we have the following. (i) NX + NY = NX∪Y + NX∩Y (additivity). (ii) ∂NX = 0 (NX is a cycle). (iii) NX α = 0 (NX is Legendrian). Proof Note that the additivity property (i) implies the inclusion-exclusion principle (5.2). Therefore, (ii) and (iii) are consequences of the corresponding properties of PR-sets. In order to show (i), we recall the additivity of the index function (Proposition 5.6). Hence, by the definition of the unit normal bundle, nor(X ∪ Y ) ⊂ nor X ∪ nor Y ∪ nor(X ∩ Y ) . Consequently, in view of the local (d − 1)-rectifiability of the unit normal bundles and due to Corollary 4.45, for Hd−1 -almost all (x, n) ∈ nor(X ∪ Y ) we have aX∪Y (x, n) = aZ (x, n) for some Z ≡ Z(x, n) ∈ {X, Y, X ∩ Y } and, moreover, aZ (x, n) = aZ  (x, n) whenever (x, n) ∈ nor Z  for another Z  ∈ {X, Y, X ∩ Y }.

92

5 Unions of Sets with Positive Reach

From this we infer NX∪Y = H d−1 = H d−1 = H d−1 = H d−1 − H d−1 = H d−1 − H d−1

nor(X ∪ Y ) ∧ iX∪Y aX∪Y

(nor X ∪ nor Y ∪ nor(X ∩ Y )) ∧ iX∪Y aZ

(nor X ∪ nor Y ∪ nor(X ∩ Y )) ∧ (iX + iY − iX∩Y )aZ



nor X ∧ iX aZ + H d−1 nor Y ∧ iY aZ

nor(X ∩ Y ) ∧ iX∩Y aZ



nor X ∧ iX aX + H d−1 nor Y ∧ iY aY

nor(X ∩ Y ) ∧ iX∩Y aX∩Y

= NX + NY + NX∩Y .

 

The index function is not defined uniquely in the literature. The following one was introduced by Fu [Fu89a]. We use the notation Hv,t := {y ∈ Rd : y · v ≤ t},

v ∈ S d−1 , t ∈ R,

for halfspaces in Rd . Definition 5.12 For any X ⊂ Rd , x ∈ Rd and n ∈ S d−1 , we define



ιX (x, n) := lim lim χ(X ∩B(x, ε)∩H−n,−x·n+δ −χ X ∩B(x, ε)∩H−n,−x·n−δ , ε→0 δ→0

whenever the right hand side is determined (Fig. 5.2). Remark 5.13 1. The index ιX (x, n) need not be defined even if reach X > 0. As an example, consider X to be the graph of f (t) = t 4 sin 1t , t ∈ [−1, 1], x = (0, 0) and n = (0, 1). 2. Fu [Fu89a, Corollary 6.7] showed that for a set X ⊂ Rd with positive reach, for Hd−1 -almost all n ∈ S d−1 and all x ∈ Rd , ιX (x, n) = (−1)λX (x,n) , where λX (x, n) is the number of negative principal curvatures of X at a regular point (x, n) ∈ nor X (cf. Theorem 4.54). 3. It is easy to see that ιX (x, n) exists and agrees with iX (x, n) for all (x, n) if X is a compact polyconvex set. In such a case, we can also write it in the form ιX (x, n) = 1 − lim lim χ(X ∩ B(x, ε) ∩ ∂Hn,n·x+δ ), ε→0 δ→0

(5.6)

5.2 Curvature Measures, Generalized Principal Curvatures and Steiner Formula

93

n

H−n,−n·x−δ

x

H−n,−n·x+δ

X

Fig. 5.2 The index function ιX (x, n) = 1 − 2 = −1

x ∈ X, n ∈ S d−1 . Indeed, decomposing H−n,−n·x+δ = H−n,−n·x−δ ∪ Ln,δ (x) with the layer Ln,δ (x) := {y ∈ Rd : −δ ≤ (x − y) · n ≤ δ}, we can use additivity of the Euler characteristic on the convex ring and the fact that χ(X ∩ B(x, δ) ∩ Ln,δ (x)) = 1 if x ∈ X and ε, δ are small enough. In this version the index function has first been considered by Schneider [Sch80].

5.2 Curvature Measures, Generalized Principal Curvatures and Steiner Formula Using the normal cycle (Definition 5.9) and the Lipschitz-Killing curvature forms from Definition 4.41 we now introduce similarly as for PR-sets the associated curvature notions. Definition 5.14 The curvature-direction measure of order k ∈ {0, . . . , d − 1} of k (X, ·) on Rd × Rd given by X ∈ UPR is the signed Radon measure C k (X, E) := (NX C  =

1E )(ϕk ) iX (x, n)aX (x, n), ϕk (n) Hd−1 (d(x, n)),

E∩nor X

for any bounded Borel set E ⊂ Rd × Rd .

94

5 Unions of Sets with Positive Reach

The corresponding (Lipschitz-Killing) curvature measure is the projection k (B × S d−1 ) Ck (X, B) := C for any bounded Borel set B ⊂ Rd . For bounded ∂X the total curvature measure Ck (X) := Ck (X, Rd ) is called Lipschitz-Killing curvature of order k. We set again Cd (X, ·) := Ld (X ∩ (·)) and Cd (X) := Cd (X, Rd ) = Ld (X) when X is bounded. The additivity of the normal cycle (cf. Theorem 5.11) implies the following. Corollary 5.15 The curvature direction-measures for UPR -sets are the additive extensions of those for PR-sets and we have k (X ∩ Y, ·) = C k (X, ·) + C k (Y, ·) k (X ∪ Y, ·) + C C if X, Y, X ∪ Y, X ∩ Y ∈ UPR . In particular, we obtain the additivity of the Lipschitz-Killing curvature measures. The following important geometric properties of the curvature-direction measures are direct analogues to their variants for PR-sets and follow again from the definition by means of the unit normal cycle: Recall the notation g(x, ˜ n) = (g(x), O(n)), (x, n) ∈ R2d , for a Euclidean motion g ∈ Gd with g(x) = g(0) + O(x). Proposition 5.16 For any X, Y ∈ UPR , B ⊂ Rd and E ⊂ R2d bounded Borel sets, λ > 0 and g ∈ Gd we have k (gX, g(E)) k (X, E) (motion covariance), (i) C ˜ =C (ii) Ck (λX, λB) = λk Ck (X, B) (homogeneity), k (X, E) = C k (Y, E) if X, Y ∈ UPR and NX 1E = NY (iii) C

1E (locality).

As a corollary of (iii) we get again Ck (X, U ) = Ck (Y, U ) if U is a bounded open set in Rd and X ∩ U = Y ∩ U , i.e. the locality of the Lipschitz-Killing curvature measures. As in the case of sets with positive reach, we can introduce (generalized) principal curvatures κi (x, n) and principal directions bi (x, n) at almost all (x, n) ∈

5.2 Curvature Measures, Generalized Principal Curvatures and Steiner Formula

95

nor X of UPR -sets X: Recall that if X has UPR -representation (5.1), nor X is covered by the locally finite union nor X ⊂



nor

I ∈P∗ (N)



 X

i

(5.7)

.

i∈I

 Let us call a point (x, if (x, n) is regular in nor i∈I Xi n) ∈ i nor X regular whenever (x, n) ∈ nor i∈I X . Clearly, Hd−1 -almost all (x, n) ∈ nor X are regular and due to Proposition 1.19 and Corollary 4.45, we have for regular points (x, n), Tand−1 (x, n)) = Tand−1 (nor Z, (x, n)) and aX (x, n) = aZ (x, n) whenever (nor X, i Z = i∈I X and (x, n) ∈ nor(Z, (x, n)). Thus, we obtain from Proposition 4.23 and Lemma 4.24 the following. Proposition 5.17 If X ∈ UPR then for H-almost all (x, n) ∈ nor X, Tand−1 (nor X, (x, n)) is a (d − 1)-dimensional subspace and there exist vectors b1 (x, n), . . . , bd−1 (x, n) in Rd and numbers κ1 (x, n), . . . , κd−1 (x, n) ∈ (−∞, ∞] such that b1 (x, n), . . . , bd−1 (x, n), n form a positively oriented orthonormal basis of Rd and the vectors ⎛ ai (x, n) := ⎝ 

⎞ 1 1+

κi2 (x, n)

bi (x, n), 

κi (x, n) 1 + κi2 (x, n)

bi (x, n)⎠ , i = 1, . . . , d−1 ,

form an orthonormal basis of Tand−1 (nor X, (x, n)). (We set √ √∞

1 1+∞2

= 0 and

= 1.) In particular, the (d − 1)-vector field orienting nor X equals 1+∞2 aX = a1 ∧ . . . ∧ ad−1 . The κi (x, n) are uniquely determined up to their order and the subspace of Rd generated by the basis vectors bj (x, n) corresponding to one particular value κi (x, n) is unique. The dimension of this subspace is given by the multiplicity of the number κi (x, n) within {κ1 (x, n), . . . , κd−1 (x, n)}. Then we get a representation of the curvature-direction measures in terms of the corresponding symmetric functions of principal curvatures sk (X; x, n), which are introduced as in Definition 4.25. Theorem 5.18 For X ∈ UPR we have  1 k (X, ·) = iX (x, n)sd−1−k (X; x, n) Hd−1 (d(x, n)) . C (d − k)ωd−k (·)∩nor X Proof This follows from the definition and property (4.14).

 

If we take into regard the multiple points of the metric projection together with the index function we can also infer a generalized Steiner formula for UPR -sets (cf.

96

5 Unions of Sets with Positive Reach

Theorem 4.30 for PR-sets). Recall that π0 (x, n) = x is the projection mapping onto the first coordinate in Rd × Rd . Theorem 5.19 For X ∈ UPR , any bounded Borel set E ⊂ Rd × Rd and r > 0 we have       y−x y−x 1E x, Ld (dy) iX x, |y − x| |y − x| Rd x∈∂X:|x−y|≤r

=

d−1 

d−1−k (X, E). ωk r k C

k=0

Proof As in Sect. 4.3 for sets of positive reach we apply the area formula to the mapping f : nor X × [0, r] → Rd with f (x, n, t) = x + tn which is, in general, no more injective. Therefore we get instead of Eq. (4.10) for g = iX 1E ,   =

r 0

Rd

 =

 iX ((x, n) Jd f (x, n, t) Hd−1 (d(x, n)) dt E∩nor X



iX (x, n)1E (x, n) Ld (dy)

(x,n,t )∈nor X×[0,r]:x+t n=y

 Rd x∈∂X:|x−y|≤r

    y−x y−x 1E x, Ld (dy) , iX x, |y − x| |y − x|

and the sum under the integral on the right hand side is finite for Ld -a.a. y. The remaining calculations for the Jacobian function Jd f leading to the polynomial representation of the left hand side with the curvature-direction measures as coefficients are the same as in Sects. 4.3 and 4.5 for sets of positive reach.   Setting E := B × S d−1 for a bounded Borel set B ⊂ Rd we infer the generalized Steiner formula for the Lipschitz-Killing curvature measures of UPR -sets X: 

 Rd x∈E∩∂X:|x−y|≤r

  d−1  y−x d iX x, ωk r k Cd−1−k (X, E). L (dy) = |y − x|

(5.8)

k=0

Note that in the special case X ∈ PR Eq. (5.8) extends the Steiner formula from Corollary 4.33, since we do not suppose now that r < reach X.

5.3 Regular UPR -Representations and the Reflection Principle for Normal. . .

97

5.3 Regular UPR -Representations and the Reflection Principle for Normal Cycles For Y ⊂ Rd , denote the closure of the complement of Y by  := Y c . Y

(5.9)

*ε for sufficiently Recall that Xε denotes the ε-parallel set of X ⊂ Rd . The set X small ε possesses in many cases much better measure geometric properties than the *ε is a closed C 1,1 primary set X. Recall that by Corollary 4.22 for X ∈ PR, X d domain. Fu [Fu85] showed that for any compact X ⊂ R with d ≤ 3 and almost all *ε has positive reach. In higher dimensions this is not true. However, ε > 0, the set X we obtain such regularity for a large subclass of UPR sets. Definition 5.20 (i) We say that the sets X1 , . . . , Xk ∈ PR touch (at a point x) if there are n1 , . . . , nk with (x, n1 ) ∈ nor X1 , . . . , (x, nk ) ∈ nor Xk and t1 n1 +· · ·+tk nk = 0 for some t1 , . . . , tk ≥ 0 with t1 + · · · + tk = 1 (i.e., the convex hull of the unit normal vectors n1 , . . . , nk at the section point x contains the zero vector). i i1 ik (ii) A UPR -representation X = ∞ i=1 X (5.1) is called regular if X , . . . , X do not touch for any 1 ≤ i1 < · · · < ik < ∞, k ∈ N. In that case, we say that the PR-sets X1 , X2 , . . . are completely non-touching. By Theorem 4.19 the finite intersection of non-touching compact PR-sets has again positive reach. Therefore we conclude the following. Proposition 5.21 Any locally finite union of completely non-touching compact sets with positive reach provides a regular UPR -representation. The main theorem of this section, the reflection principle, establishes a relation*ε for a subclass of UPR -sets. ship between the unit normal cycle of X and that of X We define the normal reflection on Rd × Rd by ρ : (x, n) → (x, −n) .

(5.10)

In the proof, we will use the following inclusion-exclusion principle for convex hulls. For the proof, see [KLZ17]. Lemma 5.22 If x1 , . . . , xk ∈ Rd and x ∈ rel int conv {x1 , . . . , xk } then 

(−1)|I |−1 1conv {xi : i∈I } (x) = (−1)m−1 ,

∅=I ⊂{1,...,k}

where m := dim conv {x1 , . . . , xk }.

98

5 Unions of Sets with Positive Reach

 i Theorem 5.23 Let X = N i=1 X be a regular UPR -representation with closed  has positive reach and C 1,1 -domains X1 , . . . , XN . Then X NX = ρ# NX .

(5.11)

Proof Since the Xi ’s are C 1,1 -domains, there exists a unique unit normal vector ni (x) ∈ Nor(Xi , x) at each x ∈ ∂Xi . For x ∈ ∂X, denote I (x) := {i ≤ N : x ∈ ∂Xi }, k(x) := card I (x) and let Δ(x) be the convex hull of {ni (x) : i ∈ I (x)}. Denote η(x) := dist (Δ(x), 0),

x ∈ ∂X.

Since X1 , . . . , XN do not touch we have η(x) > 0 for any x ∈ ∂X. Moreover, since nor Xi is closed for any 1 ≤ i ≤ N (see Lemma 4.11), the function η is lower semicontinuous on ∂X and we infer from the compactness that η := inf η(x) > 0.

(5.12)

x∈∂X

 i N−1 by Theorem 4.19. Thus, reach ( N i=1 X ) ≥ rη *i is again a C 1,1 domain, that −ni (x) is the only unit It is easy to see that each X * i outer normal to X at x ∈ ∂Xi and, using the same reasoning as above, we obtain  *i N−1 . Since also reach ( N i=1 X ) ≥ rη

= X

N 

c Xi

=

i=1

N i=1

(Xi )c =

N

*i , X

(5.13)

i=1

 ≥ rηN−1 > 0. we get reach X We denote C(x) := sco{ni (x) : i ∈ I (x)}, where sco T denotes the spherical convex hull of a set T ⊂ S d−1 (in fact, C(x) agrees with the projection of Δ(x) onto S d−1 from the origin). Let m(x) := dim C(x). We show that for any x ∈ ∂X and n ∈ S d−1 \ rel bd C(x), iX (x, n) = (−1)m(x)−11C(x)(n).

(5.14)

We use the inclusion-exclusion formula for the (additive) index function iX (x, n) =

 ∅=I ⊂{1,...,N}

(−1)|I |−1 1Nor(i∈I Xi ,x) (n).

Theorem 4.19 yields that Nor(

i∈I

Xi , x) ∩ S d−1 =

 sco{ni (x) : i ∈ I }, 0,

I ⊂ I (x), otherwise,

5.3 Regular UPR -Representations and the Reflection Principle for Normal. . .

99

and we can apply Lemma 5.22 to obtain (5.14). (In fact, Lemma 5.22 is formulated in the Euclidean space and not on the sphere, nevertheless, the set C(x) is contained in an open hemisphere by the non-touching assumption and we may project C(x) onto a suitable hyperplane from the origin and apply Lemma 5.22 then.) Theorem 4.19 together with (5.13) yield  = {(x, −n) : x ∈ ∂X, n ∈ C(x)}, nor X whereas by (5.14), {(x, n) : x ∈ ∂X, n ∈ rel int C(x)} ⊂ nor X ⊂ {(x, n) : x ∈ ∂X, n ∈ C(x)}.  is contained in Thus, the symmetric difference of nor X and ρ(nor X) D := {(x, n) : x ∈ ∂X, n ∈ rel bd C(x)}. If (x, n) ∈ D then, clearly, (x, n) ∈ nor XI with I := I (x), and there exists a unit vector u ⊥ n such that −u ∈ Tan(Nor(XI , x), n) and u ∈ (Nor(XI , x))o = Tan(XI , x) (recall Definition 2.6 for the polar cone). The first named property implies that (0, −u) ∈ Tan(nor XI , (x, n)), whereas the second one yields (0, u) ∈ Tan(nor XI , (x, n)) by Lemma 5.24 below. Thus, Tan(nor XI , (x, n)) is not a linear space and, hence, (x, n) is a non-regular point of nor XI , and the Hd−1 -measure of such points is zero. Hence, Hd−1 (D) = 0, which implies that  up to a set of Hd−1 − measure zero. nor X = ρ(nor X) Theorem 1.49 yields the relation  d−1 ρ# NX  = H

 nor X ∧ ζ,

where ζ (x, n) = (



(x, −n), d−1 ρ)aX

(x, n) ∈ nor X.

In order to show (5.11), it will be enough to verify that iX (x, n)aX (x, n) = ζ(x, n) for Hd−1 -almost all (x, n) ∈ nor X. Let (x, n) ∈ nor X be such that (x, −n) is a  Then, by Lemma 4.44, regular point of nor X. (



d−1 (π0

+ tπ1 ))aX  (x, −n) ∧ (−n), Ωd  > 0

for sufficiently small t > 0. Hence, sgn (



d−1 (π0

= sgn (



+ tπ1 ))(



 (x, −n) ∧ n, Ωd  d−1 ρ)aX

− tπ1 ))aX  (x, −n) ∧ n, Ωd   m(x)−1 = (−1)m(x)sgn ( d−1 (π0 + tπ1 ))aX ,  (x, −n) ∧ n, Ωd  = (−1) d−1 (π0

100

5 Unions of Sets with Positive Reach

since, for Hd−1 -almost all (x, n) ∈ nor X, there are exactly m(x) − 1 infinite  at (x, −n). Indeed, if n lies in the relative interior of C(x) principal curvatures of X then the principal curvatures are clearly infinite at all directions from the (m(x)−1)dimensional space Tan(C(x), n), and are finite at any direction perpendicular to Tan(C(x), n), by Lemma 5.24 below. Thus, the proof is finished.   Lemma 5.24 Let X1 , . . . , XN be as in Theorem 5.23, ∅ = I ⊂ {1, . . . , N}, (x, n) ∈ nor XI and n ⊥ u ∈ Tan(XI , x) ∩ S d−1 , where XI := i∈I Xi Then (0, u) ∈ Tan(nor XI , (x, n)). Proof Assume, for the contrary, that (0, u) ∈ Tan(nor XI , (x, n)). Then there exist (xp , np ) ∈ nor XI and rp > 0 such that xp → x, np → n, rp (xp − x) → 0 and rp (np − n) → u, p → ∞. In particular, we have |xp − x| → 0, (np − n) · u

p → ∞.

(5.15)

 Due to Theorem 4.19, we can write np in the form np = i∈I tp,i ni (xp ) with some tp,i ≥ 0. Passing to a subsequence if necessary, we  can assume that tp,i → ti ≥ 0, p → ∞, i ∈ I . Then, since np → n, we get n = i∈I ti ni (x). Denoting J := {i ∈ I : ni (x) · u = 0}, we have ti = 0 for i ∈ I \ J , since n · u = 0 by assumption. Choose L > 0 so that x → ni (x) is L-Lipschitz, i ∈ I . Then we have ni (xp ) · u = ((ni (xp ) − ni (x)) + ni (x)) · u ≤ L|xp − x|,

i ∈ I,

since ni (x) · u ≤ 0, i ∈ I , by assumption, and |ni (xp ) · u| ≤ |(ni (xp ) − ni (x)) · u| + |ni (x)) · u| ≤ L|xp − x|, Hence, denoting n˜ p :=



i∈I ti ni (xp ),

(np − n˜ p ) · u =

we get

 (tp,i − ti )(ni (xp ) · u) i∈I

≤



|tp,i − ti ||ni (xp ) · u| +

i∈J

≤

i ∈ J.







tp,i (ni (xp ) · u)

i∈I \J

|tp,i − ti | L|xp − x|

i∈I

≤ L|xp − x| for sufficiently large p. We further have

  L |(n˜ p − n) · u| ≤ ti |ni (xp ) − ni (x)| ≤ |xp − x|, η i∈I

5.3 Regular UPR -Representations and the Reflection Principle for Normal. . .

since



i∈I ti

101

≤ η−1 by (5.12). It follows that

(np − n) · u ≤ (np − n˜ p ) · u + |(n˜ p − n) · u| ≤ L(1 + η−1 )|xp − x| fur sufficiently large p, hence lim sup p→∞

|np − n| ≤ L(1 + η−1 ), |xp − x|  

which contradicts (5.15) and finishes the proof.

The regular UPR -representation property remains valid for the parallel sets of sufficiently small distance. We obtain a slightly sharper property:  i Proposition 5.25 For any regular UPR -representation X = N i=1 X of a compact N set and all sufficiently small ε, Xε = i=1 Xεi is also a regular UPR -representation.  i Moreover, there exists an r > 0 such that reach ( kj =1 Xεj ) ≥ r whenever {i1 , . . . , ik } ⊆ {1, . . . , N}. Proof Let k ≤ N, 1 ≤ i1 < · · · < ik ≤ N be fixed. Use the notation for the simplex Δk = {(t1 , . . . , tk ) ∈ [0, 1]k : t1 + · · · + tk = 1} and consider the function Ψ defined on nor Xi1 × . . . × nor Xik × Δk by      k   Ψ : (x1 , n1 , . . . xk , nk , t1 , . . . , tk ) → |xj − x1 | +  tj nj  . j =1  j =2 k 

In view of Definition 5.20, Ψ is positive. Since it is Lipschitz continuous, by the i compactness assumption its infimum η is positive. Note that for all j , nor Xεj i converge to nor X j with ε → 0 in the Hausdorff distance. Thus there exists an ε0 > 0 such that Ψ (y1 , n1 , . . . , yk , nk , t1 , . . . , tk ) ≥ η/2 i

for all (yj , nj ) ∈ nor Xεj , ε < ε0 and (t1 , . . . , tk ) ∈ Δk . Consequently, again i

i

by Definition 5.20, Xεi1 , . . . , Xεk do not touch. Furthermore, since reach Xεj ≥ reach Xij − ε, Theorem 4.19 implies that reach (

k

i

Xεj ) ≥ (rmin − ε0 )(η/2)k−1 ,

j =1

where rmin := mini reach Xi .

 

102

5 Unions of Sets with Positive Reach

As a corollary, we obtain the following result. Then there exists Theorem 5.26 Let X1 , . . . , XN be compact sets of positive  reach. i is a regular U ε0 > 0 such that whenever for some 0 < ε < ε0 , Xε = N X PR i=1 ε *ε has positive reach and representation, then X NXε = ρ# NX *ε . Proof For all sufficiently small ε > 0, the parallel sets Xεi are C 1,1 domains by Corollary 4.22, and they do not touch by Proposition 5.25. Thus we can apply Theorem 5.23.   Recall that the curvature measures of Xε are given by Ck (X, B) = (NX 1B×S d−1 )(ϕk )  = 1B (x)iX (x, n)aX (x, n), ϕk (n) Hd−1 (d(x, n)) , nor X

for any bounded Borel set B in Rd , where the ϕk are the Lipschitz-Killing curvature forms from Definition 4.41. Applying Theorem 5.26 to ϕk and regarding the change of the sign (−1)d−1−k under the mapping ρ we get the following. Corollary 5.27 Under the conditions of Theorem 5.26 we have *ε , ·) , k = 0, . . . , d − 1 . Ck (Xε , ·) = (−1)d−1−k Ck (X Remark 5.28 In Chap. 7 (Theorem 7.3) we will show that the normal cycles of the parallel sets of small distance of any UPR -set X converge in a suitable topology to *ε instead that of X. In view of Theorem 5.26 we could also work with the sets X of Xε . The regularity condition on the UPR -representation may be reformulated in terms of some regularity property of the distance function dX in the sense of [Fu85]. Therefore such an approximation is the key for extending the notions of normal cycle and curvatures to geometric sets with a highly singular structure, in particular, to some fractals (see Chaps. 9, and 10).

5.4 Bibliographical Notes 1. Extensions of classical versions of the above curvatures to piecewise flat subspaces of Riemannian manifolds in differential geometry and Morse index theory have a long history: a. Allendoerfer and Weil [AW43] proved the Gauss-Bonnet theorem for Riemannian polyhedra. b. In mathematical physics, in particular in general relativity, the corresponding theory is known as Regge calculus, which goes back to [Reg61], where the

5.4 Bibliographical Notes

103

scalar curvature was involved. (The latter corresponds to Cd−3 in the above case.) c. Banchoff [Ban67] determined the Gauss curvature C0 of Euclidean polyhedra via index theory. d. Intrinsic constructions of the Lipschitz-Killing curvatures Ck for the Riemannian case were given in Wintgen [Win82], Cheeger [Che83], Cheeger, Müller and Schrader [CMS84]. The latter also obtained a version of the Steiner formula for such sets in [CMS86]. A purely combinatorial description of these curvatures in a more abstract setting can be found in [Bud89]. 2. For parallel developments in convex geometry see Chap. 2. The closest reference is Schneider [Sch80] who extended curvature measures from convex bodies to the convex ring, using additivity and the index function iX in the sense of Remark 5.13. He also considered a nonnegative extension of curvature measures to the convex ring. 3. UPR -sets were introduced in [Zäh87], with an additional condition concerning the intersections of tangent cones of the PR components. In particular, the UPR representation (5.1) was assumed to fulfill for any x ∈ ∂X Tan

i∈I

 i

X ,x

=



Tan(Xi , x),

∅ = I ⊂ N finite.

i∈I

General UPR -sets were investigated in [RZ01].

(5.16)

Chapter 6

Integral Geometric Formulas

Integral geometry, in general, is concerned with integrals of geometric characteristics with respect to invariant measures, usually under Euclidean motions. A classical and comprehensive reference to integral-geometric relations is the book of Santaló [San76]. In this chapter we derive integral-geometric formulas concerning curvature measures and their total values (total curvatures or intrinsic volumes). This is, in particular, the Principal kinematic formula expressing the curvature measures of the intersection of two bodies, one of them fixed and the other moving, in terms of those of the primary bodies. Similarly, the Crofton formula deals with the curvature measures of flat sections of a body, integrated with respect to the invariant measure over the sectioning flats. See Chap. 2, Theorems 2.4 and 2.5, for the versions of convex geometry. Here we consider the formulas in the setting of sets with positive reach, though their validity is much broader. The reason is that the proof techniques can be well demonstrated on PR-sets. Extensions to the UPR -settings are mentioned in the bibliographic remarks at the end of the chapter. For the validity for other set classes see the remarks in Chap. 9. We start now with the formulations of the main results. Recall that dg denotes integration with respect to the properly normalized motion invariant measure on the group Gd of all rigid motions in Rd , see Sect. 2.3. Theorem 6.1 (Principal kinematic formula for sets with positive reach) Let X, Y ⊂ Rd be sets with positive reach and A, B ⊂ Rd be bounded Borel sets. Then reach (X ∩ gY ) > 0 for almost all motions g ∈ Gd and for any integer 0 ≤ k ≤ d,  

Ck X ∩ gY, A ∩ gB dg = γ (d, r, s)Cr (X, A)Cs (Y, B), Gd

1≤r,s≤d r+s=d+k

© Springer Nature Switzerland AG 2019 J. Rataj, M. Zähle, Curvature Measures of Singular Sets, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-18183-3_6

105

106

6 Integral Geometric Formulas

where γ (d, r, s) =

Γ ((r + 1)/2)Γ ((s + 1)/2) . Γ ((r + s − d + 1)/2)Γ ((d + 1)/2)

The first proof was given by Federer in [Fed59] using approximation with smooth sets. A shorter and more direct proof was presented later by Rother and Zähle [RZ90b]. We will use here another approach based on the translative version as given in [RZ95]. (See Theorem 6.10 and Lemma 6.19.) Before doing this, note that choosing for Y in Theorem 6.1 a j -flat, since Cs (Y, B) = Hj (Y ∩ B) if s = j and Cs (Y, B) = 0 otherwise, we get the Crofton formula stated as follows. Recall that μdj is the properly normalized invariant measure on the set Adj of all j -flats in Rd , see Chap. 2. Theorem 6.2 (Crofton formula for sets with positive reach) Let X be a set with positive reach in Rd , 0 ≤ k ≤ j ≤ d, A ⊂ Rd be a bounded Borel set. Then  A(d,j )

Ck (X ∩ E, A ∩ E) μdj (dE) = γ (d, d + k − j, j )Cd+k−j (X, A).

Note that if X, Y are moreover compact we can relax the boundedness assumption for A, B in Theorems 6.1 and 6.2. In particular, we obtain: Corollary 6.3 Let X, Y ⊂ Rd be compact sets with positive reach. Then   Ck (X ∩ gY )dg = γ (d, r, s)Cr (X)Cs (Y ), Gd

 A(d,j )

1≤r,s≤d r+s=d+k

Ck (X ∩ E) μdj (dE) = γ (d, d + k − j, j )Cd+k−j (X).

6.1 Translative and Kinematic Integral Formulas for Curvature Measures If X, Y are two sets with positive reach that do not touch, i.e., there is no intersection point of the boundaries x ∈ ∂X ∩ ∂Y with opposite normal vectors, 0 = n ∈ Nor(X, x), −n ∈ Nor(Y, x), then the unit normal bundle (and, hence, also the curvature measures) is completely described by means of the relation Nor(X ∩ Y, x) = Nor(X, x) + Nor(Y, x) from Proposition 4.17. If X and Y do touch at some point x then we cannot say in general anything about the local behaviour of the intersection at x. It can even happen that X ∩ Y does not have positive reach (see Example 4.16).

6.1 Translative and Kinematic Integral Formulas for Curvature Measures

107

We will derive in this section a formula for the intersection  Ck (X ∩ (Y + z); ·) Ld (dz). Rd

In order to guarantee the regular behaviour of the intersection for almost all shifts z, we assume that Ld {z ∈ Rd : X and Y + z touch} = 0.

(6.1)

The validity of this assumption will be discussed later. In order to describe the unit normal bundles of X ∩ (Y + z), we introduce the following mappings. Let m, n ∈ Rd be two nonzero vectors with angle  (m, n) < π, let t ∈ [0, 1], and set u(m, n, t) :=

sin(t  (m, n)) sin((1 − t) (m, n)) m+ n.  sin (m, n) sin  (m, n)

(6.2)

(The values of the fractions at  (m, n) = 0 are defined by the limits as  (m, n) → 0.) Note that if m, n are unit vectors then u(m, n, ·) is a unit-speed parametrization of the geodesic arc on S d−1 with endpoints m, n. Denote further R 0 := {(x, m, y, n) ∈ (Rd )4 : m, n = 0,  (m, n) < π} and consider the mapping F : R 0 × [0, 1] → R2d × S d−1 , (x, m, y, n, t) → (x, y, u(m, n, t)).

(6.3)

Note that F is differentiable and, hence, locally Lipschitz, on R 0 × [0, 1]. We are going to push forward by F the product current (cf. Sect. 1.3.3)  (NX × NY )

 R 0 × [0, 1]

(here [0, 1] denotes simultaneously the unit interval and also the 1-current (L1 [0, 1]) ∧ 1). We first define the measure  NX,Y  := F# ((NX × NY )

 R 0 ) × [0, 1] 

as in Definition 1.48 and, provided that NX,Y  is locally finite,

(6.4)

108

6 Integral Geometric Formulas

also the current NX,Y := F# ((NX × NY )

R 0 × [0, 1])

on R3d called later joint unit normal current of X and Y . Its career will be denoted by   nor(X, Y ) := F ((nor X × nor Y ) ∩ R 0 ) × [0, 1] . Using (1.26), we can write 

 (NX × NY ) R 0 × [0, 1]   = H2d−1 ((nor X × nor Y ) ∩ R 0 ) × [0, 1] ∧ (aX  aY  1)

(see Sect. 1.3.3 for the definition of the product of currents and of the symbol ). Thus, condition (6.4) can be rewritten equivalently as  (nor X×nor Y )∩(R 0 ∩K)×[0,1]

|a˜ X,Y |dH2d−1 < ∞,

K ⊂ R4d compact,

(6.5)

where a˜ X,Y := (



2d−1 DF )(aX

 aY  1).

(6.6)

Remarks on the validity of (6.4) will be given later. In order to get an integral representation of NX,Y , we need the following auxiliary result. Let f denote the restriction of F to ((nor X × nor Y ) ∩ R 0 ) × [0, 1]. Lemma 6.4 For H2d−1-almost all (x, y, u) ∈ nor(X, Y ), card f −1 {(x, y, u)} = 1. Consequently,  NX,Y = H2d−1

 nor(X, Y ) ∧ aX,Y ,

where the unit simple (2d − 1)-vector field aX,Y is given by aX,Y :=

a˜ X,Y ◦ f −1 . |a˜ X,Y ◦ f −1 |

6.1 Translative and Kinematic Integral Formulas for Curvature Measures

109

Proof Let N := {x, y, u) ∈ nor(X, Y ) : card f −1 {(x, y, u)} > 1}. By the Area formula (Theorem 1.14), 

 F −1 (N)

card F −1 {(x, y, u)} H2d−1(d(x, y, u)).

ap J2d−1f dH2d−1 = N

We will show that the approximate Jacobian J2d−1 f vanishes on f −1 (N), which implies the first statement. Take a point (x, m, y, n, t) ∈ f −1 (N). By the definition of N, there exists  (m , n , t  ) = (m, n, t) with (x, m ) ∈ nor X, (y, n ) ∈ nor Y and u = u(m, n, t) = u(m , n , t  ). Since the normal cones Nor(X, x) and Nor(Y, y) are convex, they contain the cones spanned by m, m and n, n , respectively, and we have (0, a) ∈ Tand−1 (nor X, (x, m)) and (0, b) ∈ Tand−1 (nor Y, (y, n)) with a = m − m and b = n − n. Thus, the linear hull L of (0, a, 0, 0, 0), (0, 0, 0, b, 0) and (0, 0, 0, 0, 1) is a subspace of Tan2d−1 (nor X × nor Y × [0, 1]). It is further clear that ap Df (x, m, y, n, t) maps all the vectors from L to {0} × M for the linear hull M = Lin {m, n, m , n } and, at the same time, since u maps to the unit sphere, the image of its differential lies in u⊥ . The four vectors m, m , n, n are linearly dependent (since u = u(m, n, t) = u(m , n , t  )), thus, dim M ≤ 3 and dim M ∩ u⊥ ≤ 2. It follows that ap Df (x, m, y, n, t) cannot be injective at points from f −1 (N). Consequently, J2d−1 f vanishes at f −1 (N). The second assertion is the Area formula for currents (Theorem 1.49).   For any z ∈ Rd , we can decompose the normal current of X ∩ (Y + z) into three parts, NX∩(Y +z) = Nz1 + Nz2 + Nz3 ,

(6.7)

where Nz1 = NX∩(Y +z)

π0−1 int (Y + z) = NX

Nz2 = NX∩(Y +z)

π0−1 int X = NY +z

π0−1 int (Y + z), π0−1 int X

and Nz3 = NX∩(Y +z)

π0−1 (∂X ∩ ∂(Y + z)).

We obtain the most intricate part, Nz3 , by sectioning NX,Y with the mapping G : (x, y, u) → x − y

110

6 Integral Geometric Formulas

defined on R3d , with restriction g := G | nor(X, Y ). (See Sect. 1.3.7 for the definition of slicing.) Denote the projection πˆ : (x, y, u) → (x, u). Proposition 6.5 Let two sets X, Y have positive reach and satisfy assumptions (6.1) and (6.4). Then Nz3 = (−1)d πˆ # NX,Y , G, z for Ld -almost all z ∈ Rd . Proof First, we apply the Coarea formula for currents (Theorem 1.65) to get an expression for the section current. We obtain that for Ld -almost all z ∈ Rd ,  NX,Y , g, z = Hd−1

 g −1 {z} ∧ ζ

with ζ (x, y, u) =

aX,Y (x, y, u) G# Ωd , Jd g(x, y, u)

a unit simple (2d − 1)-vector field associated with g −1 {z}. Further, note that by Lemma 6.4, πˆ is one-to-one H2d−1-almost everywhere on nor(X, Y ). Thus, the Area formula for currents (Theorem 1.49) yields  πˆ # NX,Y , g, z = Hd−1

 π(g ˆ −1 {z}) ∧ η

with unit simple (d − 1)-vector field  η(x, u) =

−1 (x, u))



ζ(πˆ −1 (x, u)) . Jd−1 (π| ˆ nor(X, Y ))(πˆ −1 (x, u)) ˆ (πˆ d−1 π

Due to assumption (6.1) and Proposition 4.17, we know that π(g ˆ −1 {z}) agrees with 3 Nz for almost all z. It is thus sufficient to verify that the vector fields (−1)d η and aX∩(Y +z) coincide Hd−1 -almost everywhere on Nz3 , for Ld -almost all z ∈ Rd . Since both are unit simple vector fields associated with the same vector space, it suffices to verify that they have the same orientation. We have (−1)d η, ϕk  = αaX,Y

G# Ωd , πˆ # ϕk 

= αaX,Y , G# Ωd ∧ πˆ # ϕk ,

6.1 Translative and Kinematic Integral Formulas for Curvature Measures

111

with a positive factor α. It will follow from Lemma 6.15 that aX,Y , G# Ωd ∧ πˆ # ϕk0  > 0,

(6.8)

where k0 is the largest index ≤ d − 1 (depending on (x, m, y, n, t)) for which aX,Y , G# Ωd ∧ πˆ # ϕk0  is nonzero. According to Remark 4.43, this proves already that η has the same orientation as aX∩(Y +z) .   Before formulating the translative intersection formula, we need to define the mixed curvature measures. Definition 6.6 (3-product of multivectors) Let α, β, γ be multivectors in Rd of multiplicities r, s, q, respectively, and assume that r + s + q = 2d. The 3-product of α, β and γ is defined by α, β, γ  := α ∧ β ∧ γ , Ωd  (for the definition of the Hodge star operator, see Definition 1.29). In the sequel, we will need the following auxiliary result. Recall Definition 1.31 of the bracket [·, ·] for subspaces. If ξ, η are simple multivectors, we will often write briefly [ξ, η] instead of [L(ξ ), L(η)] (L(ξ ) is the linear subspace associated with ξ ). Lemma 6.7 If {a1 , . . . , ad }, {b1, . . . , bd } are two positively oriented orthonormal bases of Rd , then for all 1 ≤ r, s ≤ d − 1 with r + s ≥ d, +

r 

ai ,

s 

bj ,

j =1

i=1

d 

ai ∧

i=r+1

,

d 

⎡

bj = ⎣

j =s+1

r 

ai ,

i=1

s 

⎤2 bj ⎦ .

j =1

Proof Denote A := Lin {a1 , . . . , ar }, B := Lin {b1 , . . . , bs }, k := dim(A ∩ B). If k > r + s − d then A + B  Rd and both sides of the equation vanish. Assume thus that k = r + s − d. We can assume without loss of generality that ai = bi for i = 1, . . . , k. We will use the identity d  i=r+1

ai ∧

d  j =s+1

+ bj =

k  i=1

ai ∧

d  i=r+1

ai ∧

d  j =s+1

, bj , Ω

d 

ai .

(6.9)

i=k+1

To see this, note that on both sides there are simple multivectors associated with the  same linear subspace, (A ∩ B)⊥ , and that the wedge product with k1 ai from left yields the same d-vector on both sides. Using this identity and the definition of the

112

6 Integral Geometric Formulas

3-product, we obtain +

r 

s 

ai ,

+ =

bj ,

j =1

i=1 k 

ai ∧

=

k 

ai ∧

i=1

⎡ =⎣

r 

d 

i=1

d 

ai ∧

j =s+1

d 

d 

s 

, bj

j =s+1

i=r+1

ai ∧

,+

bj , Ω

d 

d 

ai ∧

i=r+1

bj ∧ (−1)

k(d−k)

j =s+1

k 

, ai , Ω

i=1

,2 bj , Ω

j =s+1

i=r+1

ai ,

d 

ai ∧

i=r+1

i=1

+

d 

⎤2 bj ⎦ .

j =1

  Definition 6.8 (ψr,s -forms) Given r, s ∈ {1, . . . , d}, d ≤ r +s ≤ 2d −1, we define the (2d − 1)-form ψr,s on R3d by +2d−1 ,  (ai , bi , ci ), ψr,s (x, y, u) i=1

= (−1)

d+s(d−r)



Oq−1

σ ∈Sh(r,s,q)

+ sgn σ

r  i=1

a σi ,

r+s 

b σj ,

j =r+1

2d−1 

, cσ k ∧ u ,

k=r+s+1

where q = 2d − r − s − 1, Oq := Hq (S q ) = (q + 1)ωq+1 is the Hausdorff measure of the unit q-sphere in R q+1 , and Sh(r, s, q) is the set of all permutations σ of {1, . . . , 2d − 1} which are increasing on {1, . . . , r}, {r + 1, . . . , r + s} and {r + s + 1, r + s + q}. Since ψr,s (x, y, u) depends only on the third component u, we will write ψr,s (u) in the sequel. Definition 6.9 (Mixed curvature measures) Let X, Y be two sets in Rd with positive reach and assume that (6.4) is fulfilled. We define the mixed curvature r,s (X, Y, ·) of sets X, Y and orders r, s ∈ {1, . . . , d}, d ≤ r + s ≤ 2d − 1, measure C as follows. If r = d we set d,s (X, Y, ·) := Cd (X, ·) ⊗ C s (Y, ·), C

6.1 Translative and Kinematic Integral Formulas for Curvature Measures

113

and if s = d, r,d (X, Y, ·) := C r (X, ·) ⊗ Cs (Y, ·). C If r, s < d we set r,s (X, Y, E) := (NX,Y C

1E )(ψr,s ),

r,s (X, Y, · × Rd ) we for any bounded Borel subset E ⊂ R3d . By Cr,s (X, Y, ·) = C  denote the projection of Cr,s (X, Y, ·) into the first two vector coordinates. Theorem 6.10 (Translative intersection formula) Let X, Y be two subsets of Rd with positive reach and assume that (6.1) and (6.4) are fulfilled. Then, for any bounded measurable function h on (Rd )3 such that spt h ∩ ((X − Y ) × X × S d−1 ) is bounded, and for any k ∈ {0, . . . , d − 1},  

k (X ∩ (Y + z), d(x, u)) Ld (dz) h(z, x, u) C 



=

r,s (X, Y, d(x, y, u)). h(x − y, x, u) C

r+s=k+d

k and the decomposition (6.7), we can write the left Proof Using the definition of C side as 

NX∩(Y +z)

3    h(z, ·, ·) (ϕk ) L (dz) = Nzi



d

 h(z, ·, ·) (ϕk ) Ld (dz).

i=1

The first summand can be calculated as    Nz1 h(z, ·, ·) (ϕk ) Ld (dz)   NX =   =   =   =

 (1π −1 int (Y +z) h(z, ·, ·)) (ϕk ) Ld (dz) 0

k (X, d(x, u)) Ld (dz) 1int (Y +z) (x)h(z, x, u) C k (X, d(x, u)) Ld (dy) 1int Y (y)h(x − y, x, u) C k (X, d(x, u)) Cd (Y, dy), h(x − y, x, u) C

114

6 Integral Geometric Formulas

where we have used the substitution z = x − y, Ld (dz) = Ld (dy) for the outer integral. Similarly we obtain   Nz2

 h(z, ·, ·) (ϕk ) Ld (dz)

  =

k (Y, d(y, u)). h(x − y, x, u) Cd (X, dx) C

For the third summand, we apply Proposition 6.5 and get   Nz3

 h(z, ·, ·) (ϕk ) Ld (dz)  

(1.30)

h(z, ·, ·) (ϕk ) Ld (dz)

πˆ # NX,Y , G, z

= (−1)d

 πˆ # NX,Y , G, z

= (−1)d

 πˆ # h(z, ·, ·) (ϕk ) Ld (dz)



(1.27)

= (−1)

h ◦ Γ, G, z(πˆ # ϕk ) Ld (dz)

NX,Y

d

(1.33)

= (−1)d ((NX,Y

(1.22)

= (−1)d (NX,Y

h◦Γ)

G# Ωd )(πˆ # ϕk )

h ◦ Γ )(G# Ωd ∧ πˆ # ϕk ),

with Γ : (x, y, u) → (x −y, x, u). The right side of the Translative integral formula to be proved can be rewritten, using the definition of mixed curvature measures, as 

h ◦ Γ )(ψr,s ).

(NX,Y

r+s=k+d 1≤r,s≤d−1

 

The proof is thus finished by Lemma 6.11. Lemma 6.11 For all u ∈

S d−1 ,

G# Ωd ∧ πˆ # ϕk (u) = (−1)d



ψr,s (u).

r+s=k+d 1≤r,s≤d−1

Proof Consider a multivector τ=

r  i=1

(ai , 0, 0) ∧

s  j =1

(0, bj , 0) ∧

q  l=1

(0, 0, cl )

6.1 Translative and Kinematic Integral Formulas for Curvature Measures

115

with 0 ≤ r, s ≤ d, q := 2d − r − s − 1 and vectors ai , bj , ck from a positively oriented orthonormal basis  {e1 , . . . , ed } of Rd such that u = ed . Since such multivectors form a basis of 2d−1 (Rd )3 , it is sufficient to show that 

τ, G# Ωd ∧ πˆ # ϕk (u) = (−1)d

τ, ψr  ,s  (u).

(6.10)

r  +s  =k+d 1≤r  ,s  ≤d−1

Due to the special form of τ , the left side of (6.10) equals τ, ψr,s (u) if 1 ≤ r, s ≤ d and r + s = d + k, and zero otherwise. If r + s = d + k then both sides of (6.10) vanish. If r + s = d + k we can assume without loss of generality that ai = bi for 1 ≤ i ≤ k (this can be achieved by reordering the components). Since  (u, 0, 0) ∧ (0, u, 0) =

   u u u u √ , −√ , 0 ∧ √ , √ , 0 2 2 2 2

for any vector u, we can rewrite τ in the form τ =

  k  r k  s     aj aj ai ai √ , −√ , 0 ∧ (ai , 0, 0) ∧ √ , √ ,0 ∧ (0, bj , 0) 2 2 2 2 i=1 i=k+1 j =1 j =k+1 ∧

q 

(0, 0, cl ).

l=1

In this expression, exactly d wedge factors do not lie in the kernel of G. Thus, by the shuffle formula (1.3) τ, G# Ωd ∧ πˆ # ϕk (u) + k  ,  r s  ai   ai k(s−k) # (ai , 0, 0) ∧ (0, bj , 0), G Ωd = (−1) √ , −√ , 0 ∧ 2 2 i=1 i=k+1 j =k+1 +

  q k   aj aj × √ , √ , 0 ∧ (0, 0, cl ), πˆ # ϕk (u) 2 2 j =1 l=1 + r , s   = (−1)k(s−k)2k/2(−1)s−k ai ∧ b j , Ωd i=1

+ × + = (−1)

s(d−r)

Oq−1

j =k+1 k 

2−k/2 (aj , 0) ∧

j =1 r  i=1

ai ∧

s  j =k+1

b j , Ωd

q 

, (0, cl ), ϕk (u)

l=1

,+

k  j =1

aj ∧

,

q  l=1

, cl ∧ u, Ωd .

116

6 Integral Geometric Formulas

If cl = u for some l ≤ q then the last expression vanishes, but then, τ, ψr,s (u) = 0 as well. Assume thus that cl = u for all l ≤ q, and complete the vector sequences (ai ), (bj ) and (cl ) to positively oriented orthonormal bases of Rd such that cd = u. We have , + d , + k q d−1     aj ∧ cl ∧ u, Ωd = aj ∧ cl , Ω d . j =1

j =k+1

l=1

l=q+1

Moreover, applying (6.9) and the fact that +

r  i=1

ai ∧

s 

, b j , Ωd =

j =k+1

+

k 

d 

ai ∧

i=1

d 

ai ∧

, b j , Ωd ,

j =s+1

i=r+1

we get τ, G# Ωd ∧ πˆ # ϕk (u) + d , d d−1    s(d−r) −1 Oq ai ∧ bj ∧ cl , Ω d = (−1) + = (−1)

s(d−r)

Oq−1

j =s+1

i=r+1 r  i=1

ai ,

s 

bj ,

j =1

q 

l=q+1

,

cl ∧ u

l=1

= (−1)d τ, ψr,s (u),  

as required. For the particular choice of the function h h(z, x, u) = 1A (x)1B (x − z) we infer the following.

Corollary 6.12 Let X, Y be two subsets of Rd with positive reach and assume that (6.1) and (6.4) hold. Then, for any bounded Borel sets A, B ⊂ Rd and any k ∈ {0, . . . , d − 1},   Cr,s (X, Y, A × B). Ck (X ∩ (Y + z), A ∩ (B + z)) Ld (dz) = r+s=d+k

We will give now some sufficient conditions for (6.1). (Sufficient conditions for (6.4) will be given later.)

6.1 Translative and Kinematic Integral Formulas for Curvature Measures

117

Proposition 6.13 (i) Any two closed convex sets X, Y ⊂ Rd satisfy (6.1). (ii) Any two sets X, Y ⊂ R2 with positive reach satisfy (6.1). (iii) Any two sets X, Y ⊂ Rd with positive reach such that ∂X, ∂Y are smooth submanifolds of differentiability class d satisfy (6.1). (iv) Let X, Y ⊂ Rd have positive reach. Then X, ρY satisfy (6.1) for almost all ρ ∈ SO(d). Proof Denote N = {z ∈ Rd : X and Y + z touch}. (i) If X, Y are convex then the difference set X − Y = {x − y : x ∈ X, y ∈ Y } is convex as well. Since N ⊂ ∂(X − Y ), we have Ld (N) ≤ Ld (∂(X − Y )) = 0. (ii) Consider first a general dimension d and define the mapping Ψ : (x, m, y, n) → x − y,

(x, m, y, n) ∈ nor X × nor Y.

Take any (x, m, y, n) ∈ W 0 := (nor X × nor Y ) \ R 0 (for the definition of R 0 , see the beginning of Sect. 6.1) and (a, a  , b, b ) ∈ Tan2d−2 (nor X × nor Y, (x, m, y, n)). Then a ⊥ m = −n ⊥ b, hence, a − b ⊥ m. Consequently, ap Jd Ψ (x, m, y, n) = 0. If d = 2 then the area formula gives 

 0=

W0

ap J2 Ψ dH2 =

Ψ (W 0 )

H0 (Ψ −1 {z}) L2 (dz),

hence, L2 (N) = 0 since N = Ψ (W 0 ). (iii) The assumption of (iii) implies that nor X × nor Y is of differentiability class l = d−1. Thus, applying the Sard-type theorem [Fed69, §3.4.3] to the mapping Ψ , we get Hd−1+(d−1)/ l (Ψ (W 0 )) = 0, which yields the assertion. (iv) We apply Lemma 1.73 with Z = nor X × nor Y and p = 2d − 1 and obtain that Hq (A) = 0, where A = {(x, m, y, n, ρ) ∈ nor X × nor Y × SO(d) : m + ρn = 0} and q = 2d − 1 +

d(d − 1) (d − 1)(d − 2) =d+ . 2 2

The mapping h : (x, m, y, n, ρ) → (x − ρy, ρ) is Lipschitz and maps A onto h(A) = {(z, ρ) ∈ Rd × SO(d) : X and z + ρY touch}.

118

6 Integral Geometric Formulas

We have Hq (h(A)) = 0 by Lemma 1.10 and, using the Fubini theorem, we get  0 = Hq (h(A)) =

Ld {z ∈ Rd : X and z + ρY touch} Hd(d−1)/2(dρ), SO(d)

which yields the result.  

6.2 Local Representation of Mixed Curvature Measures In this section we derive a representation of mixed curvature measures as integrals over the product of the two unit normal bundles. This will give a better insight in the Translative integral formula and, in particular, we will derive the Principal kinematic formula as a corollary. The basic idea is to apply the Area formula to the mapping F from (6.3). In order to do this, we need to know the differential. Recall the mapping u defined in (6.2). Lemma 6.14 Let (x, m, y, n) ∈ (Rd × S d−1 )2 ∩ R 0 be fixed. Then, for any vectors v, w ∈ Rd , we have sin((1 − t)θ ) v + p, sin θ sin tθ w + q, Du(m, n, t)(0, w, 0) = sin θ Du(m, n, t)(0, 0, 1) = θ u∗ , Du(m, n, t)(v, 0, 0) =

for certain vectors p, q from Lin {m, n}, where θ :=  (m, n), u := u(m, n, t), u∗ := pu⊥ (n − m)/|pu⊥ (n − m)| if m = n and u∗ = 0 otherwise (pu⊥ denotes the orthogonal projection to u⊥ ). Proof The first two equations are obtained by using the product rule for differentiation. The differential at (0, 0, 1) is obtained from the fact that t → u(m, n, t) is a unit-speed parametrization of the shortest path on S d−1 from m to n.   We find now a representation of the simple multivector a˜ X,Y defined in (6.6). Let (x, m) ∈ nor X and (x, n) ∈ nor Y be regular points and t ∈ [0, 1]. We can write aX  aY  1(x, m, y, n, t) in the form aX aY 1 = (KL)−1

d−1 

d−1 

i=1

j =1

(ai , κi ai , 0, 0, 0)∧

(0, 0, bj , λj bj , 0)∧(0, 0, 0, 0, 1),

6.2 Local Representation of Mixed Curvature Measures

119

where κi = κi (x, m), λj = λj (y, n) are the principal curvatures and ai = ai (x, m), bj = bj (y, n) the corresponding principal directions of X, Y at (x, m), (y, n), respectively, and K=

d−1 

1 + κi2 ,

L=

d−1 

1 + λ2j

(6.11)

j =1

i=1

(the case κi , λj = ∞ is treated as usually). Now we have, using Lemma 6.14, a˜ X,Y

 d−1  sin((1 − t)θ ) 1  ai + pi = ai , 0, κi KL sin θ

(6.12)

i=1

∧

d−1 

0, bj , λj

j =1

sin(tθ ) bj + qj sin θ



∧ (0, 0, θ u∗ )

with pi , qj ∈ Lin {m, n} and, using the fact that pi ∧ u ∧ u∗ = qj ∧ u ∧ u∗ = 0 for all i, j , we get &



   sin((1 − t)θ ) d−1−r sin tθ d−1−s sin θ sin θ ⎞

⎛  1    × κi ⎝ λj ⎠ KL c c

' −1 a˜ X,Y , ψr,s = O2d−r−s−1 θ

|I |=r |J |=s

×(−1)

d+s(d−r)

i∈I

sgn σ

+

j ∈J

 i∈I

ai ,

 j ∈J

bj ,



ai ∧

i∈I c



, ∗

bj ∧ u ∧ u ,

j ∈J c

where the summation is carried over subsets I, J of {1, . . . , d − 1} of given cardinalities, I C , J C denote the complements of I, J , respectively, in {1, . . . , d −1}, and σ is the permutation of {1, . . . , 2d−1} that maps the first r numbers increasingly onto I , the following s numbers increasingly onto d − 1 + J , and the remaining numbers again increasingly onto the remaining numbers. An easy calculation yields sgn σ = (sgn I )(sgn J )(−1)(d−1−r)s , hence, (−1)d+s(d−r)sgn σ = (−1)d−s (sgn I )(sgn J ),

120

6 Integral Geometric Formulas

where sgn I, sgn J denotes the sign of the permutation of {1, . . . , d − 1} mapping the first r, s elements increasingly into I, J , and the remaining elements again increasingly onto I C , J C , respectively. If θ = 0, we can substitute u∗ ∧ u = −(sin θ )−1 m ∧ n and, using Lemma 6.7, we obtain +



ai ,



bj ,

j ∈J

i∈I



ai ∧



, ∗

bj ∧ u ∧ u

j ∈J c

i∈I c

+ ,     1 d−1−s (−1) ai , bj , ai ∧ m ∧ bj ∧ n =− sin θ C C j ∈J

i∈I

j ∈J

i∈I

⎡

= (−1)d−s (sgn I )(sgn J )

⎤2

 1 ⎣ ai , bj ⎦ . sin θ i∈I

j ∈J

Lemma 6.15 Let X, Y be sets with positive reach. Then we have for H2d−1 -almost all (x, m, y, n, t) ∈ ((nor X × nor Y ) ∩ R 0 ) × [0, 1], a˜ X,Y = 0 if m = n and &

'

a˜ X,Y , ψr,s =

   sin((1 − t)θ ) d−1−r sin tθ d−1−s sin θ sin θ ⎛ ⎞⎛ ⎞⎡ ⎤2       ⎝ κi ⎠ ⎝ λj ⎠ ⎣ ai , bj ⎦

−1 O2d−r−s−1

×

1 KL

θ sin θ

|I |=r |J |=s



i∈I C

j ∈J C

i∈I

j ∈J

otherwise, where θ =  (m, n). In both cases, we have  2d−2

  0 ≤ θ ≤ π2 , a˜ X,Y  ≤ h(θ ) := d θ sin θ, 2d−2 3−d θ, π2 < θ < π. d θ sin Proof The equality follows from the above computation. In order to show the inequality, we use (6.12). Note that a˜ X,Y is orthogonal to (0, 0, u), thus, its norm does not change when being multiplied by (0, 0, u). Using again the identity u∗ ∧ u = −(sin θ )−1 m ∧ n for m = n, we get   d−1  d−1    

1 θ  |a˜ X,Y | = 0, bj , βλj bj ∧ (0, 0, n) (ai , 0, ακi ai ) ∧ (0, 0, m) ∧  KL sin θ   i=1 j =1 with α :=

sin(tθ ) sin((1 − t)θ ) and β := . sin θ sin θ

6.2 Local Representation of Mixed Curvature Measures

121

Since the mappings P : (x, y) → (x, 0, y) and Q : (x, y) → (0, x, y) are isometric immersions, the mapping 

d−1 P

∧



d−1 Q

:



2d (R

2d

)→



2d (R

3d

)

is 1-Lipschitz and we have 1 θ |a˜ X,Y | ≤ KL sin θ

  d−1  d−1   

 bj , λj βbj ∧ (0, n) . (ai , κi αai ) ∧ (0, m) ∧   i=1  j =1

We can apply the shuffle formula for the last norm and obtain 1 θ |a˜ X,Y | ≤ KL sin θ

 2                ai ∧  bj   κi α   λj β     j ∈J c  i∈I c  |I |+|J |=d  i∈I j ∈J 

  (here, as above, I, J ⊂ {0, . . . , d − 1}). Since | i∈I ai ∧ j ∈J bj | ≤ |m ∧ n| = %  π −1 if θ > π and   ≤ K, sin θ , α, β ≤ 1 if θ ≤ , α, β ≤ (sin θ ) c κi i∈I 2 2 %       j ∈J c λj  ≤ L by (6.11), the result follows. Remark 6.16 For given (x, m, y, n, t) ∈ nor X × nor Y × (0, 1), let r0 , s0 be the number of finite principal curvatures of X, Y at (x, m), (y, n), respectively (assuming, of course, that the principal curvatures exist). It follows from Lemma 6.15 that (−1)d aX,Y , ψr0 ,s0  > 0, and that (−1)d aX,Y , ψr,s  ≥ 0 whenever r + s ≥ r0 + s0 . Thus, using Lemma 6.11, we get (6.8), which completes the proof of Proposition 6.5. Proposition 6.17 If X, Y are sets with positive reach satisfying (6.4) and 1 ≤ r, s ≤ d − 1, r + s ≥ d, then, for any bounded Borel subsets A ⊂ Rd × Rd and B ⊂ S d−1 , r,s (X, Y, A × B) = C

 (nor X×nor Y )∩R 0

×

d−1 



i=1

⎡ ×⎣

 i∈I

1A (x, y)μ(m, n; B)

1 

1 + κi2 1 + λ2i

ai ,

 j ∈J

   |I |=r |J |=s i∈I c

κi

 j ∈J c

⎤2 bj ⎦ H2d−2(d(x, m, y, n)),

λj

122

6 Integral Geometric Formulas

where −1 μ(m, n; B) = O2d−r−s−1



θ sin θ

1

×

1B (u(m, n, t)) 0



sin((1 − t)θ ) sin θ

d−1−r 

sin tθ sin θ

d−1−s dt

with θ =  (m, n). Proof We have by definition, r,s (X, Y, A × B) = (NX,Y 1A×B )(ψr,s ) C

= F# ((NX × NY ) 1R 0 ) × [0, 1] (ψr,s ∧ 1A×B )

= ((NX × NY ) 1R 0 ) × [0, 1] (F # (ψr,s ∧ 1A×B ))   1 & ' = a˜ X,Y , ψr,s ∧ 1A×B dL1 dH2d−2, (nor X×nor Y )∩R 0

0

 

and the result follows from Lemma 6.15.

Now we will prove that condition (6.4) is fulfilled for almost all rotations or reflections of one of the sets with positive reach. Lemma 6.18 Let X, Y be sets with positive reach, let A, B ⊂ Rd are Borel sets and let h be the function from Lemma 6.15. Then, NX,Y (A × B × S d−1 )  ≤

(nor X×nor Y )∩R 0

(6.13) 1A (x)1B (y) h( (m, n)) H2d−2 (d(x, m, y, n)).

Further, there exists a finite constant C such that  NX,ρY (A × ρB × S d−1 ) ϑd (dρ)

(6.14)

SO(d)

≤ CHd−1 (nor X ∩ π0−1 (A))Hd−1(nor Y ∩ π0−1 (B)). Consequently, the pair X, ρY satisfies (6.4) for ϑd -almost all ρ ∈ SO(d). Proof Since  NX,Y (·) =

F −1 (·)

|aX,Y | d((NX × NY )

R 0 ) × [0, 1],

6.2 Local Representation of Mixed Curvature Measures

123

Equation (6.13) follows from the inequality in Lemma 6.15. Using the fact that (y, n) → (ρy, ρn) is an isometry between nor Y and nor ρY , we get NX,ρY (A × ρB × S d−1 )  1A (x)1B (y) h( (m, ρ −1 n)) H2d−2(d(x, m, y, n)). ≤ (nor X×nor Y )∩R 0

Integrating over SO(d) and applying the Fubini theorem, we get (6.14), since h(θ ) ≤ sin3−d θ and   −1 3−d  −1 sin (m, ρ n) ϑd (dρ) = Od−1 sin3−d  (u, v) Hd−1 (dv), S d−1

SO(d)

and the last integral can be shown to be finite by direct computation.

 

We can compute now the rotational integral of the mixed curvature measures which will imply, together with Theorem 6.10, the Principal kinematic formula (Theorem 6.1). Lemma 6.19 For any sets X, Y ⊂ Rd with positive reach, for any 1 ≤ r, s ≤ d −1, r + s ≥ d, and for any bounded measurable sets A, B ⊂ Rd , we have  Cr,s (X, ρY, A ∩ ρB) ϑd (dρ) = γ (d, r, s)Cr (X, A)Cs (Y, B). SO(d)

Proof Using Lemma 6.18 and Proposition 6.13(iv), we can write  Cr,s (X, ρY, A × ρB)ϑd (dρ) SOd



=

1A (x)1ρB (y)μ(m, n; S d−1) nor X×nor ρY



×



%

%

|I |=r |J |=s i∈I c κi (X; x, m) j ∈J c λj (ρY ; y, n)  %d−1  1 + κi (X; x, m)2 1 + λi (ρY ; y, n)2 i=1

⎡ ⎤2   × ⎣ ai (X; x, m), bj (ρY ; y, n)⎦ H2d−2 (d(x, m, y, n))ϑd (dρ). i∈I

j ∈J

Since the principal curvatures are rotation invariant and the principal directions are rotation covariant, we have λj (ρY ; y, n) = λj (Y ; ρ −1 y, ρ −1 n)

124

6 Integral Geometric Formulas

and bj (ρY ; y, n) = ρbj (Y ; ρ −1 y, ρ −1 n). Thus, using the substitution y˜ = ρ −1 y, n˜ = ρ −1 n, and the rotational invariance of the Hausdorff measure, we get  Cr,s (X, ρY, A × ρB)ϑd (dρ) SOd



=

1A (x)1B (y) nor X×nor Y



×



%

%

˜ n) ˜ |J |=s i∈I C κi (X; x, m) j ∈J C λj (Y ; y,  %d−1  2 2 1 + κi (X; x, m) 1 + λi (Y ; y, ˜ n) ˜ i=1

|I |=r

 ×

⎡ ⎤2   μ(m, ρ n; ˜ S d−1 ) ⎣ ai (X; x, m), ρbj (Y ; y, ˜ n) ˜ ⎦ ϑd (dρ)

SOd

i∈I

j ∈J

˜ n)). ˜ ×H2d−2(d(x, m, y, We claim that the inner integral over SO(d) is a constant, say  γ¯ (d, r, s), depending on d, r, s only. Indeed we can write it in the form Φ(B1 , ρB2 ) ϑd (dρ) with two positively oriented orthonormal bases B1 = (a1 , . . . , ad−1 , m), B2 = (b1 , . . . , bd−1 , n) ˜ and invariant function Φ (Φ(ρB1 , ρB2 ) = Φ(B1 , B2 )), and the constancy of the integral follows from the invariance property of ϑd . Therefore, Definitions 4.28 and 4.31 imply that  Cr,s (X, Y, A ∩ ρB) ϑd (dρ) = γ˜ (d, r, s)Cr (X, A)Cs (Y, B) SO(d)

for another constant γ˜ (d, r, s). Applying now Theorem 6.10, we get the Principal kinematic formula with coefficients γ˜ (d, r, s). Applying this to two unit balls gives the final result with γ˜ (d, r, s) = γ (d, r, s).   Finally, we will derive a translative version of the Crofton formula (Theorem 6.2) as a corollary of Theorem 6.10. As above, κi and ai , i = 1, . . . , d − 1, denote the principal curvatures and principal directions of a set X with positive reach. If X ⊂ Rd has positive reach, F ∈ A(d, j ) and X ∩ F has also positive reach, the curvature-direction measure of X ∩ F can be considered either in the whole Rd , i.e., as a measure on Rd ×S d−1 , or relatively in F , i.e., as a measure on F ×S j −1 . where S j −1 stands for the unit sphere in the j -space parallel to F . In the second case, we (F ) (X ∩ F, ·). will use the notation C k

6.2 Local Representation of Mixed Curvature Measures

125

Recal that νjd denotes the normalized invariant measure on the Grassmannian v G(d, j ), and for L ∈ G(d, j ), πL : v → |v| , v ⊥ L, is the spherical projection onto L. Theorem 6.20 Let X ⊂ Rd be a set with positive reach and 0 ≤ k < j < d be integers. Then, for νjd -almost all L ∈ G(d, j ) and for any bounded measurable function g defined on Rd × (L ∩ S d−1 ) and with compact support, we have  L⊥

=

(L+z) (X ∩ (L + z), d(x, v)) Ld−j (dz) g(x, v) C k

Oj−1 −1−k





 nor X

1

%



κi g(x, πL (n)) |I |=d+k−j i∈I c i∈I  j −k % |pL n| d−1 1 + κ2 i=1

22 ai , L Hd−1 (d(x, n)).

i

In particular, if A ⊂ Rd is a bounded Borel set then  L⊥

Ck (X ∩ (L + z), A ∩ (L + z)) Ld−j (dz)

= Oj−1 −1−k





 nor X

1A (x) |pL n|j −k

|I |=d+k−j

% i∈I c

1 κi

%d−1  i=1



22 ai , L

i∈I

1 + κi2

Hd−1 (d(x, n)).

Proof Let B1 be a ball of unit j -volume in L. We apply Theorem 6.10 with X, Y = L and h(˜z, x, u) = 1B1 (pL z˜ )g(x, πL (n)). Note that, due to Proposition 6.13(iv) and Lemma 6.18, the assumptions of Theorem 6.10 are satisfied for almost all L ∈ G(d, j ) and, for such subspaces L and for any (x, n) ∈ nor X, n ⊥ L by (6.1) and, hence πL (n) = |ppLL nn| is defined. r,s (X, L, ·) vanishes unless s = j (note It follows from Theorem 6.17 that C that, at each point of L, exactly j principal curvatures are zero, and the remaining d − 1 − j are infinite). Hence we obtain, using Theorem 6.10 and Proposition 6.17, 

 g(x, v) C k

(L+z)

L⊥



=  =

L⊥

Rd

(X ∩ (L + z), d(x, v)) Ld−j (dz)

k (X ∩ (L + z), d(x, n)) Ld−j (dz) g(x, πL (n)) C k (X ∩ (L + z˜ ), d(x, u)) Ld (d˜z) h(˜z, x, u) C

126

6 Integral Geometric Formulas



d+k−j,j (X, L, d(x, y, u)) h(x − y, x, u) C

= 

 

=

g(x, πL (n)) nor X

×

L⊥ ∩S d−1

L

1

0

−1 1{m+n=0} Od−1−k

j −1−k 

 sin tθ d−1−j sin((1 − t)θ ) dt Hd−1−j (dn) Lj (dy) sin θ sin θ %

3 42 |I |=d+k−j I C κi I ai , L Hd−1 (d(x, m)) %d−1  2 1 + κi i=1

θ sin θ 

×





3 4 (recall that θ =  (m, n)). Note that if m + n = 0 then I ai , L = 0. Thus, we can omit the indicator function of {m + n = 0} in the last integral. We will show that  L⊥ ∩S d−1

θ sin θ





1 0

=

sin((1 − t)θ ) sin θ

j −1−k 

sin tθ sin θ

d−1−j dt Hd−1−j (dn)

Od−1−k |pL m|k−j . Oj −1−k

(6.15)

This will complete the proof. Fix m ∈ S d−1 \ L⊥ and consider the mapping Φ : (n, t) → u(m, n, t). The differential of Φ is (see Lemma 6.14) DΦ(n, t)(w, 0) =

sin tθ w + q, sin θ

DΦ(n, t)(0, 1) = θ u∗ ,

with some q ∈ Lin {m, n}. Let φ be the restriction of Φ to (L⊥ ∩ S d−1 ) × (0, 1) and let {b1 , . . . , bd−1−j , n} be an orthonormal basis of L⊥ . Then we have with some q1 , . . . , qd−1 ∈ Lin {m, n},  ⎞ ⎛    d−1−j      ⎠ ⎝ Jd−j φ(n, t) =  d−j DΦ(n, t) (bi , 0) ∧ (0, 1)    i=1   d−j     −1 sin tθ  =  bi + qi ∧ θ u∗  sin θ   i=1

6.2 Local Representation of Mixed Curvature Measures

127

  d−j     −1 sin tθ  bi + qi ∧ θ u∗ ∧ u =  sin θ  i=1     d−1−j    sin tθ d−1−j   ∗ =θ bi ∧ u ∧ u  sin θ   i=1 (we have used the fact that the image of the differential of Φ is always perpendicular to u). Using again the fact that u ∧ u∗ = (m ∧ n)/|m ∧ n|, we get |b1 ∧ · · · ∧ bd−1−j ∧ u∗ ∧ u| =

|b1 ∧ · · · ∧ bd−1−j ∧ m ∧ n| |pL m| = . |m ∧ n| sin θ

Hence,  Jd−j φ(n, t) = θ

sin tθ sin θ

d−1−j

|pL m| . sin θ

Observing that sin(1 − t)θ u · pL m u · pL m = = , sin θ m · pL m |pL m|2 and applying the Area formula to φ, we get  L⊥ ∩S d−1

θ sin θ 

1  sin((1 − t)θ ) j −1−k

 0

sin θ 





sin tθ sin θ

d−1−j

sin((1 − t)θ ) Jd−j φ(n, t) ⊥ d−1 sin θ 0 L ∩S j −1−k   u · pL m 1 Hd−j (du). = j −k |pL m| |pL m| im φ

1 = |pL m|

1

dt Hd−1−j (dn) j −1−k

dt Hd−1−j (dn)

The image if φ, im φ, is a hemisphere with pole m of dimension d − j , and a routine calculation yields    1 Od−1−k pL m j −1−k d−j H (du) = |u · e|j −1−k Hd−j (du) = u· d−j |p m| 2 Oj −1−k L im φ S



(e ∈ S d−j is arbitrarily chosen). Substituting into the above integral, we obtain (6.15), and the proof is complete.  

128

6 Integral Geometric Formulas

6.3 Absolute Curvature Measures The curvature measures are signed measures and for certain applications, nonnegative variants of these are needed. The first obvious idea is to consider the total variation measures, Ckvar (X, ·), of Ck (X, ·) (for sets X for which the curvature measures are defined, of course). It turns out, however, that these measures do not satisfy the Crofton formula. Therefore, other nonnegative variants of curvature measures are considered in integral geometry, see [San74] (where the case of smooth submanifolds is considered). These measures are defined by using the “touching affine subspaces” of given dimension. Let us assume that reach X > 0 (though, the results can be further extended, see Sect. 6.4.4). The total variation of the curvature-direction measures can be expressed, due to Definition 4.28, as integral of the absolute value of the generalized symmetric function of principal curvatures: var (X, E) = C k

1 Od−1−k

 |sd−1−k (X, x, n)| Hd−1 (d(x, n)),

(6.16)

E∩nor X

for any bounded Borel set E ⊂ Rd × S d−1 , k = 0, . . . , d − 1. Remark 6.21 var (X, A × S d−1 ) is not true in general. 1. Note that the equality Ckvar (X, A) = C k 1 Indeed, if X = S is the unit circle in R2 then C0var (S 1 , ·) = 0, while var (S 1 , S 1 × S 1 ) = 2. C 0 var (X, A × S d−1 ) is independent of the chosen embedding 2. The measure A → C k space of X (see Lemma 4.36(iii)). In order to introduce nonnegative variants of curvature measures that do satisfy the Crofton formula, it is natural to define absolute curvature measures by means of the Crofton formula (cf. Theorem 6.2). Recall that μdj denotes the standard invariant measure on the family of j -flats A(d, j ) (see Sect. 2.3). Definition 6.22 Let X be a set with positive reach in Rd and k ∈ {0, 1, . . . , d − 1}. The absolute curvature measure of X of order k is defined by Ckabs (X, B) = γ (d, k, d − k)−1

 A(d,d−k)

var (X ∩ F, (B ∩ F ) × S d−1 ) μd (dF ), C 0 d−k

(6.17) where B is any bounded Borel subset of Rd . var (X, · × S d−1 ), but this is not true for all k > 0, as will Clearly, C0abs (X, ·) = C 0 be seen later.

6.3 Absolute Curvature Measures

129

Immediately from the definition we obtain the analogue of Theorem 6.2: Theorem 6.23 (Crofton formula for absolute curvature measures) Let X be a set with positive reach in Rd , 0 ≤ k ≤ j ≤ d, B ⊂ Rd a bounded Borel set. Then  A(d,j )

abs Ckabs (X ∩ F, B ∩ F ) μdj (dF ) = γ (d, d + k − j, j )Cd+k−j (X, A).

Proof By definition, we have  A(d,j )

Ckabs (X ∩ F, B ∩ F ) μdj (dF )

= γ (j, k, j − k)−1



 C0abs (X ∩ G, A ∩ G) μFj−k (dG) μdj (dF ).

A(d,j ) A(F,j −k)

Since μFj−k (dG) μdj (dF ) defines a motion invariant measure on Adj−k , this must be equal to μdj−k (dG) up to a constant factor (depending only on d, k, j ), and we get 

 A(d,j )

Ckabs (X ∩ F, B ∩ F ) μdj (dF ) = const

Aj −k

C0abs (X ∩ G, A ∩ G) μdj −k (dG)

abs = const Cd+k−j (X, B),

using Definition 6.22 again. If X is a convex body then Ckabs (X, ·) = Ck (X, ·) and a comparison with Theorem 6.2 yields the correct value of the constant.   We will derive now explicit representations of  absolute curvature  measures. To save space, we will use the abbreviated notation I κi in place of i∈I κi (and similarly with products or exterior products). Fix a set X with positive reach and k ∈ {0, 1, . . . , d − 1} and a bounded Borel set B ⊂ Rd . Applying Theorem 6.20, we obtain for almost all L ∈ G(d, d − k) and any bounded measurable function g on Rd × S d−k−1  0 (X ∩ (L + z), d(x, v)) Lk (dz) 1B (x)g(x, v)C L⊥

=

−1 Od−1−k

 nor X

1B (x)g(x, πL (n)) |pL n|d−k



3 42 κi I ai , L Hd−1 (d(x, n)). %d−1  2 1 + κi i=1

|I |=k

%

Ic

130

6 Integral Geometric Formulas

Maximizing the above expression with respect to all measurable functions g with |g| ≤ 1, we obtain for a generic subspace L ∈ G(d, d − k)  L⊥

0var (X ∩ (L + z), (B ∩ (L + z)) × S d−k−1 ) Lk (dz) C

−1 = Od−1−k

 nor X

 %

3 42   κ a , L   c i i |I |=k I I 1B (x)  Hd−1 (d(x, n)). %d−1 |pL n|d−k 2 1 + κi i=1

Taking into account Definition 6.22, we get −1 γ (d, k, d − k)Ckabs (X, B) = Od−1−k

 1B (x)Ψk (x, n)Hd−1 (d(x, n)) nor X

with  42  %

3  κ a , L   c |I |=k I i I i 1 d  νd−k (dL) Ψk (x, n) = d−k %d−1 2 G(d,d−k) |pL n| 1 + κ i=1 i  %

3 42    κ a , L(ξ )   c i i |I |=k I I 1  Hk(d−k)(dξ ) = % 2cd,d−k G0 (d,d−k) d−1 2 |pL(ξ ) n|d−k i=1 1 + κi 

(recall that cd,j = Hj (d−j )(G(d, j ))). Denoting V := L ∩ n⊥ ∈ G(d, d − 1 − k) if d -almost all L), we can write n ⊥ L (which is true for νd−k 5 I

6 5 6 5 ai , L = ai ∧ n, V |pL n| = I

IC

6 ai , V ⊥ |pL n|.

We will apply now the Coarea formula for the mapping p : ξ → ζ := n⊥

ξ |ξ

n∗ n∗ |

from

G0 (d, d − k) to G0 (d − 1, d − 1 − k) (note that L(ζ ) = V if n ⊥ V ) with Jacobian J(d−1−k)(d−1)p(ξ ) = |pL n|−(d−k−1) (see Lemma 1.72): Ψk (x, n) =

1



I(n, L(ζ )) 2cd,d−k Gn0⊥ (d−1,d−1−k)  % 3 4   ⊥ 2 κ a , L(ζ )   |I |=d−1−k i i I I H(d−1−k)(d−1)(dζ ) × %d−1  2 1 + κi i=1

6.3 Absolute Curvature Measures

=

131

 cd−1,d−1−k I(n, V ) ⊥ cd,d−k Gn (d−1,d−1−k)  % 3 4   ⊥ 2   |I |=d−1−k I κi I ai , V d−1 × νd−1−k (dV ), %d−1  2 1 + κ i=1 i

where  I(n, V ) =

p −1 {V }

|pL n| Hk (dL).

k The mapping L → πL n maps isometrically p−1 {V } onto S+ , the unit hemisphere ⊥ in V with pole n, and we obtain after a straightforward calculation

 I(n, V ) =

k S+

|v · n| Hk (dv) =

Ok−1 . k

After evaluation of the constants (use (1.12))   cd−1,d−1−k Ok−1 d −1 −1 γ (d, k, d − k) = , cd,d−k k k we arrive at the following expression. Proposition 6.24 For a set X with positive reach and k ∈ {0, 1, . . . , d = 1}, we have     d − 1 −1 Od−1−k 1B (x) Ckabs (X, B) = ⊥ k nor X Gn (d−1,d−1−k)  %

3 4   ⊥ 2  |I |=d−1−k  I κi I ai , V d−1 × νd−1−k (dV ) Hd−1 (d(x, n)), %d−1  2 1 + κi i=1 for any bounded Borel set B ⊂ Rd . Remark 6.25 1. If X is a convex body then all principal curvatures are nonnegative, the absolute value in the formula can be omitted and we can can integrate (see Example 1.18) 7 

 ⊥

Gn (d−1,d−1−k)

I

82 ai , V

⊥

d−1 νd−1−k (dV )

  d − 1 −1 = . k

Thus, we obtain Ckabs (X, ·) = Ck (X, ·) in this case, cf. Definition 4.28.

132

6 Integral Geometric Formulas

2. Using the inequality we obtain



 |g| ≥ | g| for the inner integral over the Grassmannian, Ckvar (X, ·) ≤ Ckabs (X, ·).

Example 6.26 Let X be a full-dimensional body in R3 with sufficiently smooth boundary, and let κ1 (x), κ2 (x) be the principal curvatures, x ∈ ∂X. Then 1 π

C1abs (X, B) =

 B∩∂X

H1abs(X; x) H1 (dx),

where H1abs (X; x)

1 = π



π 0

   κ (x) cos2 θ + κ (x) sin2 θ  2   1  dθ.    2

   2 (x)  It is easy to see that H1abs(X; x) ≥  κ1 (x)+κ  and a strict inequality holds if 2 κ1 (x)κ2 (x) < 0. In the sequel, we will derive another representation of absolute curvature measures, namely as measures of “locally colliding planes”. Denote ⊥

G(X, d, k) := {(x, n, V ) : (x, n) ∈ nor X, V ∈ Gn (d − 1, k)}. We call G(X, d, k) the kth Grassmann bundle of X. We will need also its “oriented version” ⊥

G0 (X, d, k) := {(x, n, ξ ) : (x, n) ∈ nor X, ξ ∈ Gn0 (d − 1, k)}. Applying Lemma 1.74, we get the following. Lemma 6.27 G0 (X, d, k) (and, hence, also G(X, d, k)) is locally p-rectifiable and Hp -measurable with p = d − 1 + k(d − 1 − k). Further, for Hp -almost all (x, n, ξ ) ∈ G0 (X, d, k), Tanp (G0 (X, d, k), (x, n, ξ )) is is the linear hull of the vectors (0, 0, η),

⊥

η ∈ Tan(Gn0 (d − 1, k), ξ ),

and u, v, n ∧ (ξ

v∗ ) ,

(u, v) ∈ Tand−1 (nor X, (x, n)).

6.3 Absolute Curvature Measures

133

Definition 6.28 Let the mapping φ : G(X, d, k) → A(d, k) be defined by φ : (x, n, V ) → (pV ⊥ x, V ). Its image A(X, d, k) := φ(G(X, d, k)) will be called the affine tangent Grassmannian of X. The measure μX k on A(X, d, k) is defined as follows. If h is a measurable nonnegative function on A(X, d, k) then    X h(z, V ) μk (d(z, V )) = h(z, V ) Hd−k−1 (dz) νkd (dV ), G(d,k) V ⊥

A(X,d,k)

where we apply the usual representation of affine subspaces in the form E = z + V with a linear subspace V and z ⊥ V . For given V ∈ G(d, k), we call TV X := {z ∈ V ⊥ : (z, V ) ∈ A(X, d, k)} the tangential projection of X in direction V . We also denote φ0 : (x, n, ξ ) → (pL(ξ )⊥ x, ξ ),

(x, n, ξ ) ∈ G0 (X, d, k),

and A0 (X, d, k) := φ0 (G0 (X, d, k)). Let further P : (z, ξ ) → ξ be the projection defined on A0 (X, d, k). Applying the Coarea formula (Theorem 1.15) to P , we obtain a relation of μX k to the Hausdorff measure:  f (z, V ) μX k (d(z, V )) A(X,d,k)

= (2cd,k )−1

 A0 (X,d,k)

f (z, L(ξ ))Jk(d−k) P (z, ξ )Hp (d(z, ξ )),

where f is any measurable function with bounded support defined on A(X, d, k). In the following we express absolute curvature measures as measures over the affine tangent Grassmannian. Theorem 6.29 For a set X with positive reach, k = 0, 1, . . . , d − 1 and for a bounded Borel set B ⊂ Rd we have,   1B (x) μX Ckabs (X, B) = βd,k d−1−k (d(z, V )), A(X,d,d−1−k)

(x,n,V )∈φ −1 {z,V }

with βd,k =

  1 d −1 γ (d − 1, k, d − 1 − k). 2 k

134

6 Integral Geometric Formulas

Proof Denoting  % 3 4   ⊥ 2   |I |=d−1−k I κi I ai , V Dk (x, n, V ) := , %d−1  2 1 + κ i=1 i

(6.18)

we have using Proposition 6.24 and the Coarea formula for the projection Π : (x, n, ξ ) → (x, n) defined on G0 (X, d, d − 1 − k), Ckabs (X, B)    d − 1 −1 = Od−1−k 1B (x) k nor X  d−1 × Dk (x, n, V )νd−1−k (dV ) Hd−1 (d(x, n)) ⊥

Gn (d−1,d−1−k)

  d − 1 −1 = Od−1−k (2cd−1,d−1−k )−1 k  × 1B (x)Jd−1Π(x, n, ξ )Dk (x, n, L(ξ )) Hp (d(x, n, ξ )). G0 (X,d,d−1−k)

Using now Lemma 6.30 below and the Area formula for the mapping φ0 , we get Ckabs (X, B)   d − 1 −1 = Od−1−k (2cd−1,d−1−k )−1 k  × 1B (x)Jp φ0 (x, n, ξ )Jq P (φ0 (x, n, ξ )) Hp (d(x, n, ξ )) G0 (X,d,d−1−k)

  d − 1 −1 = Od−1−k (2cd−1,d−1−k )−1 k   × 1B (x) Jq P (φ0 (z, ξ )) Hp (d(z, ξ )) A0 (X,d,d−1−k)

(x,n,ξ )∈φ0−1 {z,ξ }

  d − 1 −1 2cd,d−1−k = Od−1−k k 2cd−1,d−1−k   × A(X,d,d−1−k)

1B (x) μX d−1−k (d(z, V )),

(x,n,V )∈φ −1 {z,V }

and an evaluation of the constant yields the assertion.

 

6.3 Absolute Curvature Measures

135

Lemma 6.30 For Hp -almost all (x, n, ξ ) ∈ G0 (X, d, k), Jp φ0 (x, n, ξ )Jq P (φ0 (x, n, ξ )) = Jd−1 Π(x, n, ξ )Dk (x, n, L(ξ )), where Π : (x, n, ξ ) → (x, n) is defined on G0 (X, d, k) and Dk (x, n, V ) is given by (6.18). Proof Assume that (x, n) is a regular point of nor X, aX (x, n) the associated unit simple (d − 1)-vector, (x, n, ξ ) ∈ G0 (X, d, k), and let V denote the k-subspace associated with ξ . Let {w1 . . . , wd−1 , n} be an orthonormal basis of Rd such that w1 , . . . , wk ∈ V and wk+1 , . . . , wd−1 ∈ V ⊥ . Denote ξv := n ∧ (ξ ηpq := wp ∧ (ξ

v ∗ ), wq∗ ),

v ∈ Rd , 1 ≤ p ≤ k, k + 1 ≤ q ≤ d − 1.

 Denote L := Dφ0 (x, n, ξ ) for brevity. If aX = d−1 i=1 (ui , vi ) then (ui , vi , ξvi ) and (0, 0, ηpq ) are tangent vectors to G0 (X, d, k) at (x, n, ξ ) by Lemma 6.27, and we have  

k d−1   d−1  i=1 L ui , vi , ξvi ∧ p=1 q=k+1 L(0, 0, ηpq )  Jp φ0 (x, n, ξ ) = 

 k d−1   d−1  i=1 ui , vi , ξvi   p=1 q=k+1 (0, 0, ηpq ) and     d−1  i=1 (ui , vi ) Jd−1 Π(x, n, ξ ) = 

 .  d−1  i=1 ui , vi , ηvi  Using Lemma 1.70, we get L(ui , vi , ξvi ) = (pV ⊥ ui + (pV vi · pV x)n + (n · pV ⊥ x)pV vi , ξvi ), i = 1, . . . , d − 1. Note that ηv = 0 whenever v ⊥ V . We can choose the vectors (ui , vi ) so that vk+1 , . . . , vd−1 ⊥ V and that (pV ⊥ ui ) · (pV ⊥ uj ) = 0 whenever 1 ≤ i ≤ k < j ≤ d − 1. Then ξvj = 0 if j > k and L(uj , vj , 0) = (pV ⊥ uj , 0),

j = k + 1, . . . , d − 1,

are orthogonal to L(ui , vi , ξvi ), i = 1, . . . , k. Further, let βpq be the orthogonal projection of L(0, 0, ηpq ) onto the orthogonal complement to Lin {(pV ⊥ uj , 0), j = k + 1, . . . , d − 1}.

136

6 Integral Geometric Formulas

(Note that the last component of βpq is again ηpq .) Then    

k d−1  d−1   k  j =k+1 (pV ⊥ uj , 0)  i=1 L ui , vi , ηvi ∧ p=1 q=k+1 βpq    Jp φ0 (x, n, ξ ) =

 k d−1  d−1   i=1 ui , vi , ηvi   p=1 q=k+1 (0, 0, ηpq ) Further, L(ui , vi , ξvi ), i = 1, . . . , k, and βpq , p = 1, . . . , k, q = k + 1, . . . , d − 1, form a basis of the orthogonal complement to the kernel of P in the space Tanp (G0 (X, d, k), (x, n, ξ )) and, hence,   k d−1  k   i=1 ξvi ∧ p=1 q=k+1 ηpq  . Jq P (φ0 (x, n, ξ )) =  k d−1  k   i=1 L(ui , vi , ξvi ) ∧ p=1 q=k+1 βpq  Putting this together, we have (omitting the arguments of the Jacobians)       d−1   k  i=1 ξvi   j =k+1 pV ⊥ uj  Jp φ0 Jq P   = .   d Jd−1 Π (ui , vi )  i=1

Since the mapping ξv → pV v is an isometry (see Lemma 1.30) and aX (x, n) is a unit multivector, we can write   k   d−1      Jp φ0 Jq P   pV ⊥ uj  = |aX (x, n), ψ| =  pV vi     Jd−1 Π  i=1

j =k+1

with the (d − 1)-form ψ given by +d−1  i=1

, (ui , vi ), ψ =

+   |I |=k

,+ pV vi , w1∗

i∈I

∧ · · · ∧ wk∗



, ∗ pV ⊥ ui , wk+1

∧

∗ · · · ∧ wd−1

.

i∈I c

Writing aX in the form given in Proposition 4.23, we obtain that the last expression equals Bk (x, n, V ), and the proof is finished.  

6.4 Bibliographical Notes 1. The first version of a translative intersection formula for curvature measures was presented by Schneider and Weil in [SW86] in the setting of convex bodies. The proof was done by approximation with polytopes and the mixed functionals were not given explicitly.

6.4 Bibliographical Notes

137

2. The intersection formula (Theorem 6.10) can be extended to more than two bodies. These can be found in [Wei90] for the case of convex bodies and in [Rat96] for sets with positive reach. An explicit integral representation of mixed curvature measures of more than two bodies with positive reach can be found in [Hug99] for convex bodies, and in [HR18] for sets with positive reach. 3. The translative Crofton formula (Theorem 6.20) and some related results were proved in [Rat99]. 4. Absolute curvature measures were treated by Santaló in [San76], in the setting of smooth submanifolds. The Crofton formula was proved by Baddeley in [Bad80]. An extension to sets with positive reach was carried out in [Zäh89], see also [RZ90a]. Absolute curvature measures for unions of sets with positive reach were introduced in [Rat02]. 5. The total absolute curvature measures abs d Cabs j (X) = Cj (X, R ),

j = 0, . . . , d − 1, are related to the total measure of tangent planes as follows. Theorem 6.31 Let X be a set with positive reach and compact boundary, and let the following full-dimensionality condition be satisfied: Hd−1 {x ∈ ∂X : ∃n ∈ S d−1 , (x, n) ∈ nor X, (x, −n) ∈ nor X} = 0.

(6.19)

Then X Cabs k (X) = μk (A(X, d, d − 1 − k)).

This results follows from Theorem 6.29 and from the following lemma which can be found in [RZ02, Lemma 1]. Lemma 6.32 For Hp almost all (z, V ) ∈ (X, d, k), φ −1 is either a singleton, or a pair of points of the type (x, n, V ), (x, −n, V ). 6. Curvature measures can be extended to tensor-valued valuations. An extension of the Principal kinematic formula to this setting was obtained by Hug and Schneider [HS08]. 7. Rotational integral geometry concerns integrals over the special orthogonal group SO(d) only (not over translations). This theory was developed by Jensen in [Jen98]. The following rotational integral formula for curvature measures was proved in [JR08]:

138

6 Integral Geometric Formulas

Theorem 6.33 Let X ⊂ Rd be a set with positive reach such that 0 ∈ ∂X and assume that for νjd -almost all L ∈ G(d, j ), there is no point (x, n) ∈ nor X with x ∈ L and n ⊥ L. Then, for all 0 ≤ k < j < d,  G(d,j )



1 d−j |x| nor X %  κi (x, n)  Hd−1 (d(x, n)), × Qj (x, n, AI c ) % i∈I d−1 2 1 + κi (x, n) |I |=j −1−k i=1

Ck (X ∩ L) νjd (dL) = Oj−1 −1−k

provided that the integral on the right side exists. Here AI c denotes the linear subspace spanned by the principal direction ai (x, n), i ∈ I , and the weight function Qj is given by  Qj (x, n, A ) = Ic

G(d,j ;x)

[L, AI c ]2 d ν (dL), |pL n|j −k j ;1

with G(d, j ; x) denoting the submanifold of G(d, j ) of j -subspaces containing x, and νjd;1 is its invariant measure. An overview on rotational integral formulas can be found in [JK17].

Chapter 7

Approximation of Curvatures

7.1 Approximation by Parallel Sets Recall that the curvature measures Ck (Xr , ·) of the r-parallel sets to a set X with positive reach converge vaguely to those of X itself (see Corollary 4.35). This stability result motivates a natural question whether curvature measures of more general sets can be introduced through approximation with parallel sets. This will indeed be the case, as it will be clear in Chap. 9. However, not only parallel sets may be used for approximation. Classically, approximations by polyhedral (piecewise linear) sets are frequently used in differential geometry, or approximation by smooth sets in different ways. Here we will use flat convergence of normal cycles as an appropriate tool for convergence of the curvature-direction measures and their total values by means of the Lipschitz-Killing curvature forms ϕk . Note that, in general, convergence with respect to the Hausdorff distance dH is not sufficient. Recall that (F) lim means convergence of (d −1)-currents in Rd ×Rd with respect to the flat norm 9 F(T ) = sup T (ϕ) : ϕ ∈ Dd−1 (Rd × Rd ), sup(x,n)∈Rd ×Rd ||ϕ(x, n)|| ≤ 1, sup(x,n)∈Rd ×Rd ||dϕ(x, n)|| ≤ 1

:

and M is the mass norm of the currents (cf. Sect. 1.3.2). Proposition 7.1 Suppose that X, Xε ∈ UPR , X, Xε ⊂ K, ε > 0, for some compact K ⊂ Rd , then for k = 0, 1, . . . , d − 1, (F) lim NXε = NX implies lim Ck (Xε ) = Ck (X) . ε→0

ε→0

© Springer Nature Switzerland AG 2019 J. Rataj, M. Zähle, Curvature Measures of Singular Sets, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-18183-3_7

139

140

7 Approximation of Curvatures

If additionally sup M(NXε ) < ∞ , ε

then we get k (X, ·) . k (Xε , ·) = C w − lim C ε→0

Proof For any smooth functions f and h on Rd × Rd such that h has compact support and is equal to 1 on K × S d−1 we get hf ϕk ∈ Dd−1 (Rd × Rd ) and for k = 0, 1, . . . , d − 1, ck (f ) := max



sup (x,n)∈Rd ×Rd

||hf ϕk (x, n)||,

sup (x,n)∈Rd ×Rd

||d(hf ϕk )(x, n)|| < ∞ .

Here we have used that h, f, ϕk as well as the corresponding derivatives are smooth and therefore bounded on spt h. Then we infer     NXε (f ϕk ) − NX (f ϕk ) = NXε (hf ϕk ) − NX (hf ϕk ) and the right-hand side does not exceed ck (f ) F(NXε −NX ) in view of the definition of the flat norm F. Consequently,   lim NXε (f ϕk ) − NX (f ϕk ) = 0 .

ε→0

Choosing f ≡ 1 we obtain the first assertion, since Ck (·) = N(·) (ϕk ). Let now g be any continuous function on Rd × S d−1 . It has a continuous extension f to Rd × Rd . Furthermore, f can be approximated by smooth functions fj , j = 1, 2, . . ., uniformly on compact sets. Then we get       NXε (f ϕk ) − NX (f ϕk ) ≤ NXε ((f − fj )ϕk ) + NX ((f − fj )ϕk )   + NXε (fj ϕk ) − NX (fj ϕk ) . Recall that the norm of the form ϕk is bounded on the compact set K × S d−1 containing spt NXε and spt NX . Then the first two summands tend to 0 as j → ∞ uniformly in ε. For the first one we have used the assumption that the mass norms of the NXi are uniformly bounded. The third summand goes to 0 as ε → 0 for every j by the above arguments for smooth functions. Hence, we obtain   lim NXε (f ϕk ) − NX (f ϕk ) = 0 .

ε→0

7.1 Approximation by Parallel Sets

141

k (·, g) = N(·) (gϕk ) = N(·) (f ϕk ) for g and f as before this yields Using that C k (Xi , g) − C k (X, g)) = 0 lim (C

i→∞

 

for all continuous g, i.e., the second assertion.

This will be applied below to several approximation problems. Turning back to the problem of approximation by parallel sets we first show that the reach of a set is upper semicontinuous with respect to the Hausdorff distance (see (2.3)). Lemma 7.2 Let Xε , X be subsets of Rd , ε ∈ (0, ε0 ), such that reach Xε ≥ r for all dH

ε ∈ (0, ε0 ) and Xε → X. Then reach X ≥ r. Proof Assume, for the contrary, that reach X < r, and let z be a point with s := dist (z, X) < r and with two different nearest points x, y ∈ X, |x − z| = |y − z| = s. For given ε > 0, denote sε := dist (z, Xε ) and let wε ∈ Xε be a point with |wε − z| = sε . Let εi → 0 be a sequence for which w := limi→∞ wεi ∈ X exists. We can assume that w = x (otherwise, we interchange x and y). Then choose a sequence of points xi ∈ Xεi such that limi→∞ xi = x. Since reach Xε ≥ r, we have by Corollary 4.6 (xi − wεi ) · (z − wεi ) ≤

|xi − wεi |2 |z − wεi | . 2r

Letting i → ∞, we get (x − w) · (z − w) ≤

|x − w|2 |z − w| . 2r

In the isosceles triangle zxw, the cosine of the angle at vertex w equals |w−x|/(2s). Thus we have (x − w) · (z − w) =

|x − w|2 |z − w| . 2s

This, however, contradicts the last displayed inequality, since s < r.

 

A basic tool for approximations is the following theorem saying that Hausdorff convergence of sets with uniformly positive reach implies already flat convergence of normal cycles. dH

Theorem 7.3 For ε ∈ (0, ε0 ) let Xε , X be compact subsets of Rd such that Xε → X and inf0 0 and for all sufficiently small ε > 0, and (7.10) implies |Hd−1(∂Yδ ) − Hd−1 (∂P ε )| ≤ L εHd−1 (∂P ε ). Since this surely implies the uniform boundedness of Hd−1 (∂P ε ) in ε, (vii) follows.   Proof (of Theorem 7.10) Let PR+ denote the family of all compact sets X ⊂ Rd which are parallel sets X = Yδ of a set with positive reach and 0 < δ < reach Y . Let a compact set X ∈ UPR and a regular sequence (εj ) w.r.t. a UPR -representation of X be given. We first show that there exists a sequence Xj ∈ PR+ such that lim dH (Xj , X) = 0, (F) lim NXj = NX and sup M(NXj ) < ∞.

j →∞

j →∞

(7.11)

j

Theorem 7.6 guaranties that (7.11) is true if we replace Xj with Z j := Xεj . The *j , do, sets Z j need not have positive reach, but the closures to their complements, Z

154

7 Approximation of Curvatures

*j , and we have NZ*j = ρ# NZ j , see Theorem 5.26. Applying Lemma 7.5 to the sets Z * and using the reflection principle (Theorem 5.26) for the parallel sets (Z j )δj with some small δj > 0, we obtain that (7.11) is true with Xj being the closure of the *j )δ , and it is easy to see that these sets belong to PR+ . complement to (Z j In order to prove the theorem, it will thus be sufficient to show that for any compact set X ∈ PR+ there exists a sequence (P j ) of d-polytopes such that lim dH (P j , X) = 0,

j →∞

(F) lim NP j = NX , and sup M(NP j ) < ∞. j →∞

j

Recall that X is a compact C 1,1 -domain in Rd (Corollary 4.22) and let νX : ∂X → S d−1 be its (Lipschitz) unit outer normal vector field. The unit normal bundle nor X is a Lipschitz submanifold of Rd × Rd with orienting (d − 1)-vector field aX and NX = (Hd−1

nor X)) ∧ aX

(cf. Sect. 4). In view of positive reach the components of the manifold ∂X, and thus of nor X, have distances greater than some positive constant. Therefore we can reduce the problem to the case when nor X is connected. In this situation the Constancy theorem for Lipschitz manifolds (Theorem 1.53) implies the following: If the current T := (Hd−1

nor X) ∧ φ aX

for some integrable function φ is a cycle, i.e., ∂T = 0, then φ is constant, hence T = φNX . Below T will be specified for our purposes. So let X = Yδ be such that 0 < δ < reach Y , let ε0 , θ > 0 and P ε : 0 < ε < ε0 be the approximating d-polytopes from Lemma 7.11 and let Q, L > 0 be the constants from Proposition 7.12; assume (without loss of generality) that ε0
0 (independent of ε). As any facet F of Σ ε has surface area at least Hd−1 (F ) ≥ θ d−2 εd−1 by the fatness condition, we get ε ) ≤ C, #Fd−1(Σ ε )θ d−2 εd−1 ≤ Hd−1 (Σd−1

which implies #Fd−1(Σ ε ) ≤ Cθ 2−d ε−(d−1) .

j -faces and each j -face has j Further, for 0 ≤ j ≤ d − 1, each facet F has d−1 j j dimensional measure at most cj ε for some constant cj > 0 (since the edge lengths are bounded by 5ε). Consequently, Hj (Σjε ) ≤ Cθ 2−d ε−(d−1) ·

  d −1 cj εj = βj ε−d−1−j j

for some constants βj > 0. On the other hand, using the property (vi) from Proposition 7.12, we see that for any 0 ≤ j ≤ d − 1 and for any face F ∈ Fj (Σ ε ), the set ΓF has diameter less or equal to Qε, hence, Hd−1−j (ΓF ) ≤ ωd−1−j Qd−1−j εd−1−j . Taking into account (7.4) we obtain Hd−1 (nor P ε ) ≤

d−1  j =0

proving (i).

βj Qd−1−j ωd−1−j ,

156

7 Approximation of Curvatures

Note that, since X = Yδ , we have Yr \ Y ⊂ Unp (∂X) whenever 0 < r < reach Y . Further, for 0 < ε < ε0 we define the mappings f ε : nor P ε → Rd × S d−1 by f ε : (x, n) → (Π∂X (x), νX (Π∂X (x))). We get from Lemma 7.11(v) that ∂P ε ⊂ Y(reach Y +δ)/2 \ Yδ/2 if 0 < ε < ε0 , and the mapping Π∂X is Lipschitz on Y(reach Y +δ)/2 \ Yδ/2 by Lemma 4.7. Since clearly νX is Lipschitz on ∂X = ∂Yδ , we get that the mappings f ε are uniformly Lipschitz in ε, which is condition (ii) of Proposition 7.9. Condition (iii) of Proposition 7.9 follows immediately from the construction since we have |x − Π∂X (x)| < qε whenever x ∈ ∂P ε , by Lemma 7.11(v), and νX is continuous on ∂X. It remains to show that (f ε )# NP ε = NX . Recall that NP ε = (Hd−1 nor P ε ) ∧ iP ε aP ε (for the index function iP ε and the orienting (d − 1)-vector field aP ε ) is a cycle. From this we infer ∂((f ε )# NP ε ) = (f ε )# ∂NP ε = 0, i.e., T ε := (f ε )# NP ε is also a cycle. Furthermore, the area formula for rectifiable currents yields T ε = (Hd−1

nor X) ∧ ηε ,

with η (y, m) = ε

 (x,n)∈(f ε )−1 {(y,m)}

=

 ( d−1 ap Df ε (x, n))aP ε (x, n) iP ε (x, n)  |( d−1 ap Df ε (x, n))aP ε (x, n)|

φ ε (y, m)aX (y, m)

where φ ε (y, m) :=

 (x,n)∈(f ε )−1 {(y,m)}

iP ε (x, n)sgn(f ε ; x, n)

7.3 Bibliographical Notes

157

and sgn(f ε ; x, n) is the corresponding sign of orientation of the (d − 1)ε ε vector  associated with Tan(nor X, f (x, n)), i.e., sgn(f ; x, n) equals 1 or −1 ε

( ap Df (x,n))aP ε (x,n) agrees with aX (y, m) or −aX (y, m), respectively. if d−1 ε |(

d−1 ap Df

(x,n))aP ε (x,n)|

According to the arguments at the beginning of the proof the function φ ε must be constant. Therefore it suffices to show that φ ε (y, m) = 1 for a set of (y, m) with positive Hd−1 -measure. Then we obtain T ε = NX . To this aim we consider an open subset Δ of some facet μ of P ε with outer unit normal n, such that the mapping f ε on Δ × {n} is bijective. For x ∈ Δ we have iP ε (x, n) = 1 and aP ε (x, n) = a1 ∧ . . . ∧ ad−1 with ai = (bi , 0) for an orthonormal basis b1 , . . . , bd−1 in the (constant) tangent space to P ε at x such that b1 ∧ . . . ∧ bd−1 ∧ n, Ωd  = 1. Then we obtain for these pairs (x, n), φ ε (y, m) = sgn(f ε ; x, n) is the sign of the scalar product 

d−1 ap Df

ε



(x, n) aP ε (x, n) · aX (y, m)

and, applying the same procedure as in the proof of Theorem 7.3, we get (7.3) again, now with si = 0, i = 1, . . . , d − 1, and the sign will be positive. At the same time the area theorem implies that the set f ε (Δ × {n}) ⊂ nor X has positive Hausdorff measure. Thus, all conditions of Proposition 7.9 are fulfilled and we get the desired convergence.  

7.3 Bibliographical Notes 1. A more direct proof of a consequence of Theorem 7.3 was given by Federer [Fed59, §5.9]. He showed, under the same assumptions, the vague (in his notation weak) convergence of curvature measures using the Steiner formula. 2. Approximation of C 1,1 surfaces by polyhedral ones and convergence of the associated curvature measures was shown by Cheeger, Müller and Schrader in [CMS84]. They triangulate the smooth surfaces and refine these triangulations in order to get “fat triangulations”, where the approximation works. Fu [Fu93] obtained the corresponding approximation result for the associated normal cycles. Approximation of the normal cycles of C 2 -submanifolds of Rd by polyhedra is also treated in [Mor08, §20.3]. 3. Cohen-Steiner and Morvan [CSM06] proved a stability result for normal cycles, which yields for a smooth compact domain X ⊂ Rd and a compact d-polytope P ⊂ Rd such that ∂P lies within the distance reach ∂X from ∂X, F(NX − NP ) ≤ (dH (X, P ) + α(X, P )) CX (M(NP ) + M(∂NP )),

158

7 Approximation of Curvatures

where α(X, P ) is the supremum of angular differences | (νX (Π∂X (y)), v)| over all (y, v) ∈ nor P , and CX is a constant depending on the second fundamental form of X. In fact, the result is proved in a significantly higher generality, in the Riemannian setting and for a much more general set P admitting a normal cycle, roughly in the sense of Definition 9.5. 4. The result of Fu & Scott [FS13] implies that if X ⊂ R3 is a parallel set to a compact set with positive reach (X ∈ PR+ ) then there exist approximating 3dH

polytopes P j such that P j → X, (F) limj →∞ NP j = NX , and sup M(NP j ) ≤ C M(NX ), j

with a constant C (independent of X). Using this in the proof of Theorem 7.10 we can get approximating polytopes P j to a compact UPR set X in R3 with a regular sequence (εj ) (in particular, to a compact set with positive reach) such that, in addition, supj M(NP j ) < ∞. This implies, in view of Proposition 7.1, that k (X, ·) k (P j , ·) = C w − lim C j →∞

k = 0, 1, . . . , d − 1.

An extension to higher dimension remains open.

Chapter 8

Characterization Theorems

8.1 Characterization of Lipschitz-Killing Curvatures In the classical case of convex geometry the Lipschitz-Killing curvatures and related measures can be characterized as certain basic Euclidean invariants, which underlines their geometric importance. Recall that a well-known result of Hadwiger (Theorem 2.1) states that any motion invariant continuous valuation on the space of compact convex sets is a linear combination of Minkowski’s quermasssintegrals. This was localized by Schneider to the case of their curvature measures and surface area measures, see Theorems 2.2 and 2.3. The aim of the present chapter is to extend such characterizations to compact UPR -sets. Here the main idea is to combine continuity with respect to the Hausdorff distance with flat convergence of the associated normal cycles. This enables us to reduce everything to the special case of polytopes via approximation. Moreover, in this way we can show that functionals or measures with such properties on polytopes have unique extensions to compact UPR -sets. First recall some classical notions: Definition 8.1 A mapping Ψ defined on a space S of subsets of Rd and with values in a vector space is said to be a valuation or additive if Ψ (A ∪ B) = Ψ (A) + Ψ (B) − Ψ (A ∩ B) , provided A, B, A ∩ B, A ∪ B ∈ S, and Ψ (∅) = 0 whenever ∅ ∈ S. By induction we infer the so-called inclusion-exclusion principle for a valuation Ψ : Ψ

n  i=1

n

 Ai = (−1)k−1 k=1



Ψ (Ai1 ∩ . . . ∩ Aik ) .

(8.1)

1≤i1 0. Definition 8.11 A mapping Ψ : S → M(Y) is (i) motion covariant if Ψ (gX, g(·)) = Ψ (X, ·), g ∈ Gd , X ∈ S, (ii) homogeneous of degree k if Ψ (λX, λ(·)) = λk Ψ (X, ·), λ > 0, X ∈ S, for Y = Rd × S d−1 or Y = Rd , (iii) locally determined if for all Borel sets B ⊂ Y and X, Y ∈ S, NX

1B = NY

1B implies Ψ (X, B) = Ψ (Y, B) ,

 := B for Y = Rd × S d−1 , B  := B × S d−1 for Y = Rd , and where B d d−1  := R × B for Y = S B . We first consider the version Y := Rd × S d−1 concerning base points and normal k (X, ·), k = 0, . . . , d −1. For the directions, i.e., the curvature-direction measures C special case of convex polytopes, where S = P, the well-known characterizations of Lebesgue measure and spherical Lebesgue measure can be used and no additivity or continuity assumption is needed. Lemma 8.12 (Characterization of curvature-direction measures for polytopes) (i) Let Ψ : P → M(Rd × S d−1 ) be motion covariant and locally determined. Then it has the unique representation Ψ (P , ·) =

d−1 

k (P , ·) ck C

k=0

for some real constants ck . If Ψ is homogeneous of degree k, then Ψ (P , ·) = k (P , ·). ck C (ii) Ψ has a unique extension to a motion covariant locally determined valuation on L(P). It has the same representation as in (i), hence it is P-continuous. Proof (i) In order to show that Ψ (∅, ·) = 0 we use that the marginal measure Ψ (∅, (·) × S d−1 ) on Rd is translation invariant and hence, a constant multiple of Lebesgue measure. Since it is finite, the constant must be 0, which implies the above assertion. For arbitrary P ∈ P the locality of Ψ yields that spt Ψ (P , ·) ⊂ nor P . Fix now an affine subspace L ∈ A(d, k). Then for Borel sets A ⊂ L, bounded, and B ⊂

166

8 Characterization Theorems

S d−1 ∩ L⊥ the values Ψ (P , A × B) do not depend on P such that A × B ⊂ nor P , again by the locality of Ψ . Denote μL (A × B) := Ψ (P , A × B) for such P . From the motion invariance of Ψ we obtain μL (A × B) = cL Lk (A)Hd−1−k (B) , for some cL ∈ R, and that cL =: ck is independent of L ∈ A(d, k). For a k-face F of the convex polytope P denote the set of unit vectors from the normal cone of P at arbitrary points of rel int F by nor(P , F ). Then we infer Ψ (P , A × B) = ck Ld (A)Hd−1−k (B) if A ⊂ rel int F and B ⊂ nor(P , F ). Moreover, nor P =

d−1 



rel int F × nor(P , F )

k=0 F ∈Fk (P )

is a disjoint union and consequently, Ψ (P , A × B) =

d−1  k=0

=

d−1 

ck



Lk (F ∩ A)Hd−1−k (nor(P , F ) ∩ B)

F ∈Fk (P )

k (P , A × B) ck C

k=0

for all Borel sets A ⊂ Rd and B ⊂ S d−1 . Since such product sets form an intersection stable generator of the Borel σ -algebra on Rd × S d−1 we get Ψ (P , ·) =

d−1 

k (P , ·) . ck C

k=0

If Ψ is homogeneous of degree k we infer cj = 0 for j = k because of the scaling j . properties of the C (ii) The right-hand side of the above representation is defined for all polytopes, which provides an extension to a P -continuous motion covariant locally determined valuation on L(P). Uniqueness follows then from the convex case by the inclusionexclusion principle applied to an arbitrary extension with these properties.   c -sets Using the approximation results from the last section the general case for UPR + is a consequence. Recall that PR denotes the set of X ⊂ Rd such that X = Yδ for some Y ∈ PRc and 0 < δ < reach Y .

Theorem 8.13 (Characterization of curvature-direction measures) Let Ψ : P → M(Rd × S d−1 ) be motion covariant and locally determined. Then it admits c a unique extension to a motion covariant valuation on UPR being P -continuous on

8.2 Characterization of Associated Measures

167

PR+ and F -continuous on PRc . It has the representation Ψ (X, ·) =

d−1 

k (X, ·), ck C

c X ∈ UPR ,

k=0

for some uniquely determined real constants ck and the curvature-direction meak (X, ·). Therefore Ψ is F -continuous on U c . sures C PR k (X, ·) for all X. If Ψ is homogeneous of degree k, we get Ψ (X, ·) = ck C Proof Here the arguments for measures are similar as in the proof of Theorem 8.8 for functionals. Lemma 8.12 provides the corresponding extension Ψ (P , ·) =

d−1 

k (P , ·) ck C

k=0

for general polytopes P ∈ L(P). According to Theorem 7.10 for any X ∈ PR+ there exists a sequence P j ∈ L(P) converging to X in the Hausdorff distance such k (P j , ·) = that supj M(NP j ) < ∞, (F) limj →∞ NP j = NX and w − limj →∞ C k (X, ·), k = 0, . . . d − 1. By P -continuity of Ψ we infer w − limj →∞ Ψ (P j , ·) = C Ψ (X, ·). This yields the required extension for X ∈ PR+ . Next one uses the approximation of arbitrary compact PR-sets by parallel sets in the sense of Lemma 7.5 together with F -continuity in order to obtain the corresponding extension of Ψ to PRc . The latter together with the inclusionc . exclusion principle resulting from additivity leads to the general version for UPR   Remark 8.14 1. The above proof shows that the condition on F -continuity of the extension of Ψ to X ∈ PRc could be replaced by the version where the approximating sets Xn in the definition are parallel sets of of X with small distances. F -continuity is in this case a consequence. 2. For the total values Ψ (Rd × S d−1 ) the results are consistent with those from the previous section. As in Corollary 8.9 the last theorem implies the following. Corollary 8.15 Let Ψ : S → M(Rd × S d−1 ) be a motion covariant locally determined valuation on S and suppose that one of the following three additional conditions is fulfilled: (i) S = C and Ψ is weakly continuous with respect to the Hausdorff distance. (ii) S = PRc , Ψ is F -continuous, and it is P -continuous at all sets from PR+ (for the unique additive extension of Ψ from P to L(P)). c and Ψ is F -continuous. (iii) S = UPR

168

8 Characterization Theorems

Then Ψ admits the unique representation Ψ (X, ·) =

d−1 

k (X, ·) , X ∈ S , ck C

k=0

with constants ck as above. Next we consider the marginal Lipschitz-Killing curvature measures k (X, (·) × S d−1 ) , k = 0, . . . , d − 1 , Ck (X, ·) = C and include again the Lebesgue measure, restricted to X, in form of Cd (X, ·). Recall that for convex polytopes weak continuity with respect to the Hausdorff distance agrees with F -continuity. Theorem 8.16 (Characterization of curvature measures) Let Ψ : P → M(Rd ) be a motion covariant locally determined valuation, which is weakly continuous with respect to the Hausdorff distance. Suppose additionally, that Ψ (P , ·) ≥ 0, P ∈ c P. Then it admits a unique extension to a valuation on UPR being P -continuous on + c PR and F -continuous on PR . It has the representation Ψ (X, ·) =

d 

ck Ck (X, ·),

c X ∈ UPR ,

k=0

for some uniquely determined constants ck ≥ 0. Therefore Ψ is an M(Rd )-valued c F -continuous motion covariant locally determined valuation on UPR . If Ψ is homogeneous of degree k, we get Ψ (X, ·) = ck Ck (X, ·) for all X. Proof Theorem 2.2 provides the unique representation Ψ (P , ·) =

d 

ck Ck (P , ·)

k=0

for nonempty convex polytopes P . (For the empty set this is trivial, since both sides of the equation vanish.) Its proof is more involved than that of Lemma 8.12, see [Sch78, 6.1]. Note that the notion of locality used there is equivalent to that from Definition 8.11 (iii). For arbitrary polytopes the equality follows by additive extension on both sides. The remaining part concerning approximations and additive extension is as in the proof of Theorem 8.13.   Finally, the direction versions, i.e., the marginal measures with respect to the normal components, are denoted by k (X, Rd × (·)) , k = 0, . . . , d − 1 . Sk (X, ·) := C Recall that in the convex case they agree with the k-th order surface area measures.

8.3 Bibliographical Notes

169

Theorem 8.17 (Characterization of direction measures) Let Ψ : C → M(Rd ) be a motion covariant locally determined valuation, which is weakly continuous with respect to the Hausdorff distance. Then it admits a unique extension to a c being being P -continuous on PR+ and F -continuous on PRc . It valuation on UPR has the representation Ψ (X, ·) =

d−1 

ck Sk (X, ·),

c X ∈ UPR ,

k=0

for some uniquely determined real constants. Therefore Ψ is an M(S d−1 )-valued F -continuous motion covariant locally determined valuation. Proof Here the representation for convex bodies is given by Theorem 2.3 (see Schneider [Sch75]). The remaining arguments are as above.   Remark 8.18 1. As in Corollary 8.15 the corresponding characterizations of the curvature c measures Ck or the direction measures Sk on the spaces C, PRc and UPR are consequences of Theorems 8.16 and 8.17, respectively. 2. Theorem 8.16 is formulated under the additional condition of positivity of Ψ on P. The question, whether arbitrary linear combinations of the LipschitzKilling curvature measures can be characterized in such a way, may be reduced to the following classical problem, which is still unsolved: Is any simple rotation invariant real valuation on spherical polytopes P a multiple of the corresponding (d − 1)-volume Hd−1 (P )? (Simple means here vanishing on lower-dimensional polytopes. See proof of Theorem 6.1 in Schneider [Sch78].) For the case of nonnegative functionals a proof of this relationship was given [Sch78, 6.2]. c 3. All results of this chapter concerning UPR -sets can be extended to more general classes of sets from the next chapter with associated normal cycles, provided they admit polytopal approximations as in Theorem 7.10. Therefore the LipschitzKilling curvatures and the curvature-direction measures as well as both their marginal variants can be considered as complete systems of Euclidean invariants, which are F -continuous (locally determined measure-valued) valuations. (For the case of the curvature measures the above positivity is still assumed.)

8.3 Bibliographical Notes 1. Hadwiger’s characterization Theorem 2.1 is a special case of Corollary 8.9. It has been used in the auxiliary Proposition 8.5 (i). 2. The characterization of the direction measures Sk on the space C in the sense of Remark 8.18(i) can be found in Schneider [Sch75].

170

8 Characterization Theorems

3. The corresponding version for the curvature measures Ck on C was shown in Schneider [Sch78]. 4. Federer [Fed59, 5.17] already posed the problem of a suitable characterization of the curvature measures for PR-sets, where a corresponding notion of continuity was not yet developed. 5. The unique additive extendability of a valuation on P (Proposition 8.4) was shown by Volland [Vol57]. Later, Groemer [Gro78] proved that any continuous valuation on C has a unique extension to a valuation on the convex ring L(C). See also [Sch14, §6.2] for an overview and further references. 6. The characterization of the curvature-direction measures for convex polytopes ( cf. Lemma 8.12 (i)) was given in Glasauer [Gla97].

Chapter 9

Extensions of Curvature Measures to Larger Set Classes

So far we have introduced curvature measures for sets with positive reach and their locally finite unions (such that any finite intersection has positive reach). This setting is still not satisfactory since it does not encompass some natural set classes as closures of complements to convex bodies, or boundaries of convex bodies, though these sets should apparently admit a natural definition of curvatures. The aim of this chapter is to show how the technique of normal cycles and the compactness theorem for currents can be used to extend curvature measures in a unique way to more general classes, covering the examples mentioned above. Different extensions appeared up to now in the literature and they are usually technically demanding. We aim rather to present the method than to give an extension as wide as possible (which seems to be an open question up to now). The basic idea is as follows. We approximate a given set X ⊂ Rd (and we will restrict ourselves to compact sets in this chapter) by appropriate “nice” sets Xi (smooth sets, sets with positive reach etc.) for which the normal cycles NXi are defined. If one can show that the normal cycles are bounded as currents in the mass norm sup M(NXi ) < ∞,

(9.1)

i

then the Compactness theorem for currents (Theorem 1.57) implies that there exists a subsequence NXik converging to an integral current T in the flat seminorm. Moreover, T retains some other important geometric properties of the normal cycles NXi , namely it is a cycle (∂T = 0) and it is Legendrian (T α = 0, cf. Proposition 4.40). Of course, if this limit would change from subsequence to subsequence, we could not expect it to be a good candidate for a normal cycle of X. Nevertheless, there is a uniqueness theorem due to J. Fu [Fu89b] (cf. Theorem 9.4 below with another proof) saying that there is at most one integral cycle with the Legendrian property and with another property linking to topological properties of the set (Euler-Poincaré characteristic of intersection with halfspaces). This makes it © Springer Nature Switzerland AG 2019 J. Rataj, M. Zähle, Curvature Measures of Singular Sets, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-18183-3_9

171

172

9 Extensions of Curvature Measures to Larger Set Classes

possible to introduce normal cycles (and, hence, also curvature measures, through the relation given in Theorem 4.42) in a unique way, and guaranteeing even a local version of the Gauss-Bonnet formula.

9.1 Legendrian Cycles Definition 9.1 A Legendrian cycle in Rd is a locally integral (d − 1)-current T ∈ d d Iloc d−1 (R × R ) with the following properties: spt T ⊂ Rd × S d−1 ,

(9.2)

∂T = 0 (T is a cycle),

(9.3)

T

α = 0 (T is Legendrian),

(9.4)

where α is the contact 1-form in R2d acting as (u, v), α(x, n) = u · n,

u, v, x, n ∈ Rd .

d d Any locally integral current T ∈ Iloc d−1 (R × R ) is integer-multiplicity locally (d − 1)-rectifiable, hence, it has an integral representation of the form

T = (Hd−1

WT ) iT ∧ aT ,

(9.5)

with a Hd−1 -measurable and locally (d − 1)-rectifiable set WT ⊂ Rd × S d−1 , a unit simple measurable (d − 1)-vector field aT on WT (prescribing an orientation of WT ) and an integer-valued Hd−1 -integrable function iT on WT (index function), cf. Definitions 1.55 and 1.43. The exterior derivative of the contact form α is the symplectic form ω := dα which is a constant 2-form in Rd × Rd acting as (u, v) ∧ (u , v  ), ω = u · v  − v · u .

(9.6)

The standard description using the notation dxi = (ei , 0)∗ , dyi = (0, ei )∗ , i = 1, . . . , d, is ω=

d 

dxi ∧ dyi .

i=1

A Legendrian cycle T clearly satisfies T

ω=T

dα = ∂(T

α) = 0,

(9.7)

9.1 Legendrian Cycles

173

and this is called Lagrangian property. Using the representation (9.5) of T , we get that ω =μ∧ξ

T is representable by integration, with dμ = iT |aT

ω| d(Hd−1

WT )

and ξ=

aT |aT

ω ω|

if |aT ω| > 0, and ξ = 0 otherwise. Thus, (9.7) and Proposition 1.36 imply that μ = 0, which means that aT (x, n)

ω = 0,

T  − a.a.

Similarly we obtain from the Legendrian property T aT (x, n)

α = 0,

(9.8)

α = 0 that

T  − a.a.

(9.9)

Recall that π0 , π1 denote the two component projections in Rd × Rd , i.e., π0 (x, n) = x,

π1 (x, n) = n.

For a general Legendrian cycle we can find a representation of its approximate tangent spaces as in the case of the normal cycle of a set with positive reach (Proposition 4.23). Theorem 9.2 Let T be a Legendrian cycle in Rd . Then, for T -almost all (x, n), Tand−1 (WT , (x, n)) is a (d −1)-dimensional linear subspace of R2d and there exists a positively oriented orthonormal basis {b1 (x, n), . . . , bd−1 (x, n), n} of Rd and numbers κ1 (x, n), . . . , κd−1 (x, n) ∈ (−∞, ∞] such that the vectors ⎛ ai (x, n) := ⎝ 

⎞ 1 1 + κi2 (x, n)

bi (x, n), 

κi (x, n) 1 + κi2 (x, n)

bi (x, n)⎠ ,

i = 1, . . . , d − 1,

form an orthonormal basis of Tand−1 (WT , (x, n)). (We set √ √∞

1+∞2

1 1+∞2

= 0 and

= 1.) The numbers κi (x, n) are uniquely determined, up to the order, and

174

9 Extensions of Curvature Measures to Larger Set Classes

the subspace spanned by the vectors bj (x, n) belonging to a fixed value among the κi (x, n) (1 ≤ i ≤ d − 1) is uniquely determined. In analogy to sets with positive reach, we will call the numbers κi (x, n) principal curvatures and vectors bi (x, n) principal directions of T at (x, n). Proof By rectifiability, for T -almost all (x, n), Tx,n := Tand−1 (WT , (x, n)) is a (d − 1)-dimensional subspace of Rd associated with aT (x, n) and aT (x, n)

ω = aT (x, n)

α=0

by (9.8) and (9.9). Note that the last equality implies that Tx,n ⊂ n⊥ × n⊥ . Fix such an (x, n) ∈ WT and an ε > 0, and consider the linear mapping Lε : Tx,n → n⊥ ,

(u, v) → u + εv.

Decomposing Tx,n = ker Lε ⊕ (ker Lε )⊥ , if p := dim ker Lε > 0 we can find and orthonormal basis of ker Lε of the form  √

ε 1 + ε2

bi , − √

1 1 + ε2

 bi ,

1 ≤ i ≤ p,

hence, we can put κ1 = · · · = κp = −1/ε. Any vector (u, v) ∈ (ker Lε )⊥ satisfies 0 = (u, v) · (εbi , −bi ) = ε(u · bi ) − (v · bi ), From the Lagrangian property aT

i = 1, . . . , p.

ω = 0 and (9.6), we get additionally

0 = (u, v) ∧ (εbi , −bi ), ω = −(u · bi ) − ε(v · bi ),

i = 1, . . . , p.

These two linear equations imply that u · bi = v · bi = 0, i = 1, . . . , p. Thus, denoting by V the ((d − p − 1)-dimensional) orthogonal complement to Lin {b1 , . . . , bp , n}, we get that (ker Lε )⊥ ⊂ V × V . Hence, the linear mapping L := π1 ◦ (Lε |(ker Lε )⊥ )−1 acts from V to V and it is self-adjoint. This follows again from the Lagrangian property since for any (u, v), (u , v  ) ∈ (ker Lε )⊥ , v · (u + εv  ) − (u + εv) · v  = u · v − u · v  = (u , v  ) ∧ (u, v), ω = 0.

9.1 Legendrian Cycles

175

Thus, we can choose an orthonormal basis {bi = ui + εvi , i = p + 1, . . . , d − 1} of eigenvectors of L and the corresponding real eigenvalues λi satisfy vi = λi (ui + εvi ), i = 1, . . . , d − 1, hence ((1 − λi ε)bi , λi bi ),

i = p + 1, . . . , d − 1,

are basis vectors of (ker Lε )⊥ . Setting κi = λi /(1 − ελi ) (which is defined as ∞ if λi ε = 1), we get the assertion on existence. The uniqueness (and, thus, independence of ε) can be proved exactly as in Lemma 4.24.   If we orient the carrier WT by the unit (d − 1)-vectorfield aT (x, n) := a1 (x, n) ∧ · · · ∧ ad−1 (x, n), there must exist an integrable integer-valued index function iT on WT such that (9.5) holds. Let λ(x, n) be the number of negative principal curvatures an (x, n) (defined Hd−1 -almost everywhere on WT ) and set ιT (x, n) := (−1)λ(x,n)iT (x, n).

(9.10)

This particular form of the index function in (9.5) will often be used in the sequel. For the uniqueness theorem we will use the restriction of a Legendrian current T to the Gauss curvature form ϕ0 (see Definition 4.31) a1 ∧ · · · ∧ ad−1, ϕ0 (x, n) = (dωd )−1 π1 (a1 ) ∧ · · · ∧ π1 (ad−1 ) ∧ n, Ωd , a1 , . . . , ad−1 ∈ Rd × Rd . We can also describe ϕ0 in a shorter way by using the operation of interior multiplication of multivectors and multicovectors (see Sect. 1.2.1): ϕ0 (x, n) = (dωd )−1 (π1 )# (n

Ωd ),

(x, n) ∈ Rd × Rd .

To any unit vector u, the Hodge star operator assigns the unit simple (d − 1)vector u associated with Tan(S d−1 , u) (see Definition 1.29). Let us denote this mapping by σ : u → u which orients the unit sphere. Recall also that (π1 )# T is the push-forward of the current T by π1 (see (1.27)). It can be described as follows.

176

9 Extensions of Curvature Measures to Larger Set Classes

Theorem 9.3 For any compactly supported Legendrian cycle T we have T (ϕ0 )(H S d−1 ) ∧ σ , (i) (π1 )# T = (ii) T (ϕ0 ) = x∈Rd ιT (x, n) for almost all n ∈ S d−1 , and (iii) for any T -integrable function g on Rd × S d−1 ,  dωd T (gϕ0 ) =

 S d−1

g(x, n)ιT (x, n) Hd−1 (dx).

x∈Rd

Proof Since T is a cycle, we get ∂(π1 )# T = (π1 )# ∂T = 0, hence, (π1 )# T is a cycle as well. The Constancy theorem (Theorem 1.53) applied to the manifold S d−1 oriented by σ yields (π1 )# T = c(H

S d−1 ) ∧ σ

with a constant c ∈ Z. Further, by the definition of ϕ0 we have for any smooth function h on S d−1 dωd T ((h ◦ π1 )ϕ0 ) = T ((h ◦ π1 )π1# (id

Ωd ))

= ((π1 )# T )(h(id Ωd ))  =c h(n)σ (n), n Ωd  Hd−1 (dn)  =c

S d−1

h dHd−1.

(9.11)

S d−1

Setting h ≡ 1, we get c = T (ϕ0 ) and (i) is proved. In order to show (ii), note that, by Theorem 9.2, the Jacobian of π1 |WT equals Hd−1 - almost everywhere d−1    κi (x, n)    Jd−1 (π1 |WT )(x, n) =  . 2  1 + κi (x, n)  i=1

On the other hand, we have dωd aT (x, n), ϕ0 (x, n) = π1 a1 (x, n) ∧ · · · ∧ π1 ad−1 (x, n), n =

d−1  i=1

Ωd 

κi (x, n)  b1 (x, n) ∧ · · · ∧ bd−1 (x, n) ∧ n, Ωd  1 + κi (x, n)2

= (−1)λ(x,n) Jd−1 (π1 |WT )(x, n).

9.1 Legendrian Cycles

177

Now, the Area formula (Theorem 1.49) applied to π1 |WT and a T -integrable function g implies  dωd T (gϕ0 ) =

g(x, n)iT (x, n)aT (x, n), ϕ0 (n) Hd−1 (d(x, n)) WT

 =

g(x, n)iT (x, n) 

WT

=



i=1

κi (x, n) 1 + κi (x, n)2

Hd−1 (d(x, n))

(9.12)

g(x, n)ιT (x, n)Jd−1 (π1 |WT )(x, n)Hd−1 (d(x, n)) 

=

d−1 

WT

 S d−1

g(x, n)ιT (x, n) Hd−1 (dn),

x∈Rd

which proves (iii). Comparing the last formula with (9.11), we get ⎛

 S d−1



g(n) ⎝T (ϕ0 ) −

⎞ ιT (x, n)⎠ Hd−1 (dn) = 0

x∈Rd

for any bounded measurable function g on S d−1 , and this implies (ii).

 

Now we can formulate and prove the uniqueness theorem for Legendrian cycles. Theorem 9.4 Let T be a compactly supported Legendrian cycle such that T ϕ0 = 0. Then T = 0. Proof Using (9.12), we get for any smooth function g on Rd × S d−1 (T

ϕ0 )(g) = T (gϕ0 ) = (dωd )

−1

 ιT g sd−1 dHd−1 , WT

where sd−1 (x, n) =

d−1  i=1



κi (x, n) 1 + κi (x, n)2

(“generalized Gauss curvature” defined Hd−1 -almost everywhere on WT ). The assumption T ϕ0 = 0 thus implies sd−1 = 0 almost everywhere on WT . In particular, denoting f = π1 |WT , we have rank Df < d − 1 almost everywhere. Assume, for the contrary, that T = 0, and let m := ess sup rank Df < d − 1.

178

9 Extensions of Curvature Measures to Larger Set Classes

We can then assume, without loss of generality (removing a subset of (d − 1)dimensional measure zero), that at all (x, n) ∈ WT , Tand−1 (WT , (x, n)) has the basis given in Theorem 9.2 with principal curvatures κi (x, n) = 0 whenever i > m. (In the case d = 3 and m = 1 it might happen that, in order to preserve the prescribed orientation, we have to choose κ1 = 0 instead of κ2 = 0, and the role of the indices must be exchanged in the whole proof.) Further, the set Y := {n ∈ S d−1 : Hd−1−m (f −1 {n}) > 0} is countably Hm -rectifiable (see Theorem 1.24) and, applying the Area formula (Theorem 1.21), we get for Hd−1 -almost all (x, n) ∈ f −1 (Y ), either ap Jm f (x, n) = 0 or Tanm (f (WT ), n) = Lin {b1 (x, n), . . . , bm (x, n)}. Note that if (x, n) ∈ f −1 (Y ) and rank Df (x, n) = m then the subspaces U (n) := Lin {bi (x, n) : 1 ≤ i ≤ m} and V (n) := Lin {bi (x, n) : m + 1 ≤ i ≤ d − 1} do not depend on x for which (x, n) ∈ f −1 (Y ), due to the property above. Let p be the orthogonal projection from Rd into an m-dimensional subspace Lm of Rd , and let Ωm denote the volume form in Lm . Due to the choice of m, we have T

(p ◦ π1 )# Ωm = 0

for some fixed subspace Lm . Slicing T by p ◦ π1 , we get for the masses M of the currents (see (1.34))  # 0 = M(T (p ◦ π1 ) Ωm ) = MT , p ◦ π1 , y Hm (dy). Lm

Hence, there exists a measurable subset Z of Lm of positive measure such that for y ∈ Z: (i) T , p ◦ π1 , y = 0 is a cycle (cf. Proposition 1.66); (ii) (see Theorem 1.65) (f −1 (p−1 {y}))) ∧ ζ(x, n),

T , p ◦ π1 , y = (Hd−1−m where for (x, n) ∈ Wm ∩ f −1 (p−1 {y}), ζ (x, n) = ιT (x, n)aT (x, n)

(p ◦ π1 )# Ωm /ap Jm (p ◦ f )(x, n)

= ιT (x, n)(bm+1 (x, n), 0) ∧ · · · ∧ (bd−1 (x, n), 0). Take a point y ∈ Y , let R be an indecomposable component (cf. Definition 1.67) of T , p ◦ π1 , y, and take a point (x, n) ∈ spt R. Applying Theorem 1.69 for the

9.2 Normal Cycles

179

projection π1 , we get spt R ⊂ f −1 {n}. It follows that n ∈ Y (otherwise, R = 0). Denoting by q the orthogonal projection of Rd × Rd to (V (n) × {0})⊥ , we have from the description above q = 0 Hd−1−m − a.e. on f −1 {n},

ζ

which implies that q|spt R is constant (apply Theorem 1.69 again, this time for the projection q ◦ π1 ). Hence, spt R is contained in the affine (d − 1 − m)-subspace (x, n) + V (n) × {0}. The Constancy theorem yields that R must be a multiple of the Lebesgue measure. Since T (and, hence, also R) is compactly supported, we get R = 0, a contradiction.  

9.2 Normal Cycles Let us come back to the approximation idea described at the beginning of this chapter. In order to get uniqueness for the Legendrian cycles obtained as flat limits of normal cycles of approximating sequences, we have to assure that all the limit currents have the same restriction to the Gauss curvature form ϕ0 . It is natural to relate this quantity to geometric properties of the set X. Assume for a moment that X is a compact domain with C 2 (or even only C 1,1 ) boundary and let NX := (Hd−1

nor X) ∧ aX

be its normal cycle. Then, clearly the index function satisfies ιX (x, n) := ιNX (x, n) = (−1)λX (x) for Hd−1 -almost all (x, n) ∈ nor X. Consider the height function hv : x → v · x,

x ∈ Rd ,

whose restriction to X is a Morse function for Hd−1 -almost all v ∈ S d−1 (see Sect. 3.4). Moreover, x is a critical point of hv |X if and only if (x, −v) ∈ nor X and, due to (9.10), Corollary 3.10 yields the formula for the Euler characteristic χ(X) =

 x∈Rd

ιX (x, −v)

for almost all v ∈ S d−1 .

180

9 Extensions of Curvature Measures to Larger Set Classes

Given v ∈ S d−1 and t ∈ R, let us denote by Hv,t := {y ∈ Rd : y · v ≤ t} the closed halfspace with outer normal vector v and distance t of the boundary hyperplane from the origin. Corollary 3.10 yields χ(X ∩ Hv,t ) =



ιX (x, −v)

for a.a. (v, t) ∈ S d−1 × R.

(9.13)

x: x·v≤t

(Here and in the sequel, we write “for a.a. (v, t) ∈ S d−1 × R” to shorten “for Hd−1 almost all v ∈ S d−1 and L1 -almost all t ∈ R”.) Equation (9.13) can be taken as a principal relation connecting a Legendrian cycle with a compact set. In order to express the right-hand side in its “current form”, recall that the slice of a Legendrian cycle T by the projection π1 T , π1 , −v ∈ I0 (Rd × Rd ),

v ∈ S d−1 ,

is an integral 0-current at Hd−1 -almost all v ∈ S d−1 , see Definition 1.63. Applying Theorem 1.65, we get  T , π1 , −v = H0

 (π1 )−1 {−v} ∧ ιT ,

in other words, for any T -integrable function g on Rd × S d−1 , T , π1 , −v(g) =



ιT (x, −v)g(x, −v)

for a.a. v ∈ S d−1 .

x∈Rd

Applying the formula with the indicator function of Hv,t , we get T , π1 , −v(1Hv,t ×S d−1 ) =



ιT (x, −v)

for a.a. (v, t) ∈ S d−1 × R.

x: x·v≤t

(9.14) Using the mapping p : (x, v) → −x · v,

(x, v) ∈ Rd × S d−1 ,

we can rewrite the left-hand side as p# T , π1 , −v(1(−∞,t ] ). Motivated by (9.13) and (9.14), we can define normal cycles for rather general sets. Definition 9.5 We say that a compact set X ⊂ Rd admits a normal cycle if there exists a compactly supported Legendrian cycle T such that p# T , π1 , −v(1(−∞,t ] ) = χ(X ∩ Hv,t ) for a.a. (v, t) ∈ S d−1 × R.

(9.15)

9.2 Normal Cycles

181

In such a case, we write NX := T , call NX the normal cycle of X, and we define the curvature(-direction) measures of X of orders k = 0, . . . , d − 1 as follows (cf. Theorem 4.42): k (X, E) := (NX C

E ⊂ Rd × S d−1 Borel,

E)(ϕk ),

k (X, B × S d−1 ), Ck (X, B) := C

B ⊂ Rd Borel,

Ck (X) := NX (ϕk ). Proposition 9.6 Any compact set X ⊂ Rd admits at most one normal cycle. Proof Let T1 and T2 be two normal cycles of a compact set X ⊂ Rd . Then T := T1 − T2 is again a Legendrian cycle; we will show that T = 0. By (9.15), we have p# T , π1 , −v(1(−∞,t ] )) = 0 for a.a. (v, t) ∈ S d−1 × R, thus (as any Radon measure on R is determined by its distribution function) p# T , π1 , −v = 0 for a.a. v ∈ S d−1 . This can be written equivalently as Λ# T , π, v = 0 for a.a. v ∈ S d−1 with the mappings Λ : (x, v) → (v, x · v),

(x, v) ∈ Rd × S d−1 ,

(9.16)

and π : (v, t) → v, (v, t) ∈ S d−1 × R (note that π1 = π ◦ Λ). Integrating, and applying the coarea formula for currents, we obtain  0=

S d−1

Λ# T , π, v(g) Hd−1 (dv)

= (Λ# T )

π # (id

= Λ# (T

Λ π (id

= Λ# (T

ϕ0 )

# #

Ωd )(g) Ωd ))(g)

for any smooth function g on S d−1 . Since Λ is injective Hd−1 -almost everywhere on spt (T ϕ0 ) by Lemma 9.7 below, we get also T ϕ0 = 0, and the assertion follows by Theorem 9.4.  

182

9 Extensions of Curvature Measures to Larger Set Classes

Lemma 9.7 Let T be a Legendrian cycle with representation (9.5). Then, Hd−1 ({n ∈ S d−1 : ∃x = y, (x, n), (y, n) ∈ WT , (x − y) · n = 0}) = 0. Proof Consider the quadratic surface Q := {(x, m, y, n) ∈ R4d : m = n, (x − y) · n = 0} ⊂ R4d and note that (x − y) · v = (x − y) · v  = 0 whenever (x, m, y, n) ∈ Q and (u, v, u , v  ) ∈ Tan(Q, (x, m, y, n)). Denote N := Π((WT × WT ) ∩ Q), where Π : (x, m, y, n) → (x, m). We have to show that Hd−1 (π1 (N)) = 0. To this end, we will show that the Jacobian Jd−1 (π1 |N) vanishes Hd−1 -almost everywhere on N. If (x, n, y, n) ∈ (WT × WT ) ∩ Q and (u, v) ∈ Tan(N, (x, n)) then it follows from the above described tangent property of Q that (x − y) · v = 0. But then, dim π1 (Tan(N, (x, n)) < d − 1 and, hence, Jd−1 (π1 |N)(x, n) = 0 whenever it exists.   Next we will show that the above definition is consistent with the former approach for special set classes. Proposition 9.8 If X is compact and reach X > 0 (or X ∈ UPR ) then X admits a normal cycle NX which agrees with that from Definition 4.38 (or Definition 5.9, respectively). Proof Let X be a compact set with reach X > 0 and let NX be its normal cycle from Definition 4.38. We will show that NX satisfies (9.15). First, note that at almost all (x, v) ∈ nor X, ιX (x, v) := ιNX (x, v) = (−1)λX (x,v). Consider the parallel sets Xr which are compact C 1,1 -domains for r > 0 and Corollary 3.10 implies that χ(Xr ∩ Hv,t ) =



λXr (y)

−1 y∈Hv,t ∩νX (−v) r

for almost all (v, t) and all r > 0. We let now r tend to 0 on both sides. If X and Hv,t do not touch (which is the case for almost all (v, t)) we get using Proposition 4.17 that reach (X ∩ Hv,t ) ≥ η reach X with η = η(v, t) := inf{|u + v| : (x, u) ∈ nor X, x · v = t} > 0.

9.2 Normal Cycles

183

Since nor X is closed (see Lemma 4.11), the function η is lower semicontinuous in t and we infer that lim infr→0+ reach (Xr ∩ Hv,t ) > 0. Since clearly Xr ∩ Hv,t → X ∩ Hv,t in the Hausdorff distance, we get from Theorem 7.3 and Proposition 7.1 that χ(Xr ∩ Hv,t ) = C0 (Xr ∩ Hv,t ) → C0 (X ∩ Hv,t ) = χ(X ∩ Hv,t ),

r → 0+ .

−1 (−v) if and only if (y + rv, −v) ∈ nor X. For the right hand side, note that y ∈ νX r If X does not touch Hv,t , again from the closedness of nor X we get that there exists a δ > 0 such that there is no (x, −v) ∈ nor X with |x · v − t| < δ. But then we get for any 0 < r < δ



λXr (y) =

−1 y∈Hv,t ∩νX (−v) r



(−1)λX (x,−v) =

x: x·v≤t



ιX (x, −v),

x: x·v≤t

hence, χ(X ∩ Hv,t ) =



ιX (x, −v) = p# NX , π1 , −v(1(−∞,t ] ),

x: x·v≤t

for almost all (v, t), proving thus (9.15).  If X ∈ UPR we choose a UPR representation X = i Xi and apply the additivity for both NX (Theorem 5.11) and χ(X ∩ Hv,t ).   Example 9.9 Let X be convex compact with nonempty interior and contained in the interior of a ball B. Then, both B \ int X and ∂X admit normal cycles. In particular, NB\int X = ρ# NX + NB

and

N∂X = NX + ρ# NX ,

where ρ : (x, n) → (x, −n). Example 9.10 Assume that reach X > 0, Rd \ X is bounded and that (x, n) ∈ nor X implies (x, −n) ∈ nor X.

(9.17)

(Note that this is a kind of “full-dimensionality” condition saying that there is no pair of outer normals at a point lying in opposite directions.) Then Rd \ int X admits a normal cycle, in particular, NRd \int X = ρ# NX . Remark 9.11 Condition (9.17) cannot be avoided in Example 9.10. Indeed, consider an infinite compact totally disconnected set K ⊂ [0, 1] (embedded in the x-axis) and set X := {(x, y) ∈ R2 : x 2 + y 2 ≥ 4 or dist (x, K)2 ≤ |y|}.

184

9 Extensions of Curvature Measures to Larger Set Classes

then reach X > 0, XC is bounded but XC has infinitely many components, hence it cannot admit a normal cycle (cf. [RZ17, Example 7.12] for more details). Lemma 9.12 (Motion covariance of normal cycles) If a compact set X ⊂ Rd admits a normal cycle then also gX admits a normal cycle for any Euclidean motion g ∈ Gd , and the normal cycles are related by NgX = g˜ # NX , where g˜ : (x, n) → (gx, g0 n), g0 being the linear component of g. Proof Assume that NX is the normal cycle of a compact set X and let g ∈ Gd be a Euclidean motion of the form g(x) = b + g0 (x), x ∈ Rd , with a linear isometry g0 . We will show that the current T := g˜ # NX is a normal cycle of gX. Clearly, T is a Legendrian cycle which satisfies (i) from Definition 9.5. It remains to verify condition (ii), i.e., p# T , π1 , −v(1(−∞,t ] ) = χ(gX ∩ Hv,t )

for a.a. (v, t).

Note that, since χ is motion invariant, χ(gX ∩ Hv,t ) = χ(g(X ∩ g −1 Hv,t )) = χ(X ∩ Hv, ˜ t˜ ), where v˜ := g0 (v) and t˜ := t − b · v. Further, ˜ −v(1(−∞,t ] ) p# T , π1 , −v(1(−∞,t ] ) = p# g˜# NX , π1 ◦ g, ˜ (−∞,t ] ) = p# g˜# NX , π1 , −v(1 = p# NX , π1 , −v(1 ˜ (−∞,t˜] ) = χ(X ∩ Hv, ˜ t˜), and the equality follows.

 

Remark 9.13 It is not known whether the normal cycle from Definition 9.5 is locally determined as that of a set with positive reach. In particular, the following question seems to be important: Assume that X ⊂ Rd is compact and that for any x ∈ X there exists δ > 0 and a compact set Y ⊂ Rd admitting normal cycle NY and such that X ∩ B(x, δ) = Y ∩ B(x, δ). Does it follow that X admits a normal cycle, NX , and that, for x, δ, Y as above, NX 1B(x,δ) = NY 1B(x,δ)?

9.3 Lipschitz Domains

185

9.3 Lipschitz Domains In the sequel, we will limit ourselves to the case when X is a closed Lipschitz domain in Rd , i.e., X coincides locally with the subgraph of a Lipschitz function defined on a hyperplane. A general compact Lipschitz domain does not admit a normal cycle, but under additional assumptions the normal cycle exists, see Theorem 9.22. At the end we also prove a Principal kinematic formula for a class of closed Lipschitz domains. Definition 9.14 (i) Let L > 0, v ∈ S d−1 and an open set U ⊂ Rd be given. A closed set X ⊂ Rd is an (L, v)-Lipschitz subgraph in U if there exists an L-Lipschitz function g defined on the orthogonal complement v ⊥ such that X ∩ U = {x ∈ Rd : x · v ≤ g(pv ⊥ x)} ∩ U. In the case U = Rd , we call X simply an (L, v)-Lipschitz subgraph. (ii) A closed set X ⊂ Rd is a closed Lipschitz domain if for any x ∈ X there exist an L > 0, unit vector v, and neighbourhood U of x (in Rd ) such that X is an (L, v)-Lipschitz subgraph in U . The unit vector v with the above properties will be called admissible for X at x, or admissible for X on U . Remark 9.15 If X is a closed Lipschitz domain and x, v, g, U are as in the above definition then φ : x → pv ⊥ x + (x · v − g(pv ⊥ x))v,

x ∈ U,

is a bi-Lipschitz chart of X in the usual sense. Abusing the terminology, we will call the triple (v, g, U ) a (Lipschitz) chart of X, and a collection of charts whose domains U cover the whole X will be called an atlas of X. Lemma 9.16 (i) If X is an (L, v)-Lipschitz subgraph in an open set U and x ∈ ∂X then the cone ; < VL,v := u ∈ Rd : u · v ≥ (sin γ )|u| , with γ := arctan L satisfies (x − VL,v ) ∩ U ⊂ X and int (x + VL,v ) ∩ U ⊂ Rd \ X. (ii) If X is an (L, v1 )- and (L, v2 )-Lipschitz subgraph in U and v ∈ S d−1 is a spherical convex combination of v1 and v2 then X is an (L, v)-Lipschitz subgraph in U .

186

9 Extensions of Curvature Measures to Larger Set Classes

Fig. 9.1 X is an (L, v)-Lipschitz subgraph in U and VL,v is the cone from Lemma 9.16

x + VL,v v γ x

X ∩U x − VL,v

Proof Assertion (i) follows from the L-Lipschitz property using standard methods. In order to show (ii), note first that (a) implies (in the case U = Rd ) that also x − VL,v ⊂ X whenever x ∈ X. This implies that if X is an (L, v1 )- and (L, v2 )Lipschitz subgraph then x − VL,v1 − VL,v2 ⊂ X if x ∈ ∂X, and since VL,v ⊂ VL,v1 + VL,v2 if v is a spherical convex combination of v1 and v2 , assertion (ii) follows using (i) again (Fig. 9.1).   The Clarke tangent cone of X at x is defined as C(X, x) = lim inf Tan(X, xi ) xi →x,xi ∈X

(i.e., u ∈ C(X, x) if and only if whenever xi → x, xi ∈ X, there exist tangent vectors ui ∈ Tan(X, xi ) with ui → u). (The definition given in [Cla83, §2.4] is equivalent, see [RW04, Theorem 6.26].) The Clarke normal cone N(X, x) of X at x is defined as the polar cone N(X, x) = C(X, x)o . The Clarke normal bundle of X is the sphere bundle norCl X = {(x, n) : x ∈ ∂X, n ∈ N(X, x) ∩ S d−1 }. We will need some properties of these cones. Lemma 9.17 If X ⊂ Rd is a closed Lipschitz domain and x ∈ Rd then (i) both C(X, x) and N(X, x) are closed convex cones, (ii) u ∈ int C(X, x) if and only if there exists ε > 0 such that [y, y + εv] ⊂ X whenever y ∈ X ∩ B(x, ε) and v ∈ B(u, ε),

9.3 Lipschitz Domains

187

(iii) if xi → x, vi ∈ N(X, xi ) and vi → v then v ∈ N(X, x) (hence, norCl X is closed), (iv) v ∈ S d−1 is admissible for X at x if and only if −v ∈ int C(X, x), (v) if X is an (L, v)-Lipschitz subgraph in an open set U then −VL,v ⊂ C(X, x) whenever x ∈ X ∩ U , (vi) if v ∈ S d−1 is admissible for X at x then v ∈ C(X, x) (hence, C(X, x) is contained in a halfspace), (vii) if x ∈ ∂X, v ∈ S d−1 and B(x −εv, ε) ⊂ X for some ε > 0 then v ∈ N(X, x), (viii) if reach (X, x) > 0 then C(X, x) = Tan(X, x) and N(X, x) = Nor(X, x). Proof The closedness and convexity of the cone C(X, x) follow from the properties of the subgradient of f at x, see [Cla83, Proposition 2.1.2]. The normal cone N(X, x) is always closed and convex as a polar cone. Hence, (i) is proved. Properties (ii) and (iii) are known results due to Rockafellar, see [Cla83, §2.5.8]. Properties (iv), (v) and (vi) follow from (ii) and Lemma 9.16 (i). Let the assumptions of (vii) hold. If there would be some u ∈ int C(X, x) with u· v > 0 then we would obtain using (ii) that x is an interior point of X, a contradiction. Thus u · v ≤ 0 whenever u ∈ C(X, x) and, hence, v ∈ N(X, x). If reach (X, x) > 0 we know that the set-valued function y → Nor(X, y) ∩ S d−1 is upper semicontinuous at x (i.e., if xi ∈ X, xi → x, ni ∈ Nor(X, xi ) ∩ S d−1 and ni → n then n ∈ Nor(X, x) ∩ S d−1 , cf. Lemma 4.11). Since Tan(X, x) = Nor(X, x)o , it follows that y → Tan(X, y) ∩ S d−1 is lower semicontinuous at x, which implies that Tan(X, x) = C(X, x), and the equality Nor(X, x) = N(X, x) follows immediately.   We say that two closed Lipschitz domains X, Y ⊂ Rd touch if there exists (x, n) ∈ norCl X such that (x, −n) ∈ norCl Y . Lemma 9.18 If X, Y are closed Lipschitz domains that do not touch then X ∩ Y and X ∪ Y are closed Lipschitz domains as well. Proof Take x ∈ X ∩ Y and observe that int C(X, x) ∩ int C(Y, x) = ∅. (Indeed, if the interiors would not intersect then the two convex cones could be separated by a hyperplane and each of the two unit vectors perpendicular to this hyperplane would belong to the Clarke normal bundle of one of the sets X, Y , which would violate the non-touching assumption at x.) Thus, by Lemma 9.17 (iv), there exists a unit vector n admissible for both X and Y at x and if g, h are the Lipschitz functions on n⊥ that determine X, Y , respectively, locally at x, then min{g, h}, max{g, h} determine X ∩ Y , X ∪ Y , respectively, locally at x.   In the following we consider “interior parallel sets” (ε > 0) X−ε := {x ∈ X : d∂X (x) ≥ ε}.

188

9 Extensions of Curvature Measures to Larger Set Classes

Lemma 9.19 If X ⊂ Rd is a closed Lipschitz domain and K ⊂ Rd compact then there exist η, ε0 > 0 such that for all 0 < ε < ε0 and x ∈ K ∩ X−ε , reach (X−ε , x) ≥ η(ε + r), where r := infy∈K∩X reach (X, y). Proof If X ∩ K = ∅ there is nothing to prove. If X ∩ K = ∅ and r = ∞ then X must be already convex. Hence, X−ε is convex as well, and its reach is infinite everywhere. We will assume that r < ∞ in the rest of the proof. Let (v, g, U ) be a Lipschitz chart of X (see Definition 9.14) and let ε > 0 and x, y ∈ X−ε be such that B(x, 2ε) ⊂ U . We will show that dTan(X−ε ,x)(y − x) ≤ (2η(ε + r))−1 |y − x|2,

(9.18)

where η := (1 + (Lip g)2 )−1/2 . If d∂X (x) > ε then Tan(X−ε , x) = Rd and (9.18) clearly holds. Assume thus that d∂X (x) = ε and denote ΣX (x) := {z ∈ ∂X : |z − x| = ε}. Observe that if a vector u belongs to the interior of the polar cone to ΣX (x) − x then the distance to ∂X increases in direction u at x, hence (ΣX (x) − x)o ⊂ Tan(X−ε , x). We will show that the vector u := y − x − (2η(ε + r))−1 |y − x|2 v ∈ Tan(X−ε , x), which will imply (9.18). Let z ∈ ΣX (x). If r > 0 then z − x ∈ Nor(X, z) and ˚ ˚ B(w, r) ∩ X = ∅, where w := z + rε−1 (z − x). Thus B(w, ε + r) ∩ X−ε = ∅, which clearly holds also if r = 0, and we get |w − y| ≥ ε + r, thus (ε + r)2 ≤ |w − y|2 = |w − x|2 + |x − y|2 + 2(w − x) · (x − y) = (ε + r)2 + |y − x|2 − 2(w − x) · (y − x), which implies (w − x) · (y − x) ≤ 12 |y − x|2 . From the Lipschitz property of g we infer that (since w − x is a normal direction to the graph of g) (w − x) · v ≥ η(ε + r).

9.3 Lipschitz Domains

189

Taking these two inequalities into account, we get u · (w − x) = (y − x) · (w − x) − (2η(ε + r))−1 |y − x|2 v · (w − x) ≤

1 |y − x|2 − (2η(ε + r))−1 |y − x|2 η(ε + r) = 0, 2

hence u ∈ (ΣX (x) − x)o and u ∈ Tan(X−ε , x) follows, proving (9.18). Since K is compact there exists an ε0 > 0 and a finite family ˚ i , 3ε0 )) : i = 1 . . . , k} {(vi , gi , B(x of Lipschitz charts of X such that K1 ⊂ Denoting

k

i=1 B(xi , ε0 ),

where K1 := K ⊕ B(0, 1).

η := inf (1 + (Lip gi )2 )−1/2 , 1≤i≤k

we have using (9.18) dTan(X−ε ,x) (y − x) ≤ (2η(ε + r))−1 |y − x|2 whenever x ∈ K1 ∩ X−ε , y ∈ X−ε . This already implies that reach (X−ε , x) ≥ η(ε + r) if x ∈ K ∩ X−ε . Indeed (proceeding as in the proof of Proposition 4.14), if reach (X−ε , x) < η(ε + r) for ˚ η(ε +r)) having two different some x ∈ K ∩X−ε there would exist a point z ∈ B(x, nearest neighbours y1 , y2 in X−ε . Clearly s := |y1 − z| = |y2 − z| < η(ε + r), and, ˚ s) does not intersect X−ε , we have as B(z, dTan(X−ε ,y1 ) (y2 − y1 ) ≥ (2s)−1 |y2 − y1 |2 > (2η(ε + r))−1 |y2 − y1 |2 , violating the above property. This completes the proof.

 

In the following, the notation X ∼ Y will denote that the sets X, Y have the same homotopy type (in particular, χ(X) = χ(Y ) if it exists). Proposition 9.20 If X ⊂ Rd is a compact Lipschitz domain then for all sufficiently small ε > 0, X ∼ X−ε . Proof We construct first a transversal unit vector field on the boundary ∂X, i.e., a Lipschitz mapping v : ∂X → S d−1 such that v(x) is admissible for X at x, x ∈ ∂X. The construction can be done as ˚ i , 5δ)) : i = 1, . . . , k} be a finite family of Lipchitz charts follows. Let {(vi , gi , B(x

190

9 Extensions of Curvature Measures to Larger Set Classes

of X (cf. Definition 9.14) such that ∂X ⊂

k 

˚ i , δ), B(x

i=1

and let (hi : i = 1, . . . , k) be a smooth partition of unity subordinated to the cover  k ˚ i=1 B(xi , δ). Set v(x) ˜ :=

k 

hi (x)vi ,

x ∈ ∂X.

i=i

For x, y ∈ ∂X we have |v(x) ˜ − v(y)| ˜ ≤

k 

|hi (x) − hi (y)| ≤

i=1

k 

 Lip hi |x − y|,

i=1

thus v˜ is Lipschitz. Set L := max1≤i≤k Lip gi , γ := arctan L and consider a point x ∈ ∂X. Then, by Lemma 9.17 (v), −VL,vi ⊂ C(X, x) whenever x ∈ B(xi , δ), and by Lemma 9.17 (vi), C(X, x) is contained in a halfspace {z : z · w ≥ 0} with some w ∈ S d−1 . We claim that vi · w ≥ cos γ whenever x ∈ B(xi , δ). Indeed, otherwise the orthogonal projection pw⊥ vi of vi into w⊥ would still belong to the interior of VL,vi , which would contradict the above inclusion in the halfspace. Thus we obtain  v(x) ˜ · w = ki=1 vi · w ≥ cos γ , hence |v(x)| ˜ ≥ cos γ . We define now v(x) :=

v(x) ˜ , |v(x)| ˜

x ∈ ∂X,

and notice that v is Lipschitz. ˚ 2δ) ⊂ B(x ˚ i , 5δ) whenever x ∈ Take a point x ∈ ∂X and notice that B(x, ˚ i , δ). Hence, X is an (L, vi )-Lipschitz subgraph in B(x, ˚ 2δ) for all such i and, B(x using Lemma 9.16 (ii), we obtain that ˚ 2δ), X is an (L, v(x))-Lipschitz subgraph in B(x,

x ∈ ∂X.

(9.19)

In particular, ˚ 2δ) ⊂ X and int (x + VL,v(x)) ∩ B(x, ˚ 2δ) ⊂ Rd \ X (x − VL,v(x)) ∩ B(x,

(9.20)

whenever x ∈ ∂X by Lemma 9.16 (i), hence (cos γ )s ≤ d∂X (x + sv(x)) ≤ s,

0 < |s| < δ.

(9.21)

9.3 Lipschitz Domains

191

Consider the continuous mapping φ : (x, s) → x + sv(x),

(x, s) ∈ ∂X × (−δ, δ),

and denote     1 c 1 1 and τ := min ,δ . c := √ min cos γ , 2 2 Lip v 2 Let x, x  ∈ ∂X, s, s  ∈ (−τ, τ ) be given, and assume that |x − x  | < δ. Equation (9.20) implies that |φ(x, s − s  ) − x  | ≥ max{cos γ |x − x  |, cos γ |s − s  |} ≥ c|(x, s) − (x  , s  )|. If |x − x  | ≥ δ then |φ(x, s − s  ) − x  | ≥

1 1 |x − x  | ≥ √ |(x, s) − (x  , s  )| ≥ c|(x, s) − (x  , s  )| 2 2 2

again. Since |v(x) − v(x  )| ≤ (Lip v)|x − x  | ≤ (Lip v)|(x, s) − (x  , s  )|, we get |φ(x, s) − φ(x  , s  )| = |(φ(x, s − s  ) − x  ) + s  (v(x) − v(x  ))| ≥ |φ(x, s − s  ) − x  | − τ |v(x) − v(x  )| ≥ (c − τ Lip v)|(x, s) − (x  , s  )| c ≥ |(x, s) − (x  , s  )|. 2 Thus φ is injective on ∂X × (−τ, τ ) and the inverse φ −1 is Lipschitz on the open set φ(∂X × (−τ, τ )). Let p, s denote the components of φ −1 , i.e., z → (p(z), s(z)) ∈ ∂X × (−τ, τ ),

z ∈ φ(∂X × (−τ, τ )),

is a continuous mapping. Fix some 0 < ε < 12 (cos γ )τ and denote hε (x) := inf{h > 0 : d∂X (x − hv(x)) = ε}, The above observation (9.21) implies that 0 < hε < ε/ cos γ < τ/2.

x ∈ ∂X.

192

9 Extensions of Curvature Measures to Larger Set Classes

If hε is continuous then the mapping F : (z, s) →

 (1 − s)z + s(hε (p(z)) − s(z))v(p(z)),

z ∈ X \ X−ε , z ∈ X−ε ,

z,

s ∈ [0, 1],

is a deformation retraction of X onto X−ε , proving X ∼ X−ε . It remains thus to show that hε is continuous on ∂X. We will show that hε is even locally q-Lipschitz with Lipschitz constant q := 2 sin γ + 1 + τ Lip v. Let x, x  ∈ ∂X be given and assume that |x  − x| < τ/(2q) < τ/2. Denote t := hε (x), t  := t + q|x  − x|, y := x − tv(x) and y  := x  − t  v(x  ). Note that t < τ/2 and t  < τ/2 + q|x  − x| < τ . We have (y − y  ) · v(x) = ((x − tv(x)) − (x  − t  v(x  ))) · v(x) = (t  − t)v(x) · v(x) + (x − x  ) · v(x) + t  (v(x  ) − v(x)) · v(x) ≥ q|x − x  | − |x − x  | − t  |v(x  ) − v(x)| ≥ (q − 1 − τ Lip v)|x − x  | = 2 sin γ |x − x  |, and since |y − y  | ≤ |x − x  | + |t − t  ||v(x) − v(x  )| ≤ |x − x  |(1 + τ Lip v) ≤ 2|x − x  |, we get (y − y  ) · v(x) ≥ sin γ |y − y  |, hence y − y  ∈ VL,v(x) and, consequently, B(y  , ε) ⊂ B(y, ε) − VL,v(x). ˚ 2δ), Since |y  − x| + ε < t  + |x  − x| + ε < 2δ, we have also B(y  , ε) ⊂ B(x, hence ˚ 2δ). B(y  , ε) ⊂ (B(y, ε) − VL,v(x)) ∩ B(x, ˚ 2δ) we have B(y, ε) ⊂ X. Thus we deduce As d∂X (y) = ε and B(y, ε) ⊂ B(x, (using again Lemma 9.16 (i)) ˚ 2δ) ⊂ X, (B(y, ε) − VL,v(x)) ∩ B(x,

9.3 Lipschitz Domains

193

which, together with the previous inclusion, yields that B(y  , ε) ⊂ X. Thus d∂X (y  ) ≥ ε, which implies that hε (x  ) ≤ t  ≤ t + q|x  − x| = hε (x) + q|x  − x|. Interchanging the role of x and x  , we thus obtain that hε is locally q-Lipschitz and the proof is finished.   We consider now a compact Lipschitz domain X. Since the sets X−ε have positive reach if ε > 0 is small enough, they admit normal cycles NX−ε . It is thus natural to try to define the normal cycle of X by approximation with NX−ε . This is, however, not possible in general; one can easily imagine examples in the plane of Lipschitz domains with unbounded Gauss curvature. Therefore, additional restrictions must be imposed. Recall that Hv,t = {y : y · v ≤ t}, Λ : (x, v) → (v, x · v), cf. (9.16), and that norCl X is the Clarke normal bundle of X. Note that Λ(norCl X) can be interpreted as the set of all hyperplanes tangent to X. Proposition 9.21 Let X be a compact Lipschitz domain and assume that Hd (Λ(norCl X)) = 0.

(9.22)

Then, the following assertions hold for a.a. (v, t) ∈ S d−1 × R: (i) X ∩ Hv,t is a Lipschitz domain, (ii) X ∩ Hv,t ∼ X−ε ∩ Hv,t for all sufficiently small ε > 0. Proof First, note that X, Hv,t do not touch whenever (−v, t) ∈ Λ(norCl X), which is the case for a.a. (v, t) ∈ S d−1 × R by (9.22). Assertion (i) follows thus by Lemma 9.18. If X, Hv,t do not touch then Y := X ∩ Hv,t is a compact Lipschitz domain. We construct a homotopy similarly as in the proof of Proposition 9.20. Let v : ∂Y → S d−1 be the transversal vector field and let the continuous mappings p, s be defined on some neighbourhood of ∂Y , as in the previous proof. Set h˜ ε : x → min{h ≥ 0 : d∂X (x − hv(x)) ≥ ε},

x ∈ ∂Y

(i.e., h˜ ε measures the distance from ∂Y to ∂X−ε along v). Again, h˜ is continuous and the mapping F˜ : (z, s) →

 (1 − s)z + s(h˜ ε (p(z)) − s(z))v(p(z)),

z ∈ Y \ X−ε , z ∈ Y ∩ X−ε ,

z,

s ∈ [0, 1],

is a deformation retraction of Y onto Y ∩ X−ε , proving X ∩ Hv,t ∼ X−ε ∩ Hv,t .

 

We are now ready to define normal cycles for compact Lipschitz manifolds. Theorem 9.22 Let a compact Lipschitz domain X ⊂ Rd satisfy (9.22) and assume that lim inf M(NX−ε ) < ∞. ε→0

(9.23)

194

9 Extensions of Curvature Measures to Larger Set Classes

Then X admits a normal cycle NX . Furthermore, NX = (F) lim NX−εi i→∞

whenever εi → 0, (M(NX−εi )) is bounded and (F) limi→∞ NX−εi exists. Remark 9.23 1. Buczolich [Buc98] presented an example of a C 1 function of two variables whose set of tangent planes has positive three-dimensional measure. Hence, condition (9.22) is not fulfilled automatically. 2. Note that since reach X−ε > 0 we have M(NX−ε ) = Hd−1 (nor X−ε ) by the definition of the normal cycle, see Sect. 4.6. Proof By (9.23), there exists a sequence εi → 0+ such that (M(NX−εi )) is bounded. The Compactness theorem for currents (Theorem 1.57) and Lemma 1.60 imply that there exists a subsequence εi(j ) such that N−εi(j) converges in the flat norm to some current T . We will show that T is the normal cycle of X. Since the normal cycle is unique, the assertion will follow. It remains thus to show that if (F) limi→∞ NX−εi = T for some εi → 0 with (M(NX−εi )) bounded then T is the normal cycle of X. Since NX−εi is the normal cycle of X−εi , we have p# NX−εi , π1 , −v(1(−∞,t ] ) = χ(X−εi ∩ Hv,t )

(9.24)

for a.a. (v, t) ∈ S d−1 × R. Proposition 1.66 implies that (F) lim NX−εi , π1 , −v = T , π1 , −v, i→∞

for a.a. v ∈ S d−1 and it follows easily that also (F) lim p# N−Xεi , π1 , −v = p# T , π1 , −v i→∞

(9.25)

for a.a. v ∈ S d−1 . Using subsequently Fatou’s lemma, (1.34) and (9.23), we obtain  lim inf MNX−εi , π1 , −v Hd−1 (dv)

S d−1 i→∞

 ≤ lim inf i→∞

S d−1

MNX−εi , π1 , −v Hd−1 (dv)

= lim inf M(NX−εi i→∞

(π1 )# ΩS d−1 )

9.3 Lipschitz Domains

195

≤ lim inf M(NX−εi ) < ∞, i→∞

where ΩS d−1 denotes the natural (d − 1)- form on the sphere. Consequently, for a.a. v ∈ S d−1 , a subsequence of (MNX−εi , π1 , −v)i is bounded, and, passing to a subsequence, we can assume that this is true for the whole sequence. Together with (9.25), this yields the weak convergence of signed measures p# NX−εi , π1 , −v to p# T , π1 , −v, i → ∞, on the real line and it is well-known that in such a case, the associated distribution functions converge almost everywhere: lim p# NX−εi , π1 , −v(1(−∞,t ] ) = T , π1 , −v(1(−∞,t ] )

i→∞

for a.a. (v, t) ∈ S d−1 × R. Hence, the left-hand side of (9.24) converges to the lefthand side of (9.15) in Definition 9.5. But the right-hand side converges as well by Proposition 9.21. This verifies condition (9.15) and the proof is finished.   Our next aim is to extend the definition of the normal cycle to closed (unbounded) Lipschitz domains. If X ⊂ Rd is a closed Lipschitz domain and ε > 0 is small enough then X−ε has “locally positive reach” by Lemma 9.19. Hence, the unit normal bundle nor X−ε and normal cycle NX−ε are defined as for sets with positive reach (in fact, both are defined locally). Instead of assumption (9.23), we will consider its local variant. For convenience, we introduce a notation for the family of closed Lipschitz domains for which normal cycles will be introduced. Definition 9.24 Let LD d denote the family of all closed Lipschitz domains X ⊂ Rd for which Hd (norCl X) = 0

(9.26)

and lim inf M(NX−ε ε→0

π0−1 (K)) < ∞,

K ⊂ Rd compact

(9.27)

(recall that π0 : (x, n) → x). Remark 9.25 Clearly (9.26) implies (9.22). Also, note that (9.27) is equivalent to NX−εi have locally uniformly bounded mass norm for some εi → 0, and that M(NX−ε

π0−1 (K)) = Hd−1 (nor X−ε ∩ π0−1 (K)).

Remark 9.26 An important subclass of closed Lipschitz domains are closed DC domains, i.e., sets that can be locally represented as subgraphs of DC functions

196

9 Extensions of Curvature Measures to Larger Set Classes

(differences of convex functions). Normal cycles of closed DC domains were constructed in [PR13]. Any closed DC domain satisfies (9.26) (norCl X has even locally finite (d − 1)-dimensional Minkowski content) and an analogon to (9.27) holds, namely 

ε0 0

M(NX−ε

π0−1 (K)) dε < ∞,

K ⊂ Rd compact, ε0 > 0.

It is not known whether any closed DC domain belongs to LDd . Let us recall some notions and facts from Sect. 6.1. If X, Y are sets with positive reach then the current NX,Y is defined as NX,Y = F# (((NX × NY )

R 0 ) × [0, 1])

(see (6.3) for the definition of F and R 0 ), provided that the mass norm of the pushforward is locally bounded. Note that F is Lipschitz on R δ := {(x, m, y, n) ∈ (Rd )4 : m · n ≥ (−1 + δ)|m||n|},

δ > 0.

Let γ δ ∈ C ∞ (R 0 ) be equal 0 on R 0 \ R δ/2 and 1 on S d−1 ∩ R δ ; the currents δ := NX,Y NX,Y

γ δ,

δ>0

(9.28)

have locally bounded mass norm (since F is Lipschitz on spt γ δ ) and δ (F) lim NX,Y = NX,Y whenever NX,Y has locally bounded mass norm. We also observe that δ M(NX,Y

(K1 × K2 × S d−1 )) δ

≤ (Lip (F |R 2 ))2d−1 M(NX

π0−1 K1 ) M(NY

π0−1 K2 )

whenever K1 , K2 ⊂ Rd are compact. The following technical lemma quantifies in certain way the important nontouching property. Lemma 9.27 If two closed Lipschitz manifolds X, Y do not touch and K ⊂ Rd is compact then there exist δ > 0 such that m · n ≥ −1 + δ whenever x ∈ K, (x, m) ∈ norCl X and (x, n) ∈ norCl Y or (x, m) ∈ nor X−r , (x, n) ∈ nor Y−r and 0 < r < δ.

9.3 Lipschitz Domains

197

Proof Since norCl X and norCl Y are closed sets (see Lemma 9.17 (iii)), there exists δ > 0 such that  (m, n) < π − δ whenever x ∈ K, (x, m) ∈ norCl X and (x, n) ∈ norCl Y . We further claim that lim sup nor X−r ⊂ norCl X,

(9.29)

r→0

i.e., if ri → 0, (xi , ni ) ∈ nor X−ri and (xi , ni ) → (x, n), i → ∞, then (x, n) ∈ norCl X. This already implies the assertion. In order to show (9.29), denote ΣX (x) := {y ∈ ∂X : |y − x| = r},

x ∈ ∂X−r ,

and note that ni belongs to the spherical convex hull of ri−1 (ΣX (xi ) − xi ) (cf. the proof of Lemma 9.19). Using the Caratheodory’s theorem and turning to a subsequence, we can achieve the existence of yi(0), . . . , yi(d) ∈ ΣX (xi ) such that n(k) := ri−1 (yi(k) − xi ) → n(k) ∈ S d−1 , i → ∞, 0 ≤ k ≤ d, and ni is a i (0) (d) spherical convex combination of ni , . . . , ni , i ∈ N. Lemma 9.17 (vii) implies (k) that ni ∈ N(X, yi ), k = 0, . . . , d, i ∈ N, hence also n(k) ∈ N(X, x) by Lemma 9.17 (iii). Since n lies in the spherical convex hull of n(0) , . . . , n(d) , also n ∈ N(X, x). This proves (9.29).   The following proposition states some generic properties of an intersection of two Lipschitz domains. Note that the assumptions on X, Y are not symmetric. Proposition 9.28 Let X, Y ∈ LDd , reach Y > 0. Then, for almost all Euclidean motions g ∈ Gd , (i) (ii) (iii) (iv)

X and gY do not touch, X ∩ gY is a closed Lipschitz domain, Hd (norCl (X ∩ gY )) = 0, X ∩ gY ∈ LDd .

Proof In order to show (i), we proceed similarly as in the proof of Proposition 6.13 (iv). Since Y has positive reach, norCl Y = nor Y is locally (d − 1)rectifiable and, using (9.27), we obtain H2d−1 (norCl X × norCl Y ) = 0. Lemma 1.73 implies that Hq (A) = 0 for A = {(x, m, y, n, ρ) ∈ norCl X × norCl Y × SO(d) : m + ρn = 0} and q = 2d − 1 +

d(d − 1) (d − 1)(d − 2) =d+ . 2 2

Hence, also Hq (h(A)) = 0 for h : (x, m, y, n, ρ) → (x − ρy, ρ), and since h(A) = {(z, ρ) ∈ Rd × SO(d) : X and z + ρY touch}, the conclusion follows.

198

9 Extensions of Curvature Measures to Larger Set Classes

(ii) follows from (i) using Lemma 9.18. (iii) Take a point x ∈ ∂X ∩ ∂Y . Using Lemma 9.17 (ii) one can easily show that int C(X, x) ∩ int C(Y, x) ⊂ int C(X ∩ Y, x). Further, if X and Y do not touch then it was shown in the proof of Lemma 9.18 that int C(X, x) ∩ int C(Y, x) = ∅. Since C(X, x) and C(Y, x) are closed conves cones, we deduce that C(X, x) ∩ C(Y, x) ⊂ C(X ∩ Y, x). Thus, using the properties of polar cones (2.7), we get N(X, x) + N(Y, x) = C(X, x)o + C(Y, x)o = (C(X, x) ∩ C(Y, x))o ⊃ C(X ∩ Y, x)o = N(X ∩ Y, x). Thus, similarly as in Sect. 6.1, if X and Y + z do not touch for some z ∈ Rd then norCl (X ∩ (Y + z)) ⊂ norCl X ∪ norCl (Y + z)   ∪(πˆ ◦ F ) ((norCl X × norCl (Y + z)) ∩ R 0 ) × [0, 1] ∩ G−1 {z}, where the mapping F : R 0 × [0, 1] → R2d × S d−1 is defined in (6.3), G : (x, y, u) → x − y and πˆ : (x, y, u) → (x, u). Both norCl X and norCl (Y + z) and have d-dimensional measure zero by assumptions, and also H2d−1 (norCl X × norCl (Y + z)) = 0. Since F is locally Lipschitz,    H2d (πˆ ◦ F ) ((norCl X × norCl (Y + z)) ∩ R 0 ) × [0, 1] = 0. An application of the Coarea formula then implies that ; < Hd z ∈ Rd : X, Y + z do not touch and Hd (norCl (X ∩ (Y + z))) > 0 = 0. Together with assertion (i), this already implies (iii). (iv) In order to show (iv), it remains to verify (9.27) for X ∩ gY . Let εi → 0+ be a sequence for which the normal cycles NX−εi have locally uniformly bounded mass norms, and denote for brevity Xi := X−εi , Y i := Y−εi . Any Euclidean motion g ∈ Gd can be uniquely decomposed into the shift z ∈ Rd and orthogonal part ρ ∈ SO(d), so that g(·) = z + ρ(·). Using (6.7), Propositions 6.5 and 6.13 (iv), we can decompose the normal cycle of Xi ∩ gY i for any K ⊂ Rd compact, almost all ρ ∈ SO(d), almost all z ∈ Rd as NXi ∩gY i

π0−1 (K) = NXi +NgY i

π0−1 (K ∩ int gY i ) π0−1 (K ∩ int Xi )

+(−1)d πˆ # NXi ,ρY i , G, z

π0−1 (K).

The mass norm of the first summand is uniformly bounded by assumption. We shall show that NY i (and, hence, also NgY i ) have locally uniformly bounded mass

9.3 Lipschitz Domains

199

norms. Lemma 9.19 implies that lim infi→∞ reach Y i > 0. If B ⊂ Rd is a ball not touching Y (which is the case for almost all balls), we get from Lemma 9.27 and Proposition 4.17 that lim inf reach (Y i ∩ B) > 0. i→∞

Since Y i ∩ B → Y ∩ B in the Hausdorff metric, Theorem 7.3 implies that the ˚ = NY i ∩B ˚ the mass norms of NY i ∩B are uniformly bounded. Since NY i B B, currents NY i have locally uniformly bounded mass norms. Using (1.34) we obtain that for almost all ρ ∈ SO(d) and any Borel subset Z ⊂ Rd ,    M πˆ # NXi ,ρY i , G, z π0−1 K dz ≤ NXi ,ρY i (K × (K − Z) × S d−1 ). Z

This, using (6.14), we get  lim sup i→∞

H

M(NXi ∩gY i

π0−1 (K)) dg < ∞

for any bounded measurable set H ⊂ Gd . The Fatou’s lemma implies that  lim inf M(NXi ∩gY i

H i→∞

π0−1 (K)) dg



≤ lim inf i→∞

H

π0−1 (K)) dg < ∞,

M(NXi ∩gY i

hence, lim inf M(NXi ∩gY i i→∞

π0−1 (K)) < ∞

for almost all g ∈ H . This completes the proof since H was any bounded measurable subset of Gd .   Recall that for a compact X ∈ LDd , the normal cycle NX was already defined (cf. Theorem 9.22). We will show that these normal cycles are locally determined. This will enable us to extend the definition to closed (unbounded) sets from LD d . Lemma 9.29 Let X, Y ∈ LD d be compact, and let U ⊂ Rd be an open set such that X ∩ U = Y ∩ U . Then NX

π0−1 (U ) = NY

π0−1 (U ).

200

9 Extensions of Curvature Measures to Larger Set Classes

Proof Let εi , δi → 0+ be two sequences such that (NX−εi ), (NY−δi ) have uniformly bounded mass norms and (F) lim NX−εi = NX ,

(F) lim NY−δi = NY .

i→∞

i→∞

Given r > 0 there exists a compact C 2 -domain Z such that U−r ⊂ Z ⊂ U . Using Lemma 9.28, we can even achieve (applying a “small” motion) that W := X ∩ Z = Y ∩ Z is a compact Lipschitz domain satisfying (9.23) and (9.26), and, turning to suitable subsequences of (εi ) and (δi ) (for which we do not change notation), NW = (F) lim NW−εi = (F) lim NW−δi . i→∞

i→∞

Note that W−ε ∩ U−r = X−ε ∩ U−r = Y−ε ∩ U−r ,

ε < r,

thus NW

π0−1 (U−r ) = NX

π0−1 (U−r ) = NY

π0−1 (U−r ),

and since this is true for all r > 0, we get also NW

π0−1 (U ) = NX

π0−1 (U ) = NY

π0−1 (U ).  

Theorem 9.30 If X ∈ LD d then there exists a unique Legendrian cycle NX such that NX π0−1 (U ) = NY π0−1 (U ) whenever Y ∈ LD d is compact and U ⊂ Rd an open set with X ∩ U = Y ∩ U . Proof Note that for any bounded open set U ⊂ Rd there is a compact Y ∈ LDd satisfying NX π0−1 (U ) = NY π0−1 (U ). Indeed, we can find such an Y by intersecting X with a finite number of halfspaces, using Proposition 9.28. We can thus define NX

π0−1 (U ) := NY

π0−1 (U ).

The uniqueness follows from Lemma 9.29. Remark 9.31 It follows from the definition and from (9.29) that spt NX ⊂ lim sup spt NXεi ⊂ norCl X. i→∞

 

9.3 Lipschitz Domains

201

Remark 9.32 A weak point of Theorem 9.22 is that it is difficult to determine which Lipchitz domains fulfill the assumption (9.27). In fact, aside of domains constructed from convex bodies or sets with positive reach and their differences, we do not know other examples for which Theorem 9.22 can be applied. The method of approximation using the uniqueness theorem and continuity of Euler characteristic is, however, canonical and used in other variants in different situation, e.g. for d.c. domains (with a different approximation), see Bibliographical note No. 2. Finally, we show a Principal kinematic formula (cf. Theorem 6.1) for intersections of sets from LDd with sets of positive reach. We start with two auxiliary lemmas. Lemma 9.33 Let X, Y be as in Proposition 9.28. Then for almost all g ∈ Gd , NX∩gY = NX

π0−1 (int gY ) + NgY

π0−1 (int X) + (−1)d πˆ # NX,ρY , G, z,

where z ∈ Rd and ρ ∈ SO(d) are the shift and orthogonal part of g. Proof Let X, Y and (εi ) be as in Proposition 9.28 and denote Xi := X−εi , Y i := Y−εi , i ∈ N. We know from (6.7) and Proposition 6.5, Proposition 6.13 (iv) and Lemma 6.18 that for all i and almost all g ∈ Gd , NXi ∩gY i = NXi

π0−1 (int gY i ) + NgY i

π0−1 (int Xi ) + (−1)d πˆ # NXi ,ρY i , G, z.

Let ψ be a smooth function with compact support K, assume that X and gY do not touch and let δ > 0 be the number from Lemma 9.27. Then ψ = (NXi

NXi ∩gY i

π0−1 int gY i + (NgY i

ψ)

+(−1)d πˆ # NXδ i ,ρY i , G, z

ψ)

π0−1 int Xi

ψ

fur sufficiently large i. Proposition 9.28 implies that, turning to a subsequence, the ψ. We will show that (again for a subsequence) left side converges to NX∩gY (NXi

ψ)

π0−1 int gY i → (NX

Denoting Ti := NXi Ti

π0−1 int gY i − T

ψ, T := NX

ψ)

π0−1 int gY weakly.

(9.30)

ψ and Wi := int Y \ int Y i , we have

π0−1 int gY = −Ti

π0−1 gWi + (Ti − T )

π0−1 int gY.

We will show that the first term tends to zero in mass norm, and the second one weakly. Concerning the first one, note that M(Ti

π0−1 gWi ) ≤ μi (gWi )

202

9 Extensions of Curvature Measures to Larger Set Classes

with μi := Ti  ◦ π0−1 . If C > 0 and Gd (C) denotes the set of all Euclidean motions from Gd with shift z ∈ B(0, C) and orthogonal part ρ ∈ SO(d), we have using Fubini’s theorem  

 Gd (C)



μi (gWi ) dg = SO(d) B(0,C)

1z+ρWi (x) dz ϑd (dρ) μi (dx)

 

Ld (B(0, C) ∩ (x − ρWi )) ϑd (dρ) μi (dx)

= SO(d)

≤ L (Wi ∩ B(0, C  )) μi (Rd ) d

with C  = C + supx∈K |x|. Since W i % ∅ and μi (Rd ) are uniformly bounded, we get  lim

i→∞ Gd (C)

μi (gWi ) dg = 0,

and Fatou’s lemma gives  lim inf μi (gWi ) dg = 0.

Gd (C) i→∞

Hence, for any C > 0 and almost all g ∈ Gd (C) (and, consequently, also for almost all g ∈ Gd ), μik (gWik ) → 0 for some subsequence (ik ). Similarly we can show that μ(∂gY ) = 0 for almost all g, thus π0−1 (int Y ) is a continuity set for T  and, since Ti → T weakly, also (Ti − T ) π0−1 int gY → 0 weakly. This verifies (9.30). By a similar procedure we can show that for almost all g, (NgY i

ψ)

π0−1 int Xi → (NgY

ψ)

π0−1 int X weakly.

δ Finally, we have (F) limi→∞ NXδ i ,ρY i = NX,ρY for any ρ ∈ SO(d) and δ > 0. Hence, using Proposition 1.66, also δ , G, z (F) lim NXδ i ,ρY i , G, z = NX,ρY i→∞

for almost all z. Thus we have NX∩gY

ψ = (NX

ψ)

π0−1 int gY + (NgY

δ +(−1)d πˆ # NX,ρY , G, z

ψ.

ψ)

π0−1 int X

9.3 Lipschitz Domains

203

Using again Lemma 9.27 (as at the beginning of the proof), we infer ψ = (NX

NX∩gY

ψ)

π0−1 int gY + (NgY

+(−1)d πˆ # NX,ρY , G, z

ψ)

π0−1 int X

ψ.

for almost all g, which is already a local version of the formula to be shown.

 

Lemma 9.34 Let X, Y be as in Proposition 9.28 and let NX−εi , NY−εi have locally uniformly bounded mass norms for some εi → 0. Then for any K ⊂ R3d compact  lim sup

δ→0 i

SO(d)

 M (NX−εi ,ρY−εi − NXδ −ε

i ,ρY−εi

)

 K ϑd (dρ) = 0

and  sup δ>0 SO(d)

 δ M NX,ρY

 K ϑd (dρ) < ∞.

δ is a well-defined integral current for Consequently, NX,ρY = (F) limδ→0 NX,ρY almost all ρ ∈ SO(d).

Proof It is sufficient to prove the statement for X, Y compact. Denote Xi := X−εi , Y i := Y−εi , i ∈ N, for brevity. Using Lemma 6.18, we have for some finite constant c   M NXi ,ρY i − NXδ i ,ρY i  ≤c 1m·n≥−1+δ (sin  (m, n))3−d H2d−2 (d(x, m, y, n))  =c

nor X i ×nor ρY i

nor X i ×nor Y i

1m·n≥−1+δ (sin  (m, ρ −1 n))3−d H2d−2 (d(x, m, y, n)).

A direct computation reveals that  lim

δ→0 SO(d)

1m·n≥−1+δ (sin  (m, ρ −1 n))3−d ϑd (dρ)

−1 lim = Od−1



δ→0 S d−1

1m·n≥−1+δ (sin  (m, v))3−d Hd−1 (dv) = 0,

and since H2d−2 (nor Xi × nor Y i ) = M(NXi )M(NY i ) is uniformly bounded, the first assertion follows. For the second statement, note that for almost all ρ ∈ SO(d), NXδ i ,ρY i → NXi ,ρY i weakly, thus δ M(NX,ρY ) ≤ lim inf M(NXδ i ,ρY i ), i→∞

204

9 Extensions of Curvature Measures to Larger Set Classes

and Fatou’s lemma yields 

 SO(d)

δ M(NX,ρY ) ϑd (dρ) ≤ lim inf i→∞

SO(d)

M(NXδ i ,ρY i ) ϑd (dρ),

which is, due to (6.14), bounded by lim inf const M(NXi )M(NY i ) < ∞. i→∞

  We are now able to formulate and prove the Principal kinematic formula for intersections of closed Lipschitz domains from LD d with sets of positive reach. Theorem 9.35 Let X, Y ∈ LDd and assume that reach Y > 0. Then X∩gY ∈ LDd for almost all Euclidean motions g ∈ Gd , and for any bounded Borel sets A, B ⊆ Rd and any integer 0 ≤ k ≤ d,  Gd





Ck X ∩ gY, A ∩ gB dg =

γ (d, r, s)Cr (X, A)Cs (Y, B).

1≤r,s≤d r+s=d+k

Proof The first statement has already been proved in Proposition 9.28. We will prove a version of the kinematic formula for smooth functions: 

 Gd



f (z)h(g −1 z) Ck X ∩ gY, dz ϑd (dg) 

=

 γ (d, p, q)

(9.31) 

f (x) Cp (X, dx)

h(y) Cq (Y, dy)

1≤p,q≤d p+q=d+k

whenever f, h are smooth function on Rd with compact supports; this is clearly equivalent to the Principal kinematic formula. Using localization, we can assume that both X and Y are compact. The left hand side of (9.31) can be written as  I := (NX∩gY ψg )(ϕk ) dg Gd

with ψg = (f ◦ π0 )(h ◦ g −1 ◦ π0 ). Using Lemma 9.33 we get  I =

 Gd

NX



+

Gd

 NgY

 (1π −1 int gY · ψg ) (ϕk ) dg 0

 (1π −1 int X · ψg ) (ϕk ) dg 0

9.3 Lipschitz Domains

205

 +

Gd

(−1)d (πˆ # NX,ρY , G, z

ψg ) (ϕk ) dg

=: I1 + I2 + I3 . Again, z and ρ are the shift and orthogonal part of g ∈ Gd . The current NX,ρY is well defined for almost all ρ ∈ SO(d) by Lemma 9.34. Using the same procedure as in the proof of Theorem 6.10, we get  I1 = (NX

(f ◦ π0 )) (ϕk )

 h(y) dy =

 I2 = (NY

(h ◦ π0 )) (ϕk )

 f dCk (X, ·)

 f (x) dx =

h dCd (Y, ·), 

f dCd (X, ·)

h dCk (Y, ·).

Applying (1.33), we obtain  (−1)d I3 =

Gd

  NX,ρY , G, z πˆ # (ψg · ϕk ) dg

 =

 NX,ρY

SO(d)

 =

G# Ω d



 πˆ # (ψg · ϕk ) ϑd (dρ)

  NX,ρY (ψg ◦ π)G ˜ # Ωd ∧ πˆ # ϕk ϑd (dρ)

SO(d)

 =:

NX,ρY (φ) ϑd (dρ). SO(d)

Let εi → 0 be a sequence for which (M(NX−εi )) is bounded and (F) lim NX−εi = NX . i→∞

Since reach Y > 0, also (M(NY−εi )) is bounded and (F) limi→∞ NY−εi = NX . We will write Xi := X−εi and Y i := Y−εi for brevity. If we replace X, Y with Xi , Y i , respectively, in the above formula for I3 , we obtain I3i =





γ (d, r, s)

 f (x) Cp (Xi , dx)

h(y) Cq (Y i , dy),

r+s=d+k 1≤r,s≤d−1

see Chapter 6, Corollary 6.12 and Lemma 6.19, and this converges to  r+s=d+k 1≤r,s≤d−1

 γ (d, r, s)

 f (x) Cp (X, dx)

h(y) Cq (Y, dy)

206

9 Extensions of Curvature Measures to Larger Set Classes

as i → ∞. Thus, the proof will be finished by showing that  lim

i→∞ SO(d)

(NX,ρY − NXi ,ρY i )(φ) ϑd (dρ) = 0.

We can estimate       ≤ (N − N )(φ) ϑ (dρ) i i X,ρY d X ,ρY   SO(d)



SO(d)

+ 

SO(d)

+ SO(d)

(9.32)

δ |(NX,ρY − NX,ρY )(φ)| ϑd (dρ)

δ |(NX,ρY − NXδ i ,ρY i )(φ)| ϑd (dρ)

|(NXδ i ,ρY i − NXi ,ρY i )(φ)| ϑd (dρ).

The first and third integrals tend to zero with δ → 0 by Lemma 9.34 uniformly in i ∈ N (we apply the Lebesgue dominated theorem for the first integral). The second integral tends to zero as i → ∞ for any fixed δ > 0. This proves (9.32).  

9.4 Bibliographical Notes 1. The uniqueness theorem for normal cycles was first proved by Fu in 1989 [Fu89b] and further developed in [Fu94] to introduce normal cycles for subanalytic sets in Rd . Fu introduced the following technique to construct normal cycles. A function on Rd is called Monge-Ampère if it admits a so-called “gradient cycle” (current corresponding to the integration over the graph of the gradient in the smooth case). The gradient cycle is then used to define normal cycles for weakly regular sublevel sets of a Monge-Ampère function. It can be shown that weakly regular sublevel sets of semiconvex functions are exactly the sets with positive reach [Ban82, Kle81]. 2. A real function is called DC if it can be represented as a difference of two convex functions. Normal cycles for compact DC domains (sets locally representable as subgraphs of d.c. functions) were introduced in [PR13]. In fact, a larger family of sets was considered, namely weakly regular sublevel sets of DC functions on Rd , called (locally) WDC sets. Since DC functions are Monge-Ampère, the construction of Fu described in 1 can be applied. The case of compact DC domains in dimension d = 3 was already solved by Fu in [Fu00], following the approach of A.D. Alexandrov. Some properties of (locally) WDC sets, including a full characterization in R2 , were given in [PRZar]. 3. Geometry of sets in Euclidean spaces from a certain o-minimal system (including, in particular, subanalytic sets) was considered by several authors. Bröcker, Kuppe [BK00] and Bernig, Bröcker [BB02] introduced curvature measures and proved the Principal kinematic formula under certain technical assumptions, the

9.4 Bibliographical Notes

207

basic technique was the stratified Morse theory. Bernig [Ber07] defined normal cycles for sets from an o-minimal system using a technique of “definable cycles”. Kashiwara and Shapiro [KS94] used a different method of sheaves. 4. A natural question is whether the sets admitting normal cycles fulfill integral geometric formulas as Croftom formula (see (2.5)) or Principal kinematic formula (see (2.4)). For (locally) WDC sets (see 2), the Crofton formula was shown in [PR13, Theorem 1.3], and the Principal kinematic formula in [FPR17]. In fact, the kinematic formula in [FPR17] works in a much more general setting, namely in Riemannian isotropic spaces (including, i.e., spherical case). For a survey on normal cycles and kinematic formulas, see [Fu17].

Chapter 10

Fractal Versions of Curvatures

10.1 Introduction In this chapter we will give a brief introduction to certain fractal versions of Lipschitz-Killing curvature measures. We present only some of the proofs giving insight into certain techniques which are applied in this and different fields. For the other proofs we refer to the literature. Throughout this section we use the notation |A| := diam A for A ⊂ Rd . The geometric sets considered in the previous chapters admit only singularities which guarantee an extension of the notions and tools from geometric integration theory, multilinear algebra and topology. More precisely, they all possess some second order rectifiability properties expressed in terms of unit normal bundles. For the more irregular fractal structures such concepts do not work anymore. However, the well-known Hausdorff measures and other covering or packing measures of arbitrary dimensions may be interpreted as fractal extensions of the corresponding classical first order analysis and geometry. They have been studied in the literature for a long time. Special related notions are the following. Definition 10.1 The Minkowski dimension of a bounded set F ⊂ Rd is given by log Ld (Fε ) , ε→0 log ε

dimM F = D := d − lim provided the limit exists. Then the values D

M (F ) := lim sup ε→0

MD (F ) := lim inf ε→0

Ld (Fε ) , ωd−D εd−D Ld (Fε ) ωd−D εd−D

© Springer Nature Switzerland AG 2019 J. Rataj, M. Zähle, Curvature Measures of Singular Sets, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-18183-3_10

209

210

10 Fractal Versions of Curvatures

are called upper (resp. lower) Minkowski content of such a D-dimensional F . If D MD (F ) = M (F ) =: MD (F ) then MD (F ) is called Minkowski content of F . F is said to be Minkowski measurable if it has positive and finite Minkowski content. It is well known that for bounded smooth sets and rectifiable versions the Minkowski content agrees with the total Hausdorff measure of the corresponding integral dimension. This does not remain valid, in general, for more irregular sets. However, Minkowski dimension and content play an important role, e.g., in spectral analysis on fractal sets or domains with fractal boundary. If one slightly extends the above notions via average convergence, fractal versions for a large class of so-called self-similar and self-conformal sets can be obtained. Definition 10.2 1 δ→0 | ln δ|

*D (F ) := lim M

 δ

1

Ld (Fε ) 1 dε ωd−D εd−D ε

is called D-dimensional average Minkowski content of the set F , if the limit exists. Note that for Minkowski measurable sets the Minkowski content MD (F ) coincides with its average version. Furthermore, other types of averaging measures than the logarithmic one could be used. However, the latter is appropriate for the fractals considered in this chapter. This also concerns the gauge function εd−D for the volume of the parallel sets. For compact sets X ⊂ Rd with positive reach of dimension m the Minkowski content may be considered as a marginal case of the associated family of total curvatures Ck (X) from the previous chapters, namely Mm (X) = Cm (X) = Hm (X) . The first equality follows here immediately from the Steiner formula (see Corollary 4.33). Moreover, in this case we have   d − m −1 ωd−k m−k ε Ck (Xε ) ε→0 d − k ωd−m

Cm (X) = lim

(10.1)

for all m ≤ k ≤ d in view of Proposition 4.34. An essential background are the scaling properties of the associated measures. Having this in mind a natural extension of all total Lipschitz-Killing curvatures to certain fractals can be considered. Definition 10.3 We call a set F ⊂ Rd tube regular if the following conditions are fulfilled. 1. (TN) for Lebesgue almost all r > 0 the parallel sets Fr admit a normal cycle in the sense of Definition 9.5

10.1 Introduction

211

2. (TM) for all k ≤ d − 2 and bounded Borel sets B the curvatures Ck (Fr , B) are measurable functions in r, where we set Ck (Fr , ·) := 0 for the exceptional ε. For general F ⊂ Rd and r > 0 we use throughout this chapter the notation Cd (Fr , ·) := Ld (Fr ∩ (·)) , and Cd−1 (Fr , ·) :=

1 d−1 H (∂Fr , ∩(·)) , 2

For k = d this is compatible with all former variants, and for k = d − 1 see Remark 10.5 below. For these marginal cases we need the following results which complement the notion of tube regularity. Proposition 10.4 (i) For F ⊂ Rd , k ∈ {d − 1, d} and all bounded Borel sets B the mappings r → Ck (Fr , B), r > 0, are measurable. (ii) If F is bounded, then for all r > 0, ∂Fr is upper Ahlfors (d − 1)-regular. Proof For k = d the first assertion follows from continuity of Lebesgue measure with respect to the Hausdorff distance. In the case k = d − 1 we can use the measurability statement for the section integrals in the Coarea theorem 1.15 applied to the Lipschitz mapping f : FR → [0, R] with f (x) := d(x, F ) for arbitrary R > 0. In order to show (ii) assume first that r > 0 is a regular value of the distance function x → dist (x, F ). Then ∂Fr is a (d − 1)-dimensional Lipschitz manifold by the inverse function theorem √ for Lipschitz mappings (see Clarke [Cla76]). This appears, in particular, if r > d/(2d + 1)|F | (cf. Fu [Fu85, Theorem 4.1]). In such a case, ∂Fr is Ahlfors (d − 1)-regular by Proposition 1.12. Let now r > 0 be arbitrary and partition F = F 1 ∪ · · · ∪ F n into finitely many pieces with |F i | < r, i = 1, . . . , n. Then, each ∂(F i )r is upper Ahlfors (d − 1)regular by the above observation and, since clearly ∂Fr ⊂ ∂(F 1 )r ∪ · · · ∪ ∂(F n )r , ∂Fr is (d − 1)-rectifiable and upper Ahlfors (d − 1)-regular.   Remark 10.5 (i) A sufficient condition for the tube regularity of F is that a.a. r > 0 are regular√ values of the distance function d(·, F ), which is fulfilled for all r > d/(2d + 2)|F | according to Fu [Fu85, Theorem 4.1]. (To see this implication use that for such r the sets (Fr  )c have positive reach for all r  in some neighbourhood of r, see [Fu85, Theorem 4.1].  Then for any continuous function f with compact support the integrals f (z)Ck (Fr , dz) are continuous at r, see [Zäh11, Lemma 2.3.4] which refers to [RZ05, Proposition 6]. Since the indicator function of a bounded Borel set B can be approximated by such f the measurability property (TM) is a consequence.) (ii) In space dimensions d ≤ 3 any compact F is tube regular, since in this case a.a. r > 0 are regular values of the distance function d(·, F ) in view of [Fu85, Theorem 4.1].

212

10 Fractal Versions of Curvatures

(iii) If r is a regular value of the distanced function, then the measure Cd−1 (Fr , ·) in the sense of Definition 9.5 agrees with half the surface area measure on the boundary, i.e., this is compatible with the above notation. We conjecture that this remains valid in the general case when Fr admits a normal cycle. Suppose now that F is bounded, and if k ≤ d − 2 that it is also tube regular. Let Ckvar (Fε , ·) be the variation measures of the corresponding curvature measures. Concerning the scaling exponents we use the notions from Pokorný and Winter [PW14]. For k ∈ {0, . . . , d} introduce   s−k var sk := inf s ≥ 0 : ess lim ε Ck (Fε ) = 0 , ε→0

 ak := inf s ≥ 0 : lim

1 δ→0 | ln δ|



ε0

ε

s−k

δ

Cvar k (Fε )

 1 dε = 0 . ε

(10.2)

(10.3)

Here and in the sequel the essential limit is taken with respect to Lebesgue measure. Definition 10.6 For 0 ≤ s ≤ d, Csk (F ) := ess lim εs−k Ck (Fε ) ε→0

and 1 δ→0 | ln δ|

s

Ck (F ) := lim



ε0 δ

1 εs−k Ck (Fε ) dε ε

are called the s-dimensional (resp. average) total fractal (Lipschitz-Killing) curvature of order k of the set F , provided the corresponding limit exists. Remark 10.7 In order to get geometrically meaningful notions one should consider for fixed k the scaling exponents s = sk and s = ak , respectively. Below we are mainly interested in the cases where sk or ak coincide for all k with the Minkowski dimension D of the set F . The Minkowski content and its average version are then contained as marginal cases MD (F ) =

1 ωd−D

*D CD d (F ) and M (F ) =

1 ωd−D

D

Cd (F ) ,

where no curvature properties are involved. (However, we will not treat here conditions for sk = ak = D, see also the discussion in [PW14]). We conjecture that D similar properties as in (10.1) are fulfilled for the values Ck (F ) with k ≥ &D' (cf. Theorem 10.15 below). For completeness we call them for all k fractal LipschitzKilling curvatures. Similarly one can introduce the localized versions, i.e., fractal curvature measures on the Borel σ -algebra of Rd with the above total values:

10.1 Introduction

213

Definition 10.8 Cks (F, ·) := ess lim εs−k Ck (Fε , ·) ε→0

is called s-dimensional fractal curvature measure of order k, and s

1 δ→0 | ln δ|

C k (F, ·) := lim



ε0 δ

1 εs−k Ck (Fε , ·) dε ε

its average version if the corresponding limits exist in the sense of weak convergence of signed measures. Again, we are interested in the cases s = sk = D or s = ak = D. Remark 10.9 In the last definition F need not be bounded if weak convergence is replaced by vague convergence. In all definitions introduced above the curvature measures Ck (Fε , ·) of the parallel sets could be replaced by their extensions, the curvature-direction measures k (Fε , ·) on Rd ×S d−1 (cf. Definition 9.5) in order to obtain the exponents C sk , ak and s (F ) = Cs (F ) s (F, ·) with total values  the fractal curvature-direction measures C C k k k in the same manner as before, provided the limits exist. For X ⊂ Rd with positive reach and more general sets from the previous sections the k (X, ·), k = 0, . . . , d − 1, possess signed densities curvature-direction measures C with respect to Hausdorff measure Hd−1 on the associated unit normal bundle nor X. For PR-sets these are the generalized local Lipschitz-Killing curvatures sd−1−k (X; x, n) at (x, n) ∈ nor X, cf. Definitions 4.25 and 4.28. In the special case of m-dimensional smooth submanifolds this corresponds to the classical (m − k)-th order mean curvatures at x ∈ X via projections from nor X to X, see Remark 4.32. The latter are then the signed densities of the k-th curvature measures with respect to Hausdorff measure Hm on X. Therefore both versions of these pointwise curvatures can be obtained from Lebesgue’s density theorem at almost all points. It turned out that for large classes of self-similar sets and some extensions one can also obtain relationships between local and global fractal curvatures in form of densities with respect to Hausdorff measure. Moreover, the latter can be determined by means of local versions of the above approximations with parallel sets, however with different scaling behaviour. To this aim we consider a localization procedure by means of appropriate bump functions as follows. For brevity we write  Ck (Fε , f ) :=

f (z) Ck (Fε , dz) ,

if f is an integrable function. Then we introduce the following notions. Let F ⊂ Rd and μF be a Radon measure on F which is lower Ahlfors D-regular (recall Definition 1.5). In particular, if F is bounded then μF (F ) < ∞.

214

10 Fractal Versions of Curvatures

Definition 10.10 For any ε0 > 0 and 0 < a < 1 and any nonnegative bounded measurable function g with support in [−1, 1] which is bounded from below by a positive constant in a neighbourhood of 0 the functions  AF (x, ε)(z) := εD  F

g  g

|x−z| aε

|y−z| aε





, z ∈ Fε ,

μF (dy)

with parameters x ∈ F , 0 < ε < ε0 , form a localizing family of bump functions associated with F and μF . Note that lower Ahlfors regularity of μF and the properties of g imply that the localizing functions AF are uniformly bounded. The following measurability property for such families of bump functions AF will be used in the sequel. Proposition 10.11 Let F ⊂ Rd and k ∈ {0, . . . , d}. If d ≥ 4 and k ≤ d − 2 suppose additionally that the set in the sense of Definition 10.3. F is tube regular

Then the curvature integral Ck Fε , AF (x, ε) (with value 0 at the exceptional ε) is a measurable function in (x, ε) ∈ F × (0, ε0 ). Proof It suffices to consider x ∈ F ∩ K for arbitrary compact K. First note that the function AF (x, ε)(z) is product measurable in (x, ε, z). Therefore it may be approximated by a sequence of uniformly bounded functions which are representable by linear combinations of measurable functions of the form f1 (x)f2 (ε)f3 (z) with supports in K × [0, ε0] × Kaε0 . Thus, their integrals with respect to Ck (Fε , ·) converge to Ck (Fε , AF (x, ε)). Furthermore, by tube regularity (TM) and Remark 10.5 for any bounded measurable function f with compact support the integral Ck (Fε , f ) is a measurable function in ε ∈ (0, ε0 ). Combining both these arguments we obtain the assertion.   This enables us to introduce the notion of fractal curvature densities. Definition 10.12 Let F ⊂ Rd and k ∈ {0, . . . , d}. If d ≥ 4 and k ≤ d − 2 suppose additionally that F is tube regular in the sense of Definition 10.3. Then for localizing functions AF as in Definition 10.10,  ε0 1 1 Dk (F, x) := lim ε−k Ck (Fε , AF (x, ε)) dε δ→0 | ln δ| δ ε is called k-th curvature density of F w.r.t. μF at point x, provided the integrals and their limit exist. There is a close relationship between such fractal densities and associated fractal curvature measures if Dk (F, x) exists at μF -a.a. x ∈ F . In this case the densities do not depend on the choice of the constants ε0 , a and the bump function g in the localizing procedure. This will be shown in the next theorem.

10.1 Introduction

215

Recall that on a finite measure space for a family of functions that converges in measure, uniform integrability is equivalent to L1 -convergence (see, e.g., Doob [Doo94, Theorem 6.18]). Theorem 10.13 Let k ∈ {0, 1, . . . , d − 1}, F be a compact subset of Rd such that there exists a lower Ahlfors D-regular finite measure μF with spt μF = F , and let AF (x, ε) be a family of localizing functions as in Definition 10.10. For k ≤ d − 2 let F be tube regular in the sense of Definition 10.3 (for d ≤ 3 this is always fulfilled). Suppose that the corresponding curvature density Dk (F, x) exists at μF -a.a. x ∈ F and that the functions 1 x→  | ln δ|



ε0 δ

Ckvar (Fε , AF (x, ε)) 1 dε , x ∈ F , εk ε

(10.4)

with parameter δ are uniformly integrable w.r.t. μF . Then the k-th average fractal curvature measure in the sense of Definition 10.8 exists and is equal to 

D

C k (F, ·) =

1(·)(x)Dk (F, x) μF (dx) . F

Consequently, in this case the density Dk (F, x) does not depend on the choice of the localizing functions AF (x, ε) with property (10.4). For k = d − 1, d the uniform integrability is always fulfilled. Proof According to Proposition 10.11 the function Ck (Fε , AF (x, ε)) is measurable in (x, ε) ∈ F × (0, ε0 ). Then in view of the integrabilitycondition (10.4) Fubini’s theorem can be applied to the integrals below. Using that F AF (x, ε)(z) μF (dx) = εD we obtain for an arbitrary continuous function f and ε < ε0 ,  εD−k

f (z) Ck (Fε , dz) = ε−k  = F



ε−k

 AF (x, ε)(z) μF (dx) Ck (Fε , dz)

f (z) F



f (z) AF (x, ε)(z) Ck (Fε , dz) μF (dx) . Fε

We are interested in the average limit of the left hand side with respect to the ε logarithmic measure | ln1 δ| δ 0 1(·)(ε) 1ε dε. Since spt AF (x, ε) ⊂ B(x, aε) and f is uniformly continuous on compact sets, the last expression can be replaced asymptotically by 

f (x) ε−k F

 AF (x, ε)(z) Ck (Fε , dz) μF (dx) , Fε

216

10 Fractal Versions of Curvatures

i.e., the average limit of the difference vanishes (see the arguments below). Therefore the primary average limit exists and equals 

 ε0  1 1 f (x) ε−k Ck (Fε , AF (x, ε)) μF (dx) dε δ→0 | ln δ| δ ε F   ε0 1 1 = f (x) lim ε−k Ck (Fε , AF (x, ε)) dε μF (dx) δ→0 | ln δ| ε F δ  = f (x) Dk (F, x) μF (dx) ,

D

f (z) C k (F, dz) = lim

F

since a.e. convergence and uniform integrability imply L1 -convergence. Using that f is an arbitrary continuous function we get CkD (F, ·) = Dk (F, ·) μ and hence, Dk (F, ·) does not depend on the choice of AF . In order to complete the above arguments, for given ε > 0 choose 0 < δ  = δ  (ε) < ε0 small enough so that |f (z)−f (x)| ≤ ε for all z ∈ B(x, aδ  ) uniformly in x ∈ F . Then for δ < δ  the above difference, say R(δ), can be estimated as follows.  R(δ) ≤ +

| ln δ  | | ln δ|



F

F

1 | ln δ| 1 | ln δ  |

 

δ

ε−k

δ ε0

δ

ε

−k

 

1 |f (z) − f (x)|AF (x, ε)(z)Ckvar (Fε , dz) dε μF (dx) ε 1 |f (z) − f (x)|AF (x, ε)(z)Ckvar (Fε , dz) dε μF (dx) ε

| ln δ  |

≤ c ε + | ln δ|

for some constant c, since the integrals  F

1 | ln δ|



ε0 δ

Ckvar (Fε , AF (x, ε)) 1 dεμf (dx) εk ε

are uniformly bounded. Taking first δ → 0 and then ε → 0, one obtains that limδ→0 R(δ) = 0, i.e., the above assertion. Finally, for k = d the uniform integrability condition (i) is obviously fulfilled, since Cd (Fε , ·) agrees with Lebesgue measure on Fε , AF is uniformly bounded and spt AF (x, ε) ⊂ B(x, aε). Similarly, Cd−1 (Fε , ·) is a constant multiple of the (d−1)dimensional Hausdorff measure on the boundary of Fε . In view of Proposition 10.4 it is upper Ahlfors (d − 1)-regular, i.e., Hd−1 (∂Fε ∩ B(x, aε)) < const εd−1 which implies (i).   Below we will see that the conditions of the last theorem are satisfied for a large class of self-similar fractal sets with μF = HD (F ∩ (·)). Moreover, we will summarize

10.2 Self-Similar Sets

217

different methods for determining their fractal curvature measures introduced above. The marginal cases k = d − 1, d will be considered in the more general context of Minkowski content. Remark 10.14 A special bump function is associated with small balls. Namely, AF (x, ε)(z) := εD (μF (B(x, aε))−1 1B(x,aε)(z) is the case where g(u) = 1[−1,1](u). The continuous tent function g(u) := max(1 − |u|, 0) first suggested by T. Bohl is also applied to related problems in the self-conformal case [Boh12]. Furthermore, for self-similar sets satisfying the Open Set Condition (see Sect. 10.2 below) in [RZ12] for ε0 := HD (F )1/D instead of AF (x, ε) the indicator function of the following set (denoted by the same symbol) was used, which yields the same limits for μF := HD (F ∩ (·)): AF (x, ε) := {z ∈ Fε : |x − z| ≤ ρF (z, ε)} where < ; ρF (z, ε) := min ρ : HD (F ∩ B(z, ρ)) = εD .

10.2 Self-Similar Sets We now briefly introduce some basic notions and results from the literature on self-similar sets needed in the sequel. A first general approach to such sets, which included the special cases and famous examples from the former literature like Cantor set, Koch curve, Sierpinski gasket and carpet, Menger sponge etc., was given in Hutchinson [Hut81]. (For subsets on the real line it was already developed in Moran [Mor46].) The facts mentioned below without a reference can be found e.g. in [Hut81] together with Falconer [Fal90], [Fal97] and Kigami [Kig01]. For N ≥ 2 let S1 , . . . , SN be an iterated function system (IFS) consisting of contracting similitudes in Rd with contraction ratios r1 , . . . , rN . It is called lattice, if there exists a constant a such that ln ri ∈ aZ for all i. (This is, for example, the case when all contraction ratios are the same.) In the general case the associated selfsimilar set F may be constructed by means of the code space W := {1, . . . , N}N , i.e., the set of all infinite words over the alphabet {1, . . . , N}. We write  W0 := ∅ and Wn := {1, . . . , N}n for the set of all words of length |w| = n, W∗ := ∞ n=0 Wn for the set of all finite words and w|n := w1 w2 . . . wn , if w = w1 w2 . . . wn wn+1 . . ., for the restriction of a (finite or infinite) word to the first n letters. If w = w1 . . . wn ∈ Wn , we also use the abbreviations Sw := Sw1 ◦ . . . ◦ Swn and rw := rw1 . . . rwn for

218

10 Fractal Versions of Curvatures

the contraction ratio of this mapping. In these terms the set F is determined by F =

∞ 

Sw K

n=1 w∈Wn

for an arbitrary non-empty compact K such that S1 K ∪. . .∪SN K ⊂ K. (For brevity the corresponding brackets are omitted.) F is characterized as the unique non-empty compact set with the self-similarity property F = S1 F ∪ . . . ∪ SN F .

(10.5)

It can also be obtained by a contraction principle for the mapping K → SK :=

N 

Si K

i=1

on the space of nonempty compact subsets of Rd equipped with the Hausdorff metric. Alternatively, the self-similar set F is the image of the code space W under the projection π given by π(w) := lim Sw|n x0 n→∞

for an arbitrary starting point x0 . We now assume the strong open set condition (SOSC) for the IFS, i.e., there exists a non-empty bounded open set O such that for all i = j , Si O ⊂ O , Si O ∩ Sj O = ∅ , which is called open set condition (OSC), and F ∩ O = ∅ for the associated self-similar set F . (According to a result of Schief (see [Sch94]), (OSC) for some O already implies (SOSC) for a possibly different open set. Basing on this paper a characterization of (SOSC) in algebraic terms of the S1 , . . . , SN was given in Bandt and Graf [BG92].) An open set O from (OSC) or (SOSC) for F is called feasible or strong feasible, respectively.

10.2 Self-Similar Sets

219

(OSC) implies that the Hausdorff dimension D of F is determined by N 

rjD = 1

(10.6)

j =1

and that it coincides with the Minkowski dimension and other types of fractal dimensions of F . Moreover, 0 < HD (F ) < ∞ . (In fact, the last property is equivalent to (SOSC), see [BG92].) The self-similar set F is even Ahlfors D-regular, i.e., there exist positive constants cF and CF such that cF r D ≤ HD (F ∩ B(x, r)) ≤ CF r D , x ∈ F, r ≤ diam F . Furthermore, under (SOSC) the above projection mapping w → x = π(w) from the code space W onto F is biunique at HD -a.a. of points x ∈ F . Therefore we identify almost every point x ∈ F with its coding sequence and write x1 x2 . . . := π −1 (x) for this infinite word. If ν denotes the infinite product measure on W determined by the probability measure on the alphabet {1, . . . , N} which assigns to i the probability riD ,i = 1, . . . N, then the normalized D-dimensional Hausdorff measure on F equals HD (F )−1 HD (F ∩ (·)) = ν ◦ π −1 .

(10.7)

It agrees with the unique normalized self-similar measure μ on F such that μ=

N 

riD μ ◦ Si−1 .

(10.8)

i=1

In particular, μ(∂O) = 0 for any strong feasible open set O. Iterated applications of (10.5) and (10.8) lead to F =

 w∈Wn

Sw F and μ =

 w∈Wn

rwD μ ◦ Sw−1 ,

220

10 Fractal Versions of Curvatures

respectively, for all n ∈ N. This extends to arbitrary Markov stoppings, i.e., mappings τ : W → N such that the value τ (w) = n is determined by the first n letters w1 , . . . , wn of the infinite word w. For such τ denote 9 : W (τ ) := w1 . . . wτ (w) : w ∈ W . Then one infers 

F =

Sw F ,

(10.9)

rwD μ ◦ Sw−1 ,

(10.10)

rwD = 1 .

(10.11)

w∈W (τ )

μ=

 w∈W (τ )

in particular,  w∈W (τ )

An example for a Markov stopping, which will be used in Sect. 10.3.3, is the following. Recall that r1 , . . . , rN denote the contraction ratios of the similarities S1 , . . . , SN . Then for given 0 < r < diam F the value τ (w) := τr (w) is determined by rw1 . . . rwτr (w) |F | < r ≤ rw1 . . . rwτr (w)−1 |F | . Finally, for any strong feasible open set O and all p ∈ N one obtains  | ln d(y, O c )|p μ(dy) < ∞ .

(10.12)

(10.13)

(See Graf [Gra95, proof of Proposition 3.4] for p = 1. The proof for general p is similar.) These notions and results for dimensions and measures can also be obtained as special linear cases from the so-called thermodynamic formalism in the sense of Bowen and Ruelle. (We refer to Falconer [Fal97] for an introduction and to Mauldin and Urbanski [MU03] as well as the references therein for more general versions concerning self-conformal sets.) In particular, if we consider the mapping T : F → F with the inverse branches S1 , . . . , SN defined for μ-a.a. x by T (x) := Si−1 (x) , if x ∈ Si F , i = 1, . . . , N , taking into regard that μ(Si F ∩ Sj F ) = 0 for i = j , then (F, T , μ) is an ergodic dynamical system. This follows here directly from the relationship μ = ν ◦ π −1 , since the product measure ν on the code space is shift invariant and ergodic.

10.2 Self-Similar Sets

221

In the sequel we also consider tilings associated with the self-similar set F and their generators. Recall that A denotes the closure of a set A in Rd . Suppose that for a feasible open set O the open set G := O \ SO is not empty. (For example, G is the inner open triangle in the first construction step of the Sierpinski gasket.) Then the family of iterates of G under the corresponding IFS, T = T (O) := {Sw G : w ∈ W∗ } is called tiling of O with generator G. The tiles, i.e., the elements of T are pairwise disjoint and O=



R.

R∈T

Such a tiling is called compatible if ∂G ⊂ F , which is equivalent to ∂O ⊂ F (see Pearse and Winter [PW12, Theorem 6.2]. (For these and further relationships and examples see [PW12] and Winter [Win15].) Classical examples of self-similar sets are e.g. the Sierpinski triangle (Fig. 10.1), the Sierpinski carpet (Fig. 10.2) and its modified version (Fig. 10.3), or the Koch curve (Fig. 10.4) together with a strong feasible open set (Fig. 10.5).

Fig. 10.1 First iterations of the Sierpinski triangle generated by N = 3 similitudes with contraction factors r = 1/2. The interior of the primary black triangle is a strong feasible open set O and the inner white triangle in the first iteration is the generator G of a compatible tiling. The Hausdorff dimension of the limit set F equals D = log 3/ log 2

222

10 Fractal Versions of Curvatures

Fig. 10.2 First iterations of the Sierpinski carpet generated by 8 similitudes with contraction factors 1/3. The interior of the primary black square is a feasible open set O and the inner white square G in the first iteration generates a compatible tiling. The Hausdorff dimension of the limit set F equals log 8/ log 3

Fig. 10.3 First iterations of the modified Sierpinski carpet. Here the interior of the primary square is also a feasible open set. In order to get a compatible tiling one should consider the interior of the polygon in the first iteration as feasible open set O and the inner white set in the second iteration as associated generator. The Hausdorff dimension is again log 8/ log 3

10.3 (Average) Minkowski Content and Localized Versions

223

Fig. 10.4 First iterations of the Koch curve. Here the primary interval with the same boundary points as the curves is omitted. The iterations by 4 similitudes with contraction factors 1/3 are presented. The following consideration allows to include this in the above general construction

Fig. 10.5 The interior of the primary black √ triangle is a strong feasible open set for the IFS of 2 similitudes with contraction factors 1/ 3, for which the first three iterates are presented. In the above construction only every second iterate of this type is considered, i.e., one has an IFS of 4 similitudes with contraction factors 1/3 generating the same limit set. Note that here the associated tilings are not compatible. The Hausdorff dimension of the Koch curve equals log 4/ log 3

10.3 (Average) Minkowski Content and Localized Versions 10.3.1 Characterization in Terms of Surface Area Recall that the measures Cd (Fε , ·) and Cd−1 (Fε , ·) correspond to volume and surface area of the parallel sets. They are included as marginal cases, though they do not reflect curvature properties. However, the Lipschitz-Killing curvature (measures) Ck , k = 0, . . . , d, form a complete system of Euclidean invariants with certain properties, cf. Chap. 8, which underlines their geometric meaning. Since the marginal cases k = d − 1, d concern only the first order infinitesimal

224

10 Fractal Versions of Curvatures

behaviour, their fractal versions exist under much weaker conditions than for the other fractal curvatures which are obtained by analytically second order properties of the approximating sets. We start with the observation that in the approaches mentioned above the (average) limits of the appropriately rescaled version for arbitrary bounded F with Ld (F ) = 0 in the cases k = d − 1 and k = d coincide provided one of them exists. This means that the (average) Minkowski content can be obtained by means of the volume as well as of the surface area of the parallel sets: Theorem 10.15 Let F be a bounded subset of Rd , D ∈ [0, d), or D = d and Ld (F ) = 0, M ∈ (0, ∞) and ε0 > 0. Then we have the following. (i) Ld (Fε ) =M ε→0 ωd−D ε d−D lim

if and only if Hd−1 (∂Fε ) =M. ε→0 (d − D)ωd−D ε d−1−D lim

(ii) If lim sup δ→0

1 | ln δ|



ε0 δ

Ld (Fε ) 1 dε < ∞ εd−D ε

then 1 lim inf δ→0 | ln δ|



ε0 δ

Ld (Fε ) 1 1 dε = lim inf d−D δ→0 | ln δ| ε ε



ε0

Hd−1 (∂Fε ) 1 dε (d − D)εd−1−D ε

ε0

Hd−1 (∂Fε ) 1 dε . (d − D)εd−1−D ε

δ

and lim sup δ→0

1 | ln δ|



ε0 δ

Ld (Fε ) 1 1 dε = lim sup d−D ε ε δ→0 | ln δ|

 δ

Part (i) is shown in [RW10] for self-similar sets and in [RW13, Theorem 2.4] for the general case. The average limits in part (ii) are treated in [RW13, Lemma 4.6].

10.3.2 Densities and Measures via Ergodic Theory For the case of self-similar sets with (OSC) both the average Minkowski content and its density with respect to Hausdorff measure always exist. Recall Definition 10.12

10.3 (Average) Minkowski Content and Localized Versions

225

for the pointwise localizing function AF with related constants a, ε0 and the notations from Sect. 10.2. In particular, y1 := (π −1 (y))1 is the first letter in the coding sequence of a.a. y ∈ F . Theorem 10.16 Let F ⊂ Rd be a self-similar set satisfying (OSC) with Hausdorff dimension D and let O be an associated strong feasible open set. (i) At HD -a.a. x ∈ F the average Minkowski densities ((d −D)ωd−D )−1 Dd−1 (F, x) and (ωd−D )−1 Dd (F, x) exist and are both equal to a constant positive value DM (F ). Moreover, at these x, 1 Dd (F, x) = η 1 Dd−1(F, x) = η

 

b−1 d(y,∂O)

b−1 d(y,∂S

 

y1 O)

b−1 d(y,∂O)

b−1 d(y,(∂S

y1 O)



1 AF (y, ε)(z)Ld (dz) dε μ(dy) ε Fε

1 εd



1 εd−1

1 AF (y, ε)(z)Hd−1 (dz) dε μ(dy) ε ∂Fε

 −1 D where η := N i=1 ri | ln ri | and b := max(2a, ε0 |O|). (ii) The average Minkowski content of F exists and satisfies *D (F ) = DM (F )HD (F ) . M Proof In Sect. 10.4 we will show in the more general context of fractal curvatures the existence of the average densities Dd−1 (F, x) and Dd (F, x) at a.a. x ∈ F . By means of an embedded dynamical system the problem can be reduced to the application of Birkhoff’s ergodic theorem, which provides the desired integral representations for the constant values. Choosing then in the proof of Theorem 10.13 μF := HD (F ∩ (·)), k = d − 1, d, and f ≡ 1 we infer that the average Minkowski content defined by means of volume or surface area of the parallel sets exists and equals the product of the corresponding constant average density and HD (F ). Therefore Theorem 10.15 for the total values implies that the average Minkowski densities in (i) coincide.   Remark 10.17 It is not difficult to see that HD (F ) does not exceed the lower Minkowski content of F , hence DM (F ) ≥ 1. For the classical examples of the Cantor set and the Sierpinski gasket one can show the strict inequality. More generally, Theorem 10.13 for the measure μF := HD (F ∩ (·)) and k = d (or k = d − 1), together with Theorem 10.16 yield the following measure version, i.e., the localized Minkowski content. Corollary 10.18 Under the conditions of Theorem 10.16 the weak limit 1 δ→0 | ln δ|

*D (F, ·) := lim M



ε0 δ

Ld (Fε ∩ (·)) 1 dε ωd−D εd−D ε

226

10 Fractal Versions of Curvatures

exists and satisfies *D (F, ·) = DM (F )HD (F ∩ (·)) . M Like for the average densities the statement remains valid up to the constant (d − D)−1 if the Lebesgue measure on the parallel sets is replaced by the surface area measure Hd−1 on their boundaries.

10.3.3 Total Values, Minkowski Measures and the Renewal Theorem *D (F ) was first obtained by D. Gatzouras [Gat00] The average Minkowski content M using methods from classical renewal theory. This approach leads to an integral representation which is more useful for numerical computations (see Theorem 10.21 below). Moreover, under an additional condition the case of ordinary limits can be treated. For d = 1 the latter was already considered in Lapidus [Lap93] and Falconer [Fal95]. We now briefly present these results. Throughout this section we use the notions and results from Sect. 10.2. Let S1 , . . . , SN be an IFS satisfying (OSC) with contraction ratios r, . . . , rN generating the self-similar set F with Hausdorff dimension D as before. Recall that μ denotes the unique self-similar probability measure on F , which coincides with the normalized Hausdorff measure, i.e., in the above notations we have μF = HD (F ) μ. Recall that the IFS is said to be non-lattice if the set {ln r1 , . . . , ln rN } has this property, i.e., these numbers do not concentrate on an additive subgroup of R. Classical renewal theory and some extensions provide useful tools for determining Minkowski content and fractal curvatures for a large class of self-similar sets and some self-conformal versions. To this aim we first need an adapted version of the Renewal theorem in Feller [Fel71, p. 363] substituting there the variable t by − ln ε and the distribution function F by that of a discrete random variable with values in {| ln r1 |, . . . , | ln rN |} and corresponding probabilities (r1D , . . . , rND ). (For more details see Gatzouras [Gat00] or Winter [Win08].) Theorem 10.19 Suppose that for r1 , . . . , rN , D as above the function Z(ε) with ε ∈ (0, 1] satisfies the renewal equation Z(ε) =

N 

riD 1(0,ri ] (ε)Z(ε/ri ) + z(ε) ,

i=1

and the error term t → z(e−t ) is directly Riemann integrable on (0, ∞). Then the function Z is bounded and the following holds:

10.3 (Average) Minkowski Content and Localized Versions

(i) The average limit limε→0

1 1 1 | ln δ| δ Z(ε) ε 1 η



1 0

227

dε exists and equals

1 z(ε) dε , ε

 D where η := N i=1 ri | ln ri |. (ii) If the IFS is non-lattice, then the ordinary limit limε→0 Z(ε) exists and agrees with the average limit. Remark 10.20 According to a result of Asmussen [Asm87, Proposition 4.1, p.118] any a.e. continuous function which is bounded by a directly Riemann integrable function is also directly Riemann integrable. In particular, if the above z(ε) is continuous at Lebesgue-a.a. ε ∈ (0, 1) and |z(ε)| ≤ c εγ for some constants c and γ > 0 these conditions are fulfilled. This can be applied to both the Minkowski content and its average version. Theorem 10.21 Let the IFS and the associated self-similar set F be as above. Then in the non-lattice case the Minkowski content Ld (Fε ) Hd−1 (∂Fε ) = lim ε→0 ωd−D ε d−D ε→0 (d − D)ωd−D ε d−1−D

MD (F ) = lim exists and is equal to M (F ) = N D



1

D i=1 ri | ln ri |

= N

1

1

Rd (ε) 1 dε ωd−D εd−D ε

1

Rd−1 (ε) 1 dε , d−1−D (d − D)ωd−D ε ε

0



D i=1 ri | ln ri |

0

where Rd (ε) := Ld (Fε ) −

N 

1(0,ri ] (ε)Ld ((Si F )ε ) ,

i=1

Rd−1 (ε) := Hd−1 (∂Fε ) −

N 

1(0,ri ] (ε)Hd−1 (∂(Si F )ε ) .

i=1

*D (F ), which always exists, In the lattice case the average Minkowski content M also agrees with the above integral representations. Proof We first consider the function Z(ε) := 1(0,1](ε)

Ld (Fε ) . ωd−D εd−D

228

10 Fractal Versions of Curvatures

It satisfies the renewal equation in the last theorem with 1 z(ε) : = ωd−D εd−D 1 = ωd−D εd−D

1(0,1](ε)L (Fε ) − d

N 

 riD 1(0,ri ] (ε)Ld (Fε/ri )

i=1

1(0,1](ε)L (Fε ) − d

N 

 d

1(0,ri ] (ε)L ((Si F )ε )

.

i=1

The last equality follows from the scaling property of Lebesgue measure. First observe that this function z is piecewise continuous. Furthermore, below after the proof of the next theorem, which uses similar estimates, we will show that |z(ε)| ≤ cεγ for some constants c and γ > 0 .

(10.14)

Therefore the above Renewal theorem together with Remark 10.20 implies the assertion for the volume part. The proof for the limit of the surface area version is similar, and equality follows from Theorem 10.15.   For the volume this was proved in Falconer [Fal95] for d = 1 and in Gatzouras [Gat00] for general d using a result of Lalley [Lal88]. The surface area was considered in Winter [Win08], Rataj and Winter [RW10] in a more general context of the above curvature measures. Furthermore, in [Win08] and [WZ13] ordinary convergence of the corresponding measures for the non-lattice case was treated in this sense. We will give now a slightly shorter proof for the special case of the Minkowski measure. (The average version is already shown in Theorem 10.18.) Recall that μ is the normalized Hausdorff measure on F . Theorem 10.22 Under the above conditions in the non-lattice case the limit 1 Ld (Fε ∩ (·)) ε→0 ωd−D ε d−D

M D (F, ·) = lim

= lim

ε→0

1 Hd−1 (∂Fε ∩ (·)) (d − D)ωd−D εd−1−D

exists in the sense of weak convergence of measures and satisfies M D (F, ·) = MD (F ) μ(·) . Proof Denote με :=

1 Ld (Fε ∩ (·)) . ωd−D εd−D

10.3 (Average) Minkowski Content and Localized Versions

229

Since the self-similar set F is compact, {με , ε < |F |} is a family of tight Borel measures. Theorem 10.21 implies that these measures are uniformly bounded, i.e., sup με (Fε ) < ∞ .

(10.15)

ε 0, it follows that lim Σ4 (ε) = 0 .

ε→0

This estimate can be derived similarly as at the beginning of the proof. Let D  be the unique number such that 



rvD = 1 .

v∈W (τε ),u≺v

Since



D v∈W (τε ) rv

1=

= 1 (see (10.11)), we get D  < D and 





rvD ≥ cεD #({v ∈ W (τε ), u ≺ v}) ,

v∈W (τε ),u≺v

i.e., the desired estimate for the above cardinality setting γ := D − D  . The proof for the summand Σ2 (ε) instead of Σ4 (ε) is analogous if considering the word wu instead of u and the set Sw O instead of O.

232

10 Fractal Versions of Curvatures

This completes the proof for the volume part in the assertion of the theorem. The arguments for the boundary version are similar, when replacing Ld on Fε by Hd−1 on ∂Fε and regarding the corresponding scaling behaviour.   Now we will complete the proof of Theorem 10.21 for the Minkowski content. Proof of (10.14) At the end of the last proof we have seen for sufficiently small ε the estimate με ((∂O)ε ) ≤ cεγ

(10.20)

and the same (with different constants c and γ > 0) for με ((∂Sw O)ε ), where με =

Ld (Fε ∩ (·)) . ωd−D εd−D

Similarly we can get the auxiliary inequality με ((∂SO)ε ) ≤ cεγ

(10.21)

for such constants by the following arguments. (Denote SA = ∪N i=1 Si A, for general A ⊂ Rd .) From (10.18) we infer (SSu F )ε ⊂ SO \ (∂SO)ε for all sufficiently small ε. Like in the estimate of the summand S4 (ε) in the previous proof we conclude με ((∂SO)ε ) ≤



με ((Sv F )ε ) ,

where the sum runs over all words v ∈ W (τε ) which are not representable as v = iuv  for some i ∈ {1, . . . , N} and v  ∈ W∗ . The remaining arguments for (10.21) are similar as in the above case with a different γ > 0. We now consider z(ε) from the assertion (10.14). In terms of με we get z(ε) = 1(0,1]με (Fε ) −

N 

1(0,ri ] (ε)με ((Si F )ε ) .

i=1

It suffices to choose ε ≤ rmin := min(r1 , . . . , rN ), since z(ε) is bounded on [b, 1] for any 0 < b ≤ 1. Denote U (ε) :=

 i=j

(Si F )ε ∩ (Sj F )ε and Bi (ε) := (Si F )ε \ U (ε) .

10.3 (Average) Minkowski Content and Localized Versions

Then Fε =

N

i=1 Bi (ε) ∪ U (ε)

233

is a disjoint union and hence,

με (Fε ) =

N 

με (Bi (ε)) + με (U (ε)) .

i=1

Similarly, μ((Si F )ε ) = με (Bi (ε)) + με ((Si F )ε ∩ U (ε)). Substituting these expressions in the above equation for z(ε) we infer |z(ε)| ≤ με (U (ε)) +

N 

με ((Si F )ε ∩ U (ε)) ≤ (N + 1)με (U (ε)) .

i=1

From (OSC) we will see that U (ε) ⊂ (∂SO)ε . Consequently, for sufficiently small ε we can use (10.21) and obtain the desired upper estimate cεγ . In order to show the above set inclusion note that for any y ∈ U (ε) there exist i = j and xi ∈ Si F ⊂ Si O, xj ∈ Sj F ⊂ Sj O with |xi − y| ≤ ε, |xj − y| ≤ ε. Because of (OSC) the segment [xi , xj ] between these points intersects ∂Si O ∪ ∂Sj O. Since any point x ∈ [xi , xj ] fulfills |x − y| ≤ ε, and since ∂Si O ∪ ∂Sj O ⊂ ∂SO, we get y ∈ (∂SO)ε .   In Freiberg and Kombrink [FK12] the last two theorems are extended to images of self-similar sets with (OSC) under conformal C 1+α -mappings. Relationships between the (average) Minkowski contents of a self-similar set F as above and its associated tiling T (O) for certain feasible open sets O are studied in Winter [Win15] with similar methods. The following particular result for compatible tilings T , which has been known for special cases before, provides a nice formula for computing the (average) Minkowski content of F in terms of the generator G. We write G−ε := {x ∈ G : d(x, ∂G) ≤ ε} for the inner parallel sets of G. Theorem 10.23 (cf. [Win15, Theorem 3.10]) If the self-similar set F with Minkowski dimension D ∈ (d − 1, d) admits a strong feasible open set O such that ∂G ⊂ F for the corresponding generator G = O \SO, then the average Minkowski content, and in the non-lattice case the Minkowski content, of F exists and is equal to 1 η where η =

 0

∞

Ld (G−ε ) 1 dε ωd−D εd−D ε

N

D i=1 ri | ln ri |.

The proof of existence via the Renewal theorem for the corresponding tiling is here more direct than in the general situation of Theorem 10.21. The result can be applied to many classical cases like the Cantor set, the Sierpinski gasket or the

234

10 Fractal Versions of Curvatures

Sierpinski carpet with evident choices of G. (See also the references in [Win15] to former literature.) In Kombrink [Kom11] under some additional conditions on the generator such a formula is shown even for certain self-conformal sets, where symbolic dynamics and a related Renewal theorem of Lalley [Lal12] is used. S. Winter also derived a generator type expression for the (average) Minkowski content of a general self-similar set F with (OSC) and D < d for any strong feasible O satisfying the so-called projection condition (PC) (see [Win15, Theorem 3.17]): *D (F ) = 1 M η



∞ 0

Ld (Fε ∩ Γ ) 1 dε ωd−D εd−D ε

(10.22)

(= MD (F ) in the non-lattice case), where Γ := O \ SO. The conditions D < d and (PC) are also necessary for the validity of this formula. E. Pearse observed that the central open set studied in Bandt, Hung and Rao [BHR06] provides an example for such an O (cf. [Win15, Proposition 3.17]). However, its concrete construction may be complicated.

10.3.4 Relationships to Fractal Zeta Functions A different approach to the (average) Minkowski content has been developed by M. Lapidus and his collaborators within a very general theory of fractal zeta functions and complex dimensions, see Lapidus, Radunovi´c, Žubrini´c [LRŽ16, Chapter 2] and related references to the former literature therein. Applied to self-similar sets with (OSC) and dimension D under some additional conditions on the asymptotic behaviour of ε−(d−D) Ld (Fε ) as ε → 0 one obtains MD (F ) =

res(ζ˜F , D) ωd−D

(10.23)

*D (F )) in the lattice case. Here in the non-lattice case and the same formula for M ˜ res(ζF , D) denotes the residue of the tube zeta function ζ˜F (s) :=



δ

εs−d−1 Ld (Fε ) dε , s ∈ C , Re s > dimH F ,

0

of F at the pole s = D, where δ > 0 is a fixed number (cf. [LRŽ16, Theorems 2.3.18, 2.4.3].) Similar results are derived in terms of the distance zeta function  ζF (s) :=

d(x, F )s−D dx Fδ

10.4 Extension to Fractal Curvature Measures

235

(cf. [LRŽ16, Theorems 2.2.3, 2.3.37]). Moreover, under further conditions in [LRŽ16, Theorem 4.1.14] the relative Minkowski content MD (F, G) for open G ⊂ Rd is expressed by the residue of the associated relative distance zeta function at pole D. Note that again more general sets F are considered. The authors conjecture that the corresponding conditions are fulfilled for all self-similar sets with (OSC) in the non-lattice (resp. lattice) case.

10.4 Extension to Fractal Curvature Measures 10.4.1 Application of Renewal Theory We will present now extensions of the above results for the (average) Minkowski content of self-similar sets F as above and its localized versions to fractal curvatures and associated measures CkD as well as their densities in the sense of Definitions 10.6, 10.8 and 10.12, respectively. Here we additionally assume √ the regularity of the parallel sets for almost all distances less than R|F | for an R > 2 and some integrability properties of their curvature measures. In contrast to the previous section we start with the approach to the total fractal curvatures by means of renewal theory. Some ideas of the proof of Theorem 10.21 are used for the case of curvatures taking into regard Proposition 10.24 below. For completeness we include the former in the formulations as marginal cases k = d or k = d − 1. However, since the curvature measures Ck (Fε , ·) of the parallel sets are determined by the corresponding unit normal bundles, one has to be more careful here concerning intersections. For the geometric relationships only the following properties of the curvature measures Ck (Fε , ·) at regular ε are involved. They are motion invariant, scaling with exponent k and locally determined: Ck (g(Fε )), g(·)) = Ck (Fε , ·) for any Euclidean motion g , Ck (r(Fε ), r(·)) = r k Ck (Fε , ·) , r > 0 , Ck (Fε , G) = Ck ((Sw F )ε ), G) if G is open and Fε ∩ G = (Sw F )ε ∩ G , w ∈ W∗ , provided that ε is also regular for Sw F . (The last property is already specified to the smaller copies of F under the similarities Sw .) An important tool is the following nice behaviour of the curvature measures of parallel sets of sufficiently large distances. √ 2 and k ∈ Proposition 10.24 ([Zäh11, Theorem 4.1]) For any R > {0, 1, . . . , d} there exists a constant ck (R) such that for all compact K ⊂ Rd

236

10 Fractal Versions of Curvatures

and r ≥ R |K|,  reach (Kr )c ≥ |K| R 2 − 1 and Ckvar (Kr , Rd ) ≤ ck (R) . rk r≥R |K| sup

The proof of the first statement is based on Fu [Fu85]. For k = d, i.e. the case of Lebesgue measure, the second inequality applied to the smaller copies of K := F with respect to the former Markov stopping corresponds to the estimate (10.19). It plays a similar role for general k: Recall that for F as above we have W (τr/R ) = {w ∈ W∗ : rw1 . . . rw|w| R |F | ≤ r < rw1 . . . rw|w|−1 R |F |} and set for a feasible open set O, 9 : WO,R (r) := w ∈ W (τr/R ) : (Sw F )r ∩ (SO)r = ∅ . Proposition 10.24 implies for w ∈ W (τr/R ), Ckvar ((Sw F )r ) = rwk Ckvar (Fr/rw ) ≤ rwk ck (R)(r k /rwk ) = c r k . This is used in the proof of the following extension of Theorem 10.21 to (average) fractal curvatures. Theorem 10.25 Let k ∈ {0, 1, . . . , d} and F ⊂ Rd be a self-similar set with Minkowski dimension D and contraction ratios r1 , . . . , rN of the corresponding similarities satisfying (OSC). If k ≤ d − 2 suppose additionally the following two conditions: √ (i) If d ≥ 4, then almost all r ∈ (0, 2|F |) are regular for F . For k ≤ 3 this is always fulfilled. √ (ii) There exist a strong feasible open set O and constants c, R > 2 such that for almost all r ∈ (0, R |F |) and all w ∈ WO,R (r), ⎛



Ckvar ⎝Fr , ∂(Sw F )r ∩ ∂

⎞ (Sv F )r ⎠ ≤ c r k .

v∈WO,R (r)\{w}

Set Rk (ε) := 1(0,R|F |](ε)Ck (Fε ) −

N  i=1

1(0,ri R|F |] Ck ((Si F )ε ) .

10.4 Extension to Fractal Curvature Measures

237

Then the average fractal curvatures exist and are given by 1 δ→0 | ln δ|

D



Ck (F ) = lim where η = given by

N

D i=1 ri | ln ri |,

1 δ

1 1 εD−k Ck (Fε ) dε = ε η



R|F | 0

1 r D−k Rk (r) dr , r

and in the non-lattice case the fractal curvatures are D

CD k (F ) = ess lim Ck (Fε ) = Ck (F ) . ε→0

Remark 10.26 In Winter [Win11, Corollary 4.9] it is shown that Condition (ii) is independent of the choice of O. The corresponding measure version as a refinement is a complement of Theorem 10.22. Because of the self-similarity all fractal curvature measures F are constant multiples of the normalized Hausdorff measure μ on F : Theorem 10.27 Under the conditions of Theorem 10.25 we get D C k (F, ·)

1 = lim δ→0 | ln δ|



1 δ

1 D εD−k Ck (Fε , ·) dε = Ck (F ) μ ε

and in the non-lattice case CkD (F, ·) = ess lim Ck (Fε , ·) = CD k (F ) μ ε→0

in the sense of weak convergence. The whole approach was worked out in Winter [Win08] for the special case, when the parallel sets Fε are polyconvex for one and thus all ε > 0. In this case it can be shown that all conditions of the theorems are fulfilled. Moreover, for some classical examples the total values are computed. Theorem 10.25 is the special deterministic version of [Zäh11, Theorem 2.3.8, Corollary 2.3.9], where such curvatures for random self-similar sets are considered. Theorem 10.27 is derived in [WZ13, Theorem 2.3]. As in the case of the Minkowski measure the proof essentially relies on Theorem 10.25. The techniques for applying the Renewal theorem in the proofs of Theorems 10.25 and 10.27 are similar to the Minkowski case, but much more extensive, since signed measures and geometric boundary properties are involved. As in the case of the Minkowski content (cf. Theorem 10.23), for compatible tilings (the boundary of their generator G belongs to F ) the assumptions and the formulas for the (average) fractal curvatures can be simplified: Theorem 10.28 ([Win15, Theorem 4.5]) Let F be a self-similar set satisfying (OSC) and let O be a strong feasible open set for F such that the associated tiling T (O) with generator G = O \SO is compatible. Let k ∈ {0, . . . , d −1}. If k ≤ d −2

238

10 Fractal Versions of Curvatures

assume additionally the following two conditions: √ (i) For d ≥ 4 almost all ε < 2|F | are regular values for F . (For d ≤ 3 this condition is always fulfilled.) √ (ii’) There are constants c, γ > 0 and R > 2 such that for almost all 0 < ε < R|F |, Ckvar (Fε , ((SO)c )ε ) ≤ cεk−D+γ . Then the average fractal curvatures of F exist and are given by D Ck (F )

1 = η



ρ 0

εD−k Ck (G−ε ) dε ,

where ρ is the inradius of G. In the non-lattice case the fractal curvatures CD k (F ) D

exist and coincide with Ck (F ). Remark 10.29 Condition (i) agrees with that from Theorem 10.25. Furthermore, from the arguments in the proof of this theorem it follows that the integrability condition (ii) from there implies Condition (ii’). A measure version of Theorem 10.28 can also be obtained. Under slightly stronger assumptions such a result was proved in Kombrink [Kom11, Theorem 2.37] with methods from Perron-Frobenius theory and renewal theory for shift dynamical system in the sense of Lalley [Lal12]. This approach is more complicated, however it was used in [Kom11] for extensions to certain self-conformal fractals with compatible tilings. For self-similar tilings without the compatibility condition in Winter [Win15, Theorem 4.9] an analogous formula is proved by choosing any strong feasible open set O satisfying the projection condition (PC) and assuming additionally that Ckvar (Fε , ∂O) = 0 for almost all ε < ρ. If for k ≤ d − 2 the conditions (i) and (ii’) from the last theorem are fulfilled, then the average fractal curvatures (and in the non-lattice case the fractal curvatures) for k ≤ d − 1 exist and can be expressed in terms of the generator G = O \ SO: D

Ck (F ) =

1 η



ρ 0

1 εD−k Ck (Fε , G) dε . ε

(10.24)

This is the complement to formula (10.22) for the Minkowski content. The representation formulas for the fractal curvatures from Theorems 10.25 and 10.28 as well as 10.24 admit to compute numerical values for some classical cases. Example 10.30 In Winter [Win08] the following values have been obtained. D

D

D

C0 (F ) = −0, 016, C1 (F ) = 0, 0725, C2 (F ) = 1, 352

10.4 Extension to Fractal Curvature Measures

239

for the Sierpinski carpet (Fig. 10.2) and D

D

D

C0 (F ) = −0, 014, C1 (F ) = 0, 0720, C2 (F ) = 1, 344 for the modified Sierpinski carpet (Fig. 10.3). This shows that self-similar sets with equal Hausdorff dimension and different geometric features can be distinguished by means of these parameters. In particular, the corresponding fractal Euler-type D number C0 (F ) reflect the topological property that the Sierpinski carpet has more holes than the modified version.

10.4.2 Methods from Ergodic Theory: Curvature Densities and Measures In the previous sections we have seen that in the self-similar case the (average) Minkowski measure and fractal curvature measures can be obtained from the classical Renewal theorem. Next we will show how Birkhoff’s ergodic theorem (see, e.g., Walters [Wal82]) for the embedded shift dynamical system can be used in order to prove that all these measures possess a.e. constant densities with respect to the normalized Hausdorff measure. Moreover, measure convergences can also be derived from this approach. First recall the notion of a localizing family of bump functions (cf. Definition 10.12) specified to a self-similar set F with a strong feasible open set O and μF := HD (F ∩ (·)):  AF (x, ε)(z) := εD  F

g  g

|x−z| aε

|y−z| aε





HD (dy)

, z ∈ Rd ,

x ∈ F , ε ∈ (0ε0) for some ε0 > 0, a > 1 and any nonnegative bounded measurable function g with support in [−1, 1] which is bounded from below by a positive constant in a neighbourhood of 0. Such a family fulfills the following three conditions: Denote b := max(2a, ε0−1 |O|). spt AF (x, ε) ⊂ Fε ∩ B(x, aε) ;

(10.25)

AF (x, ε)(z) = AF (Sj−1 x, rj−1 ε)(Sj−1 z) ,

(10.26)

if x ∈ Sj F , ε < b −1 d(x, ∂Sj O) and 1 ≤ j ≤ N (Covariance); AF (x, ε)(z) is measurable in (x, ε, z) ∈ F × (0, ε0 ) × Rd .

(10.27)

240

10 Fractal Versions of Curvatures

The covariance property follows from the strong open set condition and the selfsimilarity of the measure μ. (Note that in 10.26 the condition on ε implies rj−1 ε < ε0 because of the choice of b.) Now we can formulate the main result of this section concerning fractal curvature densities of self-similar sets including the Minkowski densities as marginal case. This was proved in [RZ12] for the special choice of the localizing functions from the end of Sect. 10.1. The general case is completely analogous. In this approach we identify a.a. points of a self-similar set satisfying (SOSC) with their coding sequences using the corresponding notations from Sect. 10.2. Theorem 10.31 Let k ∈ {0, 1, . . . , d} and suppose that the self-similar set F in Rd with contraction ratios r1 , . . . , rN and Hausdorff dimension D satisfies the strong open set condition w.r.t. O. If d ≥ 4 and k ≤ d −2, we additionally suppose that a.a. ε < ε0 are regular for F in the sense of Definition 10.3. Let {A(x, ε) : x ∈ F, ε < ε0 } be a localizing family

of bump functions with constants a > 1, ε0 > 0, and b = max 2a, ε0−1 |O|) as above. Then for HD -a.a. x ∈ F the curvature density 1 Dk (F, x) := lim δ→0 | ln δ|



b−1 d(x,∂O)



ε−k Ck Fε , AF (x, ε) ε−1 dε

(10.28)

δ

exists and equals the constant Dk (F ) := HD (F )−1

1 η

 

b−1 d(y,∂O)

F

b−1 d(y,∂Sy1 O)



ε−k Ck Fε , AF (y, ε) ε−1 dε HD (dy) , (10.29)

 D where η = N j =1 rj | ln rj |, provided the last double integral converges absolutely if k ≤ d − 2, and for k ∈ {d − 1, d} this is always true. Proof As an essential auxiliary tool we use the ergodic shift dynamical system [W, ν, θ ] on the code space W for the shift operator θ : W → W with θ (w1 w2 . . . ) := (w2 w3 . . .). According to (10.7) it induces the ergodic dynamical system [F, μ, T ], where the transformation T : F → F is defined for μ-a.a. x by T x : Sj−1 x if x ∈ Sj F , j = 1, . . . , N , taking into regard that μ(Si F ∩ Sj F ) = 0, i = j . (More general references on this subject may be found, e.g., in Falconer [Fal97], Mauldin and Urbanski [MU03].) In the above identification of a.a. points with their coding sequences we have for such x, T x = θ (x1 x2 . . .) . For brevity we write x|i := x1 . . . xi for the corresponding concatenation. Next −1 note that ε < b−1 d(x, ∂Sx|i O) implies ε < rx|i ε < ε0 , since d(x, ∂Sx|i O) =

10.4 Extension to Fractal Curvature Measures

241

rx|i d(T i x, ∂O). From this and AF (x, ε) ⊂ B(x, aε) we obtain for Lebesgue-a.a. ε, μ-a.a. x, and i ∈ N satisfying the first condition the equalities



Ck Fε , AF (x, ε) = Ck (Sx|i F )ε , AF (x, ε) −1 −1

−1

k = Ck (Sx|i F )ε , AF (T i x, rx|i ε) ◦ Sx|i Ck Fr −1 ε , AF (T i x, rx|i ε) . = rx|i x|i

Here we have used the locality of the curvature measure Ck , the covariance property (10.26) of the functions AF (x, ε), and the scaling behaviour of Ck under similarities.

In Proposition 10.11 it is shown that Ck Fε , AF (x, ε) (with value 0 if ε is not regular) is a measurable function in (x, ε) ∈ F × (0, ε0 ). Now we can verify the limit 1 lim δ→0 | ln δ|

b−1 d(x,∂O) 



ε−k Ck Fε , AF (x, ε) ε−1 dε

δ

n(x, δ) 1 = lim δ→0 | ln δ| n(x, δ)

 n(x,δ)−1  

b−1 d(x,∂Sx|i O) b−1 d(x,∂S

i=0



x|(i+1) O)

b−1 d(x,∂Sx|n(x,δ) O)

+



ε−k Ck Fε , AF (x, ε) ε−1 dε 

ε−k Ck Fε , AF (x, ε) ε−1 dε ,

δ

where n(x, δ) := max{n ∈ N : b−1 d(x, ∂Sx|n O) ≥ δ} . By the above relation the integrand in the ith integral may be replaced by −1 −1 k rx|i ε−k Ck Fr −1 ε , AF (T i x, rx|i ε) ε . x|i

For the integral bounds we use d(x, ∂Sx|i O) = rx|i d(T i x, ∂O), d(x, ∂Sx|(i+1)O) = rx|i d(T i x, ∂S(T i x)1 O) . −1 Substituting then in the i-th integral rx|i ε by ε, we obtain the expression



b−1 d(T i x,∂O) b−1 d(T i x,∂S(T i x) O) 1



ε−k Ck Fε , AF (T i x, ε) ε−1 dε .

242

10 Fractal Versions of Curvatures

Therefore it suffices to show that for μ-a.a. x ∈ F the following integrals and limit relationships exist: 1 n→∞ n n

lim

i=1

1 n→∞ n

b−1 d(T i x,∂O) b−1 d(T i x,∂S(T i x) O)

b−1 d(y,∂O)

F

b−1 d(y,∂Sy1 O)





ε−k Ck Fε , AF (T i x, ε) ε−1 dε

1

  =

lim





ε−k Ck Fε , AF (y, ε) ε−1 dε μ(dy) ,

b−1 d(T n x,∂O) b−1 d(T n x,∂S(T n x)1 O)

(10.30)



 ε−k Ck Fε ), AF (T n x, ε)  ε−1 dε = 0 ,

(10.31)

(note that under the above conditions b−1 d(T n x, ∂S(T n x)1 O) < δ) for all n, and  | ln δ| = rjD | ln rj | . δ→0 n(x, δ) N

(10.32)

lim

j =1

Under the integrability assumption of our theorem, (10.30) follows from Birkhoff’s ergodic theorem applied to the ergodic dynamical system [F, μ, T ]. Here the curvature measures may   also be replaced by their absolute values. Taking into regard that an = ni=1 ai − n−1 i=1 ai for any real sequence, (10.31) is a consequence. In order to use these arguments for (10.32) note that for δ(x, n) := b−1 d(x, ∂Sx|n O) we get lim

δ→0

| ln δ| | ln δ(x, n)| = lim n(x, δ) n→∞ n

provided the last limit exists. Since δ(x, n) = rx|n b−1 d(T n x, ∂O) =

n 

rxi b−1 d(T n x, ∂O)

i=1

and xi = (T i x)1 , i ∈ N, Birkhoff’s ergodic theorem implies for μ-a.a. x ∈ F ,  1 1 lim | ln rxi | = lim | ln r(T i x)1 | = n→∞ n n→∞ n n

i=1

n

i=1

 F

| ln ry1 | μ(dy) =

as well as 1 | ln d(T n x, ∂O)| = 0 , n→∞ n lim

N  j =1

| ln rj | rjD ,

10.4 Extension to Fractal Curvature Measures

243

 since | ln d(y, ∂O)| μ(dy) < ∞ (cf. (10.13)). This shows (10.32). It remains to consider the cases k ∈ {d − 1, d}. Here we have Ck (Fε , B(x, aε)) ≤ ck εk (see the end of the proof of Theorem 10.13). Since the last integral for the logarithmic distance is finite, the above integrability condition is fulfilled and hence, the preceding arguments for convergence remain valid.   Recall that Theorem 10.13 establishes a general relationship between curvature densities and measures. This can be used in order to conclude from Theorem 10.31 that under the stronger uniform integrability condition (10.4) the limit values Dk (F ) are always the densities of the average fractal curvature measures with respect to Hausdorff measure HD on the self-similar set F with (OSC). In particular, they do not depend on the choice of the localizing functions AF satisfying (10.4). More precisely, we infer the following. Corollary 10.32 If the integrability condition in Theorem 10.31 is replaced by the uniform integrability of the mappings 1 | ln δ|

x →



ε0 δ

Ckvar (Fε , AF (x, ε)) 1 dε , εk ε

0 < δ < ε0 , with respect to HD (F ∩ (·)) for some family of localizing functions AF as before, then the k-th average fractal curvature measure in the sense of Definition 10.8 exists and is equal to D

C k (F, ·) = Dk (F )HD (F ∩ (·)) . Remark 10.33 (i) The whole approach was extended in [BZ13] to average curvature-direction measures of self-similar sets. Moreover, in [BZ13, Proposition 3.12] it is shown that the uniform integrability condition (10.4) is also necessary for the corresponding convergence, if the curvature measures Ck (Fε , ·) are replaced by the two nonnegative measures of their Hahn decomposition, which also possess almost everywhere constant fractal densities. (ii) Sufficient conditions for the uniform integrability are  F

1 sup | ln δ| δ