Lagerungen: Arrangements in the Plane, on the Sphere, and in Space 3031217993, 9783031217999

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Grundlehren der mathematischen Wissenschaften  360 A Series of Comprehensive Studies in Mathematics

László Fejes Tóth Gábor Fejes Tóth Włodzimierz Kuperberg

Lagerungen Arrangements in the Plane, on the Sphere, and in Space

Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics Volume 360 Editors-in-Chief Alain Chenciner, Observatoire de Paris, Paris, France S.R.S. Varadhan, New York University, New York, NY, USA Series Editors Henri Darmon, McGill University, Montréal, Canada Pierre de la Harpe, University of Geneva, Geneva, Switzerland Frank den Hollander, Leiden University, Leiden, The Netherlands Nigel J. Hitchin, University of Oxford, Oxford, UK Nalini Joshi, University of Sydney, Sydney, Australia Antti Kupiainen, University of Helsinki, Helsinki, Finland Gilles Lebeau, Côte d’Azur University, Nice, France Jean-François Le Gall, Paris-Saclay University, Orsay, France Fang-Hua Lin, New York University, New York, NY, USA Shigefumi Mori, Kyoto University, Kyoto, Japan Bào Châu Ngô, University of Chicago, Chicago, IL, USA Denis Serre, École Normale Supérieure de Lyon, Lyon, France Michel Waldschmidt, Sorbonne University, Paris, France

Grundlehren der mathematischen Wissenschaften (subtitled Comprehensive Studies in Mathematics), Springer’s first series in higher mathematics, was founded by Richard Courant in 1920. It was conceived as a series of modern textbooks. A number of significant changes appear after World War II. Outwardly, the change was in language: whereas most of the first 100 volumes were published in German, the following volumes are almost all in English. A more important change concerns the contents of the books. The original objective of the Grundlehren had been to lead readers to the principal results and to recent research questions in a single relatively elementary and accessible book. Good examples are van der Waerden’s 2-volume Introduction to Algebra or the two famous volumes of Courant and Hilbert on Methods of Mathematical Physics. Today, it is seldom possible to start at the basics and, in one volume or even two, reach the frontiers of current research. Thus many later volumes are both more specialized and more advanced. Nevertheless, most volumes of the series are meant to be textbooks of a kind, with occasional reference works or pure research monographs. Each book should lead up to current research, without overemphasizing the author’s own interests. Proofs of the major statements should be enunciated, however the presentation should remain expository. Examples of books that fit this description are Maclane’s Homology, Siegel & Moser on Celestial Mechanics, Gilbarg & Trudinger on Elliptic PDE of Second Order, Dafermos’s Hyperbolic Conservation Laws in Continuum Physics ... Longevity is an important criterion: a GL volume should continue to have an impact over many years. Topics should be of current mathematical relevance, and not too narrow. The tastes of the editors play a pivotal role in the selection of topics. Authors are encouraged to follow their individual style, but keep the interests of the reader in mind when presenting their subject. The inclusion of exercises and historical background is encouraged. The GL series does not strive for systematic coverage of all of mathematics. There are both overlaps between books and gaps. However, a systematic effort is made to cover important areas of current interest in a GL volume when they become ripe for GL-type treatment. As far as the development of mathematics permits, the direction of GL remains true to the original spirit of Courant. Many of the oldest volumes are popular to this day and some have not been superseded. One should perhaps never advertise a contemporary book as a classic but many recent volumes and many forthcoming volumes will surely earn this attribute through their use by generations of mathematicians.

László Fejes Tóth • Gábor Fejes Tóth Włodzimierz Kuperberg

Lagerungen Arrangements in the Plane, on the Sphere, and in Space

László Fejes Tóth Alfréd Rényi Institute of Mathematics Budapest, Hungary

Gábor Fejes Tóth Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Budapest, Hungary

Włodzimierz Kuperberg Department of Mathematics Auburn University Auburn, AL, USA

Translated from the German language edition “Lagerungen in der Ebene auf der Kugel und im Raum” by László Fejes Tóth, volume 65 of Grundlehren der mathematischen Wissenschaften, © Springer-Verlag Berlin Heidelberg 1953, 1972 ISSN 2196-9701 (electronic) ISSN 0072-7830 Grundlehren der mathematischen Wissenschaften ISBN 978-3-031-21799-9 ISBN 978-3-031-21800-2 (eBook) https://doi.org/10.1007/978-3-031-21800-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 1953, 1972, 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

The Lagerungen belongs alongside other great twentieth-century classics in geometry such as Weyl’s Symmetry, Coxeter’s Regular Polytopes, Hilbert and CohnVossen’s Geometry and the Imagination, and Thurston’s Geometry and Topology of three-manifolds. Starting in my teenage years, I made a bedtime habit of meditating on visual math problems with closed eyes as I fell asleep. One of my favorite such problems from my youth is the Tarski Plank Problem (generalized in Chapter 15): cover a circular disk in the plane with strips of paper (each strip being the closed region between two parallel lines) in such a way that the total width of the strips is minimized. The obvious guess is to lay the strips in parallel, giving a total width equal to the diameter of the disk. The proof of optimality of this guess is not obvious without a hint that spoils the problem: first consider the simpler problem of covering a sphere instead of a disk with strips (each strip being the closed region between two vertical parallel planes) so as to minimize total width. As Archimedes observed in a stroke of genius, the area of the sphere covered by a strip is proportional to its width. Thus to minimize total width is to minimize total area by avoiding any overlap of strips, placing them in parallel. By projection of the sphere to the disk in the plane, Tarski’s original problem in the plane reduces to the spherical version. This book is more than a bedside companion to me. I discovered László Fejes Tóth through the German edition more than three decades ago. It is no exaggeration to say that I have spent close to two of those decades investigating (and occasionally solving) conjectures from this book. His direct influences on me include the Kepler conjecture on sphere packing in three dimensions, the honeycomb conjecture on the most efficient partition of the plane into cells of equal area, the dodecahedral conjecture, and the Reinhardt conjecture on packings of centrally symmetric disks in the plane. These conjectures display symmetries in the answer that are not present in the question. In the nineteenth century, Pierre Curie formulated a series of informal maxims to explain the abundance of symmetry in nature. Fejes Tóth elaborated one of Curie’s maxims as follows: it is often an extremum postulate that underlies

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symmetry in physical structure. The optimal policy tends to make an orderly selection from the chaos of possibility. The regular polygon is an illustration of this maxim. The isoperimetric inequality for convex planar polygons is one of the great theorems of antiquity. In basic form, it asserts that a convex 𝑛-gon of given area minimizes perimeter when it is regular. The book gives two proofs, and both are “proofs from the book” – Erdős’s name for proofs of utmost perfection. The proofs are a simple joy. The better known proof starts from the existence of a minimizer and shows how first-order optimality conditions can be seen in the isoperimetric properties of isosceles triangles. These conditions are intuitive and avoid equations, yet imply regularity of the polygon. The other proof shrinks an arbitrary convex 𝑛-gon by inward parallel translation of its edges, showing that the isoperimetric inequality holds by a downward induction argument as it contracts and loses edges. The general planar isoperimetric inequality reduces to convex polygons by taking a polygonal approximation of the convex hull of an arbitrary simple closed curve. In three dimensions also, isoperimetric inequalities for polyhedra are a major theme of the book. Many of these results originate with Fejes Tóth. The honeycomb also illustrates this maxim. In the two-dimensional Euclidean plane, the regular hexagonal honeycomb is the solution of many optimization problems: the honeycomb gives the densest packing of nonoverlapping congruent circular disks, the thinnest covering of the plane by overlapping circular disks, and the least perimeter partition into cells of equal area. Axel Thue solved the sphere packing problem in two dimensions in the nineteenth century, but Rogers found the proof “too brief for a complete reconstruction.” A proof published by Fejes Tóth in 1940 is perhaps the first completely satisfactory account in two dimensions. This book presents a few proofs. One is an easy observation based on the Delaunay triangulation, showing that the minimum area of a Delaunay triangle in a saturated packing is equal to that of the equilateral Delaunay triangles in the honeycomb packing. Another proof follows from a much more general result that the best packing of any centrally symmetric convex disk is obtained by tiling the plane with the smallest circumscribing centrally symmetric hexagon. In three dimensional Euclidean space, the book considers these same three questions: what is the densest packing of nonoverlapping congruent balls (the Kepler problem), what is the thinnest covering by overlapping congruent balls, and what is the surface-area minimizing partition into cells (the Kelvin problem). Among lattices, the densest packing is given by the face-centered cubic packing (the familiar pyramid arrangement) but the covering problem and partition problem are solved by its dual lattice, the body-centered cubic (the arrangement extending the cubic lattice with additional points at the centers of cubes). Here again we see symmetry in optimal solutions, although the leading role of hexagonal honeycomb is not taken over by one and the same lattice in three dimensions. Without venturing a conjecture, the book asks whether other lattice optimization problems solved by the hexagonal honeycomb in two dimensions are in general solved by one of these two lattices in three dimensions.

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Fejes Tóth regarded these problems (the densest sphere packing, the thinnest covering by balls, and most efficient partition into cells) as great unsolved problems in discrete geometry. Of the three, he regarded the sphere packing problem as the least challenging, in the sense that his book presented a concrete strategy for a solution to this problem, but not for the others. His strategy starts with the partition of space into Voronoi cells, each sphere being contained in a separate cell. If the cells all have large volume, then the density of the sphere packing is low; and conversely, small cells mean high density. Lower bounds on volumes of cells give upper bounds on densities of packings. He observed experimentally that if a single cell in a sphere packing has smaller volume than that of the face-centered cubic, then its neighboring cells tend to have greater volume. On the basis of this observation, he conjectured that the weighted average of the volumes of a cell and its nearest neighboring cells is always at least the volume of the cell in the face-centered cubic packing. This conjecture is an optimization in a finite number of variables whose affirmative solution implies the Kepler conjecture. Based on these ideas, he saw that the solution was not “hopeless by any means. We have certainly presented a specific, in principle fully implementable program, by which we have come one step closer towards the solution” to the Kepler conjecture. As he was the first to remark, in theory, his optimization problem is an inequality that might be solved by a sufficiently powerful computer. In fact, his inequality belongs to the class of statements in Tarski’s first-order theory of the real numbers, for which general decision procedures are known. Fejes Tóth’s strategy of a geometric reduction to a finite number of variables followed by a computer-assisted proof inspired my work decades later. Fejes Tóth confessed that he did not have the slightest idea how to solve the covering problem in three dimensions. Neither do I. But quoting Tennyson’s poem Ulysses, he did not abandon hope, for the human heart is “strong in will to strive, to seek, to find, and not to yield.” To continue with Tennyson’s words, “some work of noble note may yet be done, not unbecoming men that strove with Gods.” G. Fejes Tóth and W. Kuperberg have prepared an admirable English edition, which includes detailed notes to the original as well as eight self-contained new chapters. The second German edition was published in 1972. Since then much has happened, including the recent Fields medal to M. Viazovska for the solution to the sphere-packing problem in eight dimensions. In collaboration with others, she also solved the problem in twenty-four dimensions. Furthermore, they proved a universal optimality result in these two dimensions, meaning that a single lattice is the minimizer for a large assortment of different energy potentials. Surprisingly, the universal optimality of the honeycomb in two dimensions is still merely a conjecture. A survey of results on packings and other topics in discrete geometry over the past few decades appears in the new material. Fejes Tóth has fathered an entire school of discrete geometry. The pages of this book contain many open conjectures, inviting new research and a larger following. Clebsch defined a geometer to be one who finds “joy in shape in a higher sense.” Fejes Tóth once expressed the wish “to awaken this noble joy in the reader, showing that we are all, in the sense of Clebsch, geometers.” Pittsburgh, November 2022

Thomas Hales

Preface to the English Edition

1953, the year the first edition of Lagerungen appeared in print, marks the birth of Discrete Geometry, a branch of classical geometry that deals with packing, covering, as well as an array of other arrangements, for their own sake, and not just as a tool for number theory. The contributions of László Fejes Tóth to this new, exciting, and dynamically developing field are numerous and of fundamental importance. Lagerungen is a crowning summary of his most productive period in life. The book includes many open problems and conjectures, often accompanied by the Author’s ideas of possible ways of approaching them. A good number of solutions appeared between the 1st and 2nd Editions, surveyed by the Author in his Notes to the 2nd Edition. More progress followed steadily during the next five decades. Undeniably, the book had a profound influence on the development of the theory, continuing today with undiminished vigor. We translated the original text of the 1st Edition (which was unchanged in the 2nd Edition) as faithfully as we could. We corrected some typos and errors, and hope we did not generate new ones. We also translated the Notes to the second edition and started to update them. However, after a while it became apparent that the vast amount of new results goes far beyond the frame of the Notes, so we had to reorganize the material. In result, the present work consists of the following parts: a) the translation of the Book, b) the Notes that contain only comments to the text of the Book, and c) eight self-contained chapters surveying material that is related to the subject of the Book, but is not mentioned in it. We have preserved the bibliography related to the 1st Edition. It now appears at the end of the book, followed by the bibliographies of the Notes and the further new chapters. Preparing the Notes we were particularly concerned with solutions, or partial solutions, to the problems stated in the Book, most of which were raised by the Author. The system of triangular arrows (▶ and ◀) indicating connections between the problems in the text and comments related to them in the Notes is continued here in the same way as introduced in the 2nd Edition.

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The isoperimetric problem, treated in Section 1.4 of the Book, asks for the maximum area of a region of given perimeter. It is natural to consider other measurements of a region, such as the diameter and width, and ask for the extreme value of one when another is fixed. The solution of these problems is known if the competing regions are general convex disks, however several of these problems are still open if the competing regions are polygons with at most a given number of sides. Chapter 9 surveys these problems. Problems in higher dimensions are not treated in the Book. The book “Packing and covering” by Rogers, published in 1964, is a counterpart to the present work devoted to problems in 𝑛-dimensional space with 𝑛 large. Early development in the study of packing and covering in high dimension was motivated by the geometry of numbers. Subsequent results, such as the discovery of the Leech lattice and the linear programming bound, which culminated in the recent solution of the sphere packing problem in dimensions 8 and 24, were influenced by coding theory. Chapter 10 gives a brief account of these results. In hyperbolic space density cannot be defined by a limit as it is in Euclidean space. In Chapter 11 we describe the local density bounds for sphere packings and we discuss the different attempts to define optimal arrangements in hyperbolic space. Chapter 12 deals with multiple arrangements. The study of multiple packings and coverings started after the publication of Lagerungen and was restricted mostly to lattice arrangements on the plane or of general arrangements of balls. We emphasize two areas which were intensively investigated recently: decomposition of multiple coverings into simple coverings and characterization of multiple tilings. Two members of a packing are neighbors if they have a common boundary point. A multitude of problems arises in connection with neighbors in a packing. The oldest one concerns a controversy between Newton and Gregory about the maximum number of neighbors a member can have in a packing of congruent balls. Other problems ask for the average number of neighbors or the maximum number of mutually neighboring members in a packing. Chapter 13 gives a survey of these problems. Chapter 14 deals with different variants of the following problem: Given a convex set 𝐾 and a sequence {𝐶𝑖 } of convex bodies in 𝐸 𝑛 , is it possible to pack the sequence of bodies in 𝐾 or cover 𝐾 with the bodies? Algorithmic versions of these problems are on-line packing and on-line covering: The bodies of the sequence are given one at a time and the algorithm is to decide on the placement of the arriving body before the next body is revealed; once placed, the body cannot be moved. In Chapter 15 we discuss four classic problems: Borsuk’s partition problem, Tarski’s plank problem, the Kneser–Poulsen problem on the on the monotonicity of the union of balls under a contraction of their centers, and the Hadwiger–Levi problem on covering convex bodies by their smaller positively homothetic copies. Finally, in Chapter 16 we gathered a set of special problems which occur in the Notes to the second edition of the Book, but as they are not directly related to the text of the Book, we treat them here. We augmented this set with some new problems raised after the publication of the second edition.

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It is our sincere hope that this English translation of the original text with thoroughly updated Notes and surveying related new results, will facilitate further research. Also, we hope that the new edition will suit the needs of as broad a range of readers as the previous editions did. It is our pleasant duty to express our gratitude to the institutions and persons who supported and helped our work. Our collaboration on the translation of Lagerungen started with three longer visits of Włodek in the Rényi Institute made possible by the Marie Curie Actions grant “Discrete and Convex Geometry”, Project No: 14333, of the European Commission. Also, we have spent six fruitful weeks enjoying the hospitality of the Mathematisches Forschungsinstitut Oberwolfach in the program “Research in Pairs”. We owe special thanks to Ferec Fodor and Endre Makai who read a preliminary version of the manuscript. The text improved considerably by their thoughtful suggestions. Günter Rote called our attention to one of the rare incorrect statements in Lagerungen, and we appreciate a few other valuable comments received from Frank Morgan, Alexandr Polyanskii, Konrad Swanepoel and the anonymous referees. For supplying Figure 13.7, we are indebted to Sándor Bozóki. Special thanks are due also to Tom Hales who besides writing a nice foreword to the book, called our attention to some errors in the manuscript. We would like to close with a few personal words. László Fejes Tóth dedicated his book to his wife, Gábor’s mother, with whom he lived for sixty years in happy marriage. We also owe special thanks to our wives, Anna and Krystyna, for the inspiration and support and we dedicate our work to them. Finally, Gábor has to thank his family doctor Dr. Anna Magyar and two specialists, the cardiologist Dr. Sándor Kancz and the heart surgeon Dr. Boglárka Juhász who granted him some additional years to live and made him able to finish this work. Budapest, November 2022

Gábor Fejes Tóth and Włodzimierz Kuperberg

Preface to the Second Edition

Since the publication of the first edition of this book, the theory of packing and covering has been enriched with numerous results. Many of them are mentioned in the “Notes” (Chapter 8). In these notes, we have principally limited ourselves— according to the spirit of the book expressed already in its title—to the most visually accessible spaces of elementary geometry. Regarding the theory in 𝑛-dimensional space, we refer to C.A. Rogers’ foundational monograph Packing and covering (Cambridge Tracts in Mathematics and Mathematical Physics, No. 54 Cambridge University Press, New York (1964)). Several problems presented in the book were still unsolved at the time of its appearance, yet in the meantime—partly due to the influence of the book itself— have been solved. These problems are marked in the margin with a black triangle (▶). The accompanying numbers point to the corresponding pages in the Notes. I owe sincere thanks to my friend Professor A. Florian (Salzburg) for his instrumental help with the preparation of the second edition. Budapest, March 1971

L. Fejes Tóth

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Preface to the First Edition

A discrete set of points is said to be regular if any point of the set can be sent to any other by a rigid motion sending the whole set to itself, that is, in short, if no point of the set is distinguished from any other point. Such configurations of points are related to other regularly shaped objects such as polygons, polyhedra or space tilings. Regular arrangements of points or figures have perpetually fascinated the human imagination and particularly captured the interest of mathematicians. Among the countless names that could be listed here, we mention only Plato, Archimedes, Kepler, Bravais, and Schläfli. In three-dimensional space, a full understanding of regular arrangements of points was gained with the help of group-theoretical methods, providing a reasonable explanation for the shapes of crystals that occur in nature. The crowning achievement of the research in this direction was the discovery of the 230 classes of crystals by Fedorov (1885), Schoenflies (1891) and Barlow (1894). Subsequently, attention turned to certain extremum problems concerning regular arrangements of points, aiming to explain certain physical and chemical properties of crystals. One such problem is that of the densest regular packing of balls. Let us think of the molecules of a certain material as balls of equal size that can touch each other but cannot overlap. We search for the arrangement of molecules that contains the greatest number of molecules per unit volume. An enormous boost to the investigation of such extremum problems was given by Minkowski. He recognized the connection between certain questions of number theory and problems on lattice arrangements of figures, thereby founding the Geometry of Numbers, a branch of mathematics still vigorously cultivated. Both physical chemistry and the geometry of numbers deal mainly with extremum problems in which the permitted arrangements are a priori subject to certain regularity conditions. In contrast, the present work is dedicated to problems in which arbitrary irregular arrangements are brought into consideration. Here the regular shape of the extremal configuration is often a consequence of the extremum requirement.

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Let us mention two typical problems. 1. What is the arrangement in which the greatest number of equal-sized coins can be placed on a “large” table? The answer is that every coin should touch six others. Therefore the best arrangement forms a lattice by itself. 2. Consider twelve points in a ball. In which arrangement will the convex hull of the points have the greatest volume? This problem leads us to the regular icosahedron. The first problem, that is, the problem of the densest packing of circles in the plane, was solved by the great Norwegian number theorist A. Thue in his youth (1892). Then came a longer pause in the development in this direction, hence most of the results treated here are the product of the last 10 to 12 years. Problems of this kind have not yet been treated in any textbook. No advanced knowledge is needed to understand the problems phrased here. These are simple, natural and intuitive questions, which, however, by the typical difficulties hiding in them, often happen to be very serious problems. But in most cases even the solutions do not require any “advanced” knowledge, so that almost the whole book can be understood by the average reader. Yet this range of relatively elementary questions is full of unsolved problems. One of the main goals of this book is to direct the reader’s attention to such things, thereby engaging more researchers in this attractive area of geometry. Special thanks are due to Professors H. Hadwiger, G. Hajós and B.L. van der Waerden, who have read the manuscript and helped me with an array of valuable comments. For most of the numerical computations and several remarks I am indebted to J. Molnár, my collaborator. For help with proofreading I owe thanks to M. Kneser. Veszprém, March 1953

L. Fejes Tóth

László Fejes Tóth (1905–2005)

Contents

Part I Lagerungen – Arrangements in the Plane, on the Sphere, and in Space 1

Some Theorems from Elementary Geometry . . . . . . . . . . . . . . . . . . . . . . 1.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Affinity and Polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Extremum Properties of the Regular Polygons . . . . . . . . . . . . . . . . . . 1.4 The Isoperimetric Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Some Inequalities on Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Euler’s Theorem on Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 The Regular and Semiregular Polyhedra . . . . . . . . . . . . . . . . . . . . . . 1.8 Polar Triangles, Lexell’s Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Identities in Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 1.10 Some Formulae of Spherical Trigonometry . . . . . . . . . . . . . . . . . . . . 1.11 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 7 9 11 14 17 19 25 26 27 29

2

Theorems from the Theory of Convex Bodies . . . . . . . . . . . . . . . . . . . . . . 2.1 Blaschke’s Selection Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Jensen’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Dowker’s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 An Extremum Property of the Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 On the Affine Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Variational Problems Regarding Affine Length . . . . . . . . . . . . . . . . . 2.7 Rudiments of Integral Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33 37 39 42 47 53 54

3

Problems on Packing and Covering in the Plane . . . . . . . . . . . . . . . . . . . 3.1 Density of Arrangements of Domains . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Problems of Densest Packing and Thinnest Covering with Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Some Outlines of Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Packing and Covering Convex Disks with Congruent Circles . . . . .

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3.5 Dissecting a Convex Domain into Convex Parts . . . . . . . . . . . . . . . . 3.6 Packing a Convex Domain with Circles of 𝑛 Different Sizes . . . . . . 3.7 Estimates for Incongruent Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 A Further Theorem on Covering with Circles . . . . . . . . . . . . . . . . . . 3.9 Dissecting a Convex Hexagon into Convex Polygons . . . . . . . . . . . . 3.10 Packing and Covering a Convex Hexagon with Congruent Convex Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 A Packing Problem with Respect to Affine Length . . . . . . . . . . . . . . 3.12 On a Mean Value Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 72 74 81 85 86 89 91 95

4

Efficiency of Packings and Coverings with a Sequence of Convex Disks101 4.1 Extremum Properties of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2 Centrally Symmetric Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.3 Packing and Covering Efficiency of Sequences of Disks . . . . . . . . . 108 4.4 Covering with Fragmented Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5

Extremal Properties of Regular Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . 115 Packing and Covering the Sphere with Congruent Spherical Caps . 116 5.1 5.2 Some Additional Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3 Approximating a Ball by Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.4 Volume of a Circumscribed Polyhedron . . . . . . . . . . . . . . . . . . . . . . . 125 5.5 Volume of an Inscribed Polyhedron . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.6 Inequalities Linking the Inradius and Circumradius of Polyhedra . . 131 5.7 Isoperimetric Problems for Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.8 A General Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.9 On the Shortest Net Dissecting the Sphere into Convex Parts of Equal Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.10 On the Total Length of the Edges of a Polyhedron . . . . . . . . . . . . . . 143 5.11 The Thinnest Saturated Packing of Spherical Caps . . . . . . . . . . . . . . 147 5.12 Approximating a Convex Surface by Polyhedra . . . . . . . . . . . . . . . . . 149 5.13 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6

Irregular Packing on the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.1 The Graph Associated with a Family of Points . . . . . . . . . . . . . . . . . 157 6.2 The Maximal Configuration for 𝑛 = 7 . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.3 The Maximal Configuration for 𝑛 = 8 and 9 . . . . . . . . . . . . . . . . . . . 161 6.4 Some Configurations of More Than 9 Points . . . . . . . . . . . . . . . . . . . 164 6.5 A Survey Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.6 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7

Packing in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.2 The Problem of Densest Ball Packing . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.3 On an Extremal Space Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

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7.4 7.5

The Mean Value Formula in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Part II Notes and Additional Chapters to the English Edition 8

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 G. Fejes Tóth, L. Fejes Tóth and W. Kuperberg 8.1 Notes on Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 8.1.1 Notes on Section 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 8.1.2 Notes on Section 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 8.1.3 Notes on Section 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 8.1.4 Notes on Section 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 8.1.5 Notes on Section 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 8.1.6 Notes on Section 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 8.1.7 Notes on Section 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 8.1.8 Notes on Section 1.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 8.2 Notes on Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 8.2.1 Notes on Section 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 8.2.2 Notes on Section 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 8.2.3 Notes on Section 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 8.2.4 Notes on Section 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 8.2.5 Notes on Section 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8.2.6 Notes on Section 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8.2.7 Notes on Section 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8.3 Notes on Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 8.3.1 Notes on Sections 3.1–3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 8.3.2 Notes on Section 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 8.3.3 Notes on Section 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 8.3.4 Notes on Section 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 8.3.5 Notes on Section 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 8.3.6 Notes on Section 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.3.7 Notes on Section 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.3.8 Notes on Section 3.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.3.9 Notes on Section 3.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 8.4 Notes on Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 8.4.1 Notes on Section 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 8.4.2 Notes on Section 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 8.5 Notes on Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 8.5.1 Notes on Section 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 8.5.2 Notes on Section 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 8.5.3 Notes on Section 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 8.5.4 Notes on Section 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 8.5.5 Notes on Section 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 8.5.6 Notes on Section 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 8.5.7 Notes on Section 5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

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8.6

8.7

8.5.8 Notes on Section 5.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 8.5.9 Notes on Section 5.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.5.10 Notes on Section 5.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 8.5.11 Notes on Section 5.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Notes on Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 8.6.1 Notes on Sections 6.1–6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 8.6.2 Notes on Sections 6.4–6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 8.6.3 Notes on Section 6.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Notes on Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 8.7.1 Notes on Sections 7.1–7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 8.7.2 Notes on Section 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

9

Finite Variations on the Isoperimetric Problem . . . . . . . . . . . . . . . . . . . . 243 G. Fejes Tóth

10

Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 G. Fejes Tóth 10.1 Existence of Economic Packings and Coverings . . . . . . . . . . . . . . . . 249 10.2 Upper Bounds for 𝛿(𝐵 𝑛 ) and Lower Bounds for 𝜗(𝐵 𝑛 ) . . . . . . . . . . 252 10.2.1 Blichfeldt’s bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 10.2.2 The simplex bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 10.2.3 The linear programming bound . . . . . . . . . . . . . . . . . . . . . . . 254 10.2.4 Arrangements of points with minimum potential energy . . . 257 10.2.5 Lattice arrangements of balls . . . . . . . . . . . . . . . . . . . . . . . . . 259 10.3 Bounds for the Packing and Covering Density of Convex Bodies . . 259 10.4 The Structure of Optimal Arrangements . . . . . . . . . . . . . . . . . . . . . . . 261

11

Ball Packings in Hyperbolic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 G. Fejes Tóth, L. Fejes Tóth and W. Kuperberg 11.1 The Simplex Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 11.2 Hyperspheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 11.3 Solid Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 11.4 Completely Saturated Packings and Completely Reduced Coverings268 11.5 A Probabilistic Approach to Optimal Arrangements and their Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

12 Multiple Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 G. Fejes Tóth 12.1 Multiple Arrangements on the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 271 12.2 Decomposition of Multiple Arrangements . . . . . . . . . . . . . . . . . . . . . 273 12.3 Multiple Arrangements in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 12.4 Multiple Tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Contents

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13 Neighbors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 G. Fejes Tóth, L. Fejes Tóth and W. Kuperberg 13.1 The Newton Number of Convex Disks . . . . . . . . . . . . . . . . . . . . . . . . 277 13.2 The Hadwiger Number of Convex Disks . . . . . . . . . . . . . . . . . . . . . . 278 13.3 Translates of a Jordan Disk with a Common Point . . . . . . . . . . . . . . 279 13.4 The Number of Touching Pairs in Finite Packings . . . . . . . . . . . . . . . 279 13.5 𝑛-Neighbor Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 13.6 Maximal Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 13.7 Higher-Order Neighbors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 13.8 The Newton Number of Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 13.9 𝑛-Neighbor Packing of Congruent Balls . . . . . . . . . . . . . . . . . . . . . . . 284 13.10 Results About Convex Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 13.11 Mutually Touching Translates of a Convex Body . . . . . . . . . . . . . . . 287 13.12 Mutually Touching Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 13.13 Cylinders Touching a Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 13.14 Neighbors in Lattice Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 14 Packing and Covering Properties of Sequences of Convex Bodies . . . . 293 G. Fejes Tóth 14.1 Packing and Covering Cubes and Boxes . . . . . . . . . . . . . . . . . . . . . . . 294 14.2 Results for General Convex Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 14.3 On-Line Packing and Covering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 14.4 Special Convex Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 14.5 Packing in and Covering of the Whole Space . . . . . . . . . . . . . . . . . . 300 14.6 Covering with Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 15

Four Classic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 G. Fejes Tóth and W. Kuperberg 15.1 The Borsuk Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 15.2 Tarski’s Plank Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 15.3 The Kneser–Poulsen Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 15.4 Covering a Convex Body by Smaller Homothetic Copies . . . . . . . . . 310

16 Miscellaneous Problems About Packing and Covering . . . . . . . . . . . . . . 313 G. Fejes Tóth, L. Fejes Tóth and W. Kuperberg 16.1 Arranging Houses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 16.2 Packing Barrels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 16.3 Covering with a Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 16.4 Finite Packing and Covering in 2 Dimensions . . . . . . . . . . . . . . . . . . 316 16.5 Finite Arrangements in Higher Dimensions . . . . . . . . . . . . . . . . . . . . 317 16.6 Slab, Cylinder, Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 16.7 Close Packings and Loose Coverings . . . . . . . . . . . . . . . . . . . . . . . . . 319 16.8 Arranging Regular Tetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 16.9 Packing Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 16.10 Obstructing Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

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16.11 16.12 16.13 16.14 16.15 16.16 16.17 16.18 16.19 16.20

Avoiding Obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 Minkowskian Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Saturated Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Compact Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 Totally Separable Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 Point-Trapping Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Connected Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Points on the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Arrangements of Great Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

References for Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 References for Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 References for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 References for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 References for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 References for Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 References for Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 References for Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 References for Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 References for Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 References for Chapter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

Part I

Lagerungen – Arrangements in the Plane, on the Sphere, and in Space

MEINER FRAU

Chapter 1

Some Theorems from Elementary Geometry

In this chapter we compile the necessary preliminaries from elementary geometry. They are mainly about some well-known concepts and theorems, included here just for the sake of completeness. Nevertheless, the chapter contains a few more specialized theorems, such as the inequalities on triangles in Section 1.5, the spatial generalization of which will be discussed in Chapter 5.

1.1 Convex Sets A plane point set 𝑃 is said to be convex if every line segment connecting two points in 𝑃 is contained in 𝑃. Any bounded, closed, convex plane point set with an interior point we name a convex disk. The boundary points of a convex disk 𝐺 form a closed convex curve. A straight line containing at least one point in 𝐺 but no point in its interior is a supporting line of 𝐺. The boundary points of 𝐺 that belong to a given supporting line are called the points of support of the line. If only one supporting line passes through a boundary point, we say that the line is a tangent line, and if a supporting line contains just one point of support, we say that the point is the point of tangency for the line. For a given point set 𝑀, we define the convex hull of 𝑀 as the convex set containing 𝑀 which does not have a proper convex subset containing 𝑀. A convex polygon can be defined as the convex hull of a finite set of at least three coplanar, but not collinear, points. If all sides as well as all angles of a convex polygon are congruent, then the polygon is regular. A convex polygon 𝑃 is inscribed in a convex disk 𝐺, or is circumscribed about it, if the vertices of 𝑃 are on the boundary of 𝐺 or the sides of 𝑃 are supporting lines of 𝐺, respectively. A greatest circle contained in 𝐺 and the smallest circle containing 𝐺 are respectively called an incircle and the circumcircle of 𝐺. While a convex disk can have only one circumcircle, it can have many incircles. The radius of the incircle (circumcircle) will be called the inradius (circumradius) of 𝐺.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_1

5

6

1 Some Theorems from Elementary Geometry

In a quite similar way one can define in space the notion of a convex body, convex surface, convex polyhedron, supporting plane, insphere, and circumsphere. In what follows, we shall deal mainly with convex disks or bodies. Each of them possesses, in the usual sense, an area or volume, which, throughout this text, shall be denoted by the same symbol as the disk or body itself. Furthermore, each convex disk (body) has a perimeter (surface area), which we shall, likewise, denote by the same symbol as the bounding curve or surface. We denote the intersection of two disks 𝑇 and 𝑈 by 𝑇𝑈. Unless the opposite is emphasized, 𝑇𝑈, as a quantity, means the area of the intersection and not the product of the areas of 𝑇 and 𝑈. We now define the notion of the parallel domain 𝑇 ( 𝜚) of radius 𝜚 of a convex disk 𝑇 as the union of all circular disks of radius 𝜚 and center in 𝑇. The following important formula holds: (1.1.1) 𝑇 ( 𝜚) = 𝑇 + 𝐿 𝜚 + 𝜋 𝜚 2 , where 𝐿 denotes the perimeter of 𝑇. If 𝑇 is a convex polygon, then formula (1.1.1) is easy to verify. Namely, in such case 𝑇 ( 𝜚) consists of the following parts: 1. 𝑇 itself; 2. rectangles of height 𝜚 based on the sides of 𝑇; and 3. sectors of a circle that can be combined to form a complete circle of radius 𝜚. The general case follows by approximation with polygons. The corresponding formula for the parallel body 𝑉 ( 𝜚) of a convex body 𝑉 is as follows: 4𝜋 3 𝑉 ( 𝜚) = 𝑉 + 𝐹 𝜚 + 𝑀 𝜚2 + 𝜚 . (1.1.2) 3 Here 𝐹 denotes the surface area of 𝑉 and 𝑀 is the so-called integral of the mean curvature. If the surface bounding 𝑉 is a continuously curved convex surface 𝐹, then  ∫  1 1 + d𝑓, 𝑀= 𝑅2 𝐹 𝑅1 where 𝑅1 and 𝑅2 are the principal radii of curvature at a point of 𝐹, and d 𝑓 denotes the surface element at that point. Should 𝐹, however, also have some “edges”, then a natural adjustment must be made by appending the above integral with the term ∫ 1 𝛼 d𝑙, 2 where 𝛼 denotes the external angle at an edge element d𝑙, that is, the angle between the two outer normals to the two surface elements adjacent to that edge element. In the case when 𝑉 is a convex polyhedron, formula (1.1.2) can be verified directly, and the quantity 𝑀 is expressed by the sum 𝑀=

1 ∑︁ 𝛼𝑙, 2

1.2 Affinity and Polarity

7

where 𝑙 is the length of an edge, 𝛼 is the external angle at that edge, and the summation extends over all edges of 𝑉. Following J. Steiner’s terminology, the quantity 𝑀 for polyhedra will be called the edge curvature. If we now wish to derive the validity of formula (1.1.2) in general from its validity for polyhedra, we should first prove that, under approximations of an arbitrary convex body by polyhedra, the edge curvature converges to the integral of the mean curvature. However, we will not make any use of this fact, and therefore we omit its proof. The formula (1.1.2) fittingly shows the meaning of the three fundamental characteristics of convex bodies, namely the volume 𝑉, the surface area 𝐹, and the integral of the mean curvature 𝑀.

1.2 Affinity and Polarity Let 𝑂 be a fixed point in the plane and 𝜆 a non-zero number. We associate with each point 𝑃 of the plane that point 𝑃 ′ on the ray 𝑂𝑃 whose distance from 𝑂 is 𝑂𝑃 ′ = 𝜆𝑂𝑃. We call this mapping of the plane onto itself a homothety with center 𝑂. Two figures that can be transformed into each other by such a transformation or by a parallel translation are called homothetic. A similarity is composed of a homothety with respect to a point and a congruence. Consider now a straight line 𝑔 instead of 𝑂. Again, we assign to each point 𝑃 of the plane a point 𝑃 ′: if 𝐹 is the orthogonal projection of 𝑃 onto 𝑔, then let 𝑃 ′ be that point 𝑃 ′ on the ray 𝐹𝑃 whose distance from 𝐹 is 𝐹𝑃 ′ = 𝜆𝐹𝑃. We call this transformation an affinity with respect to the line 𝑔. A general affinity is composed of an affinity with respect to a line and a similarity. Analogously, we can define in space a homothety, a general similarity, an affinity with respect to a plane and, finally, a general affinity, which is a mapping of the space onto itself composed of two affinities with respect to different planes and a similarity. An affinity transforms a circle into an ellipse and a ball into an ellipsoid. Straight lines are transformed by an affinity again into straight lines and planes into planes. In addition, affinity preserves the parallel position of two segments and the proportion of volumes of two figures. It follows that the centroid of a figure is sent to the centroid of the affine image of the figure and that an affinity that maps a figure of positive volume onto a figure of the same volume preserves volumes in general. For two triangles or tetrahedra Δ and Δ′ there is always an affinity mapping Δ onto Δ′. A polygon arising from a regular polygon by an affinity is said to be affine regular. An affine regular 𝑛-gon can be obtained from a regular 𝑛-gon also by a parallel projection to another plane. Another important transformation needed in the sequel is polarity with respect to a circle or a sphere. Let 𝐾 be a circle (sphere) with center 𝑂 and radius 𝑟. The polarity with respect to 𝐾 assigns to each point 𝑃 of the plane (space) different from 𝑂 that line (plane) 𝑝 which intersects the ray OP perpendicularly in a point 𝑃 ′ for

8

1 Some Theorems from Elementary Geometry

which 𝑂𝑃 · 𝑂𝑃 ′ = 𝑟 2 . Conversely, we assign to each line (plane) 𝑝 not passing through 𝑂 that point 𝑃 whose image is 𝑝. The crucial property of polarity is that incident points and lines (planes) are mapped onto incident lines (planes) and points. From this it follows, for example, that under a polarity in the plane, to the point of intersection of two straight lines corresponds the line passing through the images of the two lines. Consequently, under polarity, to a polygon (polyhedron) corresponds a well-defined new polygon (polyhedron) so that the vertices of one polygon (polyhedron) are assigned to the sides (faces) of the other one. Moreover, according to a basic property in projective geometry, under a polarity one conic section corresponds to another one in the sense that the polarity maps points of one conic section onto tangent lines of the other. Analogously, under polarity in space, to a nondegenerate second-order algebraic surface corresponds a surface of the same sort. We now prove the following lemma, to be used later: If 𝐸 and 𝐸 ′ are two ellipsoids corresponding to each other under the polarity with respect to the unit sphere, then their volumes satisfy the inequality 𝐸 · 𝐸′ ≥



4𝜋 3

2 (1.2.1)

and equality holds only when 𝐸 and 𝐸 ′ are concentric with the unit sphere. For the proof we consider a Cartesian coordinate system whose origin is the center of the unit sphere and whose 𝑥-, 𝑦- and 𝑧-axes are parallel to the 2𝑎, 2𝑏 and 2𝑐 axes of 𝐸. Since to the ellipsoid 𝐸 corresponds another ellipsoid, that is a surface contained in a bounded region, it follows that 𝐸 contains the origin 𝑂. Otherwise a plane tangent to 𝐸 and containing 𝑂 would be sent to a point at infinity of 𝐸 ′. Thus, denoting the coordinates of the center of 𝐸 by 𝜉, 𝜂, and 𝜁, we have |𝜉 | < 𝑎, |𝜂| < 𝑏, |𝜁 | < 𝑐. Consider the planes tangent to 𝐸 at the endpoints of the axis 2𝑎. The two points corresponding to these planes lie on the 𝑥-axis and the distance between them is 1 1 2𝑎 2 + = ≥ . 𝑎 + 𝜉 𝑎 − 𝜉 𝑎2 − 𝜉 2 𝑎 Repeating this argument for the axes 2𝑏 and 2𝑐 we get three pairwise orthogonal chords of 𝐸 ′ whose lengths are at least 2/𝑎, 2/𝑏, and 2/𝑐, respectively. For the axes 𝐴𝐴 ′ = 2𝑎 ′, 𝐵𝐵 ′ = 2𝑏 ′, and 𝐶𝐶 ′ = 2𝑐 ′ of 𝐸 ′ parallel to these chords, that is, parallel to the coordinate axes, the inequalities 𝑎𝑎 ′ ≥ 1, 𝑏𝑏 ′ ≥ 1, 𝑐𝑐 ′ ≥ 1 hold a fortiori. Among the ellipsoids with given axes 𝐴𝐴 ′, 𝐵𝐵 ′, and 𝐶𝐶 ′ consider the ellipsoid ¯ 𝐸 with smallest volume. We claim that the main axes of 𝐸¯ coincide with 𝐴𝐴 ′, 𝐵𝐵 ′, and 𝐶𝐶 ′. Otherwise, the plane tangent to 𝐸¯ at 𝐴 would not be parallel to the plane 𝐵𝐵 ′𝐶𝐶 ′. Then 𝐴𝐴 ′ could be replaced by another chord 𝐷𝐷 ′ of 𝐸¯ so that the volume of the convex hull 𝐻 ∗ of 𝐷𝐷 ′, 𝐵𝐵 ′, and 𝐶𝐶 ′ would be greater than the volume of the convex hull 𝐻 of 𝐴𝐴 ′, 𝐵𝐵 ′, and 𝐶𝐶 ′. Consider now the affinity mapping 𝐻 ∗ onto 𝐻. As the affinity preserves the proportions of volumes and 𝐻 ∗ > 𝐻, this affinity would

1.3 Extremum Properties of the Regular Polygons

9

transform 𝐸¯ into a smaller ellipsoid. This, however, would contradict our assumption that 𝐸¯ is minimal. Therefore we have  2 4𝜋 ′ ′ ′ 4𝜋 1 4𝜋 1 ′ ¯ 𝐸 ≥𝐸= 𝑎𝑏𝑐 ≥ = , 3 3 𝑎𝑏𝑐 3 𝐸 which proves inequality (1.2.1). Equality can occur only if 𝜉 = 𝜂 = 𝜁 = 0. Then, however, 𝐸 ′ is concentric with the unit sphere as well. It is obvious that in this case equality does indeed hold.

1.3 Extremum Properties of the Regular Polygons Among all convex 𝑛-gons contained in a given convex disk 𝐺 let 𝑃 be one with greatest area. The existence of such an 𝑛-gon can be deduced from the well-known theorem of Weierstrass. Obviously, 𝑃 must be inscribed in 𝐺. Moreover, it is easily seen that 𝑃 has the property that at each vertex of 𝑃 there is a supporting line of 𝐺 parallel to the diagonal passing through the neighboring vertices. Indeed, if at a vertex 𝑣 the line parallel to this diagonal was not a line of support of 𝐺, then 𝑣 could be translated along this line into the interior of 𝐺, producing a polygon contained in 𝐺 with the same maximum area as 𝑃, but not inscribed in 𝐺. This would contradict the fact that all 𝑛-gons of maximum area are are inscribed. Likewise, it is easy to show that an 𝑛gon of smallest area containing 𝐺 is circumscribed about 𝐺 in such a way that each midpoint of a side is a point of support. For the analogous problems concerning perimeter, the extreme 𝑛-gons are also inscribed in and circumscribed about 𝐺, respectively. A convex 𝑛-gon of maximum perimeter contained in 𝐺 has the property that at each vertex the outer bisector of the angle is a line of support, while for the 𝑛-gon of minimum perimeter containing 𝐺, the circle tangent to the respective side and to the elongation Fig. 1.1 of the neighboring sides is tangent to 𝐺 as well (Figure 1.1). When 𝐺 is a circle 𝐾, then the above properties determine the 𝑛-gon uniquely. In each of these cases the extreme 𝑛-gon is regular. Thus, among all convex 𝑛gons contained in a circle 𝐾, the regular one inscribed in it has maximum area and perimeter as well. Likewise, among all convex 𝑛-gons containing a circle 𝐾, the regular one circumscribed about it has minimum area and perimeter as well.

10

1 Some Theorems from Elementary Geometry

As the area and perimeter of a polygon circumscribed about a circle are proportional, the statements concerning area and perimeter in the last theorem are equivalent. The extremum properties of the regular polygons stated above can be rephrased as follows: The area 𝐹, perimeter 𝐿, inradius 𝑟 and circumradius 𝑅 of a convex 𝑛-gon satisfy the inequalities 𝜋 1 2𝜋 ≤ 𝐹 ≤ 𝑛𝑅 2 sin , 𝑛 2 𝑛 𝜋 𝜋 2𝑛𝑟 tan ≤ 𝐿 ≤ 2𝑛𝑅 sin , 𝑛 𝑛 𝑛𝑟 2 tan

(1.3.1) (1.3.2)

and in each of these four inequalities equality holds for the regular 𝑛-gon only. The proofs outlined above are indirect. We shall now give a direct proof of the extremum property of the circumscribed 𝑛-gon. We do this, on one hand, to show the contrast between a direct and an indirect proof of an extremum property, and on the other hand, because this proof will provide a generalization for later use. Obviously, we can restrict our attention a priori to an 𝑛-gon 𝑃 circumscribed about a circle 𝑘. Let 𝑃¯ be a regular 𝑛-gon circumscribed about 𝑘. We show in a single step that 𝑃 ≥ 𝑃¯ and equality holds only if 𝑃 is regular. Fig. 1.2 To that end, we consider the circumcircle 𝐾 of 𝑃¯ and we denote the (congruent) circle-segments of 𝐾 determined by the sides of 𝑃 in their cyclic order by 𝑠1 , . . . , 𝑠 𝑛 (Figure 1.2). Then the area of the part of 𝑃 contained in 𝐾 can be expressed as follows: 𝑃𝐾 = 𝐾 − (𝑠1 + · · · + 𝑠 𝑛 ) + (𝑠1 𝑠2 + 𝑠2 𝑠3 + · · · + 𝑠 𝑛 𝑠1 ). To see this, we just have to notice that 𝑠1 + · · · + 𝑠 𝑛 − (𝑠1 𝑠2 + 𝑠2 𝑠3 + · · · + 𝑠 𝑛 𝑠1 ) equals the area of the part of 𝐾 outside 𝑃. Namely, if a point of this set is contained in 𝜈 circle-segments 𝑠1 , 𝑠2 , . . . , 𝑠 𝜈 , then it is contained at the same time in the 𝜈 − 1 sets 𝑠1 𝑠2 , . . . , 𝑠 𝜈−1 𝑠 𝜈 . Consequently, we have ¯ 𝑃𝐾 ≥ 𝐾 − (𝑠1 + · · · + 𝑠 𝑛 ) = 𝑃,

(1.3.3)

and equality holds only if no vertex of 𝑃 lies in the interior of 𝐾. Since this can occur ¯ the proof is complete. The generalization announced above is that in only if 𝑃 = 𝑃,

1.4 The Isoperimetric Problem

11

the inequality 𝑃 ≥ 𝑃¯ the area of 𝑃 can be replaced by the area of the intersection 𝑃𝐾. Let us now compare this direct proof with the indirect one given before. Generally, the proof of some extremum property can be regarded as direct if it shows directly, without calling on some infinite process, that the corresponding extreme configuration is better than all the other ones in comparison. In this sense we indeed have to rate the first proof as indirect: first the existence of a best polygon is established, and afterwards it is shown that every irregular polygon can be altered so as to get a better one. Disregarding aesthetic and didactic points of view, the indirect method seems to be more natural and perhaps also more practical. If the primary goal is to find a still unknown extreme figure, one puts the question of existence aside and attacks the following problem: when and how can a figure be improved? However, the indirect method sketched here is neither completely elementary nor purely geometric, since by applying the theorem of Weierstrass it makes use of the elements of analysis. Consider now the direct proof given above, which by its directness alone seems to be more satisfactory and convincing. The question of existence is not even addressed at first, but remains open and is answered automatically in the end. Moreover, in the above direct proof we use tools from elementary geometry only, while with an indirect proof this is in principle impossible. However, since a direct proof often requires greater skill, such proofs usually emerge only after a “less elegant” solution of the problem has already been found. To close, we mention the inequality 𝑅 𝜋 ≥ sec 𝑟 𝑛

(1.3.4)

that holds for the inradius 𝑟 and circumradius 𝑅 of an arbitrary convex 𝑛-gon. This follows directly from the inequality (1.3.1). Furthermore, using the properties of affinity, 𝑟 2 and 𝑅 2 can be replaced in (1.3.1) by 𝜋𝑒 and 𝐸𝜋 , where 𝑒 and 𝐸 denote an ellipse contained in, and containing the 𝑛-gon, respectively. Hence, we have the somewhat more general inequality: 𝜋 𝐸 ≥ sec2 . 𝑒 𝑛

(1.3.5)

1.4 The Isoperimetric Problem Among the isoperimetric domains, that is, among the domains of the same perimeter, which one has the greatest area? The solution of this classic, so-called isoperimetric, problem is the circle. In other words: If 𝐿 denotes the perimeter of a domain of area 𝐹 in the plane, then 𝐿 2 − 4𝜋𝐹 ≥ 0, (1.4.1) and equality holds for the circle only.

12

1 Some Theorems from Elementary Geometry

From this fundamental problem arises a wealth of further problems if the set of competing domains is restricted in various ways. In what follows we deal with the isoperimetric problem for 𝑛-gons. Thus, we take the collection of all polygons of the same given perimeter and with at most 𝑛 sides and ask which one among them has the greatest area. First we sketch the idea of an indirect proof of the claim that the best 𝑛-gon is regular. This proof has the advantage that it can be easily carried over to the corresponding problem in spherical geometry. Since the convex hull of a nonconvex polygon is a polygon of greater area, smaller perimeter, and with fewer sides, we can restrict our attention to convex 𝑛-gons. The existence of a best convex 𝑛-gon 𝑃 is ensured by the theorem of Weierstrass. It is easy to see that at every vertex of 𝑃 the outer bisector of the angle is parallel to the line through the neighboring vertices. Otherwise 𝑃 could be improved by an appropriate displacement of the respective vertex. It follows that the sides of a best 𝑛-gon must have equal length. It is also easily seen that the circle tangent to a side and to the elongation of its neighboring sides touches the respective side at its midpoint. This implies that the angles of 𝑃 are equal as well. This completes the proof of our claim. We express the obtained result in the inequality 𝐿 2 ≥ 4𝑛 tan

𝜋 𝐹, 𝑛

(1.4.2)

where 𝐿 denotes the perimeter and 𝐹 denotes the area of an arbitrary 𝑛-gon, and equality holds for the regular 𝑛-gon only. The following direct proof will supply important generalizations. Let 𝐹 be an arbitrary convex 𝑛-gon of perimeter 𝐿 and inradius 𝑟. Let 𝑓 be an 𝑛-gon circumscribed about a unit circle such that the outer normals of 𝑓 are the same as those of 𝐹. We assert that 𝐿𝑟 − 𝐹 − 𝑓 𝑟 2 ≥ 0. (1.4.3) This inequality can be restated as 𝐿 2 − 4 𝑓 𝐹 ≥ (𝐿 − 2 𝑓 𝑟) 2 . From there we get the remarkable inequality 𝐿2 − 4 𝑓 𝐹 ≥ 0 of S. L’Huilier, in which equality holds only for a polygon circumscribed about a circle, and which states that among the convex polygons with given outer normals, the ones circumscribed about a circle provide the least value of the quotient 𝐿 2 /𝐹. Denoting by 𝜑1 , . . . , 𝜑 𝑛 the angles between the consecutive outer normals, we have 𝑛 ∑︁ 𝜑𝑖 𝑓 = tan . 2 𝑖=1

13

1.4 The Isoperimetric Problem

Therefore L’Huilier’s inequality can be rewritten as follows: 𝑛 ∑︁ 𝐿2 𝜑𝑖 ≥4 tan . 𝐹 2 𝑖=1

(1.4.4)

Moreover, taking into account that 𝑓 > 𝜋, it follows from (1.4.3) that 𝐿𝑟 − 𝐹 − 𝜋𝑟 2 > 0, or, equivalently, 𝐿 2 − 4𝜋𝐹 > (𝐿 − 2𝜋𝑟) 2 .

(1.4.5)

This stronger isoperimetric inequality, which, also allowing equality, holds for arbitrary convex disks, shows directly that in the original isoperimetric inequality (1.4.1), equality holds for the circle only. For the proof of (1.4.3) we translate each side of 𝐹 parallel to itself inwards by a distance 𝑑 ≤ 𝑟 and denote the polygon bounded by the new sides by 𝐹𝑑 . Let us analyze how Fig. 1.3 this so-called inner parallel domain varies when 𝑑 grows continuously from 0 to 𝑟. For small values of 𝑑 the vertices of 𝐹𝑑 move on the bisectors of the inner angles of 𝐹. Meanwhile, the sides of 𝐹𝑑 decrease until at a certain value of 𝑑 one side of 𝐹𝑑 degenerates to a point. From this value on, we get polygons with fewer sides, and this continues until finally, for 𝑑 = 𝑟, 𝐹𝑑 shrinks to the “kernel” of 𝐹 (Figure 1.3). This kernel is nothing else but the set of centers of the incircles of 𝐹, that is, in general, a point, or, in special cases, (as for a rectangle) a segment. The polygons 𝐹𝑑 corresponding to the critical values of 𝑑 partition the collection of the nested parallel domains 𝐹𝑑 into layers. The polygons within a given layer have the same number of sides. Let 𝐹𝑑1 and 𝐹𝑑2 be two polygons in the same layer, so that 𝑑1 − 𝑑2 = 𝛿 > 0. Denoting the respective measurements simply by the corresponding indices we have 𝐹2 𝐿2 𝑟2 𝑓2

= 𝐹1 + 𝐿 1 𝛿 + 𝑓1 𝛿2 , = 𝐿 1 + 2 𝑓1 𝛿, = 𝑟 1 + 𝛿, = 𝑓1 .

14

1 Some Theorems from Elementary Geometry

It follows that 𝐿 2 𝑟 2 − 𝐹2 − 𝑓2 𝑟 22 = 𝐿 1 𝑟 1 − 𝐹1 − 𝑓1 𝑟 12 , that is, the quantity 𝐿𝑟 − 𝐹 − 𝑓 𝑟 2 is constant within a layer. Now observe that 𝑓 = 𝑓 (𝐹𝑑 ) is a stepwise increasing function of 𝑑. Indeed, 𝑓 is constant within a layer; if, however, we pass from one layer to the adjacent deeper one, then one side of 𝑓 vanishes, whereby the area of 𝑓 increases. Since 𝐹, 𝐿, and 𝑟 are continuous functions of 𝑑, it follows that 𝐿𝑟 − 𝐹 − 𝑓 𝑟 2 is a decreasing step function of 𝑑. As the value of this function is zero for 𝑑 = 𝑟, that is, for the kernel, it cannot be negative for the original polygon 𝐹, just as asserted.

1.5 Some Inequalities on Triangles For 𝑛 = 3, the inequality (1.3.4) yields: If 𝑟 and 𝑅 denote the inradius and circumradius of a triangle, then 𝑅 ≥ 2𝑟, (1.5.1) and equality holds for the regular triangle only. We prove here some analogous theorems. We begin with the following theorem: If 𝑅1 , 𝑅2 and 𝑅3 denote the distances from an arbitrary point 𝑂 in the plane to the vertices of a triangle of inradius 𝑟, then 𝑅1 + 𝑅2 + 𝑅3 ≥ 6𝑟,

(1.5.2)

and equality occurs only if the triangle is regular centered at 𝑂. This theorem is a consequence of the following, stronger theorem: If 𝑅1 , 𝑅2 and 𝑅3 denote the distances from an arbitrary point 𝑂 in the plane to the vertices of a triangle Δ, then »√ 𝑅1 + 𝑅2 + 𝑅3 ≥ 2 3Δ, (1.5.3) and equality occurs only if Δ is a regular triangle centered at 𝑂. √ The last inequality implies (1.5.2) via the special case Δ ≥ 27 𝑟 2 of the first inequality of (1.3.1). For the proof of the inequality (1.5.3) we may assume that 𝑂 does not lie outside Δ = 𝐴𝐵𝐶. Reflect 𝑂 in the lines 𝐴𝐵, 𝐵𝐶 and 𝐶 𝐴 and denote the reflected points by 𝐶 ′, 𝐴 ′ and 𝐵 ′, respectively. The hexagon 𝐴𝐶 ′ 𝐵𝐴 ′𝐶 𝐵 has area 2Δ and perimeter 2(𝑅1 + 𝑅2 + 𝑅3 ). Now, the inequality (1.5.3) is precisely the isoperimetric inequality (1.4.2) for this hexagon.

1.5 Some Inequalities on Triangles

15

Next we prove the following theorem due to Erdős and Mordell: If 𝑅1 , 𝑅2 , 𝑅3 and 𝑟 1 , 𝑟 2 , 𝑟 3 , denote the distances to the vertices and to the sides, respectively, from an arbitrary interior point 𝑂 of a triangle, then 𝑅1 + 𝑅2 + 𝑅3 ≥ 2(𝑟 1 + 𝑟 2 + 𝑟 3 ),

(1.5.4)

and equality occurs only if the triangle is regular, centered at 𝑂. If we denote the arithmetic mean of the numbers 𝑥 1 , 𝑥2 , . . . , 𝑥 𝑛 by 𝐴(𝑥1 , 𝑥2 , . . . , 𝑥 𝑛 ), then we can express the inequality (1.5.4) as 𝐴(𝑅1 , 𝑅2 , 𝑅3 ) ≥ 2𝐴(𝑟 1 , 𝑟 2 , 𝑟 3 ).

(1.5.5)

We will show that this inequality is equivalent to the inequality 𝐻 (𝑅1 , 𝑅2 , 𝑅3 ) ≥ 2𝐻 (𝑟 1 , 𝑟 2 , 𝑟 3 ),

(1.5.6)

where 𝐻 denotes the harmonic mean. Naturally, the inequalities (1.5.5) and (1.5.6) are algebraically not equivalent. Moreover, in special cases sometimes one of them and sometimes the other is stronger. We assert, however, that the validity of any of these two inequalities for an arbitrary triangle implies the validity of the other one. Indeed, consider the polarity with respect to the unit circle centered at an interior point 𝑂 of the triangle 𝐴𝐵𝐶. Let 𝑅1 , 𝑅2 and 𝑅3 and 𝑟 1 , 𝑟 2 and 𝑟 3 denote the distances from 𝑂 to the points 𝐴, 𝐵, and 𝐶 and to the lines 𝐵𝐶, 𝐶 𝐴 and 𝐴𝐵, respectively. Then the triangle 𝐴𝐵𝐶 is mapped onto another triangle, for which the corresponding distances from 𝑂 to its vertices and sides are 𝑟11 , 𝑟12 , 𝑟13 and 𝑅11 , 𝑅12 , 𝑅13 , respectively. Now, applying the inequality (1.5.5) to this new triangle we obtain the inequality (1.5.6) for the original triangle 𝐴𝐵𝐶. Likewise, (1.5.6) implies (1.5.5). Later we shall also prove the inequality 𝐺 (𝑅1 , 𝑅2 , 𝑅3 ) ≥ 2𝐺 (𝑟 1 , 𝑟 2 , 𝑟 3 ),

(1.5.7)

where 𝐺 denotes the geometric mean. But this time polarity provides no further inequality, as the polarity we just applied transforms (1.5.7) into itself. Now back to the proof of (1.5.4). Let 𝛼, 𝛽 and 𝛾 denote the angles of the triangle 𝐴𝐵𝐶, and let 𝑃, 𝑄 and 𝑅 denote the perpendicular projections of 𝑂 to the lines 𝐵𝐶, 𝐶 𝐴 and 𝐴𝐵, respectively (Figure 1.4). Since the segment 𝑂 𝐴 = 𝑅1 is a common hypotenuse of the right triangles 𝐴𝑄𝑂 and 𝐴𝑅𝑂, the points 𝐴, 𝑂, 𝑄, 𝑅 lie on a circle whose diameter is 𝑅1 . In this circle, the angle at which the chord 𝑄𝑅 is seen from 𝐴 is either 𝛼 or 180◦ − 𝛼, depending on Fig. 1.4 whether one of the points 𝑄 and 𝑅 lies

16

1 Some Theorems from Elementary Geometry

outside the side 𝐴𝐶, respectively 𝐴𝐵, or not. In either case we have 𝑅1 =

𝑄𝑅 sin 𝛼

and, quite similarly, 𝑅2 =

𝑅𝑃 , sin 𝛽

𝑅3 =

𝑃𝑄 . sin 𝛾

Moreover, since in the triangle 𝑄𝑂𝑅 the angle at 𝑂 is 180◦ − 𝛼, we have 𝑄𝑅 2 = 𝑂𝑄 2 + 𝑂𝑅 2 − 2 𝑂𝑄 · 𝑂𝑅 · cos(180◦ − 𝛼) = 𝑟 22 + 𝑟 32 + 2𝑟 2 𝑟 3 cos 𝛼. Consequently, considering that cos 𝛼 = − cos(𝛽 + 𝛾) = sin 𝛽 sin 𝛾 − cos 𝛽 cos 𝛾, we get 2

𝑄𝑅 = 𝑟 22 + 𝑟 32 + 2𝑟 2 𝑟 3 sin 𝛽 sin 𝛾 − 2𝑟 2 𝑟 3 cos 𝛽 cos 𝛾 = 𝑟 22 (sin2 𝛾 + cos2 𝛾) + 𝑟 32 (sin2 𝛽 + cos2 𝛽) + 2𝑟 2 𝑟 3 sin 𝛽 sin 𝛾 − 2𝑟 2 𝑟 3 cos 𝛽 cos 𝛾 = (𝑟 2 sin 𝛾 + 𝑟 3 sin 𝛽) 2 + (𝑟 2 cos 𝛾 − 𝑟 3 cos 𝛽) 2 ≥ (𝑟 2 sin 𝛾 + 𝑟 3 sin 𝛽) 2 , hence 𝑄𝑅 ≥ 𝑟 2 sin 𝛾 + 𝑟 3 sin 𝛽. In the same way we find that 𝑅𝑃 ≥ 𝑟 3 sin 𝛼 + 𝑟 1 sin 𝛾,

𝑃𝑄 ≥ 𝑟 1 sin 𝛽 + 𝑟 2 sin 𝛼.

With regard to the above values of 𝑅1 , 𝑅2 and 𝑅3 , this yields 1 1 (𝑟 2 sin 𝛾 + 𝑟 3 sin 𝛽) + (𝑟 3 sin 𝛼 + 𝑟 1 sin 𝛾) sin 𝛼 sin 𝛽 1 + (𝑟 1 sin 𝛽 + 𝑟 2 sin 𝛼) sin 𝛾       sin 𝛽 sin 𝛾 sin 𝛾 sin 𝛼 sin 𝛼 sin 𝛽 = 𝑟1 + + 𝑟2 + + 𝑟3 + . sin 𝛾 sin 𝛽 sin 𝛼 sin 𝛾 sin 𝛽 sin 𝛼

𝑅1 + 𝑅2 + 𝑅3 ≥

Since 𝑥 + 1𝑥 ≥ 2 for every positive number 𝑥, each of the coefficients of 𝑟 1 , 𝑟 2 and 𝑟 3 in the above inequality is at least 2, which completes the proof of the inequality (1.5.4). Moreover, in the inequality 𝑥 + 1𝑥 ≥ 2 equality occurs only for 𝑥 = 1, therefore all three coefficients can be equal to 2 at the same time only if sin 𝛼 = sin 𝛽 = sin 𝛾, that is, only if the triangle is regular. Besides that, for equality in (1.5.4) it is also necessary that all the terms vanish that were omitted in the estimates of 𝑄𝑅, 𝑅𝑃 and 𝑃𝑄:

17

1.6 Euler’s Theorem on Polyhedra

𝑟 2 cos 𝛾 − 𝑟 3 cos 𝛽 = 𝑟 3 cos 𝛼 − 𝑟 1 cos 𝛾 = 𝑟 1 cos 𝛽 − 𝑟 2 cos 𝛼 = 0. Considering that 𝛼 = 𝛽 = 𝛾, this condition is satisfied only if 𝑟 1 = 𝑟 2 = 𝑟 3 , and the case of equality is done as well.

1.6 Euler’s Theorem on Polyhedra On the surface of the unit ball, let us consider a finite number of hemispheres. If their intersection 𝐷 contains interior points, then 𝐷 is said to be a convex spherical polygon, and this is the general definition of a convex spherical polygon. We turn first to the task of determining the area of a convex spherical 𝑛-gon. Obviously, the area of a spherical digon is 2𝛼𝜋 4𝜋 = 2𝛼, where 𝛼 denotes the angle of the digon. Consider now a spherical triangle Δ with angles 𝛼, 𝛽, 𝛾, along with the three associated digons whose intersection makes Δ (Figure 1.5). Let us also keep in sight the triangle Δ′, diametrically opposite to Δ, along with its associated digons. These six digons cover Δ and Δ′ three-fold, and the rest of the sphere exactly once. Thereby we get 2(2𝛼+ 2𝛽 + 2𝛾) = 4𝜋 + 2 · 2Δ, that is, Δ = 𝛼 + 𝛽 + 𝛾 − 𝜋.

(1.6.1)

Fig. 1.5 This renowned formula asserts that the area of a spherical triangle is equal to the excess of the sum of its angles above 𝜋. Thereby it follows at once that the area 𝐹 of a spherical convex 𝑛-gon with angles 𝛼1 , . . . , 𝛼𝑛 is 𝐹 = 𝛼1 + · · · + 𝛼𝑛 − (𝑛 − 2)𝜋. (1.6.2) The formula (1.6.2) provides a very simple way of deriving Euler’s theorem on convex polyhedra. Let 𝑃 denote a convex polyhedron with 𝑓 faces, 𝑒 edges and 𝑣 vertices. Put a unit sphere 𝐸 centered at an interior point 𝑂 of 𝑃 and map the faces of 𝑃 to 𝐸 by a projection from 𝑂. Obviously, the resulting spherical polygons cover the surface 𝐸 without overlaps. Therefore, by applying formula (1.6.2) to each of these polygons and adding all of the resulting quantities we get 4𝜋 = 2𝜋𝑣 − 2𝜋𝑒 + 2𝜋 𝑓 , that is, 𝑓 + 𝑣 = 𝑒 + 2.

(1.6.3)

Now we will show some applications of Euler’s theorem (1.6.3) on polyhedra. Denote by 𝑝 1 , . . . , 𝑝 𝑓 the number of sides of the faces and by 𝑞 1 , . . . , 𝑞 𝑣 the valence of the vertices of a convex polyhedron. Then, obviously,

18

1 Some Theorems from Elementary Geometry

3 𝑓 ≤ 𝑝 1 + · · · + 𝑝 𝑓 = 2𝑒, 3𝑣 ≤ 𝑞 1 + · · · + 𝑞 𝑣 = 2𝑒. Combined with (1.6.3), these inequalities yield: 𝑒 + 6 ≤ 3 𝑓 ≤ 2𝑒, 𝑒 + 6 ≤ 3𝑣 ≤ 2𝑒.

(1.6.4) (1.6.5)

These in turn yield two important inequalities, one on the average number 𝑝 = 2𝑒𝑓 of sides of the faces, and the other one on the average valence 𝑞 = 2𝑒 𝑣 of the vertices: 12 < 6, 𝑓 12 𝑞 ≤ 6− < 6. 𝑒 𝑝 ≤ 6−

(1.6.6) (1.6.7)

If we denote the number of 3, 4,. . .-sided faces by 𝑓3 , 𝑓4 , . . . and the number of 3, 4,. . .-valent vertices by 𝑣 3 , 𝑣 4 , . . ., respectively, then, obviously 𝑓3 + 𝑓4 + · · · = 𝑓 , 𝑣 3 + 𝑣 4 + · · · = 𝑣,

3 𝑓3 + 4 𝑓4 + · · · = 2𝑒, 3𝑣 3 + 4𝑣 4 + · · · = 2𝑒

and consequently 𝑓3 − 𝑓5 − 2 𝑓6 − · · · = 4 𝑓 − 2𝑒, 𝑣 3 − 𝑣 5 − 2𝑣 6 − · · · = 4𝑣 − 2𝑒. By adding the last two inequalities and taking (1.6.3) into account, one obtains the following interesting relation: 𝑓3 + 𝑣 3 = 8 + ( 𝑓5 + 𝑣 5 ) + 2( 𝑓6 + 𝑣 6 ) + · · · ≥ 8. Consequently, every convex polyhedron must have a triangular face or a three-valent vertex. As a further application of Euler’s theorem on polyhedra we show that the average number of sides in a collection of convex polygons that can be assembled into a convex polygon 𝑆 of at most six sides is at most 6. To this end, denote these polygons by 𝑃1 , . . . , 𝑃𝑛 , and their numbers of sides by 𝑠1 , . . . , 𝑠 𝑛 . The polygons 𝑃1 , . . . , 𝑃𝑛 along with 𝑆 can be regarded as a degenerate polyhedron 𝑃 with (𝑛 + 1) faces, where, however, in general, the vertices of 𝑆 should not be counted as vertices of 𝑃. Namely, the vertices of 𝑃 belonging to the face 𝑆 are those boundary points of 𝑆 which are endpoints of an edge different from a side of 𝑆. If we denote the number of these vertices by 𝑝 0 and the number of vertices of the remaining faces 𝑃1 , . . . , 𝑃𝑛 by 𝑝 1 , . . . , 𝑝 𝑛 , then, according to (1.6.6), we get

1.7 The Regular and Semiregular Polyhedra

19

𝑝0 + 𝑝1 + · · · + 𝑝 𝑛 12 ≤ 6− . 𝑛+1 𝑛+1 On the other hand, by our assumption on the number of vertices of 𝑆, 𝑠1 + · · · + 𝑠 𝑛 ≤ 𝑝 1 + · · · + 𝑝 𝑛 + 6. This yields 𝑠1 + · · · + 𝑠 𝑛 ≤ 6𝑛 − 𝑝 0 , by which our assertion, that is, the inequality 𝑠1 + · · · + 𝑠 𝑛 ≤6 𝑛

(1.6.8)

is proved. Equality occurs here only if 𝑆 is a hexagon and 𝑛 = 1.

1.7 The Regular and Semiregular Polyhedra A convex polyhedron is said to be regular if its faces and vertices are regular. Here, saying that a vertex 𝑉 is regular means that its corresponding spherical polygon— that is, the polygon cut out from a small sphere centered at 𝑉 by the faces adjacent to 𝑉—is regular. As a simple consequence of the above definition, the faces as well as the vertices of a regular polyhedron are congruent. After L. Schläfli, we use the symbol {𝑝, 𝑞} to denote the regular polyhedron with 𝑝-sided faces and 𝑞-valent vertices. Since according to (1.6) either 𝑝 or 𝑞 must be equal to 3, and by (1.6.6) and (1.6.7) neither 𝑝 nor 𝑞 can exceed 5, only the following combinations are possible: {3, 3}, {3, 4}, {4, 3}, {3, 5}, and {5, 3}. It is easy to show that all of these possibilities are realizable. There are therefore five regular polyhedra, which are called, according to the number of faces, tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron (Figures 1.6(a)–(e)). The regular polyhedra {𝑝, 𝑞} and {𝑞, 𝑝} are mapped onto each other through a polarity with respect to their in- or circumspheres. For that reason we say that the tetrahedron is dual to itself, the hexahedron to the octahedron, and the dodecahedron to the icosahedron. It is convenient to complete the family of regular polyhedra by including degenerate ones. We consider first the “polyhedra” of the symbols {3, 6}, {4, 4} and {6, 3}. These are the regular partitions of the plane into 3, 4 and 6-gons, respectively (Figures 1.6(f)–(h)). Further, we consider the so-called dihedra of the symbols {𝑛, 2} (Figure 1.6(i)), as well as their duals {2, 𝑛} (Figure 1.6(j)) (𝑛 = 2, 3, . . .). These can be best represented by their spherical nets. We now move on and introduce further interesting polyhedra. The convex polyhedra with regular faces and congruent but not regular vertices are known as Archimedean polyhedra, or Archimedean solids. Via polarity, to each Archimedean polyhedron a dual one can be assigned, with regular vertices and congruent but

20

1 Some Theorems from Elementary Geometry

(a) {3, 3}

(b) {3, 4}

(e) {5, 3}

(d) {3, 5}

(f) {3, 6}

(c) {4, 3}

(g) {4, 4}

(i) {3, 2}

(h) {6, 3}

(j) {2, 3}

Fig. 1.6

1.7 The Regular and Semiregular Polyhedra

21

not regular faces. The Archimedean polyhedra and their duals will be collectively named semiregular polyhedra. The family of semiregular polyhedra splits therefore into two classes—the equivertical (Archimedean) polyhedra and the equifacial ones. We need to consider just the equivertical polyhedra here. name (3,6,6) (3,8,8) (3,10,10) (3,12,12) (4,4,𝑛) (4,6,6) (4,6,8) (4,6,10) (4,6,12) (4,8,8) (5,6,6)

v e f 12 18 8 24 36 14 60 90 32 – – – 2𝑛 3𝑛 𝑛 + 2 24 36 14 48 72 26 120 180 62 – – – – – – 60 90 32

(3,3,3,𝑛) (3,4,3,4) (3,4,4,4) (3,4,5,4) (3,4,6,4) (3,5,3,5) (3,6,3,6)

2𝑛 12 24 60 – 30 –

(3,3,3,3,4) (3,3,3,3,5) (3,3,3,3,6) (3,3,4,3,4) (3,3,3,4,4)

4𝑛 2𝑛 + 2 24 14 48 26 120 62 – – 60 32 – –

24 60 60 150 – – – – – –

38 92 – – –

The defining properties of an Archimedean polyhedron can be satisfied only if not all of its faces are congruent. Only two or three different types of faces can occur. More than three types of faces cannot meet at a vertex, for if they did, then even in the best case, that is, if a regular triangle, quadrangle, pentagon and hexagon met at a vertex, the meeting angles would add up to 60◦ + 90◦ + 108◦ + 120◦ = 378◦ > 360◦ , which is not possible. In a similar way, the inequality 5 · 60◦ + 90◦ = 390◦ > 360◦ prevents more than five faces meeting at a vertex. A detailed elaboration on semiregular polyhedra can be found, for example, in Brückner [22]. Here we merely enumerate all Archimedean polyhedra, denoting them by the symbols (𝑖, 𝑗, 𝑘), (𝑖, 𝑗, 𝑘, 𝑙), . . . , (𝑖, 𝑗, 𝑘, 𝑙, 𝑚). For example, (𝑖, 𝑗, 𝑘) denotes the Archimedean polyhedron such that at every vertex three faces meet in the given cyclic order: an 𝑖-gon, a 𝑗-gon and a 𝑘-gon. Besides the proper 15 Archimedean polyhedra, our table also lists the 8 degenerate ones. The latter are those tilings of the plane with two or three kinds of incongruent regular polygons in

22

1 Some Theorems from Elementary Geometry

which the vertices are congruent. For these the numbers of vertices, edges and faces 𝑣, 𝑒 and 𝑓 , respectively, are all infinite. The nondegenerate Archimedean polyhedra except for the prisms (4, 4, 𝑛) and antiprisms (3, 3, 3, 𝑛)—whose construction does not require closer scrutiny — can be derived from the regular ones by “vertex-truncation” or “vertex-and-edgetruncation”. These constructions can be easily perceived from Figures 1.7(a)–(w) below. Only the constructions of the polyhedra (3, 3, 3, 3, 4) and (3, 3, 3, 3, 5) are somewhat more complicated. These arise from the octahedron, respectively icosahedron, (or from the hexahedron, respectively dodecahedron) by suitable vertex-andedge-truncations. However, the edge-truncation at each edge is done not by one but by two planes, neither one parallel to the edge being truncated. These two polyhedra can be constructed in a more perceivable way as follows. On each face of a regular polyhedron, draw a smaller, concentric and homothetic copy of the face, rotate each of these polygons about its center by the same angle and take the convex hull 𝐻 of the rotated polygons. At the proper scale of reduction and angle of rotation, the triangular faces of 𝐻 turn out to be regular. In this way the tetrahedron produces the icosahedron. Applied to {3, 4} or {4, 3}, this construction yields (3, 3, 3, 4); when {3, 5} or {5, 3} is treated by it, (3, 3, 3, 3, 5) emerges; and if we alter in this way either {3, 6} or {6, 3}, the result is (3, 3, 3, 3, 6).

(a) (3, 6, 6)

(b) (3, 8, 8)

(c) (3, 10, 10)

(d) (3, 12, 12)

(e) (4, 4, 5)

(f) (4, 6, 6)

Fig. 1.7

1.7 The Regular and Semiregular Polyhedra

23

(g) (4, 6, 8)

(h) (4, 6, 10)

(i) (4, 6, 12)

(j) (4, 8, 8)

(k) (5, 6, 6)

(l) (3, 3, 3, 5)

(m) (3, 4, 3, 4)

(n) (3, 4, 4, 4)

(o) (3, 4, 5, 4)

(p) (3, 4, 6, 4)

(q) (3, 5, 3, 5)

(r) (3, 6, 3, 6)

Fig. 1.7: continued.

24

1 Some Theorems from Elementary Geometry

(s) (3, 3, 3, 3, 4)

(t) (3, 3, 3, 3, 5)

(u) (3, 3, 3, 3, 6)

(w) (3, 3, 3, 4, 4)

(v) (3, 3, 4, 3, 4)

Fig. 1.7: continued.

The first five degenerate Archimedean polyhedra in our table arise from the degenerate regular polyhedra {3, 6}, {4, 4} and {6, 3} through vertex-truncation or vertex-truncation combined with the usual edge-truncation. The tiling (3, 3, 4, 3, 4) arises from {4, 4} in a similar way as, e.g., (3, 3, 3, 3, 4) arises from {4, 3}. The last polyhedron, namely (3, 3, 3, 4, 4), cannot be derived from either of the regular polyhedra. Observe that in the vertices of (3, 3, 4, 3, 4) and (3, 3, 3, 4, 4) the same faces meet, but in a different order. The equivertical semiregular polyhedra considered above are all inscribed in a sphere. The polarity about the circumsphere transforms each of them into an equifacial polyhedron, each of which is circumscribed about a sphere. Naturally, these polyhedra can also be constructed in a direct way. We will meet only one of them later, namely the rhombic dodecahedron, that is, the dual to the cuboctahedron (3, 4, 3, 4). The cuboctahedron is the convex hull of the edge-centers of a cube (or of an octahedron). Thus the dual polyhedron can be constructed as follows. Through each edge of a cube place a plane that bisects the outer dihedral angle of the cube. The rhombic dodecahedron is bounded by these planes. Another construction, even easier to visualize, goes as follows. To all faces of a cube attach congruent square pyramids. If the height of the pyramids equals one-half of the cube’s edge, then the two triangular faces of the pyramids adjacent to an edge of the cube lie in one plane, and a rhombic dodecahedron is created.

25

1.8 Polar Triangles, Lexell’s Circle

1.8 Polar Triangles, Lexell’s Circle We associate with each spherical triangle Δ ≡ 𝐴𝐵𝐶 a new triangle Δ′ ≡ 𝐴 ′ 𝐵 ′𝐶 ′. The point 𝐴 ′ is defined as the pole of the great circle 𝐵𝐶 whose spherical distance from 𝐴 is smaller than 90◦ . The points 𝐵 ′ and 𝐶 ′ are defined in a similar way. The triangle Δ′ is called the polar triangle of Δ. We show that the polar triangle of Δ′ is identical to the original triangle Δ. It suffices to show that the pole of the great circle 𝐵 ′𝐶 ′ whose spherical distance from 𝐴 ′ is smaller Fig. 1.8 than 90◦ is 𝐴. By the definition of 𝐵 ′ and ′𝐴 = 𝐶 ′ 𝐴 = 90◦ . Thus 𝐴 is a Ŋ Ŋ 𝐶 ′ we have 𝐵 ′𝐶 ′ , and by the definition of 𝐴 ′ , exactly that one for which 𝐴𝐴 Ő Ŋ′ < 90◦ . pole of 𝐵 Denoting the sides and angles of Δ and Δ′, as usual, by 𝑎, 𝑏, 𝑐, 𝛼, 𝛽, 𝛾 and by 𝑎 ′, 𝑏 ′, 𝑐 ′, 𝛼 ′, 𝛽 ′, 𝛾 ′, respectively, we have 𝑎 ′ + 𝛼 = 𝜋,

𝑏 ′ + 𝛽 = 𝜋,

𝑐 ′ + 𝛾 = 𝜋,

𝛼 ′ + 𝑎 = 𝜋,

𝛽 ′ + 𝑏 = 𝜋,

𝛾 ′ + 𝑐 = 𝜋.

A glance at Figure 1.8 makes it evident that 𝑎 ′ = 𝜋 − 𝛼. The equality 𝑎 = 𝜋 − 𝛼 ′ follows from this by the reflexivity of polarity shown above. We now apply the notion of polar triangle to the proof of a remarkable theorem of Lexell. Consider a spherical triangle Δ ≡ 𝐴𝐵𝐶 with a fixed base 𝐴𝐵 and move the vertex 𝐶 so that the area of the triangle remains unchanged. What is the locus of the point 𝐶? The answer is given by the following theorem of Lexell: The locus of the points which together with the given points 𝐴 and 𝐵 determine a spherical triangle of a prescribed area is a circular arc 𝐴∗ 𝐵∗ whose endpoints 𝐴∗ and 𝐵∗ are diametrically opposite to 𝐴 and 𝐵. For the proof we consider the polar triangle Δ′ ≡ (Figure 1.9). If 𝐶 satisfies the above condition as it moves, then the great circles containing the sides 𝑎 ′ and 𝑏 ′ remain fixed, while the side 𝑐 ′ changes so that the perimeter of the triangle Δ′ remains fixed. Let Fig. 1.9 𝐾 be the excircle of Δ′ touching the side 𝑐 ′ and the extension of the sides 𝑎 ′ and 𝑏 ′ of Δ′. Denote the points ¯ 𝐵, ¯ and of tangency of 𝐾 with the great circles containing the sides 𝑎 ′, 𝑏 ′ and 𝑐 ′ by 𝐴, ¯ respectively. Then the perimeter of Δ′ is 𝐴 ′𝐶¯ + 𝐶¯ 𝐵 ′ + 𝐵 ′𝐶 ′ + 𝐶 ′ 𝐴 ′ = 𝐴𝐶 ¯ ′ + 𝐶 ′ 𝐵, ¯ 𝐶, 𝐴 ′ 𝐵 ′𝐶 ′

26

1 Some Theorems from Elementary Geometry

¯ Thus, the side 𝑐 ′ moves so that 𝐶¯ runs through therefore it does not depend on 𝐶. ¯ ¯ all points of the arc 𝐴 𝐵 of 𝐾. It follows that 𝐶, as the pole of the great circle 𝐴 ′ 𝐵 ′, also traces a circular arc. The endpoints of this arc are the poles of the great circles 𝐶 ′ 𝐵 ′ and 𝐶 ′ 𝐴 ′; that is, they coincide with the points 𝐴 or 𝐴∗ and 𝐵 or 𝐵∗ . Since, however, 𝐴 and 𝐵 do not belong to the considered locus, only the points 𝐴∗ and 𝐵∗ come into consideration. This completes the proof of Lexell’s theorem. Additionally, we note that in the limiting case, say, 𝐶 ≡ 𝐴∗ , the triangle Δ degenerates into a digon.

1.9 Some Identities in Vector Algebra By a vector we understand a directed line segment. We denote vectors by bold-face lower-case characters: a, b, c, etc. The length of a vector a will be called its absolute value or magnitude and denoted by |a|. We can declare certain operations on vectors for which various rules can be derived. In other words: a vector algebra can be established, with applications to many areas of mathematics. Here we briefly present the basics of vector algebra. Two vectors of the same direction and magnitude are considered identical, even when their initial points are distinct. If 𝜆 is a real number, then 𝜆a denotes the vector whose magnitude is |𝜆||a|, while its direction coincides with either that of a or its opposite, depending on whether 𝜆 is positive or negative. To define the sum c = a + b, translate b so that its initial point coincides with the terminal point of a. Then c is declared to be the vector that points from the initial point of a to the terminal point of b. The difference c = a − b is defined as c = a + (−1)b. Obviously, for this vector the equality a = c + b holds. By the scalar or inner product ab of the vectors a and b, we understand the number ab = |a||b| cos 𝛾, where 𝛾 denotes the angle between a and b. The vector or outer product a × b is defined as the vector whose magnitude is |a||b| sin 𝛾, and whose direction is perpendicular to both a and b in such a way that their order a, b and a × b is the same as the order of the thumb, the index- and the middle finger of our right-hand. Obviously, a × b = −(b × a). The three kinds of products defined above possess the distributive property, which is expressed in the following, easy to verify, identities: (𝜆 + 𝜇)a = 𝜆a + 𝜇a, 𝜆(a + b) = 𝜆a + 𝜆b, a(b + c) = ab + ac, a × (b + c) = a × b + a × c. A further important notion is the mixed or volume product of three vectors, defined by (abc) = (a × b)c. This can be interpreted geometrically as the volume (furnished with a corresponding sign) of the parallelepiped spanned by a, b and c. Hence the

27

1.10 Some Formulae of Spherical Trigonometry

three vectors can be cyclically permuted: (abc) = (cab) = (bca). Consequently the mixed product can be defined by (abc) = a(b × c) as well. We prove now the following important identity: a × (b × c) = (ac)b − (ab)c.

(1.9.1)

If ab = 0, then (1.9.1) is easy to verify directly. But now, apart from the trivial case b × c = 0, every vector a can be written in the form a = ab + ac , where ab b = ac c = 0. Consequently, according to the special case just considered, we have ab × (b × c) = (ab c)b = (ac)b and ac × (b × c) = −(ac b)c = −(ab)c, wherefrom by addition we get the identity (1.9.1). Taking the scalar product of both sides of (1.9.1) with the vector d results in (da(b × c)) = (d × a) (b × c) = (ac) (bd) − (ab)(cd), or (a × b)(c × d) = (ac) (bd) − (ad)(bc).

(1.9.2)

Another identity is obtained from (1.9.1) by replacing a with a × b and b × c with c × d: (1.9.3) (a × b) × (c × d) = (abd)c − (abc)d.

1.10 Some Formulae of Spherical Trigonometry Applying the identity (1.9.2) for the special case when a, b, and c are unit vectors and d = a, we get, after rearranging, bc = (ac) (ad) + (a × b) (a × c). If the vectors a, b, and c originate at the same point, their terminal points determine a spherical triangle 𝐴𝐵𝐶 (Figure 1.10). For this triangle we have Fig. 1.10 bc = cos 𝑎, ca = cos 𝑏, ab = cos 𝑐, |a × b| = sin 𝑐, |a × c| = sin 𝑏, and the angle between the vectors a × b and a × c is just 𝛼. Thus, the above identity is nothing else but the spherical law of cosines cos 𝑎 = cos 𝑏 cos 𝑐 + sin 𝑏 sin 𝑐 cos 𝛼 (1.10.1) formulated in the language of vectors. Applying this theorem to the polar triangle yields the second law of cosines: cos 𝛼 = − cos 𝛽 cos 𝛾 + sin 𝛽 sin 𝛾 cos 𝑎.

(1.10.2)

28

1 Some Theorems from Elementary Geometry

For the very same special case as considered above, identity (1.9.3) transforms into (a × b) × (a × c) = (abc)a, yielding |(abc)| = sin 𝛼 sin 𝑏 sin 𝑐. Since a cyclic permutation of the vectors leaves the left-hand side unchanged, the same is true for the right-hand side: sin 𝛼 sin 𝑏 sin 𝑐 = sin 𝛽 sin 𝑐 sin 𝑎 = sin 𝛾 sin 𝑎 sin 𝑏, that is sin 𝛼 : sin 𝛽 : sin 𝛾 = sin 𝑎 : sin 𝑏 : sin 𝑐.

(1.10.3)

This is the spherical law of sines. Given any three components of a triangle that determine it uniquely, the two laws of cosines (1.10.1) and (1.10.2) and the law of sines (1.10.3) enable us to calculate all remaining components. Below we collect the formulae for a right triangle, where 𝑐 denotes, as usual, the hypotenuse: cos 𝑐 = cos 𝑎 cos 𝑏 = cot 𝛼 cot 𝛽 cos 𝛼 = sin 𝛽 cos 𝑎 = cot 𝑐 tan 𝑏 cos 𝛽 = sin 𝛼 cos 𝑏 = cot 𝑐 tan 𝑎 sin 𝑎 = sin 𝑐 sin 𝛼 = cot 𝛽 tan 𝑏 sin 𝑏 = sin 𝑐 sin 𝛽 = cot 𝛼 tan 𝑎.

These formulae can be easily remembered using Napier’s rule. Consider the hypotenuse 𝑐, the adjoining angles 𝛼 and 𝛽, and also the complementary angles 90◦ − 𝑎 and 90◦ − 𝑏 (Figure 1.11). If these five angles are ordered in their natural cyclic way, then the cosine of any of them is equal to the product of the sines of the two opposite angles or of the cotangents of the two neighboring angles. Fig. 1.11 We close with two formulae for the area of our right triangle Δ. We have Δ = 𝛼 + 𝛽 − 𝜋2 . Now we have, on one hand, sin

𝜋 2

 − 𝛽 = cos 𝑏 sin 𝛼,

1.11 Historical Remarks

29

and, on the other hand, tan

𝜋 2

 − 𝛽 = cos 𝑐 tan 𝛼.

Hence Δ = 𝛼 − arcsin(cos 𝑏 sin 𝛼) = 𝛼 − arctan(cos 𝑐 tan 𝛼).

(1.10.4)

1.11 Historical Remarks The theory of convex bodies is an extensive branch of geometry, whose foundations are linked with the names of J. Steiner, H. Brunn, H. Minkowski and others. Many mathematicians seem to agree with Minkowski’s words “I am interested in everything that is convex”, as even today this attractive theory is still in its most vigorous development. The theory’s literature includes the excellent textbook of Bonnesen and Fenchel [21], as well as the more specialized monograph of Alexandrov [1]. Regarding the notions of differential geometry found in this chapter, or introduced later we refer to the work [14] of Blaschke. The notions of affinity, as well as polarity with respect to a second degree curve or surface are the oldest ones of projective geometry. More, and in greater detail, can be found in every textbook on analytical or projective geometry, for example, in Schoenflies [121]. The inequality (1.2.1) is contained in the papers [43, 46] of the author. The necessary conditions for the optimal 𝑛-gons inscribed in and circumscribed about a given convex disk discussed in Section 1.3 were known to Steiner already. The question of existence of a solution to an extremum problem was considered already by Dirichlet, but it was Weierstrass who addressed this problem most explicitly. The first direct, completely elementary proof of the inequalities (1.3.1) and (1.3.2) was given by Kürschák [91]. The inequality (1.3.3) is due to the author [42], who was led to it by the circle packing problem considered later in this book. I do not know of an equally simple proof of the second inequality of (1.3.1). Inequalities (1.3.4) and (1.3.5) were derived as easy consequences of (1.3.1) by the author [43]. Elaborate historic details on the isoperimetric problem can be found in the work of Bonnesen and Fenchel [21] mentioned above, in the extremely captivating small volume of Blaschke [13], and in the encyclopedia article by Steinitz [126]. The notion of inner parallel domain was introduced in 1930 by F. Riesz [114]. The fact that this notion can be used to derive Bonnesen’s inequality (1.4.5) was observed by B.v.SZ. Nagy [130] (see footnote 5). If we want to prove the isoperimetric inequality (1.4.1) using only the tools of elementary geometry, we encounter the difficulty that already the notions of area and perimeter are non-elementary. We can avoid this difficulty by proving the isoperimetric inequality for polygons. From this the general inequality follows by a limiting process, using merely the definitions of area and perimeter, thereby reducing the non-elementary tools to a minimum. According to Bol [20], who found

30

1 Some Theorems from Elementary Geometry

the proof through inner parallel domains independently of B.v.SZ. Nagy, this proof of the isoperimetric inequality for polygons is the first “really simple” one using elementary means only. Concerning this we note that for polygons the first integralgeometric proof of the isoperimetric inequality by Santaló (Blaschke [16]) can easily be translated into the language of elementary geometry. This yields perhaps a simpler proof than the one through inner parallel domains. In this way, the inequality 𝐿 2 − 4𝜋𝐹 ≥ (𝐿 − 2𝜋 𝜚) 2 ,

191 ▶

𝑟 ≤ 𝜚 ≤ 𝑅,

where 𝜚 is an arbitrary number between the inradius and circumradius of 𝐹, can be derived even for nonconvex polygons (see Fejes Tóth [56]). Yet another simple proof was given by Hadwiger [76]. The inequality (1.5.1) was observed by L. Fejér as a participant in the Loránd Eötvös Mathematical Contest in 1897 (see T. Radó [110]). However, it is likely that this inequality is much older. We present here the elegant proof of this inequality found by the short-lived Hungarian mathematician I. Ádám, which can be carried over to space as well. Consider the circle 𝐾 passing through the midpoints of the sides of the triangle Δ. The radius of 𝐾 is 𝑅/2. On the other hand, the triangle homothetic to Δ with incircle 𝐾 contains Δ. Therefore 𝐾 is larger than the incircle of Δ except when it is the incircle of Δ. Thus, indeed, we have 𝑅2 ≤ 𝑟, and equality holds only in the indicated case, that is, if Δ is equilateral. The inequality (1.5.2) is due to Schreiber [122]; its proof as a consequence of the sharper inequality (1.5.3) was given by the author [45]. The proof of the inequality (1.5.4) presented here was given by Mordell [106,107]. A simple, purely elementary proof of this pretty inequality is currently not known. The equivalence of the inequalities (1.5.5) and (1.5.6) was observed by the author [45]. For further possible generalizations in space, we mention yet another inequality on triangles: Denoting by 𝜚1 , 𝜚2 , 𝜚3 the radii of the excircles and by 𝑅 the circumradius of a triangle we have 𝑀1 ( 𝜚1 , 𝜚2 , 𝜚3 ) ≤

3 𝑅 ≤ 𝑀2 ( 𝜚1 , 𝜚2 , 𝜚3 ), 2

where 𝑀𝑘 denotes the power mean of degree 𝑘. The first inequality is easily seen; the second one was found by Hajós [81] after Baron [8] and Erdős [26] had previously proved the inequality 23 𝑅 ≤ 𝑀∞ ( 𝜚 1 , 𝜚2 , 𝜚3 ) = max( 𝜚1 , 𝜚2 , 𝜚3 ) and the author [58] had proved 23 𝑅 ≤ 𝑀4 ( 𝜚1 , 𝜚2 , 𝜚3 ). In connection with Sections 1.6 and 1.7 we refer the reader, besides the already mentioned work of Brückner, to the excellent textbooks by Steinitz and Rademacher [128], Coxeter [23] and Alexandrov [1].

Chapter 2

Theorems from the Theory of Convex Bodies

After the two well-known theorems discussed in the first two sections of this chapter some more specialized theorems will follow, mainly concerning the maximumarea 𝑛-gon inscribed in, and the minimum-area 𝑛-gon circumscribed about, a given convex disk. In connection with the problem of finding the curves that are worst approximated by an 𝑛-gon, we will learn some interesting extremum properties of the conic sections. Finally, we discuss the most important notions and theorems of Integral Geometry—a new, prosperous area of geometry, that also has some points of contact with the problems of arrangements of figures.

2.1 Blaschke’s Selection Theorem By the Hausdorff distance between two convex disks 𝐺 and 𝐻 we understand the smallest number 𝜂 = 𝜂(𝐺, 𝐻) with the property that 𝐻 is contained in 𝐺 𝜂 and 𝐺 is contained in 𝐻 𝜂 , where 𝐺 𝜂 and 𝐻 𝜂 denote the respective parallel domains at distance 𝜂. This measure of deviation satisfies the usual requirements of a metric: We have 1. 𝜂(𝐺, 𝐻) ≥ 0, and equality occurs only if 𝐺 and 𝐻 coincide. 2. 𝜂(𝐺, 𝐻) = 𝜂(𝐻, 𝐺). 3. If 𝐺, 𝐻 and 𝐼 are three convex disks, then the triangle inequality 𝜂(𝐺, 𝐻) + 𝜂(𝐻, 𝐼) ≥ 𝜂(𝐺, 𝐼) is satisfied. We say that the sequence 𝐺 1 , 𝐺 2 , . . . of convex disks converges to the (possibly degenerate) disk 𝐺 if 𝜂(𝐺, 𝐺 1 ), 𝜂(𝐺, 𝐺 2 ), . . . is a null-sequence. We now prove one of the fundamental theorems in convexity, Blaschke’s selection theorem [13]: Every infinite sequence of convex disks, all contained in a fixed square, contains a convergent subsequence.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_2

31

32

2 Theorems from the Theory of Convex Bodies

The proof proceeds in two steps. First we prove that from the considered sequence a Cauchy subsequence can be selected. Here, a sequence 𝐶1 , 𝐶2 , . . . of convex disks is called after Cauchy if, for every 𝜀 > 0, a number 𝑁 exists such that 𝜂(𝐶𝑖 , 𝐶 𝑗 ) < 𝜀 holds for every pair of indices 𝑖, 𝑗 > 𝑁. Then we prove that every Cauchy sequence is convergent. Let 𝐺 1 , 𝐺 2 , . . . be an infinite sequence of convex disks, all contained in a square 𝑄. Partition 𝑄 into four congruent sub-squares 𝑄 1 , 𝑄 2 , 𝑄 3 , 𝑄 4 and to each disk assign the squares met by it. Since the number of combinations 𝑄 1 , . . . , 𝑄 4 , 𝑄 1 𝑄 2 , . . . , 𝑄 3 𝑄 4 , 𝑄 1 𝑄 2 𝑄 3 , . . . , 𝑄 2 𝑄 3 𝑄 4 , 𝑄 1 𝑄 2 𝑄 3 𝑄 4 is finite, by the pigeonhole principle one of them must occur infinitely many times as assigned to the disks of the sequence. Select from the sequence 𝐺 1 , 𝐺 2 , . . . those disks 𝐺 11 , 𝐺 12 , . . . to which that very combination is assigned. Next, partition each of the sub-squares into four sub-squares as before, and to each disk 𝐺 𝑖1 assign the sub-squares 𝑄 𝜈 𝜇 met by it. Then, again, a combination of the 16 sub-squares 𝑄 𝜈 𝜇 (𝜈, 𝜇 ≤ 4) exists that corresponds to infinitely many of the disks from the sequence 𝐺 11 , 𝐺 12 , . . .. These disks form a new subsequence, which we denote by 𝐺 21 , 𝐺 22 , . . .. We proceed in this way and consider the 𝑖-th subsequence 𝐺 1𝑖 , 𝐺 2𝑖 , . . . defined so that each of its disks meets partition of 𝑄 into 4𝑖 sub-squares. the same members of the 𝑖-th regular √ 𝑗 𝑗 2𝑄 𝑖 𝑖 It is evident that 𝜂(𝐺 𝜈 , 𝐺 𝜇 ) ≤ 2𝑖 . Furthermore, since the sequence 𝐺 1 , 𝐺 2 , . . . √

𝑗

is a subsequence of 𝐺 𝑖1 , 𝐺 𝑖2 , . . . for every 𝑗 ≥ 𝑖, we have 𝜂(𝐺 𝑖𝜈 , 𝐺 𝜇 ) ≤ √

√

2𝑄 . 2𝑖

𝑗 𝜂(𝐺 𝑖𝑖 , 𝐺 𝑗 )

2𝑄 2𝑁

2𝑄 , 2𝑖

hence

Consequently ≤ for each pair of indices also ≤ 1 𝑖, 𝑗 > 𝑁. Thereby it is established that the subsequence 𝐺 1 , 𝐺 22 , . . . of the original sequence is a Cauchy sequence. Let now 𝐶1 , 𝐶2 , . . . be a Cauchy sequence of convex disks. By choosing arbitrarily a point 𝑃 𝜈 from each disk 𝐶𝜈 , we get a sequence of points 𝑃1 , 𝑃2 , . . ., which, by its boundedness, must contain at least one accumulation point in 𝑄. Let 𝐷 denote the set, obviously a closed one, of all points that arise in this way, as accumulation points of such sequences. Given any 𝜚 > 0, there exists a natural number 𝑛 such that, for all 𝜈 ≥ 𝑛, 𝐶𝜈 ⊂ 𝐷 𝜚 . Otherwise infinitely many disks 𝐶𝜈 would protrude from 𝐷 𝜚 , and we could choose a sequence of points 𝑃 𝜈 in the same manner as above, that would contain an infinite subsequence of points lying outside 𝐷 𝜚 . This point sequence would then have an accumulation point lying outside 𝐷, contradicting the way 𝐷 was constructed. On the other hand, there is a natural number 𝑚 such that for all 𝜈 > 𝑚, 𝐷 ⊂ (𝐶𝜈 ) 𝜚 . Indeed: Since the disks 𝐶𝜈 form a Cauchy sequence, for sufficiently large 𝜈 the set (𝐶𝜈 ) 𝜚 must contain every set 𝐶 𝜇 with 𝜇 > 𝜈; thereby, in view of the definition of 𝐷, the above assertion follows. From these two facts it follows easily that 𝜂(𝐶𝜈 , 𝐷) is a null-sequence. It remains to show that 𝐷 is convex. If 𝑃 and 𝑄 are two points in 𝐷, then, for an arbitrarily given 𝜚, a sufficiently large 𝑛 can be chosen so that, for 𝜈 ≥ 𝑛, both points belong to the parallel domain (𝐶𝜈 ) 𝜚 . Thus, for every 𝜈 ≥ 𝑛, each of the two circles 𝑃𝜌 and 𝑄 𝜌 contains a point, say 𝑃 𝜈 and 𝑄 𝜈 , respectively, from 𝐶𝜈 . As 𝐶𝜈 is convex, it contains the entire segment 𝑃 𝜈 𝑄 𝜈 . In view of the construction of 𝐷, one easily concludes now that the segment 𝑃𝑄 must also lie in 𝐷. Hence the convex 𝜂(𝐺 𝑖𝑖 , 𝐺 𝑖𝑗 )

2.2 Jensen’s Inequality

33

disks 𝐶𝜈 converge to a closed convex set, though possibly degenerated to a point or a segment. This completes the proof of the selection theorem. The proof can be immediately generalized to the 𝑛-dimensional space. Also, it is worth noting that the theorem holds not just for convex, but for arbitrary closed sets as well. A direct consequence of the selection theorem is the following: If an infinite collection of convex disks lies in a square, then for every 𝜀 > 0 a finite number of disks can be chosen from the collection so that the distance from an arbitrary disk in the collection to one of the chosen disks is smaller than 𝜀. From the given collection we successively pick certain disks, which we will denote by 𝐺 1 , 𝐺 2 , . . ., in the following manner: 𝐺 1 is arbitrary, 𝐺 2 should satisfy the inequality 𝜂(𝐺 1 , 𝐺 2 ) ≥ 𝜀, 𝐺 3 should satisfy both 𝜂(𝐺 1 , 𝐺 3 ) ≥ 𝜀 and 𝜂(𝐺 2 , 𝐺 3 ) ≥ 𝜀, and so on. This procedure must terminate after some finite number of disks 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑁 have been picked, or else the infinite sequence of disks 𝐺 1 , 𝐺 2 , . . . would not have a Cauchy subsequence. But this means now that the distance between every disk from the given collection and one of the disks 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑁 is smaller than 𝜀. Finally, we introduce two other notions of deviation. By the area-deviation 𝜏(𝐺, 𝐻) of the convex disks 𝐺 and 𝐻, we mean the area of the parts of the disks consisting of points that belong to exactly one of them. 𝜏(𝐺, 𝐻) can also be defined as the difference between the areas of the union and of the intersection of 𝐺 and 𝐻. The difference 𝜆(𝐺, 𝐻) between the perimeters of the union and the intersection is the perimeter deviation of 𝐺 and 𝐻. The quantities 𝜂, 𝜏 and 𝜆 are linked by certain inequalities. For example, 𝜏 < 𝐿𝜂 + 𝜋𝜂2 , where 𝐿 denotes the perimeter of the intersection 𝐺𝐻.

2.2 Jensen’s Inequality A function 𝑓 (𝑥) defined on an interval (𝑎, 𝑏) is said to be convex (from below) in the interval (𝑎, 𝑏) if for any two values 𝑥1 and 𝑥2 in the interval (𝑎, 𝑏) the inequality 𝑥 + 𝑥  𝑓 (𝑥 1 ) + 𝑓 (𝑥2 ) 1 2 ≥ 𝑓 2 2 holds. If such an inequality holds with the sign “ ≤ ” in place of “ ≥ ”, then the function is said to be concave (from below). If equality occurs therein only for 𝑥1 = 𝑥2 , then the function is strictly convex, respectively strictly concave. We now prove the following inequality of Jensen:

34

2 Theorems from the Theory of Convex Bodies

If 𝑓 (𝑥) is a convex function in a given interval, then for any 𝑛 values 𝑥 1 , . . . , 𝑥 𝑛 from the interval the inequality 𝑥 + · · · + 𝑥  𝑓 (𝑥1 ) + · · · + 𝑓 (𝑥 𝑛 ) 1 𝑛 ≥ 𝑓 𝑛 𝑛

(2.2.1)

holds. Moreover, if 𝑓 (𝑥) is strictly convex, then equality holds only when 𝑥1 = · · · = 𝑥𝑛. The same inequality, but with the “ ≤ ” sign, holds for concave functions. For the proof of (2.2.1) we begin by showing that the inequality holds for 𝑛 = 4:

4 2 2 𝑓 𝑥1 +𝑥 + 2 𝑓 𝑥3 +𝑥 𝑓 (𝑥1 ) + · · · + 𝑓 (𝑥4 ) 2 2 ≥ 4 4 ! 𝑥3 +𝑥4 𝑥1 +𝑥2 𝑥 + · · · + 𝑥  + 1 4 2 2 ≥ 𝑓 = 𝑓 . 2 4 Using this inequality and the definition of convexity, we verify (2.2.1) in a quite analogous way for 𝑛 = 8, and by proceeding similarly, also for 𝑛 = 16, 32, . . . . We now show that from the validity of the inequality (2.2.1) for a certain value of 𝑛 its validity for 𝑛 − 1 follows. Namely, if we set 𝑥𝑛 =

𝑥 1 + · · · + 𝑥 𝑛−1 , 𝑛−1

then

𝑥1 + · · · + 𝑥 𝑛−1 + 𝑥 𝑛 = 𝑥𝑛 . 𝑛 Therefore (2.2.1) indeed implies 𝑓 (𝑥1 ) + · · · + 𝑓 (𝑥 𝑛−1 ) ≥ 𝑛 𝑓 (𝑥 𝑛 ) − 𝑓 (𝑥 𝑛 ) = (𝑛 − 1) 𝑓

𝑥 + · · · + 𝑥  1 𝑛−1 . 𝑛−1

This completes the proof of (2.2.1) in general. The case of equality is evident. If in addition to being convex 𝑓 (𝑥) is also continuous (which it must be if it is bounded), then the more general inequality   𝑞 1 𝑓 (𝑥1 ) + · · · + 𝑞 𝑛 𝑓 (𝑥 𝑛 ) 𝑞 1 𝑥1 + · · · + 𝑞 𝑛 𝑥 𝑛 ≥ 𝑓 𝑞1 + · · · + 𝑞 𝑛 𝑞1 + · · · + 𝑞 𝑛 holds, where 𝑞 1 , . . . , 𝑞 𝑛 are arbitrary positive weights. Indeed, due to continuity, it suffices to prove this inequality for rational, or what is the same, for integral weights. In that case we deal just with the inequality (2.2.1) where some of the values 𝑥𝑖 coincide. A sufficient condition for convexity of a function 𝑓 (𝑥) is that it should possess a continuous, increasing derivative 𝑓 ′ (𝑥) over the appropriate interval. This criterion is certainly satisfied if 𝑓 ′′ (𝑥) > 0.

2.2 Jensen’s Inequality

35

For the function 𝑓 (𝑥) = log 𝑥, Jensen’s inequality yields the fact, established by Cauchy, that the geometric mean of positive quantities cannot exceed their arithmetic mean. Further, the inequalities (1.3.1) and (1.3.2) are simple consequences of (2.2.1) as well. For example, the perimeter of a convex 𝑛-gon inscribed in the unit circle is 𝐿 = 2(sin 𝛼1 + · · · + sin 𝛼𝑛 ), where 2𝛼1 , . . . , 2𝛼𝑛 denote the circular arcs corresponding to the polygon’s sides. Since sin 𝑥 is concave on the interval (0, 𝜋), we get 𝐿 ≤ 2𝑛 sin 𝑛𝜋 . We now apply Jensen’s inequality to derive (1.5.7). We prove a somewhat more general statement: If 𝑅1 , . . . , 𝑅𝑛 denote the distances from the vertices of a convex 𝑛-gon to a point 𝑂 in the interior of the 𝑛-gon, and if 𝑟 1 , . . . , 𝑟 𝑛 are the distances from the sides of the 𝑛-gon to 𝑂, then 𝐺 (𝑅1 , . . . , 𝑅𝑛 ) 𝜋 (2.2.2) ≥ sec . 𝐺 (𝑟 1 , . . . , 𝑟 𝑛 ) 𝑛 Denote the vertices in their natural cyclic order by 𝑃1 , . . . , 𝑃𝑛 , 𝑃𝑛+1 = 𝑃1 , the distance from the line 𝑃𝑖 𝑃𝑖+1 to 𝑂 by 𝑟 𝑖 , the angle 𝑃𝑖 𝑂𝑃𝑖+1 by 2𝛼𝑖 and the distance 𝑂𝑃𝑖 by 𝑅𝑖 . If the points 𝑃𝑖 and 𝑃𝑖+1 are moved along the fixed rays 𝑂𝑃𝑖 and 𝑂𝑃𝑖+1 so that the area of the triangle 𝑂𝑃𝑖 𝑃𝑖+1 remains constant, then the lines 𝑃𝑖 𝑃𝑖+1 envelop a hyperbola. Since among the tangent lines of a hyperbola those that form a right triangle with the hyperbola’s asymptotes 𝑂𝑃𝑖 and 𝑂𝑃𝑖+1 are the most distant from the hyperbola’s center 𝑂, it follows that 𝑟 𝑖2 ≤ 𝑅𝑖 𝑅𝑖+1 cos2 𝛼𝑖 . This yields 2 log 𝑟 𝑖 ≤ log 𝑅𝑖 + log 𝑅𝑖+1 + 2 log cos 𝛼𝑖 , and by adding these inequalities for 𝑖 = 1, . . . , 𝑛 we get 2 log(𝑟 1 · . . . · 𝑟 𝑛 ) ≤ 2 log(𝑅1 · . . . · 𝑅𝑛 ) + 2(log cos 𝛼1 + · · · + log cos 𝛼𝑛 ). Since log cos 𝑥 is concave in (0, 𝜋2 ), we have 𝜋 log(𝑟 1 · . . . · 𝑟 𝑛 ) ≤ log(𝑅1 · . . . · 𝑅𝑛 ) + 𝑛 log cos , 𝑛 that is 𝑟 1 · . . . · 𝑟 𝑛 ≤ 𝑅1 · . . . · 𝑅𝑛 cos𝑛

𝜋 , 𝑛

which was to be proved. Later we shall often use Jensen’s inequality for convex functions of two variables. A function 𝑓 (𝑥, 𝑦) defined on a convex domain 𝐵 in the 𝑥𝑦-plane is said to be convex if for any two points (𝑥 1 , 𝑦 1 ) and (𝑥2 , 𝑦 2 ) in 𝐵 the inequality 𝑥 + 𝑥 𝑦 + 𝑦  𝑓 (𝑥1 , 𝑦 1 ) + 𝑓 (𝑥2 , 𝑦 2 ) 1 2 1 2 ≥ 𝑓 , 2 2 2

36

2 Theorems from the Theory of Convex Bodies

holds. This definition yields, in a quite analogous way as above, that we have for 𝑛 arbitrary points (𝑥 1 , 𝑦 1 ), . . . , (𝑥 𝑛 , 𝑦 𝑛 ) in 𝐵, 𝑥 + · · · + 𝑥 𝑦 + · · · + 𝑦  𝑓 (𝑥1 , 𝑦 1 ) + · · · + 𝑓 (𝑥 𝑛 , 𝑦 𝑛 ) 1 1 𝑛 𝑛 ≥ 𝑓 , . 𝑛 𝑛 𝑛 A sufficient condition for 𝑓 (𝑥, 𝑦) to be either convex or concave is that 𝑓 (𝑥, 𝑦) should have continuous second partial derivatives and that the determinant 𝑓 𝑥 𝑥 𝑓 𝑦 𝑦 − 𝑓 𝑥2𝑦 formed by them should be positive. This condition implies that the surface described by 𝑧 = 𝑓 (𝑥, 𝑦) has an elliptic curvature. In this case, the distinction between convexity and concavity of 𝑓 (𝑥, 𝑦) is determined by the sign of 𝑓 𝑥 𝑥 or 𝑓 𝑦 𝑦 . Namely, if 𝑓 𝑥 𝑥 ≥ 0 (and consequently also 𝑓 𝑦 𝑦 ≥ 0), then 𝑓 (𝑥, 𝑦) is convex, but if 𝑓 𝑥 𝑥 , 𝑓 𝑦 𝑦 ≤ 0, then our function is concave. If the last conditions are satisfied, then the function 𝑓 (𝑥, 𝑦) is convex or concave, even if 𝑓 𝑥 𝑥 𝑓 𝑦 𝑦 − 𝑓 𝑥2𝑦 ≥ 0. Applying Jensen’s inequality to the function 𝑧 = 𝑥 𝛼 𝑦 𝛽 ; 𝑥, 𝑦 ≥ 0, 0 ≤ 𝛼, 𝛽 ≤ 1, 𝛼 + 𝛽 = 1, which, in view of 𝑧 𝑥 𝑥 𝑧 𝑦 𝑦 − 𝑧2𝑥 𝑦 = 0, 𝑧 𝑥 𝑥 ≤ 0, 𝑧 𝑦 𝑦 ≤ 0, is concave, we get 𝛽

𝛽

𝑥 1𝛼 𝑦 1 + · · · + 𝑥 𝑛𝛼 𝑦 𝑛 ≤ (𝑥 1 + · · · + 𝑥 𝑛 ) 𝛼 (𝑦 1 + · · · + 𝑦 𝑛 ) 𝛽 .

(2.2.3)

𝛽

Setting 𝑥𝑖𝛼 = 𝑎 𝑖 , 𝑦 𝑖 = 𝑏 𝑖 the inequality 𝛼  1   1 𝛽 1 1 𝑏 1𝛽 + · · · + 𝑏 𝑛𝛽 𝑎 1 𝑏 1 + · · · + 𝑎 𝑛 𝑏 𝑛 ≤ 𝑎 1𝛼 + · · · + 𝑎 𝑛𝛼 follows, which in the case 𝛼 = 𝛽 =

1 2

turns into Cauchy’s inequality

(𝑎 1 𝑏 1 + · · · + 𝑎 𝑛 𝑏 𝑛 ) 2 ≤ (𝑎 21 + · · · + 𝑎 2𝑛 ) (𝑏 21 + · · · + 𝑏 2𝑛 ). We should also mention the integral inequality  𝛼 ∫

∫

∫ 𝛼 𝛽

𝑓 𝑔 d𝑥 ≤

𝑓 d𝑥

𝛽 𝑔d𝑥

; 𝛼 + 𝛽 = 1,

obtained by applying the inequality (2.2.3) to the approximating sums of the integral. An important special case is Schwarz’s inequality 2

∫ 𝑓 𝑔d𝑥

∫ ≤

2

𝑓 d𝑥

∫

𝑔 2 d𝑥.

37

2.3 Dowker’s Theorems

2.3 Dowker’s Theorems In the next chapter we will derive important consequences of the following theorems of Dowker [25]: Let 𝑇𝑛 be an 𝑛-gon of maximum area inscribed in a given convex disk and let 𝑈𝑛 be an 𝑛-gon of minimum area circumscribed about the same disk. Then the sequence 𝑇3 , 𝑇4 , . . . is concave: 𝑇𝑛−1 + 𝑇𝑛+1 ≤ 2 𝑇𝑛 ,

𝑛 = 4, 5, . . .

(2.3.1)

𝑛 = 4, 5, . . . .

(2.3.2)

and the sequence 𝑈3 , 𝑈4 , . . . is convex: 𝑈𝑛−1 + 𝑈𝑛+1 ≥ 2 𝑈𝑛 ,

For the proof of (2.3.1) we consider two inscribed polygons 𝐴 ≡ 𝐴1 . . . 𝐴𝑛−1 and 𝐵 ≡ 𝐵1 . . . 𝐵𝑛+1 with number of vertices 𝑛 − 1 and 𝑛 + 1, respectively. We also assume that not all vertices of 𝐴 lie on an arc determined by two parallel supporting lines. Apart from that, let 𝐴 and 𝐵 be chosen arbitrarily. It suffices to show that to 𝐴 and 𝐵 two inscribed 𝑛-gons 𝐶 and 𝐷 can be assigned so that 𝐴 + 𝐵 ≤ 𝐶 + 𝐷. Namely, 𝑇𝑛−1 surely satisfies the condition posed on 𝐴, therefore taking 𝐴 ≡ 𝑇𝑛−1 and 𝐵 ≡ 𝑇𝑛+1 we get Fig. 2.1 𝑇𝑛−1 + 𝑇𝑛+1 ≤ 𝐶 + 𝐷 ≤ 2𝑇𝑛 . We distinguish two cases. Case 1. We have, say 𝐴1 ≤ 𝐵1 < 𝐵2 < 𝐵3 < 𝐴2 , where the symbol < refers to the cyclic order of vertices (Figure 2.1). Let 𝐶 ≡ 𝐴1 𝐵2 𝐴2 . . . 𝐴𝑛−1 and 𝐷 ≡ 𝐵1 𝐵3 . . . 𝐵𝑛+1 . Then we have 𝐶 + 𝐷 = 𝐴 + 𝐵 + 𝐴1 𝐵2 𝐴2 − 𝐵1 𝐵2 𝐵3 . Let us move 𝐴1 to 𝐵1 and afterwards 𝐴2 to 𝐵3 . Then, by our assumption on the polygon 𝐴, the area of the triangle 𝐴1 𝐵2 𝐴2 decreases in both steps. Thus we have 𝐴1 𝐵2 𝐴2 > 𝐵1 𝐵2 𝐵3 , consequently 𝐶 + 𝐷 > 𝐴 + 𝐵. Case 2. Under no numbering of the vertices of 𝐴 and 𝐵 is there a sequence of vertices of the above type. Then under an appropriate numbering we have 𝐴1 ≤ 𝐵1 < 𝐵2 < 𝐴2 ≤ 𝐴𝑠−1 ≤ 𝐵𝑠 < 𝐵𝑠+1 < 𝐴𝑠 (Figure 2.2). Now let 𝐶 ≡ 𝐴1 𝐵2 . . . 𝐵𝑠 𝐴𝑠 . . . 𝐴𝑛−1 and 𝐷 ≡ 𝐵1 𝐴2 . . . 𝐴𝑠−1 𝐵𝑠+1 . . . 𝐵𝑛+1 . Denoting the point of intersection of 𝐴1 𝐵2 with 𝐵1 𝐴2 by 𝑃 and the point of intersection of 𝐴𝑠−1 𝐵𝑠+1 with 𝐵𝑠 𝐴𝑠 by 𝑄 we have 𝐶 + 𝐷 = 𝐴 + 𝐵 + 𝐴1 𝑃 𝐴2 − 𝐵1 𝑃𝐵2 + 𝐴𝑠−1 𝑄 𝐴𝑠 − 𝐵𝑠 𝑄𝐵𝑠+1 .

38

2 Theorems from the Theory of Convex Bodies

If we had 𝐴1 𝐵1 ∥ 𝐴2 𝐵2 , then the triangles 𝐴1 𝑃 𝐴2 and 𝐵1 𝑃𝐵2 would have equal areas. Since, by the assumption on 𝐴, the rays 𝐴1 𝐵1 and 𝐴2 𝐵2 intersect, we have 𝐴1 𝑃 𝐴2 ≥ 𝐵1 𝑃𝐵2 , and for the same reason 𝐴𝑠−1 𝑄 𝐴𝑠 ≥ 𝐵𝑠 𝑄𝐵𝑠+1 . Hence we have 𝐶 + 𝐷 > 𝐴 + 𝐵 in this case as well. For the proof of (2.3.2) we consider two circumscribed polygons 𝑎 ≡ 𝑎 1 . . . 𝑎 𝑛−1 and 𝑏 ≡ 𝑏 1 . . . 𝑏 𝑛+1 , where 𝑎 𝑖 and 𝑏 𝑖 denote the sides of the respective polygons. It suffices to show that to 𝑎 and 𝑏 two circumscribed 𝑛-gons 𝑐 Fig. 2.2 and 𝑑 can be assigned with total area 𝑐 + 𝑑 ≤ 𝑎 + 𝑏. As in the above proof, we again consider two cases. The symbol < will now be used to denote the cyclic order of sides.

Fig. 2.3

Fig. 2.4

Case 1. There exists a sequence of sides 𝑎 1 ≤ 𝑏 1 < 𝑏 2 < 𝑏 3 < 𝑎 2 (Figure 2.3). Let 𝑐 ≡ 𝑎 1 𝑏 2 𝑎 2 . . . 𝑎 𝑛−1 and 𝑑 ≡ 𝑏 1 𝑏 3 . . . 𝑏 𝑛+1 ; then we have 𝑎 + 𝑏 = 𝑐 + 𝑑 + 𝑎 1 𝑏 3 𝑏 1 𝑏 2 + 𝑎 1 𝑎 2 𝑏 2 𝑏 3 > 𝑐 + 𝑑. Case 2. There is no sequence of sides of the above type. Then we have the sequence 𝑎 1 ≤ 𝑏 1 < 𝑏 2 < 𝑎 2 ≤ 𝑎 𝑠−1 ≤ 𝑏 𝑠 < 𝑏 𝑠+1 < 𝑎 𝑠 (Figure 2.4). Let 𝑐 ≡ 𝑎 1 𝑏 2 . . . 𝑏 𝑠 𝑎 𝑠 . . . 𝑎 𝑛−1 , 𝑑 ≡ 𝑏 1 𝑎 2 . . . 𝑎 𝑠−1 𝑏 𝑠+1 . . . 𝑏 𝑛+1 . Then we have 𝑎 + 𝑏 = 𝑐 + 𝑑 + 𝑎 1 𝑎 2 𝑏 1 𝑏 2 + 𝑎 𝑠−1 𝑎 𝑠 𝑏 𝑠 𝑏 𝑠+1 ≥ 𝑐 + 𝑑.

2.4 An Extremum Property of the Ellipse

39

2.4 An Extremum Property of the Ellipse We prove the following theorem: Let 𝑇 be a convex disk and let 𝑇𝑛 be an 𝑛-gon of maximum area inscribed in it. Then 𝑛 2𝜋 𝑇𝑛 ≥ 𝑇 sin (2.4.1) 2𝜋 𝑛 with equality only if 𝑇 is an ellipse. Thus, among the convex disks of a given area the ellipses are the most poorly filled by inscribed 𝑛-gons. Notice that the extremal disk does not depend on 𝑛, while for the dual problem concerning circumscribed 𝑛-gons this is not so. For the proof we choose a Cartesian coordinate system 𝑥 𝑦 so that the endpoints of a longest chord of 𝑇 are (−1, 0) and (1, 0). In this coordinate system the boundary 𝐾 of 𝑇 can be represented by the system of equations 𝑥 = cos 𝑡,

𝑦 = 𝑒(𝑡) sin 𝑡;

0 ≤ 𝑡 < 2𝜋,

(2.4.2)

where 𝑒(𝑡) is a positive, continuous, (2𝜋)-periodic function of 𝑡. Inscribe an 𝑛-gon in the curve 𝐾 whose vertices are given by the parameter-values 𝑡 = 𝑡1 , . . . , 𝑡 𝑛 . The area of this 𝑛-gon is 𝑛

𝑇𝑛 (𝑡1 , . . . , 𝑡 𝑛 ) =

1 ∑︁ 𝑒(𝑡𝑖 ) sin 𝑡 𝑖 (cos 𝑡 𝑖−1 − cos 𝑡 𝑖+1 ); 2 𝑖=1

By setting 𝑡1 = 𝑡, 𝑡2 = 𝑡 +

2𝜋 𝑛 , . . . , 𝑡𝑛

𝑡 0 = 𝑡 𝑛 , 𝑡 𝑛+1 = 𝑡 1 .

= 𝑡 + (𝑛 − 1) 2𝑛𝜋 we get

     𝑛−1  2𝜋 ∑︁ 2𝜋 2𝜋 2𝜋 = sin 𝑒 𝑡 +𝑖 sin2 𝑡 + 𝑖 . 𝑇𝑛 (𝑡) = 𝑇𝑛 𝑡, . . . , 𝑡 + (𝑛 − 1) 𝑛 𝑛 𝑖=0 𝑛 𝑛 The mean value of 𝑇𝑛 (𝑡) equals 1 2𝜋

∫

2𝜋

𝑇𝑛 (𝑡) d𝑡 = 0

𝑛 2𝜋 sin 2𝜋 𝑛

∫ 0

2𝜋

𝑒(𝑡) sin2 𝑡 d𝑡 = 𝑇

𝑛 2𝜋 sin . 2𝜋 𝑛

Therefore, there is a value 𝑡 = 𝑡0 such that 𝑇𝑛 (𝑡 0 ) ≥ 𝑇 2𝑛𝜋 sin 2𝑛𝜋 , which completes the proof of (2.4.1). It remains to show that in (2.4.1) equality holds only for an ellipse. Since 𝑇𝑛 (𝑡) is continuous, if it is not constant, there is a value 𝑡 0 for which 𝑇𝑛 (𝑡 0 ) > 𝑇 2𝑛𝜋 sin 2𝑛𝜋 . However, as was shown by Sas [119], 𝑇𝑛 (𝑡) can be constant even if 𝑒(𝑡) is not. We show that in this case 𝑇𝑛 (𝑡) is not an inscribed 𝑛-gon of maximum area.

40

2 Theorems from the Theory of Convex Bodies

Assume that the polygons 𝑇𝑛 (𝑡) cannot be increased for any value of 𝑡. Then at every vertex of 𝑇𝑛 (𝑡), 𝑇 has a supporting line parallel to the line connecting the neighboring vertices. This implies that the curve 𝐾 has a continuously varying tangent and, furthermore     𝑦 𝑡 + 2𝑛𝜋 − 𝑦 𝑡 − 2𝑛𝜋 𝑦 ′ (𝑡) = . 2 sin 2𝑛𝜋 Hence, 𝑦(𝑡) has a continuous derivative of bounded variation. Expanding 𝑦(𝑡) into the Fourier series ∞ ∑︁ 𝑦(𝑡) = (𝑎 𝑘 cos 𝑘𝑡 + 𝑏 𝑘 sin 𝑘𝑡) 𝑘=0

we get by differentiation ∞ ∑︁ 𝑘=1

𝑘 (𝑏 𝑘 cos 𝑘𝑡 − 𝑎 𝑘 sin 𝑘𝑡) =

∞ ∑︁ sin 𝑘 2 𝜋 𝑛

𝑘=1

sin 2𝑛𝜋

(𝑏 𝑘 cos 𝑘𝑡 − 𝑎 𝑘 sin 𝑘𝑡).

From this, it follows by Cantor’s uniqueness theorem for trigonometric series that 𝑎 𝑘 = 𝑏 𝑘 = 0 for 𝑘 ≥ 2, that is, 𝑦(𝑡) = 𝑎 0 + 𝑎 1 cos 𝑡 + 𝑏 1 sin 𝑡. Since 𝑦(0) = 𝑦(𝜋) = 0, this is possible only if 𝑦 = 𝑏 1 sin 𝑡, that is if the curve is an ellipse. We briefly recapitulate the above construction. We draw a circle whose diameter is a longest chord of the curve and inscribe in the circle a regular 𝑛-gon. Then we project the vertices perpendicularly to the longest chord onto the curve, thereby obtaining an 𝑛-gon inscribed in the curve. If not all of these 𝑛-gons have the same area, then there is one among them with area greater than 𝑇 2𝑛𝜋 sin 2𝑛𝜋 . If, however, all these 𝑛-gons do have the same area, then this common area is necessarily 𝑇 2𝑛𝜋 sin 2𝑛𝜋 . In this latter case, except when the curve is an ellipse, among the 𝑛-gons considered there is one whose area can be increased by a suitable displacement of a vertex. Consider now the analogous problem for circumscribed polygons: among the convex disks of a given area for which one does the area 𝑈𝑛 of the circumscribed 𝑛-gon of minimum area attain its maximum? The inequality 𝑈𝑛 ≤ 𝑇 𝑛𝜋 tan 𝑛𝜋 , which would mean that the extremal disks are also ellipses in this case, does not hold generally. For instance, in the case of the unit √ square we have 𝑈3 = 2, which is greater than 𝜋3 tan 𝜋3 = 𝜋27 = 1.65 . . .. Gross [72] proved that for 𝑛 = 3 the extremal disks are the square and its affine images, that is, parallelograms. For 𝑛 > 3 the extremal disks are not known, but we know that for large values of 𝑛 they are approximately ellipses. Namely, the following theorem holds:

2.4 An Extremum Property of the Ellipse

41

Let 𝑇 be a convex disk whose boundary contains two diametrically opposite arcs of a circle of perimeter 𝐿, with total length 𝑛4 𝐿. Then one can circumscribe about 𝑇 an 𝑛-gon 𝑈𝑛 of area 𝜋 𝑛 (2.4.3) 𝑈𝑛 ≤ 𝑇 tan . 𝜋 𝑛 This theorem supports the conjecture that for 𝑛 → ∞, under suitable “affine normalization”, the extremal curves converge fairly rapidly to a circle. We represent the boundary 𝐾 of 𝑇 by the system of equations (2.4.2) again, so that the points 𝑡 = 0 and 𝜋 coincide with the midpoints of the circular arcs mentioned in the theorem, and consider the circumscribed 𝑛-gon 𝑈𝑛 (𝑡) with touching points corresponding to the parameters 𝑡, . . . , 𝑡 + (𝑛 − 1) 2𝑛𝜋 . We show that    𝑛−1  2𝜋 2𝜋 𝜋 ∑︁ 2 𝑒 𝑡 +𝑖 sin 𝑡 + 𝑖 . 𝑈𝑛 (𝑡) ≤ 2 tan 𝑛 𝑖=0 𝑛 𝑛 Consider the 𝑛-gon circumscribed about the circle 𝐿 cos 𝑡, 𝜋

𝑥=

𝑦=

𝐿 sin 𝑡 𝜋

such that the sides touch the circle at the points 𝑡, . . . , 𝑡 + (𝑛 − 1) 2𝑛𝜋 , and project each side of it orthogonally to the 𝑥-axis onto the line of the corresponding side of 𝑈𝑛 . We obtain thereby 𝑛 segments tangent to the curve 𝐾. We complete these segments to a closed cycle by connecting the corresponding endpoints with segments perpendicular to the 𝑥-axis. The polygon obtained contains 𝑈𝑛 (𝑡), and the right-hand side of the above inequality is nothing else but the area of this polygon. Now we have ∫ 2𝜋 𝑛 𝜋 1 𝑈𝑛 (𝑡) d𝑡 ≤ 𝑇 tan , 2𝜋 0 𝜋 𝑛 completing the proof of (2.4.3). For an arbitrary convex disk the above proof fails, on one hand, because the cycle of segments constructed above need not be simply connected, and on the other hand, because 𝑈𝑛 (𝑡) can protrude from the polygon bounded by this cycle of segments. The analogous problems concerning perimeter are unsolved both for inscribed and for circumscribed 𝑛-gons. Therefore it is of some interest that it is very easy to prove the following sharp bound for simultaneous approximation of a closed convex curve by inscribed and circumscribed 𝑛-gons: Each closed convex curve can be enclosed between an inscribed and circumscribed 𝑛-gon of perimeter 𝑙 𝑛 and 𝐿 𝑛 such that 𝐿 𝑛 − 𝑙𝑛 𝜋 ≤ 2 sin2 . 𝐿𝑛 2𝑛

(2.4.4)

◀ 196

42

2 Theorems from the Theory of Convex Bodies

We draw an arbitrary regular 𝑛-gon containing the given curve and move its sides parallel to themselves inwards until they touch the curve. We obtain thereby an equiangular 𝑛-gon circumscribed about the curve. Connecting the points of tangency of consecutive sides we get an inscribed 𝑛-gon. We claim that this pair of 𝑛-gons satisfies the inequality (2.4.4). Indeed, let 𝐴 and 𝐵 be two consecutive vertices of the inscribed polygon and let 𝐶 be the point of intersection of the corresponding tangents. Fixing the vertices 𝐴 and 𝐵 of the triangle 𝐴𝐵𝐶 and moving 𝐶 so that the outer angle at 𝐶 remains 2𝑛𝜋 , the sum 𝐴𝐶 +𝐶𝐵 attains its maximum when 𝐴𝐶 = 𝐶𝐵. It follows that 𝜋 𝐴𝐶 + 𝐶𝐵 ≤ 𝐴𝐵 sec . 𝑛 𝜋 Adding these inequalities we get 𝐿 𝑛 ≤ 𝑙 𝑛 sec 𝑛 . The analogous inequality for the area reads as follows: For every convex disk there is an inscribed 𝑛-gon 𝑡 𝑛 and a circumscribed 𝑛-gon 𝑇𝑛 such that 𝜋 𝑇𝑛 − 𝑡 𝑛 ≤ sin2 . (2.4.5) 𝑇𝑛 𝑛 The slightly more complicated proof of (2.4.5) follows in Section 2.6.

2.5 On the Affine Perimeter The affine length of a plane arc, introduced by Blaschke and G. Pick, is an additive arc measure invariant under area-preserving affinities. This notion plays a fundamental role in affine differential geometry. In the following introduction to the affine length, we will mainly appeal to geometric intuition rather than engage in rigorous analysis. First we define the affine length of an arc of a unitary ellipse—that is, of an ellipse for which the product 𝑎𝑏 of semi-axes is 1—as the usual length of the arc of a circle that can be transformed into the given elliptic arc by an area-preserving affinity. Next, for a continuously curved plane arc 𝐾 we proceed as follows. Partition the arc into a finite number of sub-arcs and replace every sub-arc 𝑘 with a unitary-elliptic arc 𝑒 with the property that, on one hand, the usual lengths of 𝑒 and 𝑘 are equal, and, on the other hand, the curvature of 𝑒 and of 𝑘 coincide at some of their points. It can be shown that the sum 𝜆 𝑛 of affine lengths of the elliptical arcs so assigned to the partition of 𝐾, under unlimited refinements of the partition, converges to a limit value 𝜆 = lim 𝜆 𝑛 that depends on 𝐾 only, which we will call the affine length of 𝐾. 𝑛→∞ In short, we treat all arc-elements of 𝐾 as if each of them were an arc-element of a unitary ellipse, we map them into the unit circle by affinities, assembling them into a single arc, and we declare 𝜆 to be the usual length of this arc.

43

2.5 On the Affine Perimeter

It remains to show that the considered limiting process indeed leads to a welldefined limit. Since through the parametric representation 𝑥 = 𝑎 cos 𝑡, 𝑦 = 𝑏 sin 𝑡; 𝑎𝑏 = 1 the unit circle 𝑥 = cos 𝑡, 𝑦 = sin 𝑡, is mapped affinely onto the unitary ellipse, the parameter 𝑡 represents the affine length of the corresponding elliptical arc. Now, a simple computation shows that the 1 curvature 𝜅 of the above unitary ellipse satisfies 𝜅 − 3 = d𝑠 d𝑡 , where d𝑠 denotes the arc element. Consequently, the element d𝑡 of the affine length of an ellipse with 𝑎𝑏 = 1 depends on 𝜅 and d𝑠 alone, and not on 𝑎 or 𝑏. This, however, shows that 𝜆 𝑛 converges to a value independent of both the choice of the sequence of partitions, and of the choice of the osculating ellipses. Namely, we have ∫

𝑙

1

|𝜅| 3 d𝑠,

𝜆= 0

where 𝜅 = 𝜅(𝑠) (0 ≤ 𝑠 ≤ 𝑙) is the so-called natural equation of 𝐾. The additive property of the affine length, as well as its invariance under area-preserving affinities, follows directly from the above definition. We will show next that, based on our definition, the affine length of a given curve can be measured with arbitrary accuracy. To that end, following a familiar construction of an ellipse, we mark on an ellipse points corresponding to 2𝜋 2𝜋 , . . . , 100 99 and we cut the parameters 𝑡 = 0, 100 the ellipse out of paper (Figure 2.5). This way, we obtain a “gauge” that enables us to measure the affine length of any continuously curved arc whose curvature at every point falls between the Fig. 2.5 two extreme curvatures 𝑎𝑏2 = 𝑏 3 and 𝑏𝑎2 = 𝑎 3 of the ellipse. This is best done as follows. Find two markings close to each other on the gauge such that when one of them is placed on an end point 𝐴 of the arc 𝐾, and the other one is brought into coincidence with a further point 𝑃 on 𝐾, yet a third point on the elliptical arc 𝐴𝑃 falls on the arc 𝐴𝑃 of 𝐾. This means that the two arcs practically coincide. Since in this case the usual arc length and curvature of the elliptical arc and those of the arc of 𝐾 approximately coincide, they also have about the same affine length, which can be read from our gauge directly. In the same way we can measure the next piece of the curve, and so on (Figure 2.6). We mention now several properties of the affine length, each of which can be proved according to the following pattern. One first verifies the corresponding statement 𝑆 for an arc of the unit circle. From that it follows immediately that 𝑆 holds

44

2 Theorems from the Theory of Convex Bodies

for a unitary-elliptical arc, as well as for an arc of a convex curve 𝐾 𝜈 composed of a finite number 𝜈 of unitary-elliptical arcs. The validity of 𝑆 for an arbitrary continuously curved convex arc 𝐾 then follows by a limiting process, in which 𝐾 is approximated by a sequence 𝐾1 , 𝐾2 . . . of strings made of unitary-elliptical arcs. We omit the technical details of these proofs. An arc 𝐾 is convex when it lies on the boundary 𝑅 of its convex hull. As in this case the affine length of 𝐾 coincides with that of 𝑅, we can replace 𝐾 with 𝑅. It therefore suffices to focus our attention on closed convex curves. Let 𝐸 be a convex disk bounded by the curve 𝐾 of affine length 𝜆. Consider a triangle Δ ≡ 𝐴𝑂𝐵 such that the lines 𝑂 𝐴 and 𝑂𝐵 touch the curve 𝐾 at 𝐴 and 𝐵, respectively. Let Δ slide around the curve 𝐾 so that the area of the triangle Fig. 2.6 remains constant. Consider the curve traced by the point 𝑂, as well as the curve enveloped by the chords 𝐴𝐵, and denote the sets bounded by these curves by 𝐸 Δ and 𝐸 −Δ , respectively. These can be regarded as the outer and inner affine parallel domains of 𝐸, respectively. Now, the following relations hold, which stand in close analogy with Minkowski’s notion of perimeter: lim

Δ→0

𝐸 Δ − 𝐸 −Δ Δ

2 3

= 2 lim

Δ→0

𝐸Δ − 𝐸 Δ

2 3

= 2 lim

Δ→0

𝐸 − 𝐸 −Δ 2

= 𝜆.

Δ3

Further, let 𝑃𝑛𝑖 be a maximum-area 𝑛-gon inscribed in a convex disk 𝐸 and 𝑃𝑛𝑐 be a minimum-area 𝑛-gon circumscribed about 𝐸. Then lim 𝑛2 (𝐸 − 𝑃𝑛𝑖 ) = 𝑛→∞

lim 𝑛2 (𝑃𝑛𝑐 − 𝐸) = 𝑛→∞

𝜆3 , 12 𝜆3 , 24

and consequently 𝜆3 . 𝑛→∞ 8 These relations can be complemented by a further one. If 𝑃𝑛 denotes an 𝑛-gon that, among all 𝑛-gons, has the smallest area-deviation from 𝐸, then lim 𝑛2 (𝑃𝑛𝑐 − 𝑃𝑛𝑖 ) =

lim 𝑛2 𝜏(𝐸, 𝑃𝑛 ) = 𝑛→∞

𝜆3 . 32

These equalities can be written in the following form: 12 lim 𝑛2 𝜏(𝐸, 𝑃𝑛𝑖 ) = 24 lim 𝑛2 𝜏(𝐸, 𝑃𝑛𝑐 ) = 32 lim 𝑛2 𝜏(𝐸, 𝑃𝑛 ) =

∫

1

𝜅 3 d𝑠

3 .

2.5 On the Affine Perimeter

45

In addition, the following analogous formulae hold: 24 lim 𝑛2 𝜆(𝐸, 𝑃𝑛𝑖 ) = 12 lim 𝑛2 𝜆(𝐸, 𝑃𝑛𝑐 ) = 24 lim 𝑛2 𝜆(𝐸, 𝑃𝑛 ) =

8 lim 𝑛2 𝜂(𝐸, 𝑃𝑛𝑖 ) = 8 lim 𝑛2 𝜂(𝐸, 𝑃𝑛𝑐 ) = 16 lim 𝑛2 𝜂(𝐸, 𝑃𝑛 ) =

∫

∫

2

3

𝜅 3 d𝑠 1

𝜅 2 d𝑠

,

3 ,

where in the first case the corresponding 𝑛-gons of minimum perimeter deviation from 𝐸 are taken, while in the second case the corresponding 𝑛-gons with minimum Hausdorff distance from 𝐸 are taken. As a further property of affine length we mention the following: The vertices of the maximum area inscribed 𝑛-gon 𝑃𝑛𝑖 (as well as the points of tangency of the sides of the minimum area circumscribed 𝑛-gon) are, for large 𝑛, uniformly distributed along the affine perimeter. More precisely, if 𝑛1 and 𝑛2 denote the numbers of vertices of 𝑃𝑛𝑖 lying on two sub-arcs of our convex curve, of affine length 𝜆1 and 𝜆2 respectively, then 𝜆1 𝑛1 lim = . 𝑛→∞ 𝑛2 𝜆2 A simple construction is connected with this property of affine length, by which a scale along an arc of a curve can be designed, approximately proportional to the affine length. For two points 𝑃1 and 𝑃2 on the curve, draw the tangent line at 𝑃2 along with a chord 𝑃1 𝑃3 parallel to it. Similarly, beginning with 𝑃2 and 𝑃3 , construct a point 𝑃4 , and so on. Let 𝐸 and 𝑒 be two convex curves of affine perimeter Λ and 𝜆, respectively. If we inscribe a maximum area 𝑁-gon in 𝐸, respectively an 𝑛-gon in 𝑒, where 𝑁 and 𝑛 are chosen so that, under the given value of the total number of vertices 𝜈 = 𝑁 + 𝑛, the sum of the areas of the polygons reaches the greatest possible value, then lim 𝜈→∞

𝑁 Λ = . 𝑛 𝜆

To obtain from here another definition of affine length, all we need is to choose the unit circle for 𝑒 and set 𝜆 = 2𝜋. Let us now circumscribe an 𝑛-gon about the given convex curve and connect adjacent points of tangency, thereby embedding our curve in a chain of 𝑛 triangles Δ1 , . . . , Δ𝑛 . If we now take a sequence of such chains for which max(Δ1 , . . . , Δ𝑛 ) converges to 0 as 𝑛 increases, then 𝜆 = 2 lim

𝑛 ∑︁

𝑛→∞

1

Δ𝑖3 .

𝑖=1

If the triangles are of the same area, then we have 𝜆3 = 8 lim 𝑛2 𝜏𝑛 , 𝑛→∞

46

2 Theorems from the Theory of Convex Bodies

where 𝜏𝑛 denotes the total area of the triangles. This is consistent with our previous formula 𝜆3 = 8 lim 𝑛2 𝜏(𝑃𝑛𝑐 , 𝑃𝑛𝑖 ). Let us mention a further property of the affine length. Let 𝐷 𝑛 denote the minimum of the total area of 𝑛 arbitrary triangles that can entirely cover our curve. Then √ 𝜆3 = 6 3 lim 𝑛2 𝐷 𝑛 . 𝑛→∞

To see this relation for an arc of the unit circle, observe that in the case of a circular arc, the 𝑛 triangles are isosceFig. 2.7 les and congruent, their legs touching the arc at their midpoints (Figure 2.7). Therefore if Δ is such a triangle and 𝛼 is the length of the piece of the arc of the unit Δ 1 circle that lies in Δ, then for the proof we need to show the relation lim 3 = √ . 𝛼→0 𝛼 6 3 This, however, follows from the equality   2

16Δ = 27 =

𝑥 2 √︁ 1− − 1 − 𝑥2 9

162 6 𝑥 +··· ; 27

!3

𝑥 = sin

  ! √︁ 𝑥2 2 1−𝑥 +3 1− 9

𝛼 . 2

⌢ We turn now to determining the affine length of an arc 𝐾 ≡ 𝐴𝐵 of a conic section lying in a triangle Δ ≡ 𝐴𝑂𝐵 and tangent to the lines 𝐴𝑂 and 𝐵𝑂. Let 𝑆 denote the region bounded by 𝐾 and the segment 𝐴𝐵, and let 𝑞 = Δ𝑆 . If we construct for 𝐾 a chain 𝜏𝑛 consisting of 𝑛 triangles of equal area, then, obviously, Δ1 𝑛2 𝜏𝑛 approaches a limit 1 Φ(𝑞) = lim 𝑛2 𝜏𝑛 Δ 𝑛→∞ depending on 𝑞 only. In order to determine the function Φ(𝑞) we consider two cases, depending on whether 𝐾 is an ellipse or a hyperbola, that is, whether 𝑞 < 32 or 𝑞 > 23 . In the first case, transform 𝐾 by an area-preserving affinity into a circular arc 𝑥 = 𝑟 cos 𝑢,

𝑦 = 𝑟 sin 𝑢;

−𝛼 ≤ 𝑢 ≤ 𝛼.

Then, a simple calculation yields Δ = 𝑟2

sin3 𝛼 , cos 𝛼

𝑆=

𝑟2 (2𝛼 − sin 2𝛼), 2

lim 𝑛2 𝜏𝑛 = 𝑟 2 𝛼3 = 𝑛→∞

𝛼3 cos 𝛼 sin3 𝛼

𝜏𝑛 = 𝑛𝑟 2

Δ.

sin3 𝛼𝑛 , cos 𝛼𝑛

47

2.6 Variational Problems Regarding Affine Length

From there, we get a parametric equation of Φ(𝑞) for 𝑞 < 32 : Φ(𝑞) =

𝛼3 cos 𝛼

In the case 𝑞 > hyperbolic arc into

𝑞=

,

3

sin 𝛼 2 3,

(2𝛼 − sin 2𝛼) cos 𝛼 3

2 sin 𝛼

;

0 𝑄 𝑖 𝑃𝑖+1 . Replace 𝑄 𝑖 by a point 𝑄 𝑖′ on the segment 𝑃𝑖 𝑄 𝑖 whereby Δ𝑖 = 𝑃𝑖 𝑄 𝑖 𝑃𝑖−1 decreases whereas Δ𝑖+1 = 𝑃𝑖+1 𝑄 𝑖 𝑄 𝑖+1 increases; ′ = then rotate 𝑃𝑖 𝑃𝑖+1 about 𝑄 𝑖′ so that the new triangles Δ𝑖′ = 𝑃𝑖′𝑄 𝑖′𝑄 𝑖−1 and Δ𝑖+1 ′ ′ ′ 𝑃𝑖+1 𝑄 𝑖 𝑄 𝑖+1 become of equal area again (Figure 2.11). If 𝑄 𝑖 is sufficiently close ′ 𝑄 ′ . On the other hand, if 𝑄 ′ is chosen close to 𝑃 , to 𝑄 𝑖 , then 𝑃𝑖 𝑃𝑖′𝑄 𝑖′ > 𝑃𝑖+1 𝑃𝑖+1 𝑖 𝑖 𝑖 ′ ′ ′ 𝑄 ′ . By continuity, 𝑄 ′ can be chosen so that we then conversely, 𝑃𝑖 𝑃𝑖 𝑄 𝑖 < 𝑃𝑖+1 𝑃𝑖+1 𝑖 𝑖 ′ 𝑄 ′ . In this case the area of 𝑃 remains unchanged. But since get 𝑃𝑖 𝑃𝑖′𝑄 𝑖′ = 𝑃𝑖+1 𝑃𝑖+1 𝑖 ′ ′ 𝑄 𝑖−1 𝑄 𝑖 𝑄 𝑖+1 < 𝑄 𝑖−1 𝑄 𝑖 𝑄 𝑖+1 , we have Δ𝑖′ = Δ𝑖+1 > Δ𝑖 , which proves assertion 1.

Fig. 2.11

Fig. 2.12

From now on we assume that 𝑃𝑖 𝑄 𝑖 = 𝑄 𝑖 𝑃𝑖+1 (𝑖 = 1, . . . , 𝑛 − 1). Then, to prove assertion 2, it suffices to show that 𝑄 𝑖 𝑄 𝑖+1 ∥ 𝑄 𝑖−1 𝑄 𝑖+2

(𝑖 = 1, . . . , 𝑛 − 2),

𝑄 0 ≡ 𝐴, 𝑄 𝑛 ≡ 𝐵.

50

2 Theorems from the Theory of Convex Bodies

Assume, to the contrary, that the distance 𝑑 from the point 𝑄 𝑖−1 to the line 𝑄 𝑖 𝑄 𝑖+1 is smaller than the distance 𝐷 from the point 𝑄 𝑖+2 to that line (Figure 2.12). ′ along the line parallel to 𝑃 𝑃 Then we translate 𝑃𝑖+1 to 𝑃𝑖+1 𝑖 𝑖+2 so that its distance from 𝑄 𝑖−1 𝑄 𝑖+2 decreases, and we replace the points 𝑄 𝑖 and 𝑄 𝑖+1 with the intersection ′ ′ of the line 𝑄 𝑄 ′ points 𝑄 𝑖′ and 𝑄 𝑖+1 𝑖 𝑖+1 with the lines 𝑃𝑖 𝑃𝑖+1 and 𝑃𝑖+2 𝑃𝑖+1 , respectively. While under this transformation the area of 𝑃 remains unchanged, the area of 𝑄 ′ . Denoting the new increases by an amount of the order of magnitude 𝜂 = 𝑃𝑖+1 𝑃𝑖+1 ′ ′ . . . 𝑄 by 𝑄 ′ , our assertion means that 𝑄 − 𝑄 tends to a polygon 𝑄 0 . . . 𝑄 𝑖′𝑄 𝑖+1 𝑛 𝜂 𝑑−𝐷 ′ 𝜂. negative number as 𝜂 → 0, which is evident by the equality 𝑄 − 𝑄 = 4 ′ ′ ′′ ′′ If we now replace 𝑄 𝑖 and 𝑄 𝑖+1 with those points 𝑄 𝑖 and 𝑄 𝑖+1 on the sides ′ ′ 𝑃 ′′ ′′ 𝑃𝑖 𝑃𝑖+1 and 𝑃𝑖+1 𝑖+2 , respectively, for which the corresponding triangles Δ𝑖 , Δ𝑖+1 ′′ ′′ and Δ𝑖+2 become of equal area again, and if we denote the new polygon by 𝑄 , then, ′ 𝑄 ′′ is 𝜂. In view evidently, the order of magnitude of the segments 𝑄 𝑖′𝑄 𝑖′′ and 𝑄 𝑖+1 𝑖+1 of 𝑃𝑖 𝑃𝑖+1 ∥ 𝑄 𝑖−1 𝑄 𝑖+1 and 𝑃𝑖+1 𝑃𝑖+2 ∥ 𝑄 𝑖 𝑄 𝑖+2 , the angle between the lines 𝑄 𝑖′𝑄 𝑖′′ and ′ , as well as the angle between the lines 𝑄 ′ 𝑄 ′′ and 𝑄 ′ 𝑄 𝑖−1 𝑄 𝑖+1 𝑖+2 𝑄 𝑖 , is of the same 𝑖+1 𝑖+1 ′′ ′ order of magnitude 𝜂. Thus the difference 𝑄 − 𝑄 is of the order of magnitude 𝜂2 . ′′ = Δ ′′ > Δ , completing This shows that for sufficiently small values of 𝜂, Δ𝑖′′ = Δ𝑖+1 𝑖 𝑖+2 the proof of assertion 2. For the proof of assertion 3 we can assume that 𝑃1 𝑃2 ∥ 𝐴𝑄 2 . Suppose that 𝑃1 does not lie on 𝐴𝑂. Translate 𝑄 1 to the point 𝑄 1′ ≡ 𝑃2 and 𝑃1 parallel to the line 𝐴𝑃2 to the point 𝑃1′ on 𝐴𝑂. Then neither the area of 𝑃 nor that of 𝑄 changes. If we now move 𝑄 1′ from its position 𝑄 1′ ≡ 𝑃2 on the side 𝑃2 𝑃1′ , then the area of 𝑄 obviously decreases. At a certain position 𝑄 1′′ we will then have 𝐴𝑃1′ 𝑄 1′′ = 𝑄 1′′ 𝑃2 𝑄 2 > 𝐴𝑃1 𝑄 1 , which completes the proof of assertion 3. We show now that the optimal chain of triangles satisfying the above conditions contains an arc of a conic section tangent to 𝐴𝑂 and 𝐵𝑂. To see this, consider the conic section tangent to the segments 𝑃0 𝑃1 , 𝑃1 𝑃2 , 𝑃2 𝑃3 at their midpoints. To prove the existence of such a conic section we may assume, without loss of generality, that the trapezoid 𝑃0 𝑃1 𝑃2 𝑃3 is isosceles, that is, 𝑃0 𝑃1 = 𝑃2 𝑃3 . Then among all conic sections that touch 𝑃0 𝑃1 and 𝑃2 𝑃3 at their midpoints there is exactly one that touches 𝑃1 𝑃2 as well. By symmetry, the point of tangency must be the midpoint of 𝑃1 𝑃2 , which proves our assertion. We prove next that the conic section constructed above is tangent also to the segments 𝑃3 𝑃4 , . . . , 𝑃𝑛 𝑃𝑛+1 at their midpoints. Instead of the above assumption 𝑃0 𝑃1 = 𝑃2 𝑃3 , we assume now that 𝑃1 𝑃2 = 𝑃2 𝑃3 . Then, it follows by conditions 1 and 2 that 𝑃0 𝑃1 = 𝑃3 𝑃4 and 𝑃2 lies on the symmetry axis of the isosceles trapezoid 𝑃0 𝑃1 𝑃3 𝑃4 . We now consider the family of conic sections that touch the segments 𝑃1 𝑃2 and 𝑃2 𝑃3 at their midpoints. There is exactly one among them that also touches 𝑃0 𝑃1 . But this conic section is identical with the previous one, hence it touches the segment 𝑃0 𝑃1 at its midpoint. Therefore, again by symmetry, it must touch also 𝑃3 𝑃4 at its midpoint. For the remaining sides this is shown in the same way.

2.6 Variational Problems Regarding Affine Length

51

Thus, while the polygonal arc 𝐴𝑃1 . . . 𝑃𝑛 𝐵 is circumscribed about our conic section, the arc 𝐴𝑄 1 . . . 𝑄 𝑛−1 𝐵 is inscribed in it. From condition 3 it follows that the conic section is tangent to 𝐴𝑂 and 𝐵𝑂. Thereby—in view of the formula 𝜆3 = 8 lim 𝑛2 𝜏𝑛 —the extremum property of the conic sections asserted in our theorem is proved. We still need to prove uniqueness. We intend to prove that the chain of triangles 𝜏𝑛 considered above, assigned to an extreme arc 𝐾, contains an arc of a conic section tangent to 𝐴𝑂 and 𝐵𝑂. But this is possible for an arbitrary value of 𝑛 only if 𝐾 itself is an arc of a conic section. Assume that, contrary to our claim, 𝜏𝑛 contain no arc of a conic section tangent to 𝐴𝑂 and 𝐵𝑂. Then 𝜏𝑛 can be replaced by another chain 𝜏¯𝑛 of triangles, that likewise consists of 𝑛 equiareal triangles, however with greater area than before, while the area of the outer polygon 𝑃 remains unchanged. Let Δ¯ 𝑖 ≡ 𝑄¯ 𝑖−1 𝑃¯𝑖 𝑄¯ 𝑖 be the triangle in the new chain corresponding to Δ𝑖 ≡ 𝑄 𝑖−1 𝑃𝑖 𝑄 𝑖 (𝑖 = 1, . . . , 𝑛). For each 𝑖, replace ¯ ¯Ŕ¯ the arc 𝑘 𝑖 ≡ 𝑄Ŕ 𝑖−1 𝑄 𝑖 with the conic-section arc 𝑘 𝑖 ≡𝑄 𝑖−1 𝑄 𝑖 tangent to the sides ¯ ¯ ¯ ¯ ¯ ¯ 𝑄 𝑖−1 𝑃𝑖 and 𝑃𝑖 𝑄 𝑖 at the points 𝑄 𝑖−1 and 𝑄 𝑖 , respectively, so that Δ𝑖 − 𝑠𝑖 = Δ¯ 𝑖 − 𝑠¯𝑖 , where 𝑠𝑖 and 𝑠¯𝑖 denote the segment-areas of 𝑘 𝑖 and 𝑘¯ 𝑖 , respectively. We get thereby a new convex arc running in 𝐴𝑂𝐵 with unchanged segment-area and greater affine length. This becomes clear when we realize that 𝑘¯ 𝑖 has a greater affine length than 𝑘 𝑖 . We only have to show that     𝑠𝑖 𝑠¯𝑖 ¯ Δ𝑖 Φ < Δ𝑖 Φ , Δ𝑖 Δ¯ 𝑖 where Φ is the function defined in Section 2.5. If we set Δ𝑖 − 𝑠𝑖 = 𝑤, Δ𝑖

Δ¯ 𝑖 − 𝑠¯𝑖 = 𝑤, ¯ Δ¯ 𝑖

then, in view of Δ𝑖 − 𝑠𝑖 = Δ¯ 𝑖 − 𝑠¯𝑖 , the inequality that we need to prove becomes 1 1 Φ(1 − 𝑤) < Φ(1 − 𝑤). ¯ 𝑤 𝑤¯ However, because of 𝑤 > 𝑤¯ and Φ(1) = 0, this is an immediate consequence of the concavity of the function Φ. Thereby the proof of the above theorem is complete. We shall now prove the following theorem. If a convex disk of affine perimeter 𝜆 lies in a convex 𝑛-gon 𝑇, then 𝜆3 ≤ 8𝑇 𝑛2 sin2

𝜋 . 𝑛

(2.6.1)

Equality occurs only if 𝑇 is an affine-regular 𝑛-gon and the disk is bounded by 𝑛 parabolic arcs, each touching two adjacent sides of 𝑇 at their corresponding midpoints.

52

2 Theorems from the Theory of Convex Bodies

We emphasise the limiting case 𝑛 → ∞ of this theorem: The affine perimeter 𝜆 and area 𝑇 of an arbitrary convex disk satisfy the inequality 𝜆3 ≤ 8𝜋 2 𝑇 .

(2.6.2)

This is just the affine isoperimetric inequality, expressing the aforementioned extremum property of the ellipse. In the proof of (2.6.1) we need to consider only the case when 𝑇 is a circumscribed 𝑛-gon of minimum area. Then the points of contact are the midpoints of the sides. If we connect the midpoints of the adjacent sides, then we get a closed chain of triangles, which we denote by Δ1 , . . . , Δ𝑛 . According to the extremum property of the parabolic arcs, we have   1 1 𝜆 ≤ 2 Δ13 + · · · + Δ𝑛3 . Thereby our problem is reduced to determining the convex 𝑛-gon that among all 1

1

convex 𝑛-gons of the same area has the greatest value of the sum Δ13 + · · · + Δ𝑛3 . Here, again, the existence of a maximum can be derived from the theorem of Weierstrass. It is easy to show that in the extreme case all triangles must have the same area. This can be most readily explained in the following way. Let Δ1 ≡ 𝑃1 𝑄 𝑛 𝑄 1 and Δ2 ≡ 𝑃2 𝑄 1 𝑄 2 be two neighboring triangles of an extreme chain with Δ1 < Δ2 . Rotate the side 𝑃1 𝑃2 about its midpoint 𝑄 1 by an infinitesimal angle so that dΔ1 = −dΔ2 > 0. Then d𝑇 = dΔ1 + dΔ2 = 0, while 2



1 3

1 3



d Δ1 + Δ2 =

2

Δ23 − Δ13 2

3(Δ1 Δ2 ) 3

dΔ1 > 0,

whereby we have arrived at an evident contradiction. But if the triangles are of equal area, then the arguments in the previous proof show that the extreme 𝑛-gon 𝑇 must be circumscribed about an ellipse so that the points of contact bisect the sides of 𝑇. Consequently, 𝑇 must be affine regular, and the only thing that remains to complete the proof of the above theorem is to compute 1

1

the sum Δ13 + · · · + Δ𝑛3 for the regular 𝑛-gon. Finally, it should be noted that the arguments of the above proof yield a proof of the inequality (2.4.5) Namely, embed the boundary of our convex disk in a closed chain of 𝑛 triangles of equal area and consider the corresponding circumscribed and inscribed 𝑛-gons 𝑇𝑛 and 𝑡 𝑛 , respectively. The previous proof shows that, under a free 𝑛 reaches its maximum when each of the two 𝑛-gons variation of 𝑇𝑛 , the quotient 𝑇𝑛𝑇−𝑡 𝑛 is affine regular and the vertices of 𝑡 𝑛 are the midpoints of the sides of 𝑇𝑛 , which is just expressed in the inequality (2.4.5) to be proved.

2.7 Rudiments of Integral Geometry

53

2.7 Rudiments of Integral Geometry Let 𝑀 be a certain set of geometric objects, such as, e.g., the points of a domain, the lines meeting a domain, the planes intersecting a spatial curve, or all congruent domains that intersect another fixed domain, etc. We want to associate with such a set 𝑀 a measure 𝑚(𝑀). For this, we characterize the elements of 𝑀 by some coordinates 𝑥1 , . . . , 𝑥 𝑘 . Take one, arbitrary for now, positive function 𝑓 (𝑥) = 𝑓 (𝑥1 , . . . , 𝑥 𝑘 ) of 𝑘 variables and form the integral ∫ ∫ ∫ 𝑓 (𝑥1 , . . . , 𝑥 𝑘 ) d𝑥1 . . . d𝑥 𝑘 𝑓 (𝑥) d𝑥 = ··· over the set 𝑀.∫ If the set 𝑀 and the function 𝑓 (𝑥) are such that this integral exists, then 𝑚(𝑀) = 𝑓 (𝑥) d𝑥 can be considered as a measure of 𝑀. We try to choose the function 𝑓 (𝑥) so that 𝑚(𝑀) will be motion-invariant. That is, we require that 𝑚(𝑀) = 𝑚(𝑀 ′) for every two sets 𝑀 and 𝑀 ′ which can be carried onto each other by rigid motion. This can always be achieved if the elements of 𝑀 are such that any two of them can be interchanged by a pair of motions—as is the case in the above examples. It can be shown that in this case 𝑓 (𝑥) is, up to a constant factor, uniquely determined by the above invariance postulate. Then, the quotient 𝑚(𝑇) 𝑚( 𝑀) gives the probability that a randomly chosen element of 𝑀 is contained in the subset 𝑇. We call such a motion-invariant measure 𝑚(𝑀) the (integral-geometric) number of elements in 𝑀. The number of points of a domain is, of course, the area of the domain. The number of lines intersecting a convex disk was determined in 1868 by M.W. Crofton, who found the surprising result that this number is equal to the perimeter of the disk. Later, Poincaré introduced the number of certain positions of a rigidly moving domain 𝐺. Fix a line element (point and direction) in the plane of the moving domain 𝐺 and consider its coordinates 𝑥, 𝑦 and directional angle 𝜑 relative to a given coordinate system in the fixed plane. ∫ Then∫the ∫ ∫number investigated by Poincaré can be defined by the triple integral d𝐺 = d𝑥 d𝑦 d𝜑. We note that this number is independent of the choice of the line element in the moving plane, as well as the choice of the coordinate system in the fixed plane. The differential d𝐺 = d𝑥 d𝑦 d𝜑 is called the kinematic density of 𝐺. Let us move the curve 𝐾 in the plane of the fixed curve 𝐾0 and denote by 𝑠 the number of ∫ points of intersection of the two curves in a certain position. Then the integral 𝑠 d𝐾 over all positions of 𝐾 gives the number of all positions of 𝐾 weighted with the multiplicity of the number of intersection points. According to Poincaré, we have ∫ 𝑠 d𝐾 = 4𝐿 0 𝐿, (2.7.1) where 𝐿 0 and 𝐿 denote the arc length of 𝐾0 and 𝐾, respectively. We mention now the formula of Santaló that gives the number of those positions of a moving convex disk 𝑇 of perimeter 𝐿 in which it intersects a fixed convex disk 𝑇0 of perimeter 𝐿 0 :

54

2 Theorems from the Theory of Convex Bodies

∫ d𝑇 = 2𝜋(𝑇0 + 𝑇) + 𝐿 0 𝐿.

(2.7.2)

Here, the integral is taken over all positions of 𝑇 in which 𝑇0 𝑇 ≠ 0. In contrast, in (2.7.1) we can integrate over all positions of 𝐾, since we have 𝑠 = 0 for 𝑇0 𝑇 = 0. The formulae of Poincaré and Santaló can be united in a more general theorem. We consider again a fixed and a moving domain 𝑇0 and 𝑇, which this time need not necessarily be convex, of perimeter 𝐿 0 and 𝐿, respectively. We assume only that each of them is bounded by a closed curve without self-intersections, that is, each of 𝑇0 and 𝑇 is simply connected. Denoting by 𝑘 the number of simply connected components of the intersection 𝑇𝑇0 , the unified formula reads ∫ (2.7.3) 𝑘 d𝑇 = 2𝜋(𝑇0 + 𝑇) + 𝐿 0 𝐿, where the integral is taken over all positions of 𝐾. This is Blaschke’s kinematic formula for simply connected domains. It is clear that (2.7.2) is a special case of (2.7.3). Moreover, formula (2.7.3) also implies (2.7.1), as a curve of length 𝐿 can be considered as a degenerate simply connected domain of area 0 and perimeter 2𝐿. With the aid of the above formulae one can derive various expressions for the isoperimetric deficit 𝐿 2 − 4𝜋𝑇, whose positivity is obvious. These results belong to the finest gems of integral geometry. We will see such an isoperimetric equation later. Further results of integral geometry, along with various applications, can be found in the work [16] of Blaschke.

2.8 Historical Remarks

194 ▶

The first part of the presented proof of the Selection Theorem has been adopted from an article of Hadwiger [79]. Hadwiger’s proof applies to a uniformly bounded system of arbitrary closed point sets in 𝑛-dimensional space. The inequality (2.2.1), under narrower assumptions, had already been proved by Hölder [86], and therefore is often cited as the Hölder–Jensen inequality. With the notation of Section 2.5, Dowker’s theorem says that the sequences 𝜏(𝐸, 𝑃𝑛𝑖 ) and 𝜏(𝐸, 𝑃𝑛𝑐 ) are convex. For every convex disk 𝐸, one can likewise consider the sequence 𝜏(𝐸, 𝑃𝑛 ), and also the other six sequences that arise by taking the perimeter deviation and the Hausdorff distance. Are these sequences convex as well? These questions have not been answered yet. The presented proof of the inequality (2.4.1) is due to Sas [118], and the subsequent proof of uniqueness, with the application of Fourier series, to the author. For the number of vertices 𝑛 = 3, the corresponding extremum property of the ellipse had already been proved by Blaschke [15] by an application of Steiner’s symmetrization. At the same time, Blaschke proved that among all convex bodies

2.8 Historical Remarks

55

of equal volume the ellipsoid is the most poorly filled by an inscribed tetrahedron. The author noticed in [55] that Blaschke’s proof also yields the following theorem: Every convex body contains a polyhedron with an arbitrarily prescribed number of vertices whose volume is at least as large as the maximum volume of a corresponding polyhedron contained in an ellipsoid of the same volume as the body. However, the question of uniqueness for an arbitrary number of vertices has not been settled yet. The theorem stated above can also be transferred to 𝑛-dimensional space. This was proved by Macbeath [97], unaware of Blaschke’s result. In the author’s article [29], the inequality 𝑈𝑛 < 𝑇

𝑛−2 𝜋 tan 𝜋 𝑛−2

(𝑛 ≥ 5)

is stated instead of (2.4.3), where 𝑇 denotes an arbitrary convex disk. Although the proof contains an error, the quoted inequality is presumably correct. Augmenting the work of the author [28] that contains proofs of inequality (2.4.4), and of (2.4.5) for 𝑛 ≤ 6 only, the proof of (2.4.5) for arbitrary 𝑛 was accomplished by the prematurely deceased mathematician Lázár [92]. In connection with Section 2.4 we mention another series of problems: For which spherical convex disks 𝐸 do the area-deviations 𝜏(𝐸, 𝑃𝑛𝑖 ), 𝜏(𝐸, 𝑃𝑛𝑐 ), 𝜏(𝐸, 𝑃𝑛 ), 𝜏(𝑃𝑛𝑐 , 𝑃𝑛𝑖 ) attain their greatest possible values? Here, a spherical convex disk is defined by the property that it contains the shortest spherical arc connecting each pair of its points. The interesting feature of these problems is that, besides convexity, no other assumption about 𝐸 is needed, as the considered deviations are small for both small and large disks. For example, it is conjectured that every spherical convex curve can be enclosed between an inscribed triangle 𝛿 and a circumscribed triangle Δ such √ 21−1 that Δ − 𝛿 ≤ 1.596 . . . (= 6𝛼 + 6 arctan 2 cos 𝛼 − 3𝜋; sin 𝛼 = 4 ). Additional problems arise when, besides the area-deviation, other notions of deviation are considered. The treatment of the affine length in this chapter was adopted from the author’s work [60].

◀ 197

Chapter 3

Problems on Packing and Covering in the Plane

We consider a given fixed family of finitely many convex disks and we ask: 1. How small can the area of a convex disk be in which the given disks can be placed without overlapping each other? 2. How large can the area of a convex disk be which can be covered by the given disks? The problems of this chapter are either directly of this type, or are associated with these two central problems. Our main interest lies in the limit case, in which the number of the disks is infinite. As it will turn out, in many cases the best arrangement of congruent disks is lattice-like. The so-called equilateral triangular lattice plays an exceptional role. One could therefore say that this chapter is mainly about the extremum properties of the degenerate regular polyhedra {3, 6} and {6, 3}.

3.1 Density of Arrangements of Domains Let us consider a countably infinite family {𝐺 𝑖 } of simply-connected domains, arbitrarily distributed in the plane, and allowed to overlap. Further, let 𝜒 be a functional (e.g., the area or perimeter) assigning to each domain 𝐺 𝑖 a nonnegative number let 𝐾 (𝑅) denote a circle of radius 𝑅 centered at the origin 𝑂 of 𝜒𝑖 = 𝜒(𝐺 𝑖 ). Also,∑︁ the plane, and let mean the summation extending over those domains from the 𝑅

family {𝐺 𝑖 } that are contained in the closed circular disk 𝐾 (𝑅). We assume that the number of such domains is finite for every 𝑅 < ∞, and further that the limit 𝐷 ( 𝜒) = lim 𝑅→∞

1 ∑︁ 𝜒 𝜋𝑅 2 𝑅 𝑖

exists; in which case we name 𝐷 ( 𝜒) the density of the functional 𝜒 of the family {𝐺 𝑖 }. This expresses the average of the functional’s total value per unit area of the plane.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_3

57

58

3 Problems on Packing and Covering in the Plane

It is easy ∑︁to show that 𝐷 ( 𝜒) does not depend on the choice of the origin 𝑂. ∗ Namely, if denotes the summation extended over all domains from the family 𝑅

that are contained in the circular disk 𝐾 ∗ (𝑅) centered at a different origin 𝑂 ∗ , then, because of 𝜒𝑖 ≥ 0, we have ∑︁ ∑︁ ∑︁ ∗ 𝜒𝑖 ≤ 𝜒𝑖 ≤ 𝜒𝑖 , 𝑅−Δ

𝑅

𝑅+Δ

where Δ denotes the distance between the centers 𝑂 and 𝑂 ∗ . Therefore  2  2 Δ 1 − 𝑅Δ 1 + ∑︁ ∑︁ ∑︁ 𝑅 1 ∗ 𝜒 ≤ 𝜒 ≤ 𝜒, 𝑖 𝑖 𝜋(𝑅 − Δ) 2 𝑅−Δ 𝜋𝑅 2 𝑅 𝜋(𝑅 + Δ) 2 𝑅+Δ 𝑖 which, for 𝑅 → ∞, implies the desired independence 𝐷 ∗ = 𝐷. If we assume now—as from now on we always will—that the diameters of the domains are uniformly bounded, then in the definition of the density of the functional 𝑅 ∑︁ ∑︁ 𝜒 the summation can be replaced with the summation extending over those 𝑅

domains from the family that have a point in common with 𝐾 (𝑅). This follows from the inequalities 𝑅 ∑︁ ∑︁ ∑︁ 𝜒𝑖 ≤ 𝜒𝑖 ≤ 𝜒𝑖 , 𝑅

𝑅+ 𝛿

where 𝛿 denotes the supremum of the diameters of the domains. We still note that when the above limit value does not exist, then, instead of the 1 ∑︁ density of the functional, the upper density lim sup 𝜒𝑖 , or the lower density, 2 𝑅→∞ 𝜋𝑅 𝑅 1 ∑︁ lim inf 𝜒𝑖 can be considered. 𝑅→∞ 𝜋𝑅 2 𝑅 In connection with our considerations, an important functional is 𝜒 ≡ 1. The density associated with this choice of 𝜒 is called the number density of the family of domains. This expresses the average number of domains per unit area of the plane. We now consider the limit 1 ∑︁ 𝜒¯ = lim 𝜒𝑖 , 𝑅→∞ 𝑁 (𝑅) 𝑅 where 𝑁 (𝑅) =

∑︁

1 denotes the number of domains from the family that lie in the

𝑅

circular disk 𝐾 (𝑅). If this limit value exists, then it is natural to call it the average value of the functional.

3.2 The Problems of Densest Packing and Thinnest Covering with Circles

59

Then one can write 1 ∑︁ 1 ∑︁ 𝑁 (𝑅) 𝜒 = 𝜒𝑖 , 𝑖 2 𝑁 (𝑅) 𝑅 𝜋𝑅 𝑅 𝜋𝑅 2 and if we assume that both the average value 𝜒¯ of the functional and the number density 𝐴 exist, then the functional’s density 𝐷 ( 𝜒) exists as well, and from the above relation we get, as 𝑅 → ∞, 𝐷 ( 𝜒) = 𝐴 𝜒¯ . The density most often used is the area density, corresponding to the functional 𝜒(𝐺) = 𝐺. We will call it simply the density of the family of domains. This density can be interpreted as the quotient of the sum of the areas of the domains and the area of the whole plane. The density of a family of mutually non-overlapping domains is obviously at most 1, while the density of a family of domains that completely covers the plane is at least 1.

3.2 The Problems of Densest Packing and Thinnest Covering with Circles Suppose we place some coins of the same size on a large table and we ask: How should the coins be arranged so that we can put as many coins as possible on the table? The dual problem reads as follows: We want to cover a large domain in the plane with circular paper disks of equal size. How should the disks be arranged so that we use as few of them as possible? Of course, the most favorable arrangement of the circles depends significantly on the size and shape of the domain, and we cannot expect a general rule for determining the desired arrangement for an arbitrary domain. Therefore, we focus only on the asymptotic arrangement of the circles for large domains. The notion of density provides a precise formulation of the considered questions: 1. What is the upper limit for the density of a family of congruent non-overlapping circles, and which families attain this limit? 2. What is the lower limit for the density of a family of congruent circles covering the plane, and which families attain this limit? Concerning these questions we have the following theorems, to be proved later. The density 𝑑 of an arbitrary family of congruent non-overlapping circles satisfies the inequality 𝜋 (3.2.1) 𝑑 ≤ √ = 0.9069 . . . . 12

60

3 Problems on Packing and Covering in the Plane

The density 𝐷 of an arbitrary family of congruent circles covering the whole plane satisfies the inequality 2𝜋 𝐷 ≥ √ = 1.209 . . . . 27

(3.2.2)

These bounds are sharp. Equality is attained in the case when each circle is touched, respectively intersected by, the other circles at the vertices of a regular hexagon. We call these families of circles the densest circle packing and the thinnest circle covering. However, we should bear in mind that removing e.g. one circle from the densest circle packing does not affect the density of the arrangement, although the above condition no longer holds. A general characterization of the extreme families of circles follows later. Intuitively speaking, the inequalities (3.2.1) and (3.2.2) state that at most 90.69 . . . % of the plane can be filled with congruent circles and that in order to cover the plane we need a family of congruent circles whose total area is at least 120.9 . . . % of the area of the plane. We now give some other phrasings of our problems. The problem of the densest circle packing is equivalent to the following: How should a large (but fixed) number Fig. 3.1 of points be selected in a given domain so that each point is as far as possible from the others, that is, so that the least distance between the points is as large as possible? Or conversely: In a large garden we wish to plant as many trees as possible so that any two of them are at least a given distance apart (Figure 3.1). The problem of the thinnest circle covering can be formulated this way: How should a large number of points be selected in a given domain so that each point of the domain is as close as possible to at least one of the selected points, that is, so that the largest distance of a point of the domain from the nearest point of the family is as small as possible? Or conversely: We wish to set up as few as possible oases in a desert so that the distance from each point of Fig. 3.2 the desert to the nearest oasis should not exceed a given value (Figure 3.2).

61

3.3 Some Outlines of Proofs

For both problems the points should be placed at the vertices of a regular triangular lattice, that is, at the vertices of a polyhedron {3, 6}. According to these formulations of the problems, the inequalities (3.2.1) and (3.2.2) can be restated in the following equivalent form: If the distance between any two members of a set of points with number density 𝐴 is at least 𝛿, then 2 𝛿2 𝐴 ≤ √ . (3.2.3) 3 If the distance from any point of the plane to the nearest element of a set of points with number density 𝐴 is at most 𝜚, then 2 𝜚2 𝐴 ≥ √ . 27

(3.2.4)

We now introduce a notion that will also play a role later. We say that a family of congruent circles (that can overlap) is saturated if no circle of the same size can be placed in the part of the plane unoccupied by the circles. By replacing each member of a saturated family of circles with a concentric circle of twice its radius we obtain a covering of the plane. According to (3.2.2), the density of this family is at least √2 𝜋 . However, since doubling the radii Fig. 3.3 27 causes quadrupling the density of the original family, we get: The density 𝑑 of a saturated family of congruent circles satisfies the inequality 𝜋 𝑑≥ √ = 0.302 . . . . 108

(3.2.5)

Figure 3.3 shows the thinnest saturated family of circles along with the thinnest circle covering.

3.3 Some Outlines of Proofs Let 𝐾1 , 𝐾2 , . . . be a family of unit circles with centers 𝑂 1 , 𝑂 2 , . . .. Let 𝑃𝑖 denote the set of those points in the plane whose distance from 𝑂 𝑖 is less than or equal to the distance from the other circles’ centers. 𝑃𝑖 can also be defined as the intersection of the half-planes containing 𝑂 𝑖 that are bounded by the equipotential lines between

62

3 Problems on Packing and Covering in the Plane

𝐾𝑖 and the other circles. In general, 𝑃𝑖 is a convex polygon but it can also stretch to infinity. To take yet another, intuitive look at the polygons 𝑃𝑖 , think of the plane as a sphere of an infinitely large radius. Then the polygons 𝑃𝑖 can be seen as the faces of the “circumscribed polyhedron” touching the sphere at the points 𝑂 𝑖 (Figure 3.4). We will call 𝑃𝑖 the cell assigned to 𝑂 𝑖 (or to 𝐾𝑖 ). These cells cover the plane completely and without overlaps (except for the common boundary points of the cells). If the circles do not overlap, then each circle is contained in its own cell. On the other hand, if the circles cover the plane, then each circle contains its own cell. Consequently, by (1.3.1), we get, in the first case 𝑃𝑖 ≥ 𝜋𝜑( 𝑝 𝑖 );

𝜑( 𝑝) =

𝑝 𝜋 tan 𝜋 𝑝

and in the second case Fig. 3.4 𝑃𝑖 ≤ 𝜋𝜓( 𝑝 𝑖 );

𝜓( 𝑝) =

𝑝 2𝜋 sin , 2𝜋 𝑝

where 𝑝 𝑖 denotes the number of vertices of 𝑃𝑖 . Now, 𝜑( 𝑝) is decreasing for 𝑝 ≥ 3, and in view of 𝜑 ′′ ( 𝑝) =

𝜋 𝑝 𝑝 3 cos2 𝜋𝑝

2𝜋 tan

> 0,

it is convex. On the other hand, 𝜓( 𝑝) is increasing, and by 𝜓 ′′ ( 𝑝) = −

2𝜋 2𝜋 sin < 0, 𝑝 𝑝3

it is concave. Consequently, by Jensen’s theorem it follows that 𝑃¯ ≥ 𝜋𝜑( 𝑝) ¯ for packings and 𝑃¯ ≤ 𝜋𝜓( 𝑝) ¯ for coverings. Here 𝑃¯ and 𝑝¯ denote the average area and the average number of vertices of the cells in the sense of Section 3.1. According to the last interpretation of the cells as the faces of the degenerate polyhedron, and in view of (1.6.6), it is to be expected that 𝑝¯ ≤ 6 holds. This indeed happens to be so, thus for the packing problem we get √ 𝑃¯ ≥ 𝜋𝜑(6) = 12 , and for the covering problem √ 𝑃¯ ≤ 𝜋𝜓(6) =

27 . 2

3.3 Some Outlines of Proofs

63

𝜋 Since the density of a family of unit circles with the average cell area 𝑃¯ is , these 𝑃¯ inequalities are equivalent to the inequalities 3.2.1 and 3.2.2, respectively, which were to be proved. The simple arguments presented above can easily be completed to a rigorous proof. However, we do not wish to go into details, even less so, since in the sequel we will learn several strict proofs, as well as various generalizations, of the inequalities 3.2.1 and 3.2.2. Before we point out further possibilities of proofs, we discuss the question of when equality occurs in our inequalities. Let 𝜀 denote an arbitrarily given positive number and let 𝑆 be a rigidly movable regular hexagon. We consider those cells 𝑃𝑖 ¯ from every congruent copy 𝑆¯ of 𝑆 is greater than 𝜀. Let whose deviation 𝜂(𝑃𝑖 , 𝑆) 𝐹 (𝑅, 𝜀) denote the intersection of the circle 𝐾 (𝑅) considered in Section 3.1 with these cells, and assume that for every value of 𝜀 we have lim 𝑅→∞

𝐹 (𝑅, 𝜀) = 0. 𝐾 (𝑅)

Intuitively speaking, this means that almost all cells are approximately congruent to the regular hexagon 𝑆, while the total area of the remaining cells, compared to the area of the whole plane, is negligible. If there is such a hexagon 𝑆, then we say that our family of circles is hexagonal or honeycomb-like. If, moreover, the circles are congruent to the incircles of 𝑆 and do not overlap, or are congruent to the circumcircles of 𝑆 and cover the plane, then we speak of a honeycomb-like packing and honeycomb-like covering, respectively. It is easy to see that the density √𝜋 , 12

respectively, √2 𝜋 is attained by families of this kind, and only by such families. 27 The fact that for honeycomb-like packings and coverings equality occurs in (3.2.1)and (3.2.2), respectively, is clear without further explanation. Although on the basis of the previous proofs it is plausible that equality can occur only in these cases, this, however, needs closer scrutiny. Here again we omit the details until we have an opportunity for such considerations while dealing with an analogous problem in space. We now seek other ways that might lead us to inequalities (3.2.1) and (3.2.2). Remarkably, in the following arguments, the inequality 𝑝¯ ≤ 6 or any inequality equivalent to it will not be used. Such methods appear to be particularly needed for the still unsolved analogous problems in space, where for the decomposition of space into convex polyhedra, no universal upper bound can be given—neither for the average number of faces, nor for the average number of vertices of the polyhedra. The following arguments appear interesting on their own, since, in view of the large number of spontaneously arising analogous problems, we should place equal emphasis on the methods as on the results themselves. As before, we choose the unit to be the radius of the circles and we show that the “packing density” is at most √𝜋 , not only with respect to the whole plane, but 12 already with respect to each individual cell 𝑃𝑖 , that is, 𝑃𝜋𝑖 ≤ √𝜋 . Thus, we prove the 12

64

3 Problems on Packing and Covering in the Plane

√ √ inequality 𝑃𝑖 ≥ 12 instead of 𝑃¯ ≥ 12, stating that the area of a cell 𝑃𝑖 cannot fall below the area of the regular hexagon 𝑆 circumscribed about 𝐾𝑖 . We even show that √ 𝑃𝑖 𝑈 ≥ 𝑆 = 12, where 𝑈 denotes the circumcircle of 𝑆 (Figure 3.5). If the number 𝑝 𝑖 of vertices of 𝑃𝑖 is at most 6, then our assertion is a direct consequence of the inequality (1.3.3). In the case 𝑝 𝑖 > 6, observe that the distances between the feet of the perpendiculars drawn from 𝑂 𝑖 to the sides of 𝑃𝑖 are at least 1. An easy computation shows that at most 7 points with this property can be placed in the circular ring bounded by 𝐾𝑖 and 𝑈. Since the edge length of the regular 7-gon inscribed in 𝑈 is √4 sin 𝜋 = 1.00201 . . ., slightly greater than Fig. 3.5 7 3 1, it is clear that all seven points in question must lie very close to the boundary of 𝑈. Consequently, the sides of 𝑃𝑖 can cut off only an extremely small portion of 𝑈, hence the area of 𝑃𝑖 𝑈 can fall just a small amount below the area of 𝑈, so that 𝑃𝑖 𝑈 must be considerably greater than 𝑆. Naturally, the last assertion can be supported by numerical estimates. We omit such computations, as the above considerations are completely convincing on their own. In addition, we should notice that what we really need to prove is just the inequality 𝑃𝑖 ≥ 𝑆. However, in the considered case, a much larger part of 𝑃𝑖 lies outside 𝑈 than the other way around, so that even the inequality 𝑃𝑖 > 𝑈 is satisfied. Let us now turn to the covering problem. The following approach treats only the case when every point of the plane belongs to the interior of at most two circles; nevertheless, we discuss it for the reasons mentioned before. If this condition is satisfied, then, without any additional restrictions, we can assume that at most three circles can have a common boundary point, as the case of two pairs of mutually tangent circles passing through a point can be seen as a limiting case of the previous one. We pick a circle 𝐾𝑖 and denote the circles that intersect 𝐾𝑖 in their cyclic order by 𝐾1 , . . . , 𝐾 𝜈 . We consider the “lenses” 𝐾1 𝐾2 , . . . , 𝐾 𝜈 𝐾1 , and we denote by 𝐾12 , . . . , 𝐾 𝜈1 the circles that pass through a vertex of the corresponding lens but do not intersect 𝐾𝑖 (Figure 3.6). Then the sum 𝑆𝑖 = (𝐾𝑖 𝐾1 + · · · + 𝐾𝑖 𝐾 𝜈 ) + (𝐾1 𝐾2 + · · · + 𝐾 𝜈 𝐾1 ) + (𝐾1 𝐾12 + 𝐾2 𝐾12 + · · · + 𝐾 𝜈 𝐾 𝜈1 + 𝐾1 𝐾 𝜈1 )

65

3.3 Some Outlines of Proofs

attains its minimum when all of the occurring lenses are congruent, i.e., when 𝜈 = 6 and all of the thirteen circles 𝐾𝑖 , 𝐾1 , . . . , 𝐾6 , 𝐾12 , . . . , 𝐾61 belong to a thinnest circle covering of the plane. To see this, denote the perimeter of the lens 𝐾 𝑗 𝐾𝑙 by 2𝛼 𝑗𝑙 ; then 𝐾 𝑗 𝐾𝑙 = 𝛼 𝑗𝑙 − sin 𝛼 𝑗𝑙 and thus 𝑆𝑖 = (𝛼𝑖1 + · · · + 𝛼𝑖𝜈 ) − (sin 𝛼𝑖1 + · · · + sin 𝛼𝑖𝜈 ) + (𝛼12 + · · · + 𝛼𝜈1 ) − (sin 𝛼12 + · · · + sin 𝛼𝜈1 ) + (𝛼112 + 𝛼212 + · · · + 𝛼𝜈𝜈1 + 𝛼1𝜈1 ) − (sin 𝛼112 + sin 𝛼212 + · · · + sin 𝛼𝜈𝜈1 + sin1𝜈1 ). The sum of the angles between the parentheses is 2𝜋, (𝜈 − 4)𝜋, and 4𝜋, respectively. Moreover, all of these angles lie in the interval (0, 𝜋), thus, in view of the concavity of sin 𝛼, Jensen’s inequality yields 2𝜋 𝜈−4 2𝜋 + (𝜈 − 4)𝜋 − 𝜈 sin 𝜋 + 4𝜋 − 2𝜈 sin 𝜈 𝜈 𝜈  4𝜋 2𝜋 = (𝜈 + 2)𝜋 − 𝜈 sin + 3 sin = 𝑆(𝜈). 𝜈 𝜈

𝑆𝑖 ≥ 2𝜋 − 𝜈 sin

We show that 𝑆(𝜈) ≥ 𝑆(6) for 𝜈 ≥ 5. In the cases 𝜈 = 3 and 𝜈 = 4 a three-fold covered area, respectively a pair of tangent circles, occur, hence by our assumptions these cases can be excluded. We have 𝑆(5) ≈ 4.7, 𝑆(6) ≈ 4.3 and 𝑆(7) ≈ 5.0. Observe further that, for 𝜈 ≥ 8, we have sin

4𝜋 2𝜋 4𝜋 2𝜋 3 + 3 sin ≤ sin + 3 sin = 1 + √ < 𝜋, 𝜈 𝜈 8 8 2

therefore

  3 𝑆(𝜈) > 2𝜋 + 𝜈 𝜋 − 1 − √ > 2𝜋; 2 which proves our assertion. Now, place on top of each circle eight circular paper disks. Then, obviously, the lenses 𝐾 𝑗 𝐾𝑙 will be covered 16-fold, while the remaining part of the plane only eightfold. From these paper disks we cut out the lens-shaped leaves that correspond to the 4𝜈 terms in the sum for 𝑆𝑖 , one leaf per term, and we perform this operation for each individual circle 𝐾𝑖 . Thereby, from the 16 sheets covering every lens exactly eight leaves are removed. For example, the lens 𝐾𝑖 𝐾1 from Figure 3.6 would lose eight leaves corresponding to the sums 𝑆𝑖 , 𝑆1 , 𝑆2 , 𝑆3 , 𝑆7 , 𝑆6 , 𝑆12 , 𝑆71 . In conclu-

𝜈 ≥ 8,

Fig. 3.6

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3 Problems on Packing and Covering in the Plane

sion, the plane will be covered eight-fold, which can also be interpreted as the sum Í (8𝐾𝑖 − 𝑆𝑖 ), extended over all circles, producing eight times the area of the plane. More precisely, with the notation of Section 3.1, we have lim 𝑅→∞

1 ∑︁ 1 ∑︁ (8𝐾𝑖 − 𝑆𝑖 ) = lim (8𝜋 − 𝑆𝑖 ) = 8 , 𝑅→∞ 𝐾 (𝑅) 𝐾 (𝑅) 𝑅 𝑅

√ from which, in view of 𝑆𝑖 ≥ 𝑆(6) = 8𝜋 − 12 3, we get the inequality lim 𝑅→∞

2𝜋 1 ∑︁ 𝜋 ≥ √ 𝐾 (𝑅) 𝑅 27

as we wanted to prove. To close, we notice that the sum (𝐾𝑖 𝐾1 + · · · + 𝐾𝑖 𝐾 𝜈 ) + (𝐾1 𝐾2 + · · · + 𝐾 𝜈 𝐾1 ) reaches its minimum for 𝜈 = 5, therefore, if we had considered this sum only, then we would not have reached our goal.

3.4 Packing and Covering Convex Disks with Congruent Circles A domain that can fill the plane without gaps and overlaps will be called a tiling domain. Every triangle, quadrilateral or centrally symmetric hexagon is a tiling domain. For the quadrilateral one can see this best by observing that every (not necessarily convex) quadrilateral can be augmented to a centrally symmetric hexagon by its own reflection in the center of a side (Figure 3.7(a)). Figures 3.7(b)–(g) provide further examples of tiling domains. Observe now that the density of a packing of a family of circles in a tiling domain cannot exceed the density of the densest circle packing in the whole plane. More precisely: If a tiling domain 𝑃 is packed with an arbitrary number of non-overlapping congruent circles, then the sum of their areas is at most

√𝜋 𝑃. 12

Otherwise, through a tiling of the plane with replicas of the tiling domain and placing the corresponding circles in each tile, an infinite family of non-overlapping circles could be obtained with density exceeding √𝜋 , which is impossible. 12 In a quite analogous way, we get the statement: If a tiling domain 𝑃 is covered by an arbitrary number of congruent circles, then the sum of their areas is at least √2 𝜋 𝑃. 27

3.4 Packing and Covering Convex Disks with Congruent Circles

67

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Fig. 3.7

68

3 Problems on Packing and Covering in the Plane

It is interesting that the inequalities (3.2.1) and (3.2.2), which can be considered as asymptotic estimates for large domains, make it possible to state such estimates for, say, an arbitrary square. Conversely, the last stated inequalities for a single tiling domain imply the inequalities (3.2.1) and (3.2.2). Naturally, for arbitrary domains the above inequalities are no longer valid, since, for instance, a circle can be completely filled, or covered, by a single (congruent) circle. On the other hand, we show that if the number of circles used for the packing or covering is at least two, then arbitrary convex domains can be considered instead of tiling domains. Thus, the following theorems holds: If at least two congruent non-overlapping circles of total area 𝑠 are contained in a convex disk 𝑇, then 𝜋 𝑠 < √ 𝑇. (3.4.1) 12 If a convex disk 𝑇 is covered by at least two congruent circles of total area 𝑆, then 2𝜋 𝑆 > √ 𝑇. 27

(3.4.2)

As a corollary, these theorems yield: Let a convex disk 𝑇 be covered by 𝐴 unit circles and packed with 𝑎 unit circles. Then 3𝐴 > 4𝑎, unless the disk itself is a unit circle and 𝐴 = 𝑎 = 1. For example, if 100 unit circles suffice to cover a convex disk, then at most 74 unit circles can be packed in it. We consider the convex hull 𝐻 of the unit circles 𝐾1 , . . . , 𝐾 𝑎 that are packed in 𝑇, as well as the intersection of 𝐻 with the cells of 𝐾𝑖 , that is, the sets 𝑇𝑖 consisting of those points of 𝑇 whose distance from the center of 𝐾𝑖 is less than or equal to the distance from the center of any other circle. We split the domains 𝑇𝑖 into two groups, according to whether 𝑇𝑖 is bounded only by straight line segments or its boundary also contains a circular arc. We saw in the previous section that in the first case √ 𝑇𝑖 ≥ 12. We now show that in the second case 𝜋 √ 𝑇𝑖 ≥ 2 + > 12 2 holds. Namely, if we move from a common boundary point of 𝐾𝑖 and 𝐻 along the boundary of 𝑇𝑖 in both directions until we encounter the vertices 𝐸 1 and 𝐸 2 , then already the area of the convex hull of 𝐸 1 , 𝐸 2 and 𝐾𝑖 is at least 2 + 𝜋2 , since the inner angles of 𝑇𝑖 at 𝐸 1 and 𝐸 2 cannot be obtuse, and equality holds only for the so-called cap domain of 𝐾𝑖 with two right-angled caps √ (Figure 3.8). Hence, we have 𝑇 ≥ 𝐻 = 𝑇1 + · · · + 𝑇𝑎 > 12 𝑎, by which the inequality (3.4.1) is proved. We assume now that the unit circles 𝐾1 , . . . , 𝐾 𝐴 cover 𝑇. We may assume that none of these circles is superfluous, i.e. none of them can be removed leaving 𝑇 still covered. As before, we decompose 𝑇 into convex subsets 𝑇1 , . . . , 𝑇𝐴 and view

69

3.4 Packing and Covering Convex Disks with Congruent Circles

them as faces of a degenerate convex polyhedron 𝑃 with 𝐴 + 1 faces, where the additional face 𝑇𝐴+1 is 𝑇 itself. We can assume that every vertex of 𝑃 is three-valent, since a vertex belonging to 𝑚 edges (𝑚 > 3) can be viewed as the limiting case of 𝑚 − 2 coinciding three-valent vertices. Since the number of vertices of a convex polyhedron with 𝑓 faces and three-valent vertices is 2 𝑓 − 4, 𝑃 has exactly 2𝐴 − 2 vertices. We pick a domain 𝑇𝑖 and we denote the “circular segments” cut off from 𝐾𝑖 by the (extended) sides of 𝑇𝑖 cyclically by 𝑠1 , . . . , 𝑠 𝜈 . We should add here that when 𝑇𝑖 has a common edge with the face 𝑇, then by the corresponding “circular segment” we will understand the part of 𝐾𝑖 that lies outside 𝑇, or the corresponding component of that part, if it consists of two or more connected components. Evidently, a point from 𝐾𝑖 that lies neither Fig. 3.8 in 𝑇𝑖 nor on the boundary of a segment lies in circular segments that correspond to consecutive sides of 𝑇𝑖 , for otherwise there would be a superfluous circle. (Consider, for example, the point 𝑄 in Figure 3.9 that lies inside 𝑠1 ≡ 𝑠3 and 𝑠4 , but outside 𝑠2 . The circle indicated there is superfluous.) But if a point lies, say, in 𝑠1 , 𝑠2 and 𝑠3 , then it also lies in 𝑠1 𝑠2 and in 𝑠2 𝑠3 . If we now cover each of the segments 𝑠1 , . . . , 𝑠 𝜈 with a paper leaf and cut the 𝜈 “triangles” 𝑠1 𝑠2 , . . . , 𝑠 𝜈 𝑠1 out of the leaves, then the part of 𝐾𝑖 outside 𝑇𝑖 will be covered exactly one-fold. Consequently, we get 𝐾𝑖 = 𝑇𝑖 + 𝑠1 + · · · + 𝑠 𝜈 − (𝑠1 𝑠2 + · · · + 𝑠 𝜈 𝑠1 ). If we write the corresponding equalities for all circles and add them up, we get ∑︁ ∑︁ 𝜋𝐴 = 𝑇 + 𝐾𝑖 𝐾 𝑗 − 𝐾𝑖 𝐾 𝑗 𝐾 𝑘 . Here 𝑖 𝑗 denotes a pair of indices for which 𝑇𝑖 and 𝑇 𝑗 are adjacent to the same edge of 𝑃, and 𝑖 𝑗 𝑘 denotes a triple for which 𝑇𝑖 , 𝑇 𝑗 and 𝑇𝑘 are adjacent to the same vertex of 𝑃. The summation is extended over all edges and vertices of 𝑃, respectively, with the convention that for the edges and vertices of 𝑇, for the second or third missing circle we have to take the connected component of the corresponding circle lying in the complement 𝐾0 of 𝑇.

Fig. 3.9

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3 Problems on Packing and Covering in the Plane

We can give the above equality yet another form. Namely, writing 𝑆𝑖 𝑗 𝑘 = 𝐾𝑖 𝐾 𝑗 + 𝐾 𝑗 𝐾 𝑘 + 𝐾 𝑘 𝐾𝑖 − 2𝐾𝑖 𝐾 𝑗 𝐾 𝑘 and considering that each edge ends at exactly two vertices, one easily verifies the following relation: 1 ∑︁ 𝑇 = 𝜋𝐴 − 𝑆𝑖 𝑗 𝑘 , (3.4.3) 2 where the summation extends over all vertices of 𝑃 (Figure 3.10). Observe that in the general case, that is when all the three intersections 𝐾𝑖 𝐾 𝑗 , 𝐾 𝑗 𝐾 𝑘 , 𝐾 𝑘 𝐾𝑖 are connected, 𝑆𝑖 𝑗 𝑘 represents the part of the plane covered at least two-fold by 𝐾𝑖 , 𝐾 𝑗 and 𝐾 𝑘 . Now the question is: when does the area of the region 𝑆𝑖 𝑗 𝑘 of points covered at least two-fold by the three circles 𝐾𝑖 , 𝐾 𝑗 and 𝐾 𝑘 , or by two circles 𝐾𝑖 , 𝐾 𝑗 and the outer region 𝐾0 , attain its minimum? Additionally, in both cases the three sets in question should have a point in common, and in the second case, 𝐾0 should be variable as the complement of an arFig. 3.10 bitrary convex set. It is easy to see that for an extreme figure, 𝐾𝑖 𝐾 𝑗 𝐾 𝑘 is reduced to a point in both cases. Further, the six or four bounding circular arcs must be of the same length, and in the case when one of the sets is 𝐾0 the boundary arc of 𝑇 at which 𝐾0 and 𝑆𝑖 𝑗0 meet must consist of a single straight line segment. Therefore, in the extreme case, 𝑆𝑖 𝑗 𝑘 (𝑖, 𝑗, 𝑘 ≠ 0) and 𝑆𝑖 𝑗0 consists of six and four congruent circular segments, respectively. This is expressed in the following inequalities: √ 27 𝑆𝑖 𝑗 𝑘 ≥ 𝜋 − , 𝑆𝑖 𝑗0 ≥ 𝜋 − 2. 2 Consequently, in view of the fact that 𝑃 has 2𝐴 − 2 vertices, we have √ ! 1 27 1 𝑇 ≤ 𝜋𝐴 − 𝜋− (2𝐴 − 2 − 𝑛) − (𝜋 − 2)𝑛, 2 2 2 where 𝑛 is the number of vertices of 𝑃 that belong to the face 𝑇. Thus we have √ √ √ 27 27 − 4 27 𝐴 ≥ 𝑇+ 𝑛−𝜋+ ≈ 𝑇 + 0.3𝑛 − 0.54, 2 4 2 and since 𝑛 ≥ 2, at this point our inequality (3.4.2) is proved as well.

3.5 Dissecting a Convex Domain into Convex Parts

71

3.5 Dissecting a Convex Domain into Convex Parts We give here a second proof of the inequality (3.4.1), more closely related to the proof of inequality (3.4.2) just given above than our first proof. Inequality (3.4.1) will be a corollary of the following more general theorem: If a convex domain is dissected into 𝑛 ≥ 2 convex subdomains 𝑇1 , . . . , 𝑇𝑛 of perimeters 𝐿 1 , . . . , 𝐿 𝑛 , then 𝑛 √ 1 ∑︁ 𝐿 𝑖2 > 8 3. (3.5.1) 𝑛 𝑖=1 𝑇𝑖 𝐿2

As no subdomain can be a circle, by the isoperimetric inequality we have 𝑇𝑖𝑖 > 4𝜋 for each subdomain, and nothing more can be said about a single subdomain in general. On the other hand, inequality (3.5.1) states that the mean value of these quotients, for a certain number of convex domains that can be assembled into a single convex domain, is greater than the value of this quotient for a regular hexagon. For the proof we make the inessential restriction that the boundary 𝐵 of the dissected domain 𝑇 has no vertices and we regard the subdomains 𝑇1 , . . . , 𝑇𝑛 —just as above—as the faces of a trihedral polyhedron 𝑃. By a slight modification of the proof of L’Huilier’s inequality (1.4.4) we get 𝑚𝑖 ∑︁ 𝛽𝑖 𝐿 𝑖2 ≥ 2𝛼𝑖 + 4 tan 𝑘 , 𝑇𝑖 2 𝑘=1

where 𝛽1𝑖 , . . . , 𝛽𝑖𝑚𝑖 denote the outer angles at the vertices of 𝑇𝑖 , and 𝛼𝑖 = 2𝜋 − (𝛽1𝑖 + · · · + 𝛽𝑖𝑚𝑖 ) is the so-called total curvature of the possibly present curvilinear sides of 𝑇𝑖 . Obviously, 𝛼1 + · · · + 𝛼𝑛 = 2𝜋. We add the above inequalities and collect the values of tan 𝛽2 for the individual vertices of 𝑃. If 𝛽1 and 𝛽2 are two angles belonging to a vertex lying on 𝐵, then 𝛽1 + 𝛽2 = 𝜋, hence 𝛽2 𝛽1 𝜋 + tan ≥ 2 tan = 2. tan 2 2 4 If, on the other hand, 𝛽1 , 𝛽2 , 𝛽3 , are three angles belonging to a vertex in the interior of 𝑇 then we have 𝛽1 + 𝛽2 + 𝛽3 = 𝜋, and consequently tan

𝛽2 𝛽3 𝛽1 𝜋 √ + tan + tan ≥ 3 tan = 3. 2 2 2 6

Denoting the number of vertices of 𝑃 lying on 𝐵 by 𝑏, the number of the remaining vertices is 2𝑛 − 2 − 𝑏, and we have

72

3 Problems on Packing and Covering in the Plane 𝑛 ∑︁ √ 𝐿 𝑖2 ≥ 2 · 2𝜋 + 4 · 2𝑏 + 4 3(2𝑛 − 2 − 𝑏) 𝑇 𝑖=1 𝑖 √ √ √ = 8 3 𝑛 + 4[(2 − 3)𝑏 − (2 3 − 𝜋)].

Hence, because of 𝑏 ≥ 2, we get 𝑛 ∑︁ √ √ 𝐿 𝑖2 ≥ 8 3 𝑛 + 0.21 . . . > 8 3 𝑛, 𝑇 𝑖=1 𝑖

which completes the proof. Consider now 𝑛 unit circles packed in a convex domain 𝑇. Dissect 𝑇 according to the known construction into 𝑛 subdomains 𝑇1 , . . . , 𝑇𝑛 of perimeters 𝐿 1 , . . . , 𝐿 𝑛 so that each subdomain contains a unit circle. Then, obviously, we have 𝑇𝑖 ≥ 21 𝐿 𝑖 , that is 4𝑇𝑖 ≥

𝐿𝑖2 𝑇𝑖 .

From this and (3.5.1) we get

4𝑇 = 4

𝑛 ∑︁ 𝑖=1

𝑇𝑖 ≥

𝑛 ∑︁ √ 𝐿 𝑖2 > 8 3 𝑛, 𝑇 𝑖=1 𝑖

which is equivalent to (3.4.1).

3.6 Packing a Convex Domain with Circles of 𝒏 Different Sizes In the densest packing with congruent circles in which every circle is touched by  six others, the circles leave vacant 100 1 − √𝜋 = 9.30 . . . % of the plane. If we 12 fill the interstices with similarly arranged very small non-overlapping circles, then we get a packing of circles of two different sizes that leaves vacant only about 2  = 0.8669 . . . % of the plane. 100 1 − √𝜋 12 The fact that the area percentage of the resulting interstices cannot be pushed below the above value of 0.8669 . . . % by any arrangement of two kinds of circles is seemingly obvious, yet by no means trivial. In what follows, we will prove this under more general conditions. Our result is expressed in the following theorem: Let 𝑇 be a convex disk with an incircle 𝐾, and let 𝑛 be a positive integer. If arbitrarily many non-overlapping circles of 𝑛 different sizes are packed in 𝑇, then the subset 𝑡 of 𝑇 left vacant satisfies the inequality   𝑛−1    𝜋 𝜋 𝐾 𝑡 ≥ 1− √ 1 − max √ , . 𝑇 12 12 𝑇

(3.6.1)

3.6 Packing a Convex Domain with Circles of 𝑛 Different Sizes

73

The bound on the right-hand side is exact. Equality is reached only in the trivial √𝜋 and the incircle 𝐾 alone is packed in 𝑇. case when 𝐾 𝑇 ≥ 12

If we are given a domain 𝑇 with most six sides, then we have

𝐾 𝑇

≤

√𝜋 , 12

such as for example a polygon with at

 𝑛 𝑡 𝜋 ≥ 1− √ . 𝑇 12 The case 𝑛 = 1 of the above theorem is just a different phrasing of the inequality (3.4.1). The proof in the general case is based on the following lemma: Let 𝐺 be a domain obtained by removing from a convex disk 𝑇 a finite number of mutually non-overlapping circles contained in 𝑇. If we pack in 𝐺 an arbitrary number of congruent circles not larger than the smallest removed circle, then the sum of their areas is smaller than √𝜋 𝐺. 12

Let 𝐾1 , . . . , 𝐾 𝜈 be the circles lying in 𝐺 which—as before—we can assume without loss of generality to be unit circles, and let 𝐾 𝜈+1 , . . . , 𝐾 𝜇 be the circles removed from 𝑇. We decompose 𝑇 into sub-domains 𝑇1 , . . . , 𝑇𝜈 , 𝑇𝜈+1 , . . . , 𝑇𝜇 in a similar way as was done for congruent circles, that is, into the intersection of 𝑇 with the cells (Figure 3.11). In this case we use the second definition of the cells given in Section 3.3 (the one based on the potential lines) that can also be applied to Fig. 3.11 the case of non-congruent circles. The arguments in Sections 3.3 and 3.4 in connection with (3.4.1) show that for √ 𝑖 = 1, . . . , 𝜈, that is, for the small circles, 𝑇𝑖 ≥ 12 holds. Therefore we have √ 𝐺 > 𝑇1 + · · · + 𝑇𝜈 ≥ 12 𝜈 , concluding the proof of the lemma. The proof of the inequality (3.6.1) now follows by induction. As we noticed, for 𝑛 = 1 the inequality is valid. If we assume that the inequality is valid for 𝑛 − 1, then for the part 𝐺 of 𝑇 lying outside the circles with the 𝑛 − 1 largest radii, we have:   𝑛−2    𝐺 𝜋 𝜋 𝐾 ≥ 1− √ 1 − max √ , . 𝑇 12 12 𝑇

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3 Problems on Packing and Covering in the Plane

Then, for the part 𝑡 of 𝐺 left void by the smallest circles, our lemma gives 𝑡 𝜋 > 1− √ . 𝐺 12 Multiplying the last two inequalities produces the desired inequality (3.6.1).

3.7 Estimates for Incongruent Circles In what follows, we make another restriction, besides convexity, on the domain to be packed or covered. Namely, we assume that the domain is a convex hexagon; but we allow degenerate hexagons as well, that is, convex polygons with fewer than six sides. This constraint allows for generalizations in other directions and also has a small advantage that the case of a single circle need not be excluded. Most considerations of the next paragraphs can be easily transferred to non-convex hexagons. However, for the sake of simplicity and uniformity we shall only consider convex hexagons. First we prove the following theorems: If 𝑛 arbitrary circles 𝐾1 , . . . , 𝐾𝑛 are packed in a convex hexagon 𝑆, then the density of the packing satisfies 𝜋 𝜋 𝐾1 + · · · + 𝐾 𝑛 ≤ cot ; 𝑆 6𝑞 6𝑞

𝑞=

max(𝐾1 , . . . , 𝐾𝑛 ) . min(𝐾1 , . . . , 𝐾𝑛 )

(3.7.1)

If a convex hexagon 𝑆 is covered by 𝑛 arbitrary circles 𝐾1 , . . . , 𝐾𝑛 , then the density of the covering satisfies 𝜋 𝜋 𝐾1 + · · · + 𝐾 𝑛 ≥ csc ; 𝑆 3𝑞 3𝑞

𝑞=

max(𝐾1 , . . . , 𝐾𝑛 ) . min(𝐾1 , . . . , 𝐾𝑛 )

(3.7.2)

max(𝐾1 , . . . , 𝐾𝑛 ) in each These results can be strengthened by replacing 𝑞 with 𝐾 𝐾1 + · · · + 𝐾 𝑛 is the average area of the circles 𝐾1 , . . . , 𝐾𝑛 . For of them, where 𝐾 = 𝑛 𝑞 = 1, i.e. for congruent circles, the inequalities (3.7.1) and (3.7.2) yield the exact 𝜋 𝜋 𝜋 𝜋 2𝜋 𝜋 bounds cot = √ and csc = √ , respectively. 6 6 3 3 27 12 We decompose the hexagon into the polygons 𝑃1 , . . . , 𝑃𝑛 such that 𝑃𝑖 contains the circle 𝐾𝑖 in the case of packing and 𝐾𝑖 contains 𝑃𝑖 if the circles cover 𝑆. Denoting by 𝑝 𝑖 the number of vertices of 𝑃𝑖 , we have 𝑃𝑖 ≥ 𝐾𝑖 𝜑( 𝑝 𝑖 ) ; for the packing problem and

𝜑( 𝑝) =

𝜋 𝑝 tan 𝜋 𝑝

3.7 Estimates for Incongruent Circles

𝑃𝑖 ≤ 𝐾𝑖 𝜓( 𝑝 𝑖 ) ;

75

𝜓( 𝑝) =

𝑝 2𝜋 sin 2𝜋 𝑝

for the covering problem. We saw in Section 3.3 that 𝜑( 𝑝) is convex, while 𝜓( 𝑝) is concave. Consequently, we get for the density 𝐷 of the arrangement 𝑃1 + · · · + 𝑃 𝑛 1 𝐾1 𝜑( 𝑝 1 ) + · · · + 𝐾𝑛 𝜑( 𝑝 𝑛 ) = ≥ 𝐷 𝐾1 + · · · + 𝐾 𝑛 𝐾1 + · · · + 𝐾 𝑛   𝐾1 𝑝 1 + · · · + 𝐾 𝑛 𝑝 𝑛 ≥𝜑 𝐾1 + · · · + 𝐾 𝑛 or 𝑃1 + · · · + 𝑃 𝑛 1 𝐾1 𝜓( 𝑝 1 ) + · · · + 𝐾𝑛 𝜓( 𝑝 𝑛 ) = ≤ 𝐷 𝐾1 + · · · + 𝐾 𝑛 𝐾1 + · · · + 𝐾 𝑛   𝐾1 𝑝 1 + · · · + 𝐾 𝑛 𝑝 𝑛 ≤𝜓 𝐾1 + · · · + 𝐾 𝑛 depending on whether we have a packing or covering, respectively. However, in view of (1.6.8), we have 𝐾1 𝑝 1 + · · · + 𝐾 𝑛 𝑝 𝑛 max(𝐾1 , . . . , 𝐾𝑛 ) 𝑝 1 + · · · + 𝑝 𝑛 ≤ 𝐾1 + · · · + 𝐾 𝑛 𝐾 𝑛 max(𝐾1 , . . . , 𝐾𝑛 ) ≤ 6 ≤ 6𝑞 𝐾 and consequently—by the monotonicity of the functions 𝜑( 𝑝) and 𝜓( 𝑝)—we get 1 1 𝐷 ≥ 𝜑(6𝑞) and 𝐷 ≤ 𝜓(6𝑞), respectively. This is exactly what the inequalities (3.7.1) and (3.7.2) state. We now prove the following, somewhat deeper theorems: If the circles 𝐾1 , . . . , 𝐾𝑛 are packed in a convex hexagon 𝑆, then for an arbitrary √ !2 27 1 4− = 0.77 . . . we have 𝛼 ≤ 1− 24 𝜋 𝜋 𝑛𝑀 𝛼 (𝐾1 , . . . , 𝐾𝑛 ) ≤ √ 𝑆. 12

(3.7.3)

If the convex hexagon 𝑆 is covered by the circles 𝐾1 , . . . , 𝐾𝑛 , then for an arbitrary √ !2 27 1 2+ = 2.11 . . . we have 𝛼 ≥ 1+ 12 𝜋 2𝜋 𝑛𝑀 𝛼 (𝐾1 , . . . , 𝐾𝑛 ) ≥ √ 𝑆. 27

(3.7.4)

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3 Problems on Packing and Covering in the Plane

Here, consistent with previously used notation, 𝐾1𝛼 + · · · + 𝐾𝑛𝛼 𝑀 𝛼 (𝐾1 , . . . , 𝐾𝑛 ) = 𝑛 

 𝛼1

stands for the so-called power mean of degree 𝛼 of 𝐾1 , . . . , 𝐾𝑛 . For congruent circles, the inequalities (3.7.3) and (3.7.4) are equivalent to (3.7.1) and (3.7.2), respectively. Further, it should be noted that in the case 𝛼 = 1, only the trivial inequalities 𝑛𝑀1 < 𝑆 and 𝑛𝑀1 > 𝑆 can be stated. On the other hand, in the limiting cases 𝛼 → −∞ and 𝛼 → ∞, we have 𝑛𝑀−∞ = 𝑛 min(𝐾1 , . . . , 𝐾𝑛 ) ≤ √𝜋 𝑆 12

and 𝑛𝑀∞ = 𝑛 max(𝐾1 , . . . , 𝐾𝑛 ) ≥ √2 𝜋 𝑆, respectively. It is therefore expected that 27 the bounds that can be reached by 𝑀 𝛼 change monotonically from 𝑆 to √𝜋 𝑆 and 12

from 𝑆 to √2 𝜋 𝑆 as 𝛼 varies from −1 to −∞ and from 1 to ∞, respectively. However, 27 the interesting fact about inequalities (3.7.3) and (3.7.4) is that the bounds valid in the limiting cases 𝛼 → −∞ and 𝛼 → ∞ are already valid for certain specific values of 𝛼 lying relatively close to 1. The constants 0.77 . . . and 2.11 . . . are not the best possible. On the other hand, the packing with circles of two sizes (with “twice as many” small circles as large ones) obtained by filling each of the interstices in the densest, honeycomb circle  2𝛼  > 31−𝛼 , packing with a smaller circle, shows that for 𝛼 satisfying 1 + 2 √2 − 1 3 that is, for 𝛼 > 0.94 . . ., the inequality (3.7.3) is not valid anymore. The circles considered in the above example form the family of incircles of the faces of the degenerate semiregular polyhedron (3, 12, 12). In an analogous way, the family of the face-circumcircles (4, 8, 8) shows that for any value of 𝛼  polyhedron  √  of the √  𝛼

𝛼

√ 2 , that is, for 𝛼 < 1.1 . . ., the inequality (3.7.4) is satisfying 1 + 2−2 2 < 2 2+ 27 not valid anymore. We emphasise the case 𝛼 = 21 of the inequality (3.7.3):

If 𝑛 non-overlapping circles of radii 𝑟 1 , . . . , 𝑟 𝑛 are placed in a convex hexagon 𝑆, then 𝑛𝑆 (3.7.5) (𝑟 1 + · · · + 𝑟 𝑛 ) 2 ≤ √ . 12 This implies: In order to pack a large (but fixed) number of circles in a given domain so that the sum of their perimeters is as large as possible, one should take congruent circles of the appropriate size and arrange them in the densest packing. Naturally, the analogous statement for the total area of the circles is not correct. The proofs of (3.7.3) and (3.7.4) are based on the convexity of certain functions of two variables, namely 1

Φ(𝑥, 𝑦) = 𝑥 𝛼 𝜑(𝑦),

𝑥 ≥ 0,

𝑦 ≥ 3;

𝜑(𝑦) =

𝜋 𝑦 tan 𝜋 𝑦

3.7 Estimates for Incongruent Circles

77

and 1

Ψ(𝑥, 𝑦) = 𝑥 𝛼 𝜓(𝑦),

𝑥 ≥ 0,

𝑦 ≥ 3;

𝜓(𝑦) =

𝑦 2𝜋 sin . 2𝜋 𝑦

We show that Φ(𝑥, 𝑦) is convex for every 𝛼 ≤ 0.77 . . . (𝛼 ≠ 0), while Ψ(𝑥, 𝑦) is concave for 𝛼 ≥ 2.11 . . . . The condition for convexity of Φ(𝑥, 𝑦) is   Φ 𝑥 𝑥 Φ 𝑦𝑦 − Φ2𝑥 𝑦 = 𝛼−2 𝑥 2𝛼−2 (1 − 𝛼)𝜑𝜑 ′′ − (𝜑 ′) 2 ≥ 0, that is, 𝜋 2 𝑦 2 cos4

 𝜋  (1 − 𝛼)𝜑𝜑 ′′ − (𝜑 ′) 2 𝑦 = 2𝜋 2 (1 − 𝛼) sin2

 2 𝜋 𝜋 𝜋 − 𝜋 − 𝑦 sin cos ≥0 𝑦 𝑦 𝑦

or 2𝜋 2 (1 − 𝛼) ≥ [ 𝑓 (𝑦)] 2 ,

𝑓 (𝑦) = 𝜋 csc

𝜋 𝜋 − 𝑦 cos . 𝑦 𝑦

Because   𝜋 𝜋 ′ 𝜋 2 2 𝜋 2 𝑦 tan sin 𝑓 (𝑦) = 𝜋 − sin 𝑦 + 𝜋𝑦 tan 𝑦 𝑦 𝑦 𝑦   𝜋 𝜋 𝜋 < 𝜋 2 − sin2 𝑦 2 + 𝜋 2 = 𝜋 2 cos2 − 𝑦 2 sin2 < 0, 𝑦 𝑦 𝑦 2

the function 𝑓 (𝑦) is monotonically decreasing for 𝑦 ≥ 3. Therefore the convexity condition is certainly satisfied when 2𝜋 2 (1 − 𝛼) ≥ [ 𝑓 (3)] 2 . Similarly, the condition for concavity of Ψ(𝑥, 𝑦) is   Ψ𝑥 𝑥 Ψ𝑦𝑦 − Ψ𝑥2 𝑦 = −𝛼−2 𝑥 2𝛼−2 (𝛼 − 1)𝜓𝜓 ′′ + (𝜓 ′) 2 ≥ 0, that is,   −4𝜋 2 𝑦 2 (𝛼 − 1)𝜓𝜓 ′′ + (𝜓 ′) 2  2 2𝜋 2𝜋 2𝜋 − 𝑦 sin − 2𝜋 cos ≥ 0 = 4𝜋 (𝛼 − 1) sin 𝑦 𝑦 𝑦 2

2

or 4𝜋 2 (𝛼 − 1) ≥ [𝑔(𝑦)] 2 ,

𝑔(𝑦) = 𝑦 − 2𝜋 cot

2𝜋 . 𝑦

In view of sin2

2𝜋 ′ 2𝜋 4𝜋 2 𝑔 (𝑦) = sin2 − 2 < 0, 𝑦 𝑦 𝑦

the function 𝑔(𝑦) is monotonically decreasing for 𝑦 ≥ 3, and thus the above concavity condition is certainly satisfied when 4𝜋 2 (𝛼 − 1) ≥ [𝑔(3)] 2 .

78

3 Problems on Packing and Covering in the Plane

Using the above notation, we now have

𝑆=

𝑛 ∑︁

𝑛 ∑︁

𝑃𝑖 ≥

𝑖=1

𝑛

1 (𝐾𝑖𝛼 ) 𝛼 𝜑( 𝑝 𝑖 )

𝑖=1

1 ∑︁ 𝛼 𝐾 ≥𝑛 𝑛 𝑖=1 𝑖

! 𝛼1 𝜑(6);

𝛼 ≤ 0.77 . . . ,

𝜓(6);

𝛼 ≥ 2.11 . . .

for packings and

𝑆=

𝑛 ∑︁ 𝑖=1

𝑃𝑖 ≤

𝑛 ∑︁

𝑛

1 (𝐾𝑖𝛼 ) 𝛼 𝜓( 𝑝 𝑖 )

𝑖=1

1 ∑︁ 𝛼 𝐾 ≤𝑛 𝑛 𝑖=1 𝑖

! 𝛼1

for coverings. This completes the proof of the inequalities (3.7.3) and (3.7.4). We present the inequalities (3.7.3) and (3.7.4) in another form that allows us to estimate the density 𝐷 of the packing and of the covering of arbitrarily given circles 𝐾1 , . . . , 𝐾𝑛 in a convex hexagon. These inequalities are as follows: 𝜋 𝑀1 (𝐾1 , . . . , 𝐾𝑛 ) 𝐷 ≤ √ ; 12 𝑀 𝛼 (𝐾1 , . . . , 𝐾𝑛 )

𝛼 ≤ 0.77 . . .

and

2𝜋 𝑀1 (𝐾1 , . . . , 𝐾𝑛 ) 𝐷 ≥ √ ; 𝛼 ≥ 2.11 . . . . 27 𝑀 𝛼 (𝐾1 , . . . , 𝐾𝑛 ) We now direct our attention to a somewhat more special problem. We consider the case when an equal number of copies of two circles, 𝑘 and 𝐾, of different radii are packed in a convex hexagon 𝑆. For the density 𝐷 of such a packing, the arguments used in the proof of (3.7.1) yield the inequality 𝑘 𝜑( 𝑝) + 𝐾 𝜑(𝑃) 1 ≥ , 𝐷 𝑘+𝐾 where 𝑝 and 𝑃 denote the average number of vertices of the polygons assigned to the small and to the large circles, respectively. Obviously, we have 𝑝, 𝑃 ≥ 3, thus, in view of 𝑝 + 𝑃 ≤ 12, we get 𝑃 ≤ 9. Since under these conditions the sum 𝑘 𝜑( 𝑝) + 𝐾 𝜑(𝑃) reaches its minimum when 𝑝 + 𝑃 = 12, we get 𝐷≤

𝑘+𝐾 . min [𝑘 𝜑(6 − 𝑥) + 𝐾 𝜑(6 + 𝑥)]

0≤𝑥 ≤3

Quite analogously, for the covering density we have 𝐷≥

𝑘+𝐾 . max [𝑘𝜓(6 − 𝑥) + 𝐾𝜓(6 + 𝑥)]

0≤𝑥 ≤3

√ Let us consider the case 𝑘 = ( 2 − 1) 2 𝜋, 𝐾 = 𝜋, corresponding to the family of circles inscribed in the faces of the tiling (4,8,8). Here, “equally many” small and

79

3.7 Estimates for Incongruent Circles

large circles occur, in accordance with our assumption (Figure 3.12). The density of this family is √ ! 𝑘+𝐾 2 𝐷0 = = 1− 𝜋 = 0.92015 . . . . 𝑘 𝜑(4) + 𝐾 𝜑(8) 2

Fig. 3.13

Fig. 3.12

However, 𝑘 𝜑(6 − 𝑥) + 𝐾 𝜑(6 + 𝑥) reaches its minimum not for 𝑥 = 2, but for some value 1.4 < 𝑥 < 1.5. The numerical value of this minimum is approximately only 3.9869, compared to 𝑘 𝜑(4) + 𝐾 𝜑(8) = 4. Thus, our argument yields only the bound 𝐷 < 0.92317, a value greater than 𝐷 0 by about 13 %. The natural conjecture that the circles of the packing considered here are indeed arranged in the densest way has not yet been proved. On the other hand, it is easy to see that the incircles of the faces of the degenerate polyhedron (3,12,12) are indeed arranged in a densest packing. For, in this family, the large circles by themselves form the densest packing (Figure 3.13). If there were a denser packing with the two kinds of circles, then by deleting the small ones, one would obtain a packing with congruent circles, denser than √𝜋 , which cannot 12 happen. In conclusion, we would like to point out a few other unsolved problems arising from the following theorem: There is a value 𝑞 > 1 such that the density 𝐷 of every family {𝐾𝑖 } of nonoverlapping circles with the property 𝐾𝑖 ≤ 𝑞, 𝐾𝑗

𝑖, 𝑗 = 1, 2, . . . ;

𝑖≠ 𝑗

𝜋 satisfies the inequality 𝐷 ≤ √ . 12 In other words, if the circles are “not too different” from each other, then the density of their packing cannot exceed that of the densest packing with circles of equal size. This theorem is a consequence of the fact that for sufficiently small values of 𝐾𝑖 𝜋 𝑞 − 1, the inequality ≤ √ holds for all circles, where 𝑃𝑖 denotes the cell of 𝑃𝑖 12

◀ 204

80

3 Problems on Packing and Covering in the Plane

Fig. 3.14

203 ▶

𝐾𝑖 . Namely, if the number of vertices 𝑝 𝑖 of 𝑃𝑖 is at most six, then this statement is trivial, and in the case 𝑝 𝑖 > 6, it can be deduced in an analogous way as for the congruent circles. Let 𝐷 (𝑞) denote the supremum of the densities of all circle packings {𝐾𝑖 } 𝐾𝑖 satisfying ≤ 𝑞 for 𝑖 ≠ 𝑗. Figure 3.14 shows a lower bound 𝑑 (𝑞) ≤ 𝐷 (𝑞) for the 𝐾𝑗 function 𝐷 (𝑞). This bound is probably exact (𝑑 = 𝐷), and if it is, it would imply, for example, that the above theorem is true even for about 𝑞 = 2. It should be noticed, 𝐾𝑖 however, that the local density with respect to a single cell can exceed √𝜋 already 12 𝑃𝑖  2 𝜋 for smaller values of 𝑞. For example, for the value 𝑞 = csc − 1 = 1.7 . . ., the cell 7 𝑃𝑖 can be a regular heptagon circumscribed about 𝐾𝑖 . But in that case the adjacent cells must be fairly irregularly shaped, such as, say, relatively large pentagons, compensating for the small area of the regular heptagon. Concerning the problem of the thinnest covering, the existence of a corresponding constant 𝑞 > 1 has not been established yet. An upper bound for 𝐷 (𝑞) is provided by (3.7.1). It is easy to prove the following, better estimate:   𝑄+𝑞 𝐷 ≤ max : 0 ≤ 𝑥 ≤ 3, 𝑄 ≥ 0 . 𝑄𝜑(6 − 𝑥) + 𝑞𝜑(6 + 𝑄𝑥) This bound holds under the same conditions as (3.7.1). Combining the above with √ the lower bound from Figure 3.14 yields, e.g. for 𝑞 = 3 + 8 = 5.82 . . ., 0.920 . . . ≤ 𝐷 ≤ 0.925 . . . .

3.8 A Further Theorem on Covering with Circles

81

3.8 A Further Theorem on Covering with Circles Let Ω denote a finite family of congruent circles, as well as their total area. If the circles are packed in a convex hexagon 𝑆, then we have Ω 𝜋 ≤ √ . 𝑆 12

(3.8.1)

If, on the other hand, 𝑆 is covered by Ω, then Ω 2𝜋 ≥ √ . 𝑆 27

(3.8.2)

Consider now a family Ω of total area Ω = 𝑆, or more generally, consider the case √𝜋 < Ω𝑆 < √2 𝜋 . Then Ω can neither be packed in 𝑆, nor can it cover 𝑆. The 27 12 question arises: What portion of 𝑆 can be covered by Ω? The answer is contained in the following theorem, illustrated by Figure 3.15: Let 𝑆 be a convex hexagon and let Ω be a family of congruent circles. Then for the area of the part of 𝑆 covered by Ω the inequality ¯ 𝑆¯ Ω𝑆 ≤ Ω

(3.8.3)

¯ is a circle of area Ω ¯ =Ω holds, where 𝑆¯ is a regular hexagon of area 𝑆¯ = 𝑆 and Ω ¯ concentric with 𝑆.

Fig. 3.15 ¯ However, Ω 𝑆 can be Equality holds only in the case when 𝑆 ≡ 𝑆¯ and Ω ≡ Ω. ¯ ¯ arbitrarily close to Ω𝑆 if the centers of the circles are vertices of a sufficiently dense regular triangular lattice (that is, of a degenerate regular polyhedron {3, 6} with sufficiently small edge length). In  other words, a family Ω of congruent circles can cover at most  100Δ Ω𝑆 % of 𝑆, where the function Δ(𝑥), illustrated in Figure 3.16, is defined

82

3 Problems on Packing and Covering in the Plane

by  𝑥  √︃ √    Δ(𝑥) = 𝑥 + 6 𝜋3𝑥 − 3 −     1

for 0 ≤ 𝑥 ≤ 6 𝜋 𝑥 arccos

»

√𝜋 2 3𝑥

for

√𝜋 12

for 𝑥 ≥

≤𝑥

√𝜋 , 12 ≤ √2 𝜋 , 27

√2 𝜋 . 27

For example, a family of congruent circles of total area 1 can cover at most 100Δ(1) ≈ 96.3% of the unit square. Our theorem contains the inequalities (3.8.1) and (3.8.2) as special cases. For, according to (3.8.3), the family Ω can be packed into 𝑆 only ¯ is contained in 𝑆, ¯ as otherwise we if the circle Ω ¯ ¯ ¯ This, however, would have Ω𝑆 = Ω = Ω > Ω𝑆. ¯ Ω Ω 𝜋 implies that 𝑆 = 𝑆¯ ≤ √ . If, on the other hand, 12 ¯ must Ω covers the hexagon 𝑆, then, conversely, Ω ¯ contain the regular hexagon 𝑆, since otherwise Fig. 3.16 ¯ 𝑆. ¯ However, we would have Ω𝑆 = 𝑆 = 𝑆¯ > Ω ¯ this is possible only when Ω𝑆 = Ω𝑆¯ ≥ √2 𝜋 . 27 The above theorem is an immediate consequence of the following more general theorem. Let 𝑃1 , . . . , 𝑃𝑛 be 𝑛 points in the plane of a convex hexagon 𝑆; let 𝑑 (𝑃) = min(𝑃𝑃1 , . . . , 𝑃𝑃𝑛 ) denote the distance of a variable point 𝑃 to the nearest point 𝑃𝑖 and let d 𝑓 be the area-element at 𝑃. If 𝑎(𝑥) is a decreasing function defined for 𝑥 ≥ 0, then ∫ ∫ 𝑎(𝑑 (𝑃)) d 𝑓 ≤ 𝑛 𝑎(𝑂𝑃) d 𝑓 , (3.8.4) 𝑆

𝜎

where 𝜎 is a regular hexagon of area 𝜎 =

𝑆 𝑛

centered at 𝑂.

Applying (3.8.4) to the function ( 𝑎(𝑥) =

1 for 0 ≤ 𝑥 ≤ 𝑟 0 for 𝑟 < 𝑥

the left-hand side of (3.8.4) gives the area of the part of 𝑆 covered by the circles of radius 𝑟 centered at 𝑃1 , . . . , 𝑃𝑛 . Further, since the integral on the right-hand side is equal to the area of the intersection of 𝜎 and the circle of radius 𝑟 concentric with 𝜎, the inequality (3.8.4) indeed yields (3.8.3).

83

3.8 A Further Theorem on Covering with Circles

For the proof of (3.8.4), we make two preliminary remarks. Remark 1. Let 𝐾 be a circle with center 𝑂 and let 𝑠 be a segment of 𝐾. Then the function ∫ 𝐾 𝜔(𝑠) = 𝑎(𝑂𝑃) d 𝑓 ; 0≤𝑠< 2 𝑠 is convex. Proof. Let 𝑠1 , 𝑠1∗ , 𝑠2 , and 𝑠2∗ be segments of 𝐾 cut off by parallel lines such that 𝑠1 < 𝑠1∗ < 𝑠2 < 𝑠2∗ and Δ𝑠1 = 𝑠1∗ −𝑠1 = Δ𝑠2 = 𝑠2∗ −𝑠2 . Obviously there exists an area-preserving mapping of the “trapezoid” Δ𝑠1 onto Δ𝑠2 such that Fig. 3.17 each point 𝑄 1 of Δ𝑠1 is mapped onto a point 𝑄 2 of Δ𝑠2 lying closer to 𝑂 than 𝑄 1 (Figure 3.17). Therefore, by the monotonicity of 𝑎(𝑥), we have 𝑎(𝑂𝑄 1 ) ≤ 𝑎(𝑂𝑄 2 ), hence ∫ ∫ 𝜔(𝑠1∗ ) − 𝜔(𝑠1 ) = 𝑎(𝑂𝑃) d 𝑓 ≤ 𝑎(𝑂𝑃) d 𝑓 = 𝜔(𝑠2∗ ) − 𝜔(𝑠2 ). Δ𝑠1

This implies

𝜔 ′ (𝑠

1)

≤

𝜔 ′ (𝑠

Δ𝑠2

2 ).

Remark 2. Let 𝐴 and 𝐵 be two points of the circle 𝐾 different from the center 𝑂 and let 𝐴 ′, 𝐵 ′ be the points of intersection of the rays 𝑂 𝐴 and 𝑂𝐵 with the boundary of 𝐾. If 𝑡 denotes the subset of 𝐾 bounded by the segments 𝐴 ′ 𝐴, 𝐴𝐵, 𝐵𝐵 ′ and by the circular arc 𝐵 ′ 𝐴 ′, then we have ∫ 𝑎(𝑂𝑃) d 𝑓 ≥ 𝜔(𝑡). 𝑡

Proof. Let 𝐶 and 𝐷 be two points on the boundary of 𝑡 such that 𝑂𝐶 = 𝑂𝐷 and the line 𝐶𝐷 cuts off from 𝐾 a segment 𝑠 with the same area as 𝑡 (Figure 3.18). Since, with the exception of 𝐶 and 𝐷, each point of the set 𝑡 − 𝑡𝑠 is closer to 𝑂 than any point of 𝑠 − 𝑠𝑡, we have the desired inequality ∫ ∫ ∫ ∫ ∫ ∫ = + ≥ + = , 𝑡

𝑡𝑠

𝑡−𝑡 𝑠

𝑠𝑡

𝑠−𝑠𝑡

𝑠

where after each integral sign the expression 𝑎(𝑂𝑃) d 𝑓 is to be placed. Now, we turn to the proof of the inequality (3.8.4). Obviously, we may assume that the points 𝑃1 , . . . , 𝑃𝑛 lie in 𝑆 since otherwise the left-hand side of (3.8.4) could be increased by translating the points outside 𝑆 into 𝑆. We dissect 𝑆 into the convex sub-polygons 𝑇1 , . . . , 𝑇𝑛 , where 𝑇𝑖 is defined,

84

3 Problems on Packing and Covering in the Plane

according to the previously used construction, as the set of those points of 𝑆 for which 𝑑 (𝑃) = 𝑃𝑃𝑖 . Then we can write ∫ 𝑎(𝑑 (𝑃))d 𝑓 = 𝑆

𝑛 ∫ ∑︁

𝑎(𝑃𝑃𝑖 ) d 𝑓 .

𝑇𝑖

𝑖=1

S P2

A’

00 11

A C

P d(P) 11 00

t1

00 11P1

t5 000000000000000000000000000000 111111111111111111111111111111 E E5 1 000000000000000000000000000000 111111111111111111111111111111 E4

P

O

i 1111111111111111111111 0000000000000000000000 00 11 0000000000000000000000 1111111111111111111111 0000000000000000000000 E1111111111111111111111 2 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 t2 1111111111111111111111 0000000000000000000000 t3 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 00 11P3 E3

D

B

B’ Fig. 3.18

00 11P4

Fig. 3.19

We denote the vertices of 𝑇𝑖 in their cyclic order by 𝐸 1 , . . . , 𝐸 𝑝𝑖 . Let 𝑟 be the circumradius of 𝜎. We draw a circle 𝐾 ≡ 𝐾𝑖 with center 𝑃𝑖 and radius 𝑟 and denote by 𝑡 1 the subset of 𝐾𝑖 bounded by the rays 𝑃𝑖 𝐸 1 , 𝑃𝑖 𝐸 2 and the line 𝐸 1 𝐸 2 lying outside of 𝑇𝑖 . Of course, 𝑡1 can be empty. Analogously, we consider the domains 𝑡2 , . . . , 𝑡 𝑝𝑖 (Figure 3.19). Further, denoting the part of 𝑇𝑖 lying outside of 𝐾𝑖 by 𝑇𝑖′ , we have ∫ ∫ ∫ ∫ ∫ = + − −··· − , 𝑇𝑖

𝑇𝑖′

𝐾

𝑡1

𝑡 𝑝𝑖

where after each integral sign the expression 𝑎(𝑂𝑃) d 𝑓 is to be placed. Now, according to Remark 2, we have ∫ ∫ ∫ ≤ + −𝜔(𝑡1 ) − · · · − 𝜔(𝑡 𝑝𝑖 ). 𝑇𝑖

𝐾

𝑇𝑖′

Setting 𝑁 = 𝑝 1 + · · · + 𝑝 𝑛 and adding the respective inequalities for 𝑖 = 1, . . . , 𝑛 yields ∫ ∫ 𝑛 ∫ 𝑁 ∑︁ ∑︁ 𝑎(𝑑 (𝑃)) d 𝑓 ≤ 𝑛 + − 𝜔(𝑡𝑖 ). 𝑆

𝐾

𝑖=1

𝑇𝑖′

𝑖=1

On the other hand, 𝑆 = 𝑛𝐾 + 𝑇 ′ −

𝑁 ∑︁

𝑡𝑖 ,

𝑖=1

Í𝑛 ′ where 𝑇 ′ = 𝑖=1 𝑇𝑖 denotes the set of points of 𝑆 lying outside of the circles 𝐾1 , . . . , 𝐾𝑛 . Hence, in view of Remark 1 and Jensen’s inequality, we get

3.9 Dissecting a Convex Hexagon into Convex Polygons

∫

∫ 𝑎(𝑑 (𝑃)) d 𝑓 ≤ 𝑛

+

𝑆

𝐾

𝑛 ∫ ∑︁

85

 −𝑁𝜔

𝑇𝑖′

𝑖=1

 𝑛𝐾 + 𝑇 ′ − 𝑆 . 𝑁

Further, it follows from 𝜔(0) = 0 and the convexity of 𝜔(𝑥) that 1𝑥 𝜔(𝑥) is a monotonically increasing function, and thus 𝑥𝜔( 1𝑥 ) is a decreasing function of 𝑥. Since 𝑁 ≤ 6𝑛, this implies ∫

∫ 𝑎(𝑑 (𝑃)) d 𝑓 ≤ 𝑛

+

𝑆

𝐾

𝑛 ∫ ∑︁ 𝑖=1

 −6𝑛𝜔

𝑇𝑖′

 𝑛𝐾 + 𝑇 ′ − 𝑆 . 6𝑛

Denoting by 𝑠 the part of 𝐾 that supplements the segment of area ′ −𝑆 we have segment of area 𝑛𝐾+𝑇 6𝑛 ∫

∫



𝑎(𝑑 (𝑃)) d 𝑓 ≤ 𝑛

−6𝜔

𝑆

𝐾

𝑛𝐾 − 𝑆 6𝑛

 +

𝑛 ∫ ∑︁ 𝑖=1

𝑛𝐾−𝑆 6𝑛

to a

∫ −6𝑛

𝑇𝑖′

. 𝑠

However, because of the monotonicity of the function 𝑎(𝑥), we have 𝑛 ∫ ∑︁ 𝑖=1

∫

≤ 𝑇 ′ 𝑎(𝑟) − 6𝑛𝑠𝑎(𝑟) = 0.

−6𝑛

𝑇𝑖′

𝑠

= 16 (𝐾 − 𝜎), the expression in the square On the other hand, because of 𝑛𝐾−𝑆 6𝑛 ∫ brackets is just 𝜎 𝑎(𝑂𝑃) d 𝑓 . This completes the proof of the inequality (3.8.4).

3.9 Dissecting a Convex Hexagon into Convex Polygons We prove here the following theorem: If a convex hexagon is dissected into convex polygons 𝑇1 , . . . , 𝑇𝑛 with respective perimeters 𝐿 1 , . . . , 𝐿 𝑛 , then 𝑛 √4 1 ∑︁ 𝐿 𝑖 √ ≥ 2 12 . 𝑛 𝑖=1 𝑇𝑖

(3.9.1)

Considering the fact that the quadratic mean of different quantities is greater than their arithmetic mean, in the case when the dissected set is a hexagon, the inequality (3.9.1) is an improvement of the inequality (3.5.1). We emphasize the following corollary of our theorem: If 𝐿 denotes the total length of a net that dissects a convex hexagon 𝑆 of perimeter 𝑈 into convex polygons of equal areas, then 𝐿+

√4 √ 1 𝑈 ≥ 12 𝑛𝑆 . 2

(3.9.2)

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3 Problems on Packing and Covering in the Plane

Therefore, if one were to span a net with a given large number of eyelets of equal areas on a fixed frame so that the total length of the net is small, then the most appropriate way of doing it would be to knit the net in the pattern of the {6, 3} tiling of the plane. For a proof of (3.9.1), we denote the number of vertices of 𝑇𝑖 by 𝑝 𝑖 and we write the isoperimetric inequality (1.4.2) for 𝑇𝑖 : 𝐿 𝑖2 𝜋 ≥ 4𝑝 𝑖 tan . 𝑇𝑖 𝑝𝑖 … 𝜋 𝜋 Now for 𝑥 ≥ 3, not only 𝑥 tan , but already 𝑦 = 𝑥 tan is a convex function of 𝑥 𝑥 𝑥. Namely, the condition 𝑦 ′′ > 0 for convexity is equivalent to   2𝜋 𝜋 2𝜋 − sin . 4𝜋 sin > 𝑥 𝑥 𝑥 𝑥 3 2𝜋 2𝜋 < 86𝑥𝜋3 , this inequality is satisfied provided that we have 𝑥 − sin 𝑥 3 3 𝜋2 4𝜋 sin 𝜋𝑥 > 𝑥 86𝑥𝜋3 = 43𝑥𝜋2 , that is, sin 𝜋𝑥 > 3𝑥 2 . This last inequality is satisfied 𝜋 𝜋 for 𝑥 < 2 , that is, for 𝑥 > 2. Adding the inequalities √𝐿𝑇𝑖 ≥ 2𝑦( 𝑝 𝑖 ) and applying Jensen’s inequality yields, 𝑖 in view of 𝑝 1 + · · · + 𝑝 𝑛 ≤ 6𝑛, the desired inequality

Since

𝑛

1 ∑︁ 𝐿 𝑖 √ ≥ 2𝑦(6). 𝑛 𝑖=1 𝑇𝑖

3.10 Packing and Covering a Convex Hexagon with Congruent Convex Disks All the results of the previous sections originated from a common dual source, namely from the problems of the densest circle packing and the thinnest circle covering. The theorems in Sections 3.4 through 3.9 all are generalizations of (3.2.1) and (3.2.2), or of the somewhat more general inequalities (3.8.1) and (3.8.2). As a further far-reaching generalization, we prove here the following theorems: If 𝑛 congruent convex disks are packed in a convex hexagon 𝑆, then 𝑛≤

𝑆 , ℎ

where ℎ denotes the hexagon of minimum area circumscribed about a disk.

(3.10.1)

3.10 Packing and Covering a Convex Hexagon with Congruent Convex Disks

87

If 𝑛 congruent convex disks cover a convex hexagon 𝑆 so that the boundary curves of any pair of the disks have at most two points in common, then 𝑛≥

𝑆 , 𝐻

(3.10.2)

where 𝐻 denotes the hexagon of maximum area inscribed in a disk. Consequently, the density of the convex disks considered in the first theorem with respect to 𝑆 cannot exceed the quotient of the area of one disk and the minimum area of a circumscribed hexagon. Likewise, the density of the convex disks considered in the second theorem with respect to 𝑆 cannot fall below the quotient of the area of one disk and the area of the inscribed hexagon of maximum area. Since furthermore, in the case of a centrally symmetric disk 𝐵 the quotient 𝐵ℎ is just the density of the densest lattice-like packing, (3.10.1) implies the following remarkable fact: The density of an arbitrary irregular packing of congruent centrally symmetric convex disks cannot exceed the density of the densest lattice-like packing. However, as the “herringbone” pattern of parquet floors shows, the maximum density packing of the plane with rectangles (namely 1) can also be reached by a non-lattice packing. For the formulation of another consequence of (3.10.1), we define the separation of a family of sets as the greatest lower bound of the lengths of line segments connecting points from distinct members of the family. The number density of an infinite family of congruent centrally symmetric convex disks with a given separation reaches its maximum in a lattice-like arrangement. Indeed, if 2𝑎 is the prescribed separation, then the desired arrangement of the disks is determined by the densest arrangement of the (likewise centrally symmetric) outer parallel domains at distance 𝑎 from the original convex disks. To illustrate the importance of the fact mentioned above, we stress the following special case which, though very specific, is nevertheless interesting in its own right. In a plane region we place a large number of congruent small rods (line segments), affected by a mutually repelling force that causes the smallest distance between them to reach its maximum. If some frame on the boundary keeps the rods in the region, then all the rods, like magnetic needles, will spontaneously turn in one direction. We now turn to the proofs of our theorems. Let 𝐺 1 , . . . , 𝐺 𝑛 denote the disks packed in 𝑆, and for every disk 𝐺 𝑖 consider a one-parameter nested family of convex disks 𝐺 𝑖 (𝜆), 0 ≤ 𝜆 ≤ 1, that, beginning with 𝐺 𝑖 (0) = 𝐺 𝑖 , turns continuously into 𝐺 𝑖 (1) ≡ 𝑆. For example, 𝐺 𝑖 (𝜆) can be chosen as the so-called linear family joining 𝐺 𝑖 with 𝑆, or as the intersection of 𝑆 with the outer parallel domain of 𝐺 𝑖 at distance 𝐷𝜆, where 𝐷 denotes the diameter of 𝑆. If we let 𝜆 increase continuously starting at 𝜆 = 0, then at some point we encounter the first value 𝜆0 ≥ 0 of the parameter such that two of the disks, say 𝐺 𝑖 (𝜆) and 𝐺 𝑗 (𝜆), overlap each other for 𝜆 > 𝜆0 . Consider a common supporting line 𝐺 of 𝐺 𝑖 (𝜆0 ) and 𝐺 𝑗 (𝜆0 ) and also the half-plane 𝐻 determined by 𝐺 and containing 𝐺 𝑖 (𝜆), and, from this value of the parameter on, replace 𝐺 𝑖 (𝜆) with its

88

3 Problems on Packing and Covering in the Plane

part contained in 𝐻. By continuing this process, the disks 𝐺 𝑖 are finally inflated into mutually non-overlapping polygons 𝑃𝑖 (Figure 3.20). Now we get 𝑛 𝑝¯ = 𝑝 1 + · · · + 𝑝 𝑛 ≤ 6𝑛, where 𝑝 𝑖 denotes the number of sides of 𝑃𝑖 . We need to point out here that although, in general, the polygons 𝑃𝑖 do not fill the hexagon 𝑆 without gaps, they, together with the complement of 𝑆, can still be considered as faces of a degenerate polyhedron with 𝑛 + 1 faces, where the polygonal gaps in 𝑆 left uncovered by the polygons 𝑃𝑖 are considered as “vertices” of the polyhedron. Fig. 3.20 We use the congruence of the disks 𝐺 𝑖 only now. Let 𝑇 ( 𝑝) be the minimum area of a 𝑝-gon circumscribed about 𝐺 𝑖 . Since, by Dowker’s theorem, the sequence 𝑇 (3), 𝑇 (4), . . . is convex, we can extend the sequence to a convex, decreasing real-valued function 𝑇 ( 𝑝) defined for all real numbers 𝑝 ≥ 3. Consequently, 𝑆 ≥ 𝑃1 + · · · + 𝑃𝑛 ≥ 𝑇 ( 𝑝 1 ) + · · · + 𝑇 ( 𝑝 𝑛 ) ≥ 𝑛𝑇 ( 𝑝) ¯ ≥ 𝑛𝑇 (6) = 𝑛ℎ, which was to be shown. The proof of the second theorem is analogous. As before, we denote the Gi disks that satisfy the conditions of this theorem by 𝐺 1 , . . . , 𝐺 𝑛 . We assume that none of these disks can be omitted from the family without destroying the covering property, that is, each disk Gk 𝐺 𝑖 contains a point 𝑂 𝑖 from 𝑆 that does Gj not belong to any other disk. Consider Fig. 3.21 two overlapping disks 𝐺 𝑖 and 𝐺 𝑗 . Let their boundaries intersect in 𝐴 and 𝐵. We replace 𝐺 𝑖 and 𝐺 𝑗 by the non-overlapping disks 𝐺 𝑖 and 𝐺 𝑗 into which their union is divided by the chord 𝐴𝐵. It may now happen that one of the new disks, say 𝐺 𝑖 , is overlapped by another disk 𝐺 𝑘 so that their boundaries intersect in more than two points. Since this did not happen for the original disks 𝐺 𝑖 and 𝐺 𝑘 , this situation can occur only if the boundary of 𝐺 𝑘 intersects the closed segment 𝐴𝐵 in two points. It is easy to see (Figure 3.21) that then the intersection of 𝐺 𝑖 and 𝐺 𝑘 is contained in the intersection of 𝐺 𝑖 and 𝐺 𝑗 .

3.11 A Packing Problem with Respect to Affine Length

89

Therefore, applying the separation construction successively to overlapping pairs of disks whose intersection does not contain the intersection of two other disks, after b1 , . . . , 𝐺 b𝑛 finitely many steps we obtain mutually non-overlapping convex subsets 𝐺 of the original disks, that still cover 𝑆. Of course, the result depends on the order b𝑖 𝑆, 𝑖 = 1, . . . , 𝑛, form a in which we perform the separations. The sets 𝑃𝑖 = 𝐺 decomposition of 𝑆 into convex polygons (Figure 3.22). Denoting by 𝑝 𝑖 the number of sides of 𝑃𝑖 , we have 𝑛 𝑝¯ = 𝑝 1 + · · · + 𝑝 𝑛 ≤ 6𝑛. If we now denote by 𝑇 ( 𝑝) the maximum area of a 𝑝-gon inscribed in a disk 𝐺 𝑖 and extend the definition of 𝑇 ( 𝑝) to all real numbers 𝑝 ≥ 3 so as to obtain a concave increasing function, then we get 𝑆 = 𝑃1 + . . . + 𝑃𝑛 ≤ 𝑇 ( 𝑝 1 ) + . . . + 𝑇 ( 𝑝 𝑛 ) ≤ 𝑛𝑇 ( 𝑝) ¯ ≤ 𝑛𝑇 (6) = 𝑛𝐻, which was to be shown. The above proofs show directly that our theorems retain their validity even under more general assumptions: instead of requiring that the disks be congruent, it suffices that each of them can be transformed into any other by an area-preserving affinity. Hence, the best upper bound for the density of an arbitrary packing with equiareal ellipses is √𝜋 , the same as for congruent 12 circles. In closing, let it be noted that in the second theorem, the restriction concerning the Fig. 3.22 number of intersection points of the boundary curves appears to be superfluous. This restriction, on one hand, diminishes the elegance of the theorem, and on the other hand, breaks the analogy between the two theorems. It would therefore be desirable to free the second theorem from this restriction. With the two theorems proved above, the results that originated from the sources (3.2.1) and (3.2.2) mentioned in the introduction of this section have gradually swollen to a brook. We will now leave this brook for a while to look around the neighboring areas. In Chapter 5, we will continue to follow the flow of our brook.

3.11 A Packing Problem with Respect to Affine Length We consider a large but fixed number of arbitrary convex disks, packed in a given convex domain. How should the disks be chosen and arranged so that the sum of their affine perimeters is as large as possible? We answer this question in the following theorem:

◀ 207

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3 Problems on Packing and Covering in the Plane

If 𝑛 convex disks are packed in a convex hexagon 𝑆, then the sum Λ of their affine perimeters satisfies the inequality Λ3 ≤ 72𝑛2 𝑆.

(3.11.1)

Equality occurs only for an affine regular hexagon with an inscribed single affine 3 regular curvilinear hexagon whose “sides” are parabolic arcs. Still, the quotient Λ𝑛2 can be arbitrarily close to 72𝑆 when 𝑆 contains a large number of small congruent affine regular “parabolic hexagons” arranged in a densest packing. If 𝐴 denotes the number density of an infinite family of non-overlapping convex disks and 𝜆 is their average affine perimeter, then the above theorem yields the inequality 𝐴𝜆3 ≤ 72 . (3.11.2) Here, the very general assumptions are remarkable: neither the shape nor the arrangement of the disks are restricted by any regularity conditions. The congruence of the disks as well as their regular shape and arrangement are all spontaneous consequences of a single optimality requirement, namely that their average affine perimeter under a given number density should attain its largest possible value. An analogous result, the inequality (3.7.5), is already known to us. Indeed, this is equivalent to the statement that the total perimeter of the incircles of the convex disks√︃considered in the above theorem √ . cannot exceed 𝜋 2𝑛𝑆 3 A comparison of the inequalities (3.11.1) and (3.7.5) for congruent circles shows the following. The inequality (3.7.5) gives in this case the exact density 𝐷 of the circles in 𝑆, namely, 𝐷 ≤ √𝜋 . On 12 the other hand, from the inequality (3.11.1), we only get 𝐷 ≤ 𝜋92 = 0.9119 . . . . This constant exceeds √𝜋 by less than 0.6%. The fact that the inequality 12 (3.11.1) gives a good, though not the best, estimate Fig. 3.23 for the circle packing density could be expected a priori, since a regular parabolic-arc hexagon just barely deviates from a circle (Figure 3.23). For a proof of the inequality (3.11.1), we embed each of the domains 𝐺 1 , . . . , 𝐺 𝑛 lying in 𝑆—in the same way as in the proof of (3.10.1)—in a convex polygon 𝑃𝑖 . Then, by (2.6.1) we get  2 1 𝜋 3 3 , 𝜆𝑖 ≤ 2𝑃𝑖 𝑝 𝑖 sin 𝑝𝑖 where 𝜆 𝑖 is the affine perimeter of 𝐺 𝑖 and 𝑝 𝑖 denotes the number of sides of 𝑃𝑖 . We show now that the function  2 1 𝜋 3 𝐹 (𝑥, 𝑦) = 𝑥 3 𝑦 sin ; 𝑦

𝑥 ≥ 0;

𝑦≥3

3.12 On a Mean Value Formula

91

is concave. More generally: if 𝜑(𝑦) is a positive-valued concave function defined for 𝑦 ≥ 𝑎 and 𝛼, 𝛽 are two numbers with 0 < 𝛼, 𝛽 < 1, 𝛼 + 𝛽 = 1, then the function 𝑧 = 𝑥 𝛼 [𝜑(𝑦)] 𝛽 ;

𝑥 ≥ 0;

𝑦≥𝑎

is concave. Indeed, we have 𝑧 𝑥 𝑥 𝑧 𝑦 𝑦 − 𝑧 2𝑥 𝑦 = −𝛼𝛽(1 − 𝛼)𝑥 2𝛼−2 𝜑2𝛽−1 𝜑 ′′ ≥ 0 , which proves the concavity of 𝑧. Consequently, by Jensen’s inequality and the monotonicity of 𝐹 (𝑥, 𝑦) in both variables, we get   𝑆 𝜆1 + · · · + 𝜆 𝑛 ≤ 2[𝐹 (𝑃1 , 𝑝 1 ) + · · · + 𝐹 (𝑃𝑛 , 𝑝 𝑛 )] ≤ 2𝑛𝐹 , 6 , 𝑛 which is just the inequality (3.11.1). The case of equality is obvious.

3.12 On a Mean Value Formula As we saw in Section 3.4, the maximum √ number 𝑛 of unit circles that can be placed in a convex disk 𝑇 (of area at least 12) so that they do not overlap satisfies the inequality 𝑛 ≤ √𝑇 . Further, we saw that the smallest number 𝑁 of unit circles that 12

√

can cover a convex disk 𝑇 (of area at least 227 ) satisfies the inequality 𝑁 ≥ √2𝑇 . 27 Conversely, we now wish to estimate 𝑛 from below and 𝑁 from above. Such estimates can be obtained through an elegant approach due to Hadwiger, using a certain mean value formula that can be applied to a variety of problems besides those that interest us here. Suppose we toss some convex disk at random onto a plane containing a given infinite family of convex disks. The mean value formula mentioned above refers to the expected number of the domains intersected by the disk. More generally, instead of convex disks we consider a family of simply connected regions 𝑇1 , 𝑇2 , . . .. Let another simply connected region 𝑇 move in the plane in which this family is located. For a fixed position of 𝑇 we determine the number 𝑠(𝑇, 𝑇𝑖 ) of those simply connected pieces that form the intersection 𝑇𝑇𝑖 , and we consider the total number of pieces ∞ ∑︁ 𝑆(𝑇) = 𝑠(𝑇, 𝑇𝑖 ). 𝑖=1

If each of the regions 𝑇, 𝑇1 , . . . is convex, then 𝑆(𝑇) is exactly the number of those regions from the family {𝑇𝑖 } that intersect 𝑇.

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3 Problems on Packing and Covering in the Plane

¯ as the value of the limit We now define the average number of pieces, 𝑆, ∫ 𝑅 𝑆(𝑇)d𝑇 ∫ 𝑆¯ = lim , 𝑅 d𝑇 𝑅→∞ ∫ where d𝑇 means the kinematic density of 𝑇 and 𝑅 denotes the integral over all those positions of 𝑇 that intersect the circle 𝐾 (𝑅) of radius 𝑅. By this definition, it is easy to verify the following theorem: Let a simply connected region 𝑇 of perimeter 𝐿 move in the plane containing a given family 𝑇1 , 𝑇2 , . . . of simply connected regions with uniformly bounded diameters. If the number density 𝐴 of the given family exists along with its average area 𝑇¯ ¯ of the intersections ¯ then the average number of pieces, 𝑆, and average perimeter 𝐿, 𝑇𝑇1 , 𝑇𝑇2 , . . . exists as well, and can be expressed as   𝐿 𝐿¯ ¯ ¯ . (3.12.1) 𝑆 = 𝐴 𝑇 +𝑇 + 2𝜋 If we consider the functional 𝜒 that assigns the number ∫ 1 𝑠(𝑇, 𝑇𝑖 )d𝑇 𝜒𝑖 = 2𝜋 to each region 𝑇𝑖 , then by the fundamental kinematic formula (2.7.3) we get 𝜒𝑖 = 𝑇 + 𝑇𝑖 +

𝐿𝐿 𝑖 . 2𝜋

Consequently, the average value of this functional is 𝐿 𝐿¯ . 𝜒¯ = 𝑇 + 𝑇¯ + 2𝜋 Furthermore, on one hand we have 2𝜋

∑︁

∫ 𝜒𝑖 ≤

𝑅

𝑆(𝑇) d𝑇 ≤ 2𝜋

𝑅+ ∑︁𝛿

𝜒𝑖 ,

𝑅

and, on the other hand, 2

2

2𝜋 𝑅 ≤

∫ 𝑅

d𝑇 ≤ 2𝜋 2 (𝑅 + 𝛿) 2 ,

where 𝛿 denotes the diameter of 𝑇. Therefore 𝑆¯ is just the density

93

3.12 On a Mean Value Formula

𝐷 ( 𝜒) = lim 𝑅→∞

𝑅 1 ∑︁ 1 ∑︁ 𝜒 = lim 𝜒𝑖 𝑖 𝑅→∞ 𝜋𝑅 2 𝜋𝑅 2 𝑅

of the functional 𝜒. Hence, indeed, 𝑆¯ = 𝐴 𝜒. ¯ We apply the mean value formula (3.12.1) to the case in which the region 𝑇 moves in its own unit lattice. This lattice of regions arises by moving 𝑇 by those translations that send a fixed point in 𝑇 to the points whose coordinates are both integers (Figure 3.24). In view of 𝐴 = 1, 𝑇¯ = 𝑇 and 𝐿¯ = 𝐿, we get 𝐿2 𝑆¯ = 2𝑇 + . 2𝜋

Fig. 3.24

Moreover, since the boundary of 𝑇 can be considered as a simply connected region of area zero and perimeter 2𝐿, we get that the average number of intersections of the boundary curves is 2𝐿 2 𝑆¯ ′ = . 2𝜋 The last two equalities combined give rise to the very nice isoperimetric equality due to Hadwiger: 𝐿 2 − 4𝜋𝑇 = 𝜋𝑆 ′ − 2𝑆. It should be noticed here that to each of the simply connected pieces of the intersection of two congruent regions belong at least two intersection points of the boundaries. Therefore in every position of 𝑇 we have 𝑆 ′ ≥ 2𝑆, so obviously 𝑆 ′ − 2𝑆 ≥ 0. The situation becomes particularly simple in the case when the regions are convex. Here, apart from the factor 𝜋, the isoperimetric deficit is the average of the excess of the number of intersection points on the boundary of 𝑇 over the double of the number of disks intersected by 𝑇. We now return to our original problem. Let us consider the family of faces of the degenerate regular polyhedron {6, 3} of unit edge length. For this family of hexagons √ 27 2 ¯ √ we have 𝐴 = , 𝑇 = 2 , and 𝐿¯ = 6. If a simply connected region 𝑇 of perimeter 27 𝐿 moves over this family of regions, then the average number of pieces satisfies 2 2 𝑆¯ = √ 𝑇 + √ 𝐿 + 1. 3 3 𝜋 3 For example, if 𝑆¯ = 7.6, then there exists a placement of 𝑇 at which 𝑆 ≤ ⌊7.6⌋ = 7 (Figure 3.25). Further, as on one hand, the number of hexagons intersected by 𝑇 can in no placement exceed the number 𝑆 of the pieces of 𝑇, while, on the other hand, the

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3 Problems on Packing and Covering in the Plane

Fig. 3.25

Fig. 3.26

intersected hexagons cover 𝑇, it follows that 𝑇 has a position in which it is covered by at most 7 hexagons of the tiling. Observe also that, naturally, in this position 𝑇 is also covered by the circumcircles of the intersected hexagons. Hence we come to the following general conclusion: Every simply connected region 𝑇 of perimeter 𝐿 can be covered by   3 2 √ 𝑇 + √ 𝐿+1 3 3 𝜋 3 unit circles. Now let 𝑇 be a convex disk with an inradius 𝜚 ≥ 2; let 𝑇−2 denote its inner parallel domain of distance 2, and let 𝐿 −2 be the perimeter of 𝑇−2 . We move 𝑇−2 over the family of unit circles forming the densest lattice packing (Figure 3.26). Here we have 𝐴 = √1 , 𝑇¯ = 𝜋 and 𝐿¯ = 2𝜋; hence the average number of unit circles intersected by 12 𝑇−2 is 1 𝑆¯ = √ (𝑇−2 + 𝐿 −2 + 𝜋) . 12 Since the unit circles intersected by 𝑇−2 are evidently contained in 𝑇, we can assert: The number of unit circles that can be packed in a convex disk 𝑇 with an inradius 𝜚 ≥ 2 is at least 1 √ (𝑇−2 + 𝐿 −2 + 𝜋) , 12 where 𝑇−2 and 𝐿 −2 denote, respectively, the area and perimeter of the inner parallel domain of 𝑇 at distance 2. For a circle of radius 𝑅, this number is 𝜋 √ (𝑅 − 1) 2 , 12

3.13 Historical Remarks

95

and for a square of side length 𝑎, it is 1 √ (𝑎 2 − 4𝑎 + 𝜋). 12 Finally, we apply the mean value formula (3.12.1) to a family of points with number-density 𝐴. Since we have now 𝑇¯ = 𝐿¯ = 0, the average number 𝑁¯ of points from the family that lie in 𝑇 satisfies 𝑁¯ = 𝐴𝑇 . As we follow the proof for this special case step-by-step, we notice that the assumption of simple connectedness of 𝑇 can be omitted, and that the very same mean value formula holds even if we restrict the motions of 𝑇 to parallel translations. Consequently, any region 𝑇 lying in the plane of a family of points with numberdensity 𝐴 can be moved to a position in which it covers at least 𝐴𝑇 points of the family. This is a somewhat more general version of a theorem of Blichfeldt [17] concerning lattices of points. Considering the set of vertices of a regular polyhedron {6, 3} of unit edge length results in the following: √ In every region 𝑇 one can place at least 2𝑇/ 3 points with minimum distance 1.

3.13 Historical Remarks At the Scandinavian Congress of Mathematicians in 1891, A. Thue [131] proved a theorem that can be formulated as follows: If we take a finite number of faces of the polyhedron {3, 6} of edge-length 1, and if the union 𝐹 of these faces contains 𝑛 vertices of the polyhedron, then at most 𝑛 points can be packed in 𝐹 with minimum distance 1 between them. This theorem implies the inequality (3.2.1), hence it solves the problem of the densest circle packing. Eighteen years later, in [132], Thue returned to the problem of the densest circle packing. At this point we mention a result analogous to Thue’s theorem: the union of 𝑛 faces of the polyhedron {6, 3} whose faces have inradius 1 can contain at most 𝑛 non-overlapping unit circles. Otherwise, by adding to them some other faces, the union of the chosen faces could be enlarged to a tiling domain in which more unit circles could be packed than the number of faces, contradicting the remark in Section 3.4. The solution of the problem of the thinnest circle covering by the proof of the inequality (3.2.2) is found in the paper of Kershner [89]. Concerning the fact that in the most favorable covering of a region by small congruent circles, the set of their centers forms “approximately” a lattice of equilateral triangles, Kershner remarks that it is not easy to make a precise statement to that effect. Such a statement is contained in the assertion in Section 3.3, saying that the extreme family of circles is honeycomb-like.

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Independently from Thue and Kershner, the inequalities (3.2.1) and (3.2.2) were rediscovered by the author [30, 31, 34] in a different way. Other proofs of and improvements upon (3.2.1) and (3.2.2) are found in the articles of Segre and Mahler [125], Hadwiger [77] and Verblunsky [133]. The theorems of Sections 3.4–3.11 come from the author [37, 40, 50, 51, 53, 59, 60]; those from Section 3.8 are partly new. An inequality analogous to (3.10.1), but covering only the case of packings with positively homothetic convex disks, was found by Rogers [115] in a completely different way. On the other hand, in this special case the inequality of Rogers reaches somewhat farther than (3.10.1), as for disks without central symmetry it is stronger, and it refers not just to packings in hexagons, but also to packings in arbitrary convex disks. With the use of this inequality, the consequence of the inequality (3.10.1) mentioned in Section 3.10 concerning the packing density of centrally symmetric disks can be complemented so that in the case of positively homothetic disks its validity extends also over disks without a center of symmetry. Naturally, for disks that are not parallel-oriented, this does not apply. However, it can be conjectured that the density of an infinite packing with congruent replicas of an arbitrary convex disk cannot be greater than the density of the densest regular packing. Also, Rogers gives interesting applications of his inequality to number theory. Concerning the inequality (3.10.2), compare the work of Bambah and Rogers [5]. The special case of the formula (3.12.1) in which the family {𝑇𝑖 } is lattice-like, as well as the application of this very formula given in Section 3.12, comes from Hadwiger [74, 75]. In its general form, as presented here, the formula is found in the articles of the author and Hadwiger [65, 66]. To gain further perspectives, we mention some additional results and problems. We begin with the following theorem: From any family of homothetic triangles whose union is of area 𝑉 one can select a subfamily of mutually non-overlapping triangles of total area 𝑇 ≥ 𝑉6 . Here the constant 61 cannot be replaced by a larger one. This theorem, as well as a number of analogous theorems about congruent circles, homothetic squares, etc., are due to R. Rado [108]. See also Marcinkiewicz and Zygmund [101] and T. Radó [109]. We say that a family of domains is separable if there is a line avoiding the interiors of all of the domains of the family such that some two of the domains lie on opposite sides of the line. Otherwise the family is inseparable. According to a theorem proved by A.W. Goodman and R.E. Goodman [71], every inseparable family of 𝑛 circles of radii 𝑟 1 , . . . , 𝑟 𝑛 can be covered by a single circle of radius 𝑟 1 +· · ·+𝑟 𝑛 . This seemingly obvious but non-trivial theorem was previously stated by P. Erdős as a conjecture. Additional results about inseparable families can be found in Hadwiger [78]. We consider now a finite family of circles with the property that none of the circles contains the center of another one in its interior. Then, according to a theorem proved by von Reifenberg [112], as well as by Bateman and Erdős [9], a circle of the smallest radius cannot intersect more than 18 other circles of the family. The number 18 in this theorem cannot be replaced by a smaller one. Figure 3.27 shows a family of congruent circles with the above property, where the circle in the middle intersects

3.13 Historical Remarks

97

18 other circles. Besicovitch [11] proved a weaker theorem whose statement is obtained from the one above by replacing the number 18 with 21. The theorem with the number 18 is equivalent to the following: The maximum number of points at distance at least 1 from each other that can be placed in a circle of radius 2 so that one of them is put in the circle’s center, is 19. If we drop the condition that one of the points must be put in the circle’s center, then the maximum number of points coincides with the maximum number 𝑛 of circles of radius 12 that can be packed in a circle of radius 2.5. The inequality (3.4.1) yields the estimate 𝜋 𝜋 25 𝜋 𝑛 < √ 𝜋 , that is, 𝑛 < √ 25 < 23 . 4 4 12 12 Let us consider a family of circles with the property that every two circles have a Fig. 3.27 common point. Then, in general, all circles need not have a common point, or, graphically, in general, the circles cannot be pierced with one needle. Now, T. Gallai has stated a conjecture that there exists a number 𝑛 independent from the family such that for each family of circles with the above property 𝑛 needles suffice to pierce all circles of the family. P. Ungár and G. Szekeres showed that the conjecture is true, and that 𝑛 = 7 can be chosen. The relatively simple proof is left to the reader. According to a subsequent conjecture of Gallai, already 5 needles suffice. At the origin of Gallai’s problems is a general theorem of Helly [83], stating in the two-dimensional case that for each family of convex disks with the property that every three disks have a common point, there exists a point common to all disks of the family. We state the conjecture that for the lower bound 𝑙 𝑛 for the length of a polygonal chain of 𝑛 line segments connecting 𝑛 + 1 points of a family of points with number2 density 𝐴, the inequality 𝑙 𝑛 ≤ 𝑛 √ holds. The case 𝑛 = 1 is equivalent to 3𝐴 (3.2.3). However, already the case 𝑛 = 2 seems to involve some difficulties. The following further conjecture is closely connected with the previous one: Distribute 𝑛 points in a convex disk 𝑇 so that the shortest polygonal chain connecting 𝐿2 2𝑇 the points reaches its maximum length 𝐿 𝑛 . Then lim 𝑛 = √ . This means that 𝑛→∞ 𝑛 3 any 𝑛 points in a convex disk√𝑇 can be connected by a polygonal chain of length approximately equal to 1.076 𝑛𝑇. If the points belong to an equilateral triangular lattice, then a shorter path √ cannot be found. If 𝑇 is a unit square, then, according to Verblunsky [134], 𝐿 𝑛 < 2.8𝑛 + 2.

◀ 215

◀ 215

98

217 ▶

3 Problems on Packing and Covering in the Plane

An interesting covering problem was raised by Tarski and solved by Bang [6,7]: If a convex disk can be covered by 𝑛 strips of widths 𝑑1 , . . . , 𝑑 𝑛 , then the disk can be covered by just one strip of width 𝑑1 + · · · + 𝑑 𝑛 . A simplified proof is due to Fenchel [67]. The following problem has not yet been completely solved: How large a portion of the plane can be one-fold covered by congruent circles? Begin with the densest lattice-like circle packing and enlarge the circles concentrically until each circle intersects its six neighbors at the vertices √ of a regular dodecagon. The resulting family of circles covers one-fold 100( 48 − 6) = 92.8 . . . % of the plane and, presumably, this number cannot be exceeded by any family of congruent circles. We prove this conjecture under the restriction that no point of the plane is covered by more than two circles. Let 𝑆 be a convex hexagon and let Ω be a finite family of congruent circles. Denote by 𝑆𝑖 the part of 𝑆 that is covered 𝑖-fold by the circles. Then, on one hand, according to (3.8.3), we have ¯ 𝑆¯ , 𝑆1 + 𝑆2 + 𝑆3 + · · · = Ω𝑆 ≤ Ω and, on the other hand, ¯. 𝑆1 + 2𝑆2 + 3𝑆3 + · · · = Ω = Ω Therefore we get ¯ 𝑆¯ − Ω ¯. 𝑆1 − 𝑆3 − 2𝑆4 − · · · = 2Ω

217 ▶

¯ in the six sides of 𝑆, ¯ 𝑆¯ − Ω ¯ is the area of the part ¯ then 2Ω If we reflect the circle Ω ¯ ¯ of 𝑆 that is covered exactly one-fold by Ω and its reflections. It is easy to show that, ¯ the area of this region reaches its maximum when the boundaries under a fixed 𝑆, ¯ ¯ of Ω and 𝑆 intersect √ each other at the vertices of a regular dodecagon. The value of ¯ If we now assume that 𝑆𝑖 = 0 for 𝑖 > 2, then we get that maximum is ( 48 − 6) 𝑆. 𝑆1 √ ≤ 48 − 6. This inequality implies our assertion by taking the limit as 𝑆 → ∞. 𝑆 The conjecture stated above can be rephrased as follows: If a circle is thrown randomly onto a given planar family of points, then the √ probability that the circle will contain exactly one point from the family is at most 48−6. Does this conjecture remain true even if we consider an arbitrary convex disk in place of a circle? According to a conjecture of Heilbronn, from 𝑛 points contained in a unit square 𝑐 three vertices of a triangle Δ can be chosen so that Δ < 2 , where 𝑐 is some numerical 𝑛 𝑐 constant. In that direction, only the rough estimate Δ < √︁ is known, due 𝑛 log log 𝑛 to Roth [116]. What can be said about the triangle of the smallest perimeter? A point and a line passing through it will be called a line-element. For two lineelements the two points together with the intersection point of the two lines through 1 them determine a triangle Δ. The number Δ 3 is called the affine distance between the line elements (cf. Blaschke [15]). Place in a unit square 𝑛 short segments, each

3.13 Historical Remarks

99

representing a line-element. Let 𝜆 be the smallest affine distance determined by them. A further problem, analogous to Heilbronn’s problem, is to give an upper bound for 𝜆 depending on 𝑛 only. Further problems arise from the following general concept: A family of domains is saturated with respect to the domain 𝐺 if it is impossible to place in the plane a domain congruent to 𝐺 without intersecting one of the domains of the family. Then we can ask, for example: What is the thinnest family of unit circles saturated with respect to a given square?

Chapter 4

Efficiency of Packings and Coverings with a Sequence of Convex Disks

We search for those convex disks which are: 1) the least efficient for packing the plane, 2) the least efficient for covering it. Thus, in a certain sense we are interested in the opposite counterparts of the convex tiling domains. These problems appear to be quite difficult and thus far are still unsolved. In this chapter we try to take the first steps towards the solution. In Section 4.1, we solve the analogous problems for lattice-like arrangements. In Section 4.2, we turn to the corresponding problems for the centrally symmetric disks. Then we introduce the notions named in the title of this chapter that, in the case of incongruent domains, play the same role as the density of the densest packing and of the thinnest covering do in the case of congruent domains. Finally, we consider the question of how the most economical packing or covering with a large number of domains is affected when each domain is allowed to be dissected into a given number of suitable pieces.

4.1 Extremum Properties of Triangles We denote by 𝐺 + a the set obtained from 𝐺 by translating it by the vector a. Let a and b be two linearly independent vectors lying in the plane of 𝐺. We say that the collection of sets 𝐺 𝑖 𝑗 = 𝐺 + 𝑖a + 𝑗b;

𝑖, 𝑗 = . . . , −2, −1, 0, 1, 2, . . .

forms a lattice of sets. If no two of the sets of the lattice have an interior point in common, then the sets form a lattice packing. If, on the other hand, the plane is completely covered by the sets, then we have a lattice covering. We consider a fundamental domain 𝑃 associated with the lattice, characterized by the condition that the domains 𝑃𝑖 𝑗 = 𝑃 + 𝑖a + 𝑗b;

𝑖, 𝑗 = 0, ±1, ±2, . . .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_4

101

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4 Efficiency of Packings and Coverings with a Sequence of Convex Disks

cover the plane completely and without overlaps. For example, one such fundamental domain is the so-called basic parallelogram, that is, the set of endpoints of the vectors 𝜆a + 𝜇b;

0 ≤ 𝜆, 𝜇 ≤ 1

with a fixed common initial point. Then the density of the lattice of domains can 𝐺 be defined as the quotient 𝑃, . Naturally, this definition agrees with our previous definition of density. It is easy to show that for each convex domain there exists a densest lattice packing and a thinnest lattice covering. After these preliminary definitions and remarks, we prove the following beautiful theorems of Fáry [27]. The density 𝑑 (𝐺) of the densest lattice packing of a convex disk 𝐺 satisfies 𝑑 (𝐺) ≥

2 3

(4.1.1)

and equality occurs only for a triangle. The density 𝐷 (𝐺) of the thinnest lattice covering of a convex disk 𝐺 satisfies 𝐷 (𝐺) ≤

3 2

(4.1.2)

and equality occurs only for a triangle.

Fig. 4.1

Fig. 4.2

Figure 4.1 shows the densest lattice packing of a triangle. In the thinnest lattice covering of a triangle, the triangles overlap so that their boundaries produce the edges of an affine regular plane tiling {3, 6}. For the proof of (4.1.1), we consider the so-called difference body 𝑉 of the convex disk 𝐺, which consists of the vectors contained in 𝐺, all drawn from a fixed point 𝑂. 𝑉 is a centrally symmetric convex disk, centered at 𝑂. Obviously, for every boundary point 𝑄 1 of 𝑉 there is an affine regular hexagon 𝑄 ≡ 𝑄 1 , . . . , 𝑄 6 inscribed in 𝑉

4.1 Extremum Properties of Triangles

103

(Figure 4.2). To this hexagon we assign the hexagon 𝑃 ≡ 𝑃1 , . . . , 𝑃6 inscribed in 𝐺 for which −−−−−→ −−−→ 𝑃𝑖+3 𝑃𝑖 = 𝑂𝑄 𝑖 ; 𝑖 = 1, . . . , 6, 𝑃𝑖+6 = 𝑃𝑖 . We assert that the translates of 𝐺 by the vectors of the lattice generated by the −−−−→ −−−−→ vectors a = 𝑃1 𝑃4 and b = 𝑃2 𝑃5 form a packing. To see this, observe that in view −−−→ −−−→ −−−→ of the fact that 𝑄 is affine regular, we have 𝑂𝑄 3 = 𝑂𝑄 2 − 𝑂𝑄 1 , consequently −−−−→ −−−−→ −−−−→ 𝑃3 𝑃6 = 𝑃2 𝑃5 − 𝑃1 𝑃4 = b − a (Figure 4.3). On the other hand, each of the vectors −−−−→ −−−−→ −−−−→ 𝑃1 𝑃4 , 𝑃2 𝑃5 and 𝑃3 𝑃6 is the longest one among all vectors parallel to it and lying in 𝐺. Consequently, 𝐺 has a common boundary point with each of 𝐺 + a, 𝐺 + b, 𝐺 + b − a, 𝐺 − a, 𝐺 − b, 𝐺 − b + a , but no common interior point with either one of them, which proves our assertion. Next we show that 𝑄 1 can be chosen so that 𝑃2 𝑃6 ∥ 𝑃3 𝑃5 . In order to see this, we certainly may restrict our attention to a disk whose boundary contains no line segment. Then the assignment of 𝑃 to 𝑄 is unique, and the point 𝑆 at which the lines 𝑃2 𝑃6 and 𝑃3 𝑃5 intersect changes continuously as 𝑄 1 moves along the boundary of 𝑉 unless 𝑆 escapes to infinity. Thus let us move 𝑄 1 to the point 𝑄 4 of the original hexagon 𝑄. Then the hexagon 𝑃, hence the intersection point 𝑆 as well, end up in their original positions. However, since 𝑃1 and 𝑃4 have traded their places, the traveling point 𝑆 has moved from one side of Fig. 4.3 the line 𝑃1 𝑃4 to the other. Furthermore, since 𝑆 can never be on 𝑃1 𝑃4 , this side-switching can only take place if 𝑆 passes through a point at infinity verifying our assertion. We now turn to the proof of the fact that at the above-described position of the hexagon 𝑃 the lattice generated by the vectors a and b is of density 𝑑 ≥ 23 . Set 𝑃2 + a = 𝑃2′ and 𝑃6 + a = ′ 𝑃6 , and consider the octagon 𝐴 ≡ 𝑃1 𝑃2 𝑃3 𝑃2′ 𝑃4 𝑃6′ 𝑃5 𝑃6 . Since 𝐴 is a fundamental domain of our lattice, we have 𝑑 = 𝐺𝐴 . Further, since 𝑃 is inscribed in 𝐺, it suffices to show that 𝑃𝐴 ≥ 23 . If we denote the triangles 𝑃3 𝑃2′ 𝑃4 and 𝑃4 𝑃6′ 𝑃5 by Δ1 and Δ2 , respectively, then we get 𝐴 = 𝑃 + Δ1 + Δ2 , and the and the inequality 3𝑃 ≥ 2𝐴 to be proved becomes 𝑃 ≥ 2(Δ1 + Δ2 ) (Figure 4.4). Fig. 4.4 As 𝑃 is dissected along the diagonal

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4 Efficiency of Packings and Coverings with a Sequence of Convex Disks

𝑃1 𝑃4 into the quadrilaterals 𝑉1 ≡ 𝑃1 𝑃2 𝑃3 𝑃4 and 𝑉2 ≡ 𝑃4 𝑃5 𝑃6 𝑃1 , it suffices to show that 𝑉1 ≥ 2Δ1 and 𝑉2 ≥ 2Δ2 . Since these two inequalities are completely analogous, we only need to verify one of them, say 𝑉1 ≥ 2Δ1 . Since under an affinity the ratio between areas remains unchanged, we can assume that 𝑃1 𝑃4 is perpendicular to 𝑃2 𝑃6 and 𝑃3 𝑃5 . Denote the intersection points of 𝑃1 𝑃4 with 𝑃2 𝑃6 and 𝑃3 𝑃5 by 𝑆 and 𝑇, respectively, and set 𝑆𝑃2 = 𝑠, 𝑇 𝑃3 = 𝑡, 𝑆𝑇 = 21 𝑃1 𝑃4 = 𝑣. We can assume that 𝑡 ≤ 𝑠. Then we have 𝑉1 ≥ 𝑃1 𝑃2 𝑃4 = 𝑣𝑠 ≥ 2Δ1 . In the first inequality, equality occurs only if 𝑃3 lies on the segment 𝑃2 𝑃4 , and in the second one only if 𝑃1 coincides with 𝑆. If analogous conditions hold for the quadrilateral 𝑉2 as well, then the hexagon 𝑃 degenerates into the triangle 𝑃2 𝑃4 𝑃6 , and this completes the proof. Now we turn to the proof of the inequality (4.1.2). We begin by proving that in every convex disk 𝐺 there is an inscribed affine regular hexagon. Again, we may assume that the boundary of 𝐺 contains no line segment. For a given line 𝑔 we consider the longest chord 𝐴𝐵 among all chords parallel to 𝑔. If 𝑠 is a given number with 0 < 𝑠 < 𝐴𝐵, then on opposite sides of the line 𝐴𝐵 there are chords 𝑃1 𝑃2 and 𝑃4 𝑃5 , each parallel to 𝐴𝐵 Fig. 4.5 and each of length 𝑠. We choose the notation −−−−→ −−−−→ for these chords so that 𝑃1 𝑃2 + 𝑃4 𝑃5 = 0 and consider the affine regular hexagon 𝑃1 𝑃2 𝑃3 𝑃4 𝑃5 𝑃6 (Figure 4.5). If 𝑠 is small, then 𝑃3 𝑃6 lies inside 𝐺. On the other hand, if 𝑠 is close to 𝐴𝐵, then the points 𝑃3 and 𝑃6 lie outside 𝐺. Therefore, if we vary 𝑠 continuously from 0 to 𝐴𝐵, then, by reason of continuity, there is the smallest value 𝑠 = 𝑠0 for which one of the points 𝑃3 and 𝑃6 lies on the boundary of 𝐺 and the other one lies not outside 𝐺. The hexagon 𝑃(𝑠0 ), uniquely determined in this way, varies continuously with the direction of 𝑔. Assume that, say, 𝑃3 lies on the boundary of 𝐺 and 𝑃6 is in the interior of 𝐺, and rotate 𝑔 by 180◦ . Then 𝑃3 assumes the position of 𝑃6 ; therefore at some point 𝑃3 must take off from the boundary of 𝐺. But at that very point both 𝑃3 and 𝑃6 lie on the boundary of 𝐺, which proves our assertion. 3 Next we show that if 𝑃 is an affine regular hexagon inscribed in 𝐺, then 𝐺 𝑃 ≤ 2. To that end, we denote by 𝑠1 , . . . , 𝑠6 the segments cut off from 𝐺 by the sides 𝑃1 𝑃2 , . . . , 𝑃6 𝑃1 of the hexagon 𝑃 ≡ 𝑃1 . . . 𝑃6 . Further, we consider the triangles 𝐴 ≡ 𝐴1 𝐴2 𝐴3 and 𝐵 ≡ 𝐵1 𝐵2 𝐵3 determined by the three lines 𝑃1 𝑃2 , 𝑃3 𝑃4 , 𝑃5 𝑃6 and by the three lines 𝑃2 𝑃3 , 𝑃4 𝑃5 , 𝑃6 𝑃1 , respectively, where 𝐴1 is the vertex opposite to the side 𝑃3 𝑃4 , 𝐵1 is the vertex opposite to the side 𝑃4 𝑃5 , and so on (Figure 4.6).

105

4.2 Centrally Symmetric Domains

The reflection of 𝑠1 in 𝑃1 is a region 𝑠1′ that lies outside 𝑠6 but inside the triangle 𝐴1 𝑃1 𝑃6 . This is so by the convexity of 𝐺 and by the fact that 𝐴1 𝑃1 𝑃6 is the reflection of the triangle 𝐵1 𝑃1 𝑃2 , containing 𝑠1 . Consequently, 𝑠6 + 𝑠1′ = 𝑠6 + 𝑠1 is less than or equal to the area of 𝐴1 𝑃1 𝑃6 . By repeating the same argument for the points 𝑃3 and 𝑃5 we can see that 𝐺 ≤ 𝐴 = 23 𝑃. Equality occurs if and only if 𝐺 coincides with one of the triangles 𝐴 or 𝐵. The tiling of the plane Fig. 4.6 with congruent replicas of 𝑃 generates a lattice covering with 𝐺 of density less than or equal to 32 , equality occurring only if 𝐺 is a triangle.

4.2 Centrally Symmetric Domains Next we prove the following theorem: The density 𝐷 (𝑀) of the thinnest lattice covering of a convex centrally symmetric disk 𝑀 satisfies the inequality 2𝜋 𝐷 (𝑀) ≤ √ , 27

(4.2.1)

and equality occurs only if 𝑀 is an ellipse. This theorem essentially solves the second main problem posed in the introduction of this chapter for the case of centrally symmetric domains: Each centrally symmetric convex disk other than an ellipse can cover the plane more economically than the circle can. The question of uniqueness remains to be resolved, that is, to show that the extreme convex disks are ellipses. To that end, one should prove that when congruent ellipses cover the plane most economically, they are arranged in a lattice-like pattern, which, according to Section 3.10, is most likely so. The fact that for an ellipse we have equality in (4.2.1) is obvious. It therefore suffices to show that for any non-elliptical centrally symmetric convex disk a lattice covering of density smaller than √2 𝜋 can be constructed. This, however, is 27 an immediate consequence of the construction of Sas (Section 2.4), assigning to every non-elliptical centrally symmetric convex disk 𝑀 an inscribed hexagon of √ 27 area greater than 2 𝜋 𝑀 that is centrally symmetric as well. The desired lattice-like covering arises by a tiling of the plane with translates of such a hexagon.

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4 Efficiency of Packings and Coverings with a Sequence of Convex Disks

We now turn our attention to the dual problem: Which among all centrally symmetric convex disks is the one whose densest lattice packing is the least dense? The natural assumption that the extreme disk is again the ellipse has been refuted by Reinhardt [113] and Mahler [100]. The remarkable fact is that there are centrally symmetric convex disks for which the density of their densest lattice packing is smaller than √𝜋 . Mahler showed that, for example, the regular octagon 𝐴 has this 12 √ property, as in its case the density in question is 𝑑 ( 𝐴) = 47 (3 − 2) = 0.90616 . . . . The investigations of the two abovenamed authors led them to conjecture that the extreme convex disk is the smoothed octagon. This convex disk is obtained from an affine regular octagon by rounding off each of its vertices with the hyperbola tangent to the sides adjacent to the vertex and whose asymptotes are the extended next adjacent sides (Figure 4.7). The conjecture asserts that for every centrally symmetric convex disk 𝑀 the inequality √ 8 − 32 − log 2 Fig. 4.7 𝑑 (𝑀) ≥ = 0.9024 . . . √ 8−1 holds, and that equality occurs only if 𝑀 is a smoothed octagon. In view of the results of Section 3.10, a proof of the conjecture would also provide a solution to our first fundamental problem for centrally symmetric convex disks. In the following, using quite elementary and simple observations, we prove the weaker estimate: Every centrally symmetric convex disk admits a lattice packing of density √ 3 𝑑> = 0.8660 . . . . (4.2.2) 2 The constant on the right-hand side is only about 4% smaller than the conjectured best possible constant 0.9042 . . . . Let 𝑀 be a given convex disk centered at 𝑂 and let a and b be, for now arbitrary, linearly independent vectors. In the usual manner, we set 𝑀 + 𝑖a + 𝑗b = 𝑀𝑖 𝑗 and 𝑂 +𝑖a + 𝑗b = 𝑂 𝑖 𝑗 , and choose a and b such that any two of the sets 𝑀00 , 𝑀10 and 𝑀01 have common boundary points but no common interior points (Figure 4.8). This is always possible, and even the direction of a may still be chosen freely. The lattice of convex disks produced by these vectors is, obviously, a packing. Let us consider the affine regular hexagon 𝑂 10 𝑂 01 𝑂 −11 𝑂 −10 𝑂 0−1 𝑂 1−1 as well as the homothetic hexagon 𝑆 inscribed in 𝑀. We partition 𝑆 into six (equiareal) triangles that share a vertex at 𝑂. As the basic parallelogram 𝑃 ≡ 𝑂 00 𝑂 10 𝑂 11 𝑂 01 of

107

4.2 Centrally Symmetric Domains

Fig. 4.8

the lattice is composed of eight equiareal triangles, we have the density 𝑑 = 𝑀 𝑃 of the lattice packing satisfies 𝑑=

3𝑀 . 4 𝑆

𝑃 𝑆

=

8 6

= 43 . Therefore

(4.2.3)

Since 𝑀 ≥ 𝑆, we have already a lattice packing of density at least 34 for every direction a. We now choose the direction of a such that the area of 𝑆 becomes minimal. We can assume that the minimal hexagon 𝑆 ≡ 𝑄 1 . . . 𝑄 6 is regular. In addition, we choose a rectangular coordinate system 𝑥𝑦 such that its origin lies at 𝑂 and the unit point of the 𝑦-axis is at 𝑄 1 (Figure 4.9). By our assumptions, 𝑀 contains another affine regular hexagon 𝑆 = 𝑅1 . . . 𝑅6 with center 𝑂 and 𝑅1 ≡√𝑅1 (2𝜉, 0) lying on the 𝑥-axis such that 𝑆 ≥ 𝑆 = 227 . Then, the vertices 𝑆 lie outside of 𝑆 and the vertices of 𝑆 lie outside 𝑆. Let 𝐻 be the convex hull of the union of 𝑆 Fig. 4.9 and 𝑆. Moving the sides 𝑅2 𝑅3 and 𝑅5 𝑅6 , each along its own line, to a new position symmetric about the 𝑦-axis does not change the area of 𝐻, therefore we may assume that 𝑆 is symmetric about the 𝑦-axis. If we set √ √ 𝑅2 ≡ 𝑅2 (𝜉, 𝜂), then the condition 𝑆 ≥ 227 is equivalent to 4𝜉𝜂 ≥ 3. We have now 𝐻 = 𝑆 + 2Δ1 + 4Δ2 , where Δ1 and Δ2 denote the triangles 𝑅1 𝑄 5 𝑄 6 and 𝑅2 𝑄 1 𝑄 6 , respectively. Since the normal equation of the line 𝑄 1 𝑄 6 is √ √  1 𝑥 + 3𝑦 − 3 = 0, 2

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4 Efficiency of Packings and Coverings with a Sequence of Convex Disks

the distance from 𝑅2 (𝜉, 𝜂) to this line is

1 2



√ √  𝜉 + 3𝜂 − 3 . Consequently,

√ √ !    √ √  3 27 3 1 + 2𝜉 − + 𝜉 + 3𝜂 − 3 ≥ 2𝜉 + ≥ 3. 𝑀≥𝐻= 2 2 2 2𝜉 Equality in each of the last two inequalities holds only if 𝑆  𝑆 and 2𝜉 = 1, that is, when 𝐻 is a regular dodecagon. But in this case we have 𝑀 > 𝐻, for otherwise 𝑆 would not be the smallest affine regular hexagon inscribed in 𝑀. Anyway, we get 𝑀 > 3 and thereby √ 33 3 3𝑀 > = , 𝑑= 4 𝑆 4𝑆 2 which was to be shown. It should be additionally noticed that by a tiling of the plane with congruent replicas of the hexagon 𝑆 a lattice covering with replicas of 𝑀 is obtained, whose density is 𝑀 𝑆 . Therefore, in view of (4.2.3), for every centrally symmetric convex disk 𝑀, the following exact estimate holds: 3 𝑑 (𝑀) ≥ . 𝐷 (𝑀) 4

(4.2.4)

4.3 Packing and Covering Efficiency of Sequences of Disks Consider an infinite sequence 𝑆1 , 𝑆2 , . . . of convex disks whose inradii have a positive lower bound 𝑟 and whose circumradii have a finite upper bound 𝑅. We call such a sequence normal sequence. For a given convex disk 𝑇 let 𝑇𝑛 be the smallest similar copy of 𝑇 in which the first 𝑛 disks 𝑆1 , . . . , 𝑆 𝑛 can be packed and let 𝑡 𝑛 be the largest similar copy of 𝑇 that can be covered by the disks 𝑆1 , . . . , 𝑆 𝑛 . Let 𝜎𝑛 = 𝑆1 + · · · + 𝑆 𝑛 be the total area of the first 𝑛 disks. Obviously, the limits 𝑤 = lim inf 𝑛→∞

𝜎𝑛 , 𝑇𝑛

𝑊 = lim inf 𝑛→∞

𝑡𝑛 𝜎𝑛

satisfy 0 < 𝑤, 𝑊 ≤ 1. Further, it is easy to show that 𝑤 and 𝑊 do not depend on the choice of the domain 𝑇. The number 𝑤 shows which portion of the plane can be filled in the densest packing of the disks. Similarly, 𝑊 can be interpreted as the number giving the portion of the sum of the areas of the disks used in a covering of the plane. We call 𝑤 the packing efficiency and 𝑊 the covering efficiency of the sequence of disks. 1 are equal to the density of the For a sequence of congruent disks 𝑤 and 𝑊 densest packing and thinnest√covering, respectively. For congruent circles we have 𝑤 = √𝜋 = 0.9069 . . ., 𝑊 = 227 𝜋 = 0.8269 . . ., while for congruent quadrangles (or 12 more generally for congruent plane-fillers) obviously 𝑤 = 𝑊 = 1 holds.

4.3 Packing and Covering Efficiency of Sequences of Disks

109

We claim: If 𝑤 and 𝑊, respectively, denote the infimum of the packing efficiency and the infimum of the covering efficiency of sequences of congruent disks, then we have for the packing efficiency 𝑤 and for the covering efficiency 𝑊 of an arbitrary normal sequence of convex disks 𝑤 ≥ 𝑤, 𝑊 ≥ 𝑊 . (4.3.1)

The main problems raised in the introduction of this chapter ask exactly for the determination of the quantities 𝑤 and 𝑊. If these were known then, in view of (4.3.1), we would have the exact lower bounds for arbitrary normal sequences. In any case, in view of the inequalities (4.1.1) and (4.1.2), the packing efficiency 𝑤 and the covering efficiency 𝑊 of an arbitrary normal sequence satisfy 𝑤, 𝑊 >

2 . 3

(4.3.2)

For a normal sequence of centrally symmetric disks we have the exact bound √ 27 = 0.8269 . . . , 𝑊≥ (4.3.3) 2𝜋 and presumably 𝑤 ≥ 0.9224 . . . .

(4.3.4)

In the inequality (4.3.1) 𝑤 and 𝑊 can be replaced by the quantities obtained when the infimum is taken only for those disks which occur in the given normal sequence. From this, for instance, the non-trivial fact follows that for an arbitrary normal sequence of convex quadrilaterals (or of arbitrary convex tiling domains) we have 𝑤 = 𝑊 = 1. The proofs for the two inequalities (4.3.1) are analogous. We prove here the second one. The proof is based on Blaschke’s selection theorem according to which the disks of an arbitrary normal sequence—disregarding an error that can be made arbitrarily small—can be represented by a finite number of disks. We give this statement the following exact form suitable for our purpose: Let 𝑆1 , 𝑆2 , . . . be an arbitrary normal sequence of convex disks and let 𝛼 < 1 be an arbitrary number. Then one can find a finite number 𝑁 of convex disks 𝐺 1 , . . . , 𝐺 𝑁 and a classification of the disks of the sequence in 𝑁 classes such that if 𝑆 𝑗 is an element of the 𝑖-th class, then 𝐺 𝑖 ≥ 𝛼𝑆 𝑗 and 𝐺 𝑖 can be covered by 𝑆 𝑗 . As a finite number of disks has no effect on the covering efficiency, we can assume that each class contains infinitely many disks. We denote by 𝑛𝑖 the number of disks in the 𝑖-th class with an index less than or equal to 𝑛 and by 𝜎𝑛𝑖 the system of these 𝑖 disks, as well as their total area. Furthermore, let 𝑄 𝑖𝑛 and 𝑄 𝑛 be the largest square

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4 Efficiency of Packings and Coverings with a Sequence of Convex Disks

that can be covered by the disks from 𝜎𝑛𝑖 and by 𝑛𝑖 congruent copies of 𝐺 𝑖 , respec𝑖 tively. Since 𝑄 𝑖𝑛 ≥ 𝑄 𝑛 , we have by the definition of 𝑊 𝑖

lim inf 𝑛→∞

𝑄 𝑄 𝑖𝑛 ≥ lim inf 𝑛 ≥ 𝑊 , 𝑛→∞ 𝑛𝑖 𝐺 𝑖 𝑛𝑖 𝐺 𝑖

𝑖 = 1, 2, . . . , 𝑁.

Therefore there exists an index 𝜈 such that for 𝑛 > 𝜈 we have 𝑄 𝑖𝑛 ≥ 𝛼𝑛𝑖 𝐺 𝑖 𝑊 ≥ 𝛼2 𝜎𝑛𝑖 𝑊 ,

𝑖 = 1, 2, . . . , 𝑁,

thus 𝑄 ∗𝑛 = 𝑄 1𝑛 + · · · + 𝑄 𝑛𝑁 ≥ 𝛼2 𝜎𝑛 𝑊 . Now we dissect the squares 𝑄 1𝑛 and 𝑄 2𝑛 by three straight line cuts into five pieces that can be reassembled into a single square. This can be done in several ways (see e.g. Ball and Coxeter [3]). Then we merge the resulting square with 𝑄 3𝑛 to obtain a new square. Continuing this process, the squares 𝑄 1𝑛 , . . . , 𝑄 𝑛𝑁 are dissected by 3(𝑁 − 1) cuts into some pieces which can be reassembled into a single square 𝑄 ∗𝑛 . The total length 𝐿 of segments in 𝑄 ∗𝑛 corresponding to the cuts satisfies √ √︁ ∗ 𝐿 < 3(𝑁 − 1) 2𝑄 𝑁 < 3(𝑁 − 1) 2𝜋𝑅 2 𝑛, where 𝑅 is the upper limit of the circumradii of the disks. Since under the dissection of the squares 𝑄 1𝑛 , . . . , 𝑄 𝑛𝑁 some of the disks covering them are cut, in general 𝑄 ∗𝑛 cannot be covered by the disks 𝑆1 , . . . , 𝑆 𝑛 . However, all the cut disks lie in the parallel domain at distance 2𝑅 √ from the collection 𝐿 of segments. This parallel domain can be covered by 𝜇 𝑛 < 𝑐 𝑛 circles of radius 𝑟, where 𝑟 denotes the lower limit of the inradii of the disks and 𝑐 is a constant depending only on 𝑅, 𝑟 and 𝑁. Consequently, 𝑄 ∗𝑛 can be covered by the disks 𝑆1 , . . . , 𝑆 𝑛+𝜇𝑛 . Denoting by 𝑄 𝑛 the largest square that can be covered by the system 𝜎𝑛 of the disks 𝑆1 , . . . , 𝑆 𝑛 , we have lim inf

𝑄 𝑛+𝜇𝑛 𝑄∗ 𝑄𝑛 = lim inf ≥ lim inf 𝑛 ≥ 𝛼2𝑊 . 𝜎𝑛 𝜎𝑛 𝜎𝑛

𝑛 Since this inequality holds for all 𝛼 < 1, lim inf 𝑄 𝜎𝑛 ≥ 𝑊, which was to be proved. By analogous considerations, in view of (4.2.4), we get the exact bound

𝑤𝑊 ≥

3 4

(4.3.5)

for the product of the two efficiencies of an arbitrary normal sequence of centrally symmetric convex disks. The product 𝑤𝑊 can be interpreted as the ratio of the area of the largest domain that can be covered by the disk to the area of the smallest domain in which the disks can be packed.

4.4 Covering with Fragmented Disks

111

4.4 Covering with Fragmented Disks Consider a large number of small convex disks that do not differ from each other “too much”, whose total area is 1, and about which we otherwise do not know anything specific. If we try to use these disks to cover a square of area, say, 0.999, then naturally, in general, our attempt will fail. Such a covering can be achieved only if we first partition the disks by suitable cuts into a certain number of pieces. We wish to split each of the disks into 𝑘 pieces, and we ask, how large must 𝑘 be so that the desired covering can be achieved under all circumstances. In a similar way, one can ask how large the number 𝑘 should be chosen so that the disks, after being split into 𝑘 pieces each, can be packed in a square of area, say, 1.001. To give these problems a precise form, consider a normal sequence 𝑆1 , 𝑆2 , . . . of convex disks and split each disk 𝑆𝑖 into 𝑘 convex pieces 𝑆𝑖1 , . . . , 𝑆𝑖𝑘 . Thereby we obtain a new sequence 𝑆11 , . . . , 𝑆1𝑘 , 𝑆21 , . . . , 𝑆2𝑘 , . . . of disks. This sequence need not necessarily be normal anymore, but it is easy to show that it has a packing efficiency 𝑤 𝑘 and a covering efficiency 𝑊 𝑘 that are independent of the shape of the region being packed or covered. We can then ask: how high an efficiency can be achieved by appropriately chosen splittings? The following theorem is about covering: By splitting each member of an arbitrary normal family of convex disks into 𝑘 (𝑘 = 1, 2, . . .) convex pieces a new family with covering efficiency 𝑊𝑘 ≥

𝑘 +1 𝜋 sin 𝜋 𝑘 +1

(4.4.1)

can be obtained. If each of the disks is centrally symmetric and 𝑘 is odd, then the sharper estimate 𝜋 𝑘 +2 sin (4.4.2) 𝑊𝑘 ≥ 𝜋 𝑘 +2 holds. Should we, for example, desire to obtain a covering corresponding to the above question with just 0.1% of the disks’ total area wasted (that is, with 𝑊 = 0.999), then it suffices to split each of the disks into 40 pieces. Moreover, in order to achieve such an efficient covering, a “significantly” smaller number of pieces will probably not suffice. More precisely, it can be conjectured that the inequality 𝑊𝑘 > 1 −

𝜋2 , 6(𝑘 + 1) 2

derived from (4.4.1), gives an exact asymptotic estimate for large values of 𝑘. The case 𝑘 = 1 deals with the original sequence of convex disks. For 𝑘 = 1 (4.4.1) yields the inequality 𝑊1 > 0.6366 . . . – a result weaker than the estimate (4.3.2). In contrast, the inequality (4.4.2) with 𝑘 = 1 coincides with the sharp estimate (4.3.3).

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4 Efficiency of Packings and Coverings with a Sequence of Convex Disks

For a proof we convince ourselves, using the observations from the previous section, that it suffices to prove the theorem for congruent disks. Inscribe in a convex disk 𝑆 a maximum area (2𝑘 + 2)-gon 𝑃 ≡ 𝑃1 · · · 𝑃2𝑘+2 and split it into 𝑘 convex quadrilaterals, for example by cutting along the lines 𝑃1−𝑖 𝑃2+𝑖 (𝑖 = 1, . . . , 𝑘 − 1; 𝑃−𝑖 ≡ 𝑃2𝑘+2−𝑖 ). Thereby 𝑆 is also split into 𝑘 convex pieces, and we assert that the covering efficiency of the sequence resulting from such a splitting of every disk will satisfy the inequality (4.4.1). Indeed, let us denote by 𝑄 𝑛 and 𝑞 𝑛 the largest square that can be covered by the 𝑛𝑘 pieces obtained by splitting 𝑛 disks and by the 𝑛𝑘 corresponding quadrilaterals, respectively. Then, since every quadrilateral is a plane-tiling domain, we get 𝑞𝑛 lim = 1. 𝑛→∞ 𝑛𝑃 On the other hand, considering (2.4.1), we have 𝑘 +1 𝜋 𝑃 ≥ sin . 𝑆 𝜋 𝑘 +1 This yields 𝑊 𝑘 = lim inf 𝑛→∞

𝑞𝑛 𝑘 +1 𝜋 𝑄𝑛 𝑞𝑛 𝑃 𝑃 lim sin , ≥ lim = = ≥ 𝑛→∞ 𝑛𝑆 𝑛𝑆 𝑆 𝑛→∞ 𝑛𝑃 𝑆 𝜋 𝑘 +1

whereby the proof of (4.4.1) is completed. For a proof of (4.4.2), we inscribe in the centrally symmetric disk 𝑆 a concentric (2𝑘 + 4)-gon 𝑃 ≡ 𝑃1 · · · 𝑃2𝑘+4 of area 𝑃≥𝑆

𝜋 𝑘 +2 sin 𝜋 𝑘 +2

and we cut 𝑃 along the lines 𝑃1−𝑖 𝑃2+𝑖 , 𝑖 = 1, . . . ,

𝑘 +1 𝑘 +1 − 1, + 1, . . . , 𝑘 − 1; 𝑃−𝑖 ≡ 𝑃2𝑘+4−𝑖 2 2

into 𝑘 − 1 convex quadrilaterals and one convex hexagon with a center. Since such a hexagon is a tiling domain as well, everything works out just as above. In an analogous way one can show that, by splitting every disk of a normal 1 can be sequence into 𝑘 suitable convex pieces, a packing efficiency 𝑤 𝑘 ≥ 𝑈2𝑘+2 attained, where 𝑈𝑛 denotes the supremum of the area of a minimum-area 𝑛-gon circumscribed about convex disks of area 1. Because of the difficulties in determining 𝑈𝑛 exactly, we have to be satisfied with the asymptotic estimate 𝑤𝑘 ≥ 1 −

𝜋2 + 𝜀 𝑘 , 12𝑘 2

lim 𝜀 𝑘 = 0. 𝑘→∞

However, it is conjectured that even 𝑤𝑘 ≥ 1 −

𝜋2 , 12𝑘 2

𝑘 = 1, 2, . . . ,

(4.4.3)

4.5 Historical Remarks

113

from which it would follow that, in order to succeed in filling 99.9% of the plane with the disks, it always suffices to split each of them into 29 pieces.

4.5 Historical Remarks The problem of determining the best lower bound for 𝑑 (𝑀) extended over a certain class of centrally symmetric regions 𝑀 dates back to Minkowski. Later the problem was stated again, under more general conditions, by Courant (see Blaschke [15, p. 65]). The remarkable fact that the minimum of 𝑑 (𝑀) over all centrally symmetric convex disks is not attained by the ellipses was first noticed by Reinhardt [113]. Also due to him is the conjecture that the extreme region is the smoothed octagon. Later, unaware of Reinhardt’s work, and in a quite different way, Mahler [100] rediscovered the same result. Mahler introduced, among other things, the problem of determining those centrally symmetric 2𝑛-gons 𝑃2𝑛 for which 𝑑 (𝑃2𝑛 ) attains the smallest possible value, and he proved that the extreme octagon 𝑃8 is affine regular. Ledermann and Mahler [93] determined the extreme decagon 𝑃10 and they found that it is not regular. Furthermore, they compared the values of 𝑑 ( 𝑃8 ) and 𝑑 (𝑃10 ), where 𝑃2𝑛 denotes the smoothed 2𝑛-gon obtained by modifying the polygon 𝑃2𝑛 . It turned out that even though 𝑑 (𝑃8 ) > 𝑑 (𝑃10 ), we have 𝑑 (𝑃8 ) < 𝑑 (𝑃10 ), supporting Reinhardt’s conjecture. We adopted the name “smoothed octagon” from the lastquoted article. The problem of determining the centrally symmetric convex disks 𝑀 with the largest possible value of 𝐷 (𝑀), as well as the simple observation that the solution is an immediate consequence of Sas’s theorem (2.4.1), are due to the author [41]. The theorems of Fáry had previously been stated by the author as conjectures. The proof of (4.1.2) implies that every convex disk 𝐸 contains a centrally symmetric convex disk 𝑒 of area 𝑒 ≥ 32 𝐸, which was noticed about the same time by Fáry and Besicovitch [12]. The inequality (4.1.2) is an immediate consequence of the fact that 𝐸 already contains an affine regular hexagon of area at least 32 𝐸, which also follows from Besicovitch’s proof. Since the inequality (4.1.2) was first stated by Fáry, it seems appropriate to attribute the corresponding theorem to him. The inequality (4.2.2) was proved first by Mahler [99]. The proof given here comes from the article [47], written by the author independently from Mahler. The results of Sections 4.3 and 4.4 are contained in the author’s article [55]. There, 𝜋 𝜋 cot 2𝑘 is given, which, due to a flawed proof instead of (4.4.3), the estimate 𝑤 𝑘 > 2𝑘 𝜋 𝑛−2 of the inequality 𝑈𝑛 < 𝜋 tan 𝑛−2 , could be considered unproven. Still, one can confidently assume that the estimate itself is correct. Concerning our fundamental problems stated in the introduction to this chapter, we do not even have a reasonable conjecture. It would therefore be desirable to give at least an estimate corresponding to (4.3.5) which is also valid for convex disks without central symmetry.

Chapter 5

Extremal Properties of Regular Polyhedra

On the surface of a ball the problems of the densest circle packing and the thinnest circle covering with 4, 6, or 12 congruent circles lead to arrangements of circles in which their centers are vertices of a regular tetrahedron, octahedron, or icosahedron. In contrast, in the most favorable arrangement of 8 or 20 circles, their centers are not the vertices of a regular hexahedron or dodecahedron. Hence for these problems, the regular triangular polyhedra play a special role. However, since for example in the densest packing of 12 congruent circles on the sphere, their planes bound a regular dodecahedron, the same problems can be formulated so that the regular three-valent polyhedra play a special role. In this chapter, we will learn about some other extremal properties of the regular polyhedra with triangular faces and with three-valent vertices, that are expressed by certain inequalities. These inequalities are valid for polyhedra with an arbitrary number 𝑛 of vertices or faces, but are exact only for 𝑛 = 4, 6, and 12, and they give exact asymptotic estimates for large values of 𝑛. In space, the “proper” analogues to the extremum properties of regular polygons actually arise when we simultaneously specify the number of vertices and the number of faces. In this manner, in some cases we succeed in deriving inequalities that express extremum properties of all five (and in certain sense even all eight) regular polyhedra. On the other hand, regarding estimates of the total edge length of a polyhedron, we shall encounter extremum problems whose solution does not depend on the number of vertices or faces, and turns out to be a specific regular polyhedron, for example the cube. In the last two sections we consider certain arrangement problems on an arbitrary convex surface.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_5

115

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5 Extremal Properties of Regular Polyhedra

5.1 Packing and Covering the Sphere with Congruent Spherical Caps We introduce the notation:

𝑛 𝜋 𝑛−2 6 which will be often used throughout this chapter. Observe that 2𝜔 𝑛 is the angle of 4𝜋 on the unit sphere. Since an equilateral spherical triangle Δ𝑛 of area 6𝜔 𝑛 − 𝜋 = 2𝑛−4 2𝑛 − 4 is the number of faces of a triangular polyhedron with 𝑛 vertices, Δ𝑛 is the average area of the spherical triangles into which the unit sphere is partitioned by the spherical net of a triangular polyhedron with 𝑛 vertices. We will now prove the following theorems: 𝜔𝑛 =

If a sphere is packed with 𝑛 ≥ 3 congruent spherical caps, then the density 𝐷 of the packing satisfies   1 𝑛 1 − csc 𝜔 𝑛 . (5.1.1) 𝐷≤ 2 2 If a sphere is covered by 𝑛 ≥ 3 congruent spherical caps, then the density 𝐷 of the covering satisfies   1 𝑛 1 − √ cot 𝜔 𝑛 . (5.1.2) 𝐷≥ 2 3 In each of the above two inequalities, equality is attained only for 𝑛 = 3, 4, 6, and 12, namely, if the centers of the caps are the vertices of a regular triangle, tetrahedron, octahedron, or icosahedron, respectively. Further, it can be shown that, with increasing 𝑛, the bound on the right-hand side of (5.1.1) is monotonically increasing, approaching the limit √𝜋 , while the bound 12

in (5.1.2) is decreasing, approaching

√2 𝜋 . 27

Thereby we get:

The density of a family of at least three non-overlapping congruent spherical caps is smaller than √𝜋 . Similarly, the density of a family of at least three congruent 12

spherical caps covering the sphere is greater than

√2 𝜋 . 27

Consequently, the inequalities (5.1.1) and (5.1.2) provide exact asymptotic estimates for large values of 𝑛. Thus, in some sense we can say that equality occurs in (5.1.1) and (5.1.2) only for 𝑛 = 3, 4, 6, 12 and ∞, namely, if the centers of the caps are the vertices of the regular polyhedra {3, 2}, {3, 3}, {3, 4}, {3, 5} and {3, 6}. The following simple proofs of the two inequalities are quite analogous. For the case 𝑛 = 3, we can convince ourselves immediately about the validity of our inequalities. We therefore restrict our attention to the case 𝑛 > 3. Furthermore, we may assume that the centers 𝑂 1 , . . . , 𝑂 𝑛 of the caps 𝐾1 , . . . , 𝐾𝑛 do not lie in a hemisphere, hence the planes tangent to the sphere at the points 𝑂 1 , . . . , 𝑂 𝑛 bound an 𝑛-faced polyhedron 𝑈. If we project the faces of 𝑈 onto the sphere by a central projection from the center of the sphere, we get 𝑛 spherical polygons 𝑃1 , . . . , 𝑃𝑛

5.2 Some Additional Proofs

117

whose total area is 4𝜋. On the other hand, we will estimate this total area from below, respectively from above, as a quantity depending on the radius 𝑟 of the caps. In the packing problem 𝐾𝑖 is contained in 𝑃𝑖 , while in the covering problem 𝐾𝑖 contains 𝑃𝑖 . We may also assume that in the first case 𝑃𝑖 is circumscribed about 𝐾𝑖 , while in the second case 𝑃𝑖 is inscribed in 𝐾𝑖 , for otherwise the area of 𝑃𝑖 could be decreased or increased, respectively. Then 𝑃𝑖 can be partitioned into right triangles with a common vertex 𝑂 𝑖 . The total number of the right triangles so obtained is 4𝑒, where 𝑒 is the number of edges of 𝑈. We pick one such right triangle Δ and we denote by 𝛼 its angle at the center of the corresponding cap. Then, by the formula (1.10.4), in the first case we get Δ = 𝛼 − arcsin(cos 𝑟 · sin 𝛼) and in the second case, Δ = 𝛼 − arctan(cos 𝑟 · tan 𝛼). Observe now that Δ, as a function of 𝛼, in the first case is convex, and in the second
case is concave 0 < 𝛼 < 𝜋2 , hence we can apply Jensen’s inequality, resulting in: h 𝜋𝑛

Δ ≤ 4𝑒

h 𝜋𝑛

∑︁

4𝜋 =

∑︁

and

 𝜋𝑛 i − arcsin cos 𝑟 · sin 2𝑒 2𝑒

Δ ≥ 4𝑒

4𝜋 =

 𝜋𝑛 i − arctan cos 𝑟 · tan , 2𝑒 2𝑒

that is, cos 𝑟 ≥

sin 𝜋 (𝑛−2) 2𝑒 , sin 𝜋𝑛 2𝑒

and

cos 𝑟 ≤

tan 𝜋 (𝑛−2) 2𝑒 , tan 𝜋𝑛 2𝑒

respectively. Therefore, in view of 𝑒 ≤ 3(𝑛 − 2), we get the inequalities cos 𝑟 ≥

sin 30◦ 1 = csc 𝜔 𝑛 sin 𝜔 𝑛 2

and

cos 𝑟 ≤

1 tan 30◦ = √ cot 𝜔 𝑛 , tan 𝜔 𝑛 3

equivalent to the inequalities (5.1.1) and (5.1.2), respectively. The case of equalities poses no difficulty.

5.2 Some Additional Proofs The following proofs of the same inequalities (5.1.1) and (5.1.2) are in principle even easier than the previous ones, as they do not use Jensen’s inequality. We prove the inequality (5.1.1) in the following, equivalent form:

5 Extremal Properties of Regular Polyhedra

118

Among 𝑛 points on the unit sphere there always exists a pair with a distance   21 𝑑 ≤ 4 − csc2 𝜔 𝑛

(5.2.1)

between them. For the spherical distance between the pair of points in question 𝛿 ≤ arccos

cot2 𝜔 𝑛 − 1 2

(5.2.2)

holds. The right-hand side is the length 𝛿 𝑛 of a side of an equilateral spherical 2𝜋 triangle of area 𝑛−2 . The proof is based on the following lemma: Let 𝐴𝐵 be the shortest side of a spherical triangle Δ ≡ 𝐴𝐵𝐶 and let Δ′ ≡ 𝐴𝐵𝐶 ′ be an equilateral triangle. If Δ < Δ′, then the spherical circumradius of Δ is greater than 𝐴𝐵. To show this, we can assume that 𝐶 and 𝐶 ′ lie on the same side of the great circle 𝐴𝐵. Draw circles 𝑎, 𝑏 and 𝑐 ′, each of radius 𝐴𝐵, centered at 𝐴, 𝐵, and 𝐶 ′, respectively (Figure 5.1). Mark with 𝐴 ′ the intersection point of 𝑏 and 𝑐 ′ other than 𝐴, and with 𝐵 ′ the intersection point of 𝑎 and 𝑐 ′ other than 𝐵. Then the triangles 𝐴𝐵𝐴 ′, 𝐴𝐵𝐵 ′ and 𝐴𝐵𝐶 ′ are of the same area, and therefore the arc of the circle 𝐴 ′ 𝐵 ′𝐶 ′ lying “above” the great circle 𝐴𝐵 is the locus of the vertices of eqiareal triangles with Fig. 5.1 base 𝐴𝐵. By our assumptions, 𝐶 lies above the great circle 𝐴𝐵 and below the Lexell circle 𝐴 ′ 𝐵 ′𝐶 ′. Furthermore, since 𝐶 lies outside both 𝑎 and 𝑏, 𝐶 must lie outside 𝑐 ′, and our lemma is proved. Again, we assume that 𝑛 ≥ 3 and that the points 𝑃1 , . . . , 𝑃𝑛 do not lie in a hemisphere. Then the convex hull 𝐻 of these points contains the center 𝑂 of the sphere. We can further assume that every face of the polyhedron 𝐻 is a triangle, for a non-triangular face can be partitioned into triangles. Then the number of faces of 𝐻 is 2𝑛 − 4. Consider the spherical net of 𝐻 obtained by radially projecting the edges of 𝐻 onto the sphere from its center 𝑂; among the resulting spherical triangles there is one, say 𝑃𝑖 𝑃 𝑗 𝑃 𝑘 ≡ Δ, of smallest area. If now, contrary to (5.2.2), the smallest distance between the points were greater than 𝛿 𝑛 , so too would be the length of each 4𝜋 side of Δ, and as Δ ≤ 2𝑛−4 , by our lemma the circumradius of Δ would also be greater than 𝛿 𝑛 . Since the circumcircle of Δ contains none of the points 𝑃1 , . . . , 𝑃𝑛 in its interior, the circumcenter 𝑃𝑛+1 of Δ could be added to the family of points

5.2 Some Additional Proofs

119

𝑃1 , . . . , 𝑃𝑛 , while still maintaining the property that the minimum distance between the points is greater than 𝛿 𝑛 . This process can be continued. But since the number of points with the above property is obviously bounded, after a finite number of steps we arrive at a contradiction. We now turn to the inequality (5.1.2), giving it, for 𝑛 > 3, a new expression: If a convex polyhedron with 𝑛 vertices or with 𝑛 faces lies in a concentric spherical shell of inner radius 𝑟 and outer radius 𝑅, then 𝑅 √ ≥ 3 tan 𝜔 𝑛 . 𝑟

(5.2.3)

Our assumptions about the prescribed number of vertices and faces are equivalent, implying each other by polarity. Our statement for the 𝑛-faced polyhedron is connected to the covering problem. Namely, if 𝑛 congruent caps cover the sphere 𝐾, then the planes of the caps’ boundaries bound an 𝑛-faced polyhedron. The fact that the caps cannot be too small implies that the radius 𝑟 of a sphere concentric with 𝐾 and contained in the 𝑛-faced polyhedron cannot be too large. Conversely, the size of 𝑟 implies a bound for the size of the caps, and hence for the covering’s density. We prove (5.2.3) for a polyhedron 𝑃 with 𝑛 vertices. Without loss of generality, assume that 𝑅 = 1. Furthermore, assume that 𝑃 has triangular faces only. Now, since the number of faces is 2𝑛 − 4, in the spherical net of 𝑃 (on the outher sphere) (on the outher sphere) there is at least one triangle of area at 4𝜋 . The spherical circumradius of such a triangle cannot be smaller than the least 2𝑛−4 4𝜋 . Consequently, the distance circumradius 𝜚 of an equilateral triangle of area 2𝑛−4 from the plane of this triangle to the center of the sphere—hence 𝑟 as well—is at most cos 𝜚 = cot 60◦ cot 𝜔 𝑛 , and the proof of (5.2.3) is complete. The above proof can be considered as the simplest possible proof of the inequality (5.1.2). We present yet another proof for each of the inequalities (5.1.1) and (5.1.2). Draw an equilateral spherical triangle Δ whose side length 𝛿 is the smallest spherical distance between the points 𝑃1 , . . . , 𝑃𝑛 , then denote the circumradius of the triangle Δ by 𝜚 and its angle by 2𝜔. For each 𝑖, draw a circle 𝐾𝑖 of radius 𝜚 centered at the point 𝑃𝑖 , and observe that no positive-area part of the sphere can be covered by more than two of the circles. Therefore the area of the portion of the sphere covered by these circles is ∑︁ 𝑛𝐾 − 𝐾𝑖 𝐾 𝑗 , 𝐾 = 𝐾𝑖 , where the summation is extended over all pairs 𝑖 𝑗 with 1 ≤ 𝑖 < 𝑗 ≤ 𝑛. However, the number of nonempty intersections 𝐾𝑖 𝐾 𝑗 cannot exceed the maximum possible number of edges of a convex 𝑛-faced polyhedron, which is 3𝑛 − 6. On the other hand, 𝐾𝑖 𝐾 𝑗 cannot be greater than the area of the intersection of two circles of radius 𝜌 whose centers are at a distance 𝛿 from each other. Consequently,

120

5 Extremal Properties of Regular Polyhedra





2𝜔 2𝜋 𝐾𝑖 𝐾 𝑗 ≤ 2 𝐾 − 2𝜔 + −𝜋 2𝜋 3

 .

Furthermore, since the part of the sphere’s area covered by the circles cannot exceed 4𝜋, we have 𝜔 𝜋 𝐾 − 2𝜔 + ≤ 4𝜋 , 𝑛𝐾 − 6(𝑛 − 2) 𝜋 3 that is, 𝑛 𝜋 = 𝜔𝑛 , 𝜔≤ 𝑛−2 6 which proves (5.2.2) and with it also (5.1.1). For an analogous proof of (5.1.2), we assume that the circles 𝐾1 , . . . , 𝐾𝑛 cover the unit sphere, and we begin with the equality 4𝜋 = 𝑛𝐾 −

1 ∑︁ 𝑆 𝑗𝑖𝑘 , 2

where 𝑆 𝑗𝑖𝑘 = 𝐾𝑖 𝐾 𝑗 + 𝐾 𝑗 𝐾 𝑘 + 𝐾 𝑘 𝐾𝑖 − 2𝐾𝑖 𝐾 𝑗 𝐾 𝑘 means the part of the area of the sphere covered by the circles 𝐾𝑖 , 𝐾 𝑗 and 𝐾 𝑘 at least twice. Here, the summation extends over those index triples 𝑖 𝑗 𝑘 for which the triangle 𝑂 𝑖 𝑂 𝑗 𝑂 𝑘 is a face of the convex hull of the circles’ centers 𝑂 1 , . . . , 𝑂 𝑛 . This relation corresponds to the equality (3.4.3). However, this time the proof is simpler, because now the covered region has no boundary. If we now let the circles 𝐾𝑖 , 𝐾 𝑗 and 𝐾 𝑘 vary under the constraint that they have a common point, then 𝑆 𝑗𝑖𝑘 attains its minimum —just as in the planar case— when 𝐾𝑖 𝐾 𝑗 𝐾 𝑘 shrinks to a single point and the triangle 𝑂 𝑖 𝑂 𝑗 𝑂 𝑘 is equilateral. Consequently,    2𝜋 2𝜔 𝐾 − 2𝜔 + −𝜋 . 𝑆 𝑗𝑖𝑘 ≥ 6 2𝜋 3 Since the number of the intersections 𝑆 𝑗𝑖𝑘 is 2𝑛 − 4, we get 𝜔 𝜋 𝐾 − 2𝜔 + ≥ 4𝜋, 𝑛𝐾 − 6(𝑛 − 2) 𝜋 3 that is, 𝜔≥

𝑛 𝜋 = 𝜔𝑛 . 𝑛−2 6

This concludes the proof.

5.3 Approximating a Ball by Polyhedra If the Hausdorff distance of an 𝑛-faced convex polyhedron 𝑈𝑛 from the unit ball 𝐾 is 𝜂(𝑈𝑛 , 𝐾), then 𝑈𝑛 is contained in the ball of radius 1 + 𝜂 concentric with 𝐾, and contains the concentric ball of radius 1 − 𝜂. Hence, according to (5.2.3),

5.3 Approximating a Ball by Polyhedra

121

1+𝜂 √ ≥ 3 tan 𝜔 𝑛 . 1−𝜂 The same holds true for polyhedra with 𝑛 vertices. By a simple calculation we get: The Hausdorff distance of a polyhedron 𝑈𝑛 with 𝑛 faces or with 𝑛 vertices from the unit ball 𝐾 satisfies 2 𝜋 sin 𝑛−2 6 . (5.3.1) 𝜂(𝑈𝑛 , 𝐾) ≥ 𝑛−4 𝜋 cos 𝑛−2 6 Equality occurs only for 𝑛 = 4, 6 and 12 if 𝑈𝑛 is a regular polyhedron concentric with 𝐾 for which the arithmetic mean of the inradius and circumradius is equal to 1. If, instead of polyhedra with a fixed number of vertices or faces, we consider, after J. Steiner, the narrower class of mutually isomorphic polyhedra, that is, polyhedra of the same topological type, then we can assert that among polyhedra isomorphic to a given regular polyhedron the best one to approximate the ball is the regular one. As a consequence of (5.3.1), we get the inequality 2𝜋 , 𝜂(𝑈𝑛 , 𝐾) > √ 27𝑛 providing an exact asymptotic bound for large values of 𝑛. In view of the fact that the number of vertices, or faces, of a convex polyhedron with 𝑒 edges does not exceed 2𝑒 3 , the inequality (5.3.1) as well as the equivalent inequality (5.2.3), can be expressed in terms of the number of edges. By the minimal spherical shell of a polyhedron 𝑃 we mean the shell bounded by two concentric spheres, one of radius 𝑟, contained in 𝑃, and one of radius 𝑅, containing 𝑃, such that the ratio 𝑅𝑟 is as small as possible. Then the following theorem holds: If 𝑅 and 𝑟 are the radii of a minimal spherical shell of a convex polyhedron 𝑃 with 𝑒 edges, then 𝑅 √ 𝑒 𝜋 ≥ 3 tan , (5.3.2) 𝑟 𝑒−3 6 and equality occurs only if 𝑃 is a regular tetrahedron. It would be desirable to strengthen (5.3.2) for 𝑒 > 6. We now turn to certain generalizations of the inequality (5.2.3), and begin by proving the following two equivalent theorems: In a three-valent convex polyhedron with 𝑛 faces containing the unit ball, let 𝑅1 , . . . , 𝑅2𝑛−4 denote the distances from the polyhedron’s vertices to the ball’s center. Then √ 𝐴(𝑅1 , . . . , 𝑅2𝑛−4 ) ≥ 3 tan 𝜔 𝑛 . (5.3.3)

122

5 Extremal Properties of Regular Polyhedra

In a triangular convex polyhedron with 𝑛 vertices contained in a unit ball centered at an interior point of the polyhedron, let 𝑟 1 , . . . , 𝑟 2𝑛−4 denote the distances from the planes of the polyhedron’s faces to the ball’s center. Then √ 3 𝐻 (𝑟 1 , . . . , 𝑟 2𝑛−4 ) ≤ cot 𝜔 𝑛 . (5.3.4) 3 Here, 𝐴 = 𝑀1 and 𝐻 = 𝑀−1 denote the arithmetic and the harmonic means, respectively. Since the inequalities (5.3.3) and (5.3.4) imply each other by polarity, it suffices to prove (5.3.4) only. We may assume that the vertices 𝐸 1 , . . . , 𝐸 𝑛 lie on the surface of the ball. If 𝐸 𝑖 𝐸 𝑗 𝐸 𝑘 is a face of the polyhedron whose plane is at a distance 𝑟 from the ball’s center, then 1 √ Δ+𝜋 ≥ 3 tan , 𝑟 6 where Δ denotes the area of the spherical triangle 𝐸 𝑖 𝐸 𝑗 𝐸 𝑘 . This inequality expresses the fact, needed in the proof of (5.2.3) as well, that among all spherical triangles inscribed in a given circle, the equilateral one is of maximum area. Consequently, in Í view of Δ = 4𝜋 and the convexity of tan Δ+6 𝜋 for 0 < Δ < 2𝜋, we get ∑︁ 1 𝑟

√ ≥

3(2𝑛 − 4) tan

4𝜋 2𝑛−4

+𝜋

6

,

which we needed to prove. We now give further generalizations of the inequalities (5.3.3) and (5.3.4), inspired by the Erdős–Mordell inequality (1.5.5) for triangles. Let 𝑟 1 , . . . , 𝑟 4 and 𝑅1 , . . . , 𝑅4 denote the distances from the faces and from the vertices, respectively, of a tetrahedron to a point in its interior. D.K. Kazarinoff noticed the remarkable fact that the inequality 𝐴(𝑅1 , . . . , 𝑅4 ) ≥ 3𝐴(𝑟 1 , . . . , 𝑟 4 ), which would be the natural generalization to space of the Erdős–Mordell inequality, does not hold in general. On the other hand, Kazarinoff proved the inequality √ 𝐴(𝑅1 , . . . , 𝑅4 ) > 8𝐴(𝑟 1 , . . . , 𝑟 4 ), √ and that the constant 8 cannot be replaced by a larger one. Consequently, in space, an inequality analogous to (1.5.5) that expresses the extremal property of the regular tetrahedron can be given only if the arithmetic mean value 𝐴(𝑟 1 , . . . , 𝑟 4 ) is replaced by some smaller mean value. We show that the harmonic mean is suitable for this purpose.

5.3 Approximating a Ball by Polyhedra

123

We prove the following, more general theorems: For a triangular convex polyhedron with 𝑛 vertices, let 𝑅1 , . . . , 𝑅𝑛 and 𝑟 1 , . . . , 𝑟 2𝑛−4 denote the distances from the vertices and faces, respectively, to a fixed interior point of the polyhedron. Further, let 𝑞 1 , . . . , 𝑞 𝑛 denote the numbers of edges meeting at the polyhedron’s vertices. Then 𝐴(𝑅; 𝑞) √ ≥ 3 tan 𝜔 𝑛 . 𝐻 (𝑟)

(5.3.5)

For a three-valent convex polyhedron with 𝑛 faces, let 𝑟 1 , . . . , 𝑟 𝑛 and 𝑅1 , . . . , 𝑅2𝑛−4 denote the distances from its faces and vertices, respectively, to a fixed interior point of the polyhedron. Further, let 𝑝 1 , . . . , 𝑝 𝑛 denote the numbers of sides of the polyhedron’s faces. Then √ 𝐴(𝑅) ≥ 3 tan 𝜔 𝑛 . (5.3.6) 𝐻 (𝑟; 𝑝) Here, 𝐴(𝑅; 𝑞) denotes the weighted arithmetic mean of 𝑅1 , . . . , 𝑅𝑛 with weights 𝑞 1 , . . . , 𝑞 𝑛 . Similarly, 𝐻 (𝑟; 𝑝) denotes the harmonic mean weighted with 𝑝 1 , . . . , 𝑝 𝑛 . In accordance with our previous notation, 𝐻 (𝑟) = 𝐻 (𝑟; 1) and 𝐴(𝑅) = 𝐴(𝑅; 1) denotes the harmonic and arithmetic mean, respectively, taken with equal weights 1. The example of a tetrahedron shows that 𝐻 (𝑟) cannot be replaced with 𝐴(𝑟), and that 𝐴(𝑅) cannot be replaced with 𝐻 (𝑅) either. Moreover, we will see that neither 𝐴(𝑅; 𝑞) can be replaced with 𝐴(𝑅), nor can 𝐻 (𝑟; 𝑝) be replaced with 𝐻 (𝑟). Therefore our inequalities are quite sharp, and they cover almost everything that can be said on this matter. The inequalities (5.3.5) and (5.3.6) are also equivalent to each other via polarity. Therefore, it suffices to prove (5.3.5) only. Let 𝑓 ≡ 𝐸 𝑖 𝐸 𝑗 𝐸 𝑘 be a face of the polyhedron, let 𝐹 be the perpendicular projection of an interior point 𝑂 of the polyhedron to the plane of 𝑓 , and let 𝛼 be the solid angle of the tetrahedron 𝑂𝐸 𝑖 𝐸 𝑗 𝐸 𝑘 at the vertex 𝑂. It is easy to see that if we move the points 𝐸 𝑖 , 𝐸 𝑗 and 𝐸 𝑘 within their original plane, preserving the value of 𝛼, then 𝑓 reaches its minimum area when 𝐸 𝑖 𝐸 𝑗 𝐸 𝑘 becomes an equilateral triangle centered at 𝐹. In the next section we will prove the corresponding fact for a polygon with an arbitrary number of vertices. Therefore, by a simple calculation, we get √   𝛼+𝜋 2 27 3 tan2 −1 . 𝑓 ≥𝑟 4 6 Consequently, by (1.5.3) we get (𝐹𝐸 𝑖 + 𝐹𝐸 𝑗 + 𝐹𝐸 𝑘 ) 2 = √︃ 2   » » 𝛼+𝜋 𝑅𝑖2 − 𝑟 2 + 𝑅 2𝑗 − 𝑟 2 + 𝑅 2𝑘 − 𝑟 2 ≥ 9𝑟 2 3 tan2 −1 6

124

5 Extremal Properties of Regular Polyhedra

√ yielding, in view of the concavity of the function 𝑥 2 − 𝑟 2 for 𝑥 > 𝑟, that √︄ © ­3 «



𝑅𝑖 + 𝑅 𝑗 + 𝑅 𝑘 3

2

2

  𝛼+𝜋 ª − 𝑟 2 ® ≥ 9𝑟 2 3 tan2 −1 , 6 ¬

that is,

√ 𝛼+𝜋 27𝑟 tan . 6 By adding the corresponding inequalities for all 2𝑛 − 4 faces of the polyhedron, we get   −1 ∑︁ √ ∑︁ 𝛼 + 𝜋 √ ∑︁ 1 𝛼+𝜋 𝑟 tan = 27 tan . 𝑞𝑅 ≥ 27 6 𝑟 6 𝑅𝑖 + 𝑅 𝑗 + 𝑅 𝑘 ≥

Since the function Φ(𝑥, 𝑦) = 𝑥 −1 tan 𝑦; is convex due to Φ 𝑥 𝑥 Φ 𝑦 𝑦 − Φ2𝑥 𝑦 =

0 < 𝑥,

𝜋 𝜋 ≤𝑦< 6 2

4 sin2 𝑦 − 1 ≥ 0, 𝑥 4 cos4 𝑦

an application of Jensen’s inequality yields    −1 𝛼 + 𝜋 √ 1 1 ∑︁ 𝑞𝑅 ≥ 27 𝐴 . tan 𝐴 2𝑛 − 4 𝑟 6 Í This, however, is equivalent to the inequality (5.3.5), since 𝑞 = 6𝑛 − 12, 4𝜋 𝛼 + 𝜋 + 𝜋 = 2𝑛−4 = 𝜔𝑛 . [ 𝐴(𝑟 −1 )] −1 = 𝐻 (𝑟), and 𝐴 6 6 In addition, we present an example showing that the inequality 𝐴(𝑅) √ ≥ 3 tan 𝜔 𝑛 𝐻 (𝑟) does not hold in general. Consider the truncated dodecahedron » (3,10,10)√of edge length 2. This polyhedron has 60 vertices at a distance of 𝑅 = 12 (37 + 15 5) from » √ its center, 20 triangular faces at a distance 𝑟 3 = 16 (103 + 45 5), and 12 decagonal » √ faces at a distance of 𝑟 10 = 12 (25 + 11 5). Therefore √︄ √︄ √ √ 𝐴(𝑅) 20 3(37 + 15 5) 12 37 + 15 5 = √ + √ 32 25 + 11 5 𝐻 (𝑟) 32 103 + 45 5 … … 5 211.6 . . . 3 70.53 . . . = + 8 203.6 . . . 8 53.60 . . . = 0.6371 . . . + 0.4301 . . . = 1.0672 . . . ,

5.4 Volume of a Circumscribed Polyhedron

125

and this is smaller than √ √ 3 tan 𝜔32 = 3 tan 32◦ = 1.0825 . . . . On the other hand, the inequality (5.3.6) is satisfied here: … … 1 211.6 . . . 2 70.53 . . . 𝐴(𝑅) = + 𝐻 (𝑟; 𝑝) 3 203.6 . . . 3 53.60 . . . = 0.340 . . . + 0.765 . . . = 1.105 . . . > 1.0825 . . . .

5.4 Volume of a Circumscribed Polyhedron In (5.2.3), we encountered an inequality concerning polyhedra with a prescribed number of edges or faces, in which equality occurs precisely for the five regular polyhedra. An inequality of this kind can arise only in a special case, namely if the quantity whose maximum or minimum is sought takes equal values for the polyhedra {3, 4} and {4, 3}, and also for {3, 5} and {5, 3}. This was the case for the quantity 𝑅𝑟 . However, in general, an inequality that expresses an extremum property of all five regular polyhedra must refer to polyhedra in which both the number of vertices and faces are prescribed. To illustrate this fact, mentioned already in the introduction to this chapter, we give an example in the following theorem. If a polyhedron 𝑉 with 𝑣 vertices, 𝑓 faces, and 𝑒 edges contains the unit ball, then   𝜋𝑓 𝑒 2 𝜋𝑓 2 𝜋𝑣 tan tan −1 (5.4.1) 𝑉 ≥ sin 3 𝑒 2𝑒 2𝑒 and equality occurs only if 𝑉 is a regular polyhedron circumscribed about the unit ball. Since every polyhedron with 𝑛 faces can be regarded as a three-valent polyhedron, that is, as a polyhedron with 2𝑛 − 4 vertices and 3𝑛 − 6 edges, for each 𝑛-faced polyhedron 𝑉𝑛 containing the unit ball the inequality (5.4.1) implies that 𝑉𝑛 ≥ (𝑛 − 2) (3 tan2 𝜔 𝑛 − 1) sin 2𝜔 𝑛 .

(5.4.2)

Furthermore, since every polyhedron with 𝑛 vertices can be regarded as a triangular polyhedron, that is, a polyhedron with 2𝑛 − 4 faces and 3𝑛 − 6 edges, for each such polyhedron 𝑈𝑛 with 𝑛 vertices containing the unit ball, we have √ 3 (𝑛 − 2)(3 tan2 𝜔 𝑛 − 1) . (5.4.3) 𝑈𝑛 ≥ 2

126

5 Extremal Properties of Regular Polyhedra

It can be shown, however, that the two sequences   4𝜋 2 𝑛 (𝑛 − 2) (3 tan 𝜔 𝑛 − 1) sin 2𝜔 𝑛 − 3 and

"√ # 3 4𝜋 2 𝑛 (𝑛 − 2)(3 tan 𝜔 𝑛 − 1) − 2 3 √

2

√

2

are monotonically decreasing and converge to 20 273 𝜋 and 4 93 𝜋 , respectively. Consequently we obtain the following inequalities: √ √ 20 3𝜋 2 4 3𝜋 2 4𝜋 4𝜋 > , 𝑈𝑛 − > . (5.4.4) 𝑉𝑛 − 3 27𝑛 3 9𝑛 For a proof of (5.4.1), consider a face 𝑡 of the polyhedron. Let 𝑝 denote the number of vertices of 𝑡, and let 𝜏 denote the radial projection of 𝑡 to the surface of the ball from its center. We assert that   2𝜋 𝑝 2 𝜋 2 2𝜋 − 𝜏 tan cot − 1 = 𝑇 (𝜏, 𝑝). 𝑡 ≥ sin 2 𝑝 𝑝 2𝑝 This inequality expresses the fact that a polygon 𝑡 with a fixed area and a fixed number of vertices, and whose plane does not intersect the interior of the unit ball, has the greatest possible central projection on the surface of the ball when it is a regular polygon tangent to the ball at its center. It can be assumed that the plane of the polygon 𝑡 touches the ball at some point 𝐴 interior to 𝑡, since otherwise 𝜏 can be increased by a parallel translation of 𝑡. Denote the sides of 𝑡 by 𝑠1 , . . . , 𝑠 𝑝 , their midpoints by 𝑀1 , . . . , 𝑀 𝑝 , and the distances from the sides to 𝐴 by 𝑎 1 , . . . , 𝑎 𝑝 . If 𝑡 is a regular polygon centered at 𝐴, then each side 𝑠𝑖 is perpendicular to 𝐴𝑀𝑖 and all distances 𝑎 𝑖 are equal. It suffices to prove that if any of these conditions is not satisfied, the area of 𝜏 can be increased. Assume, for example, that 𝐴𝑀1 is not perpendicular to 𝑠1 . If we rotate 𝑠1 by an infinitesimal angle about 𝑀1 , then the area of 𝑡 remains unchanged. Since the projection of 𝑀1 is not the midpoint of the projection of 𝑠1 , the area of 𝜏 does change under this rotation. Thus, under the appropriate direction of the rotation, the area of 𝜏 increases. This argument shows at the same time that 𝐴 must lie in the interior of 𝑡. Assume now that 𝑠1 ⊥ 𝐴𝑀1 and 𝑠2 ⊥ 𝐴𝑀2 but, say, 𝑎 1 < 𝑎 2 . Translate each of the sides 𝑠1 and 𝑠2 in the direction perpendicular to it by an infinitesimal amount and so that the resulting changes 𝑑1 𝑡 and 𝑑2 𝑡 of area of 𝑡 satisfy 𝑑1 𝑡 + 𝑑2 𝑡 = 0. Then the changes of the areas of the corresponding projections satisfy |𝑑1 𝜏| > |𝑑2 𝜏|. Indeed, by the assumption 𝑎 1 < 𝑎 2 , the trapezoids 𝑑1 𝑡 and 𝑑2 𝑡 can be decomposed into elementary parts such that for each part of 𝑑1 𝑡 corresponds a part of 𝑑2 𝑡 of equal area but lying farther away from 𝐴. Obviously, to an area element lying farther away from 𝐴 corresponds a projection of smaller area. If one chooses 𝑑1 𝑡 > 0, then

127

5.4 Volume of a Circumscribed Polyhedron

|𝑑1 𝜏| + |𝑑2 𝜏| > 0, which proves the stated extremal property of the regular polygons, whereby the inequality 𝑡 ≥ 𝑇 (𝜏, 𝑝) is proved. Now we use the fact that, for 𝜏 ≥ 0 and 𝑝 ≥ 3, the function 𝑇 (𝜏, 𝑝) is a convex function of two variables. This immediately yields the following inequality, equivalent to (5.4.1):   ∑︁ ∑︁ 4𝜋 2𝑒 𝑇 (𝜏, 𝑝) ≥ 𝑓 𝑇 𝑡≥ 3𝑉 ≥ , . □ 𝑓 𝑓 The only difficulty in this—in principle very simple—proof is the unfortunate fact that the function 𝑇 (𝜏, 𝑝) is too complicated to directly carry out all the computations needed for the proof of the convexity. We therefore just show in Figure 5.2 a graphical representation of a few curves 𝑇 (const., 𝑝), in which the convexity is indicated empirically. The convexity is expressed by the condition that the midpoint of the line segment joining a point on the curve 𝑇 = 𝑇 (𝜏1 , 𝑝) with a Fig. 5.2 point on 𝑇 = 𝑇 (𝜏 , 𝑝) lies above the curve
2 2 𝑇 = 𝑇 𝜏1 +𝜏 2 ,𝑝 . In what follows we give another, rigorous proof of the inequality (5.4.1). Let 𝐴 be the perpendicular projection of the ball’s center to the plane of the face 𝑓 of the given polyhedron, let 𝐵𝐷 be a side of 𝑓 , and let 𝐶 be the perpendicular projection of 𝐴 on the line 𝐵𝐷. We may assume that 𝐴 lies on the face 𝑡 and 𝐶 lies on the edge 𝐵𝐷. Otherwise the face could be replaced by a face of a smaller area, still lying outside the ball, that satisfies this condition, while its number of sides, as well as the area of its projection on the ball’s surface, remains unchanged. If the above condition is satisfied for every face and every edge, then the surface of the polyhedron can be partitioned into 4𝑒 right triangles, one of which is the triangle Δ ≡ 𝐴𝐵𝐶. Denote the central projection of 𝐴𝐵𝐶 on the ball’s surface by 𝐴 ′ 𝐵 ′𝐶 ′, the angles of the right spherical triangle 𝐴 ′ 𝐵 ′𝐶 ′ at 𝐴 ′ and 𝐵 ′ by 𝛼 and 𝛽, respectively, and the hypotenuse 𝐴 ′ 𝐵 ′ by 𝑐. Then, in view of 𝐴𝐵 ≥ tan 𝑐 and cos 𝑐 = cot 𝛼 cot 𝛽, we have Δ≥

1 1 sin 2𝛼 tan2 𝑐 = sin 2𝛼(tan2 𝛼 tan2 𝛽 − 1) = 𝐹 (𝛼, 𝛽). 4 4

However, as the inequality 2 𝐹𝛼𝛼 𝐹𝛽𝛽 − 𝐹𝛼𝛽 =

2 tan4 𝛼 [1 − (sin2 𝛼 + sin2 𝛽)] 2 ≥ 0 cos6 𝛽

assures the convexity of the function 𝐹 (𝛼, 𝛽) in the region 0 < 𝛼 < 𝛼 + 𝛽 ≥ 𝜋2 , we have

𝜋 2,

00 6 3 6 3 0 < Δ < 2𝜋 ,

we have 𝑉=

∑︁

𝑣≤

∑︁

 𝑡 (Δ) ≤ (2𝑛 − 4)𝑡

 4𝜋 , 2𝑛 − 4

as claimed. Rotate one face of a cube 𝐶 by 45◦ about the face’s center and take the convex hull 𝐻 of the new face and the face opposite to the original one. It is easy to show that 𝐻 > 𝐶. Consequently, the cube is not of maximum volume among all polyhedra with 8 vertices inscribed in a given ball. Similarly, the dodecahedron is not an extremal polyhedron. More generally, it can be shown that the extremal polyhedra can be bounded by triangular faces only, all lying in distinct planes. Such a polyhedron will be called a proper triangular polyhedron. Quite generally, we prove: If a polyhedron has maximum volume among all polyhedra with a given number of vertices and inscribed in a given smooth convex surface, then it is a proper triangular polyhedron. Here a convex surface is said to be smooth if each of its points is contained in exactly one supporting plane. From a certain point of view this theorem appears to be quite interesting. We would like to illustrate this with an example. Among all polyhedra inscribed in a cube 𝐶 and with 8 vertices, obviously it is the cube itself that is of the greatest volume. Now, let us coat the cube with a thin layer of wax so that the vertices of the cube remain on the surface of the resulting convex body 𝐾. The body can even possess the very same symmetries as 𝐶, but it can be asymmetric as well. One should only make sure that the edges and vertices of 𝐶 are sufficiently rounded, so that 𝐾 turns out to be smooth. Then the original cube is no longer of maximum volume among all inscribed polyhedra with 8 vertices, no matter how thin the layer of wax may be. For the proof, we consider a set of points 𝑃1 , . . . , 𝑃𝑛 not contained in one plane. Next we will try to determine the geometric locus 𝐺 of the points 𝑃 for which the volume of the convex hull 𝐻 of 𝑃1 , . . . , 𝑃𝑛 and 𝑃 remains constant. Assume for a moment that all faces Δ1 , . . . , Δ𝑚 that meet at the vertex 𝑃 of 𝐻 are triangular. To each face Δ𝑖 we assign the outer normal vector v𝑖 of magnitude |v𝑖 | = Δ𝑖 and we set v = v1 + . . . + v𝑚 .

130

5 Extremal Properties of Regular Polyhedra

−−→ If we now translate 𝑃 by the vector r = 𝑃𝑃 ′ to a new position 𝑃 ′, so that the new polyhedron 𝐻 ′ remains isomorphic to the original one, then 3(𝐻 ′ − 𝐻) = vr. Therefore in order to preserve the volume of 𝐻, 𝑃 must move within the plane perpendicular to v. This holds only as long as 𝑃 does not encounter a plane 𝑝 containing some face of the convex hull of the points 𝑃1 , . . . , 𝑃𝑛 . At that moment a face of 𝐻 with more than three vertices meets in 𝑃. As soon as 𝑃 crosses such a plane 𝑝, other triangular faces meet at 𝑃, hence now 𝑃 must be moving in another plane than before. A glance at Figure 5.3, illustrating the analogous situation in the plane, makes clear that 𝐺 must be the boundary surface of a convex polyhedron. We now consider 𝑛 points 𝑃1 , . . . , 𝑃𝑛 on the smooth surface 𝐸 of a convex body 𝐾 and the convex hull 𝐻 of these points. Suppose that one of the vertices of 𝐻, say 𝑃𝑛 , belongs to a face of 𝐻 with four or more sides. If we vary 𝑃𝑛 while holding all other vertices of 𝐻 fixed so that the volume of 𝐻 does not change, then 𝑃𝑛 runs over the surface of a convex polyhedron. Since the original point 𝑃𝑛 lies on an edge of this polyhedron, near 𝑃𝑛 the polyhedron must penetrate inside 𝐾. This shows that the polyhedron 𝐻 canFig. 5.3 not be maximal. Our observations also show that the polyhedron of maximum volume has the property that at each vertex the vector k1 × k2 + k2 × k3 + · · · + k𝑚 × k1 points in the direction of the normal vector of the surface at that very point, where k1 , . . . , k𝑚 denote the vectors of the edges emanating from the vertex, listed cyclically. We now leave the general convex surfaces for a while and turn back to the ball. The following observations can be considered as an attempt to prove the inequality   2𝑒 𝜋𝑣 2 𝜋𝑓 2 𝜋𝑓 2 𝜋𝑣 𝑉 ≤ cos cot 1 − cot cot , (5.5.3) 3 2𝑒 2𝑒 2𝑒 2𝑒 224 ▶

where 𝑒, 𝑓 and 𝑣 denote the number of edges, faces, and vertices, respectively, of a convex polyhedron 𝑉 contained in the unit ball. At present we can only conjecture the validity of this inequality in general. Let 𝑡 denote a 𝑝-sided face of 𝑉, let 𝑣 be the convex hull of 𝑡 and the center of the ball, and let 𝜏 be the central projection of 𝑡 on the ball’s surface. It can be conjectured that   𝑝 𝜋 2𝜋 − 𝜏 𝜋 2𝜋 − 𝜏 𝑣 ≤ cos2 tan 1 − cot2 tan2 = 𝑈 (𝜏, 𝑝). 3 𝑝 2𝑝 𝑝 2𝑝 This would mean that for a prescribed number 𝑝 of vertices and area 𝜏 of the projection of the pyramid’s base 𝑡, the volume of the pyramid 𝑣 reaches its maximum

131

5.6 Inequalities Linking the Inradius and Circumradius of Polyhedra

when 𝑡 is a regular polygon inscribed in the ball. Yet the proof of this fact does not appear to be as simple as the proof of the analogous extremal property of the regular 𝑝-gon in the previous section. The function 𝑈 (𝜏, 𝑝), for 0 ≤ 𝜏 ≤ 2𝜋 and 3 ≤ 𝑝, is concave both as a function of 𝜏 when holding 𝑝 fixed, and also as a function of 𝑝 when holding 𝜏 fixed. Were it concave as a function of two variables, then one could write:   ∑︁ ∑︁ 4𝜋 2𝑒 𝑣≤ 𝑈 (𝜏, 𝑝) ≤ 𝑓 𝑈 𝑉= , , 𝑓 𝑓 which would prove (5.5.3). However, 𝑈 (𝜏, 𝑝) is not concave in the entire strip 0 ≤ 𝜏 ≤ 2𝜋, 3 ≤ 𝑝. But it suffices to use its concavity just for 0 ≤ 𝜏 ≤ 𝜋, about which we can convince ourselves from the graphical presentation of a few functions 𝑈 (𝜏, const) in Figure 5.4. Assuming the convexity of 𝑈 (𝜏, 𝑝), we first deal with the case 𝑓 ≥ 8 and we notice that, for an arbitrary value 𝑝 ≥ 3, the inequalities 𝑈 (𝜏, 𝑝) ≤ 𝑈 (𝜋, 𝑝), 𝑈 (𝜏1 , 𝑝) ≤ 𝑈 (𝜏2 , 𝑝),

𝜏≥𝜋 0 ≤ 𝜏1 ≤ 𝜏2 ≤

𝜋 2 Fig. 5.4

hold. If we now replace each value 𝜏𝑖 > 𝜋 with 𝜋 and denote the new values of 𝜏𝑖 with 𝜏1′ , . . . , 𝜏 ′𝑓 , then for 𝑓 ≥ 8 we get 𝑉 ≤

∑︁

𝑈 (𝜏, 𝑝) ≤

∑︁

𝑈 (𝜏 ′, 𝑝) ≤ 𝑓 𝑈

   𝜏 ′ 2𝑒 4𝜋 2𝑒 , ≤ 𝑓𝑈 , . 𝑓 𝑓 𝑓 𝑓

Í

Instead of a detailed discussion of the case 𝑓 < 8 we just give one simple example. Consider a polyhedron of the same type as a pentagonal prism ( 𝑓 = 7, 𝑣 = 10), or more generally, a polyhedron with 𝑓 ≥ 6, 2𝑒𝑓 ≥ 4. We just notice that for a fixed 𝑝 ≥ 4, the function 𝑈 (𝜏, 𝑝) is an increasing function of 𝜏 not only for 0 ≤ 𝜏 ≤ 𝜋2 , but for 0 ≤ 𝜏 ≤ 23𝜋 as well. Then the previous observations can be applied without any difficulties.

5.6 Inequalities Linking the Inradius and Circumradius of Polyhedra If 𝑟 and 𝑅 denote, respectively, the inradius and circumradius of a convex polyhedron 𝑉 with 𝑓 faces, 𝑣 vertices and 𝑒 edges, then (5.4.1) and (5.5.3) yield the following inequalities:

132

5 Extremal Properties of Regular Polyhedra



 𝑒 𝜋𝑓 2 𝜋𝑓 2 𝜋𝑣 sin tan tan − 1 𝑟3 ≤ 𝑉 ≤ 3 𝑒 2𝑒 2𝑒   2𝑒 𝜋𝑣 2 𝜋𝑓 2 𝜋𝑓 2 𝜋𝑣 ≤ cos cot 1 − cot cot 𝑅3 . 3 2𝑒 2𝑒 2𝑒 2𝑒 This implies the inequality 𝜋 𝜋 𝑅 ≥ tan tan , 𝑟 𝑝 𝑞

224 ▶

(5.6.1)

where 𝑝 = 2𝑒𝑓 is the average number of sides of the faces and 𝑞 = 2𝑒 𝑣 is the average valence of the vertices. Unfortunately, due to the incomplete proof of the inequality (5.5.3), strictly speaking, the above inequality can only be viewed as a conjecture. In contrast, some special cases of the inequality (5.6.1) can be proved rigorously. So, for example, by the reasoning from Section 5.1 used in the proof of (5.1.2), the following can be proved: If 𝑟 and 𝑅 are the radii of the minimal spherical shell of a convex polyhedron, then 𝜋 𝜋 𝑅 ≥ tan tan , (5.6.2) 𝑟 𝑝 𝑞 where 𝑝 is the average number of sides of the faces and 𝑞 is the average number of valence of the vertices. Moreover, equality holds exactly for the regular polyhedra. Let 𝑃4 , 𝑃5 , . . . be a sequence of convex polyhedra with 4, 5, . . . faces, whose inradii increase without bound, and let 𝐹1𝑛 , . . . , 𝐹𝑛𝑛 be the faces of 𝑃𝑛 . If the numbering of the faces can be chosen so that each column of the table 𝐹14 𝐹24 𝐹34 𝐹44 𝐹15 .. .

𝐹25 .. .

𝐹35 .. .

𝐹45 .. .

𝐹55 .. . . . .

forms a convergent sequence of polygons, then we say that the sequence 𝑃4 , 𝑃5 , . . . is convergent. The limit polyhedron provides a partition of the plane into convex polygons. It can be proved that for each of the partitions {3, 6}, {4, 4} and {6, 3} there exists a sequence of polyhedra {𝑃 𝑘 } converging to it such that   𝜋 𝜋 𝑅 − tan tan =0 (5.6.3) lim 𝑘 𝑘→∞ 𝑟 𝑝 𝑞 holds. Conversely: If a convergent sequence of polyhedra satisfies (5.6.3), then the limit polyhedron is “essentially” regular. This means that almost all faces of the polyhedron are approximately regular, in the sense of the definition of a hexagonal packing of circles. We can therefore say in short that the inequality (5.6.2) expresses an extremal property of all eight regular polyhedra {𝑝, 𝑞} with 𝑝, 𝑞 > 2.

5.6 Inequalities Linking the Inradius and Circumradius of Polyhedra

133

We now mention the following theorem: The inradius 𝑟 and the circumradius 𝑅 of a convex polyhedron with 𝑛 vertices or 𝑛 faces are linked by the inequality 𝑅 √ ≥ 3 tan 𝜔 𝑛 . 𝑟

(5.6.4)

More generally: If a convex polyhedron with 𝑛 vertices or 𝑛 faces contains the ellipsoid 𝑒 and is contained in the ellipsoid 𝐸, then 𝐸 √ ≥ 27 tan3 𝜔 𝑛 . 𝑒

(5.6.5)

If 𝑛 denotes the number of vertices, then (5.6.5) is a direct consequence of (5.4.3) and (5.5.1). The validity of the theorem for an 𝑛-faced polyhedron 𝑈 follows from this with the aid of the inequality (1.2.1). Namely, let 𝐸 be an ellipsoid containing 𝑈 and 𝑒 an ellipsoid contained in U. Without loss of generality we can assume that 𝑒 is the unit ball. Then the polarity with respect to 𝑒 transforms 𝑈 into a polyhedron with 𝑛 vertices contained in 𝑒 and it transforms 𝐸 into an ellipsoid 𝐸 ′ contained in that polyhedron. Then, using (1.2.1) and the just proven inequality (5.6.5) for 𝑛-vertex polyhedra, we get 𝑒 √ 𝐸 ≥ ′ 27 tan3 𝜔 𝑛 □ 𝑒 𝐸 The above theorem gives rise to a new problem. For a given polyhedron, consider the maximum volume ellipsoid 𝑒 contained in it, and the minimum volume ellipsoid 𝐸 containing it. These ellipsoids could be called the inscribed and circumscribed ellipsoids. We can now search for those polyhedra with a prescribed number 𝑛 of vertices or faces for which the quotient 𝐸𝑒 reaches its minimum. In other words, we want to find those polyhedra with 𝑛 vertices or 𝑛 faces that in a certain sense are best approximated by ellipsoids. For the analogous problem in the plane, the extremal polygon is affine regular. For such a polygon, the inscribed and circumscribed ellipses are concentric and homothetic. The inequality (5.6.5) provides the solution to our problem for 𝑛 = 4, 6 and 12, and in these cases the inscribed and circumscribed ellipsoids are, likewise, concentric and homothetic. We realize that the determination of the extremal polyhedra for every number of vertices and faces is not to be expected. Nevertheless, one can ask: Are the inscribed and circumscribed ellipsoids concentric and homothetic for every extremal polyhedron? We show that for an extremal polyhedron, the two ellipsoids are indeed concentric, but in general they are not homothetic. For a proof of the first part of this statement, consider, say, an 𝑛-faced polyhedron 𝑈 for which the inscribed ellipsoid 𝑒 and the circumscribed ellipsoid 𝐸 are not concentric. The polarity with respect to 𝑒 transforms 𝑈 into a polyhedron 𝑈 ′ with 𝑛 vertices contained in 𝑒 and containing the ellipsoid 𝐸 ′, where 𝐸 ′ denotes the image

134

5 Extremal Properties of Regular Polyhedra

of 𝐸. Consider now the polarity with respect to 𝐸 ′, transforming 𝑒 into 𝑒 ′ and 𝑈 ′ into an 𝑛-faced polyhedron 𝑈 ′′, then, according to (1.2.1) we have 𝐸 𝑒 𝐸′ > ′ > ′ . 𝑒 𝐸 𝑒 Since 𝑈 ′′ lies between 𝑒 ′ and 𝐸 ′, this inequality shows that 𝑈 ′′ is better than 𝑈. We show now that already in the case of the extremal polyhedron 𝑃 with 𝑛 = 5 vertices, the ellipsoids 𝑒 and 𝐸 are not homothetic. Namely, if they were homothetic, then we could assume that they are concentric balls. Consequently, 𝑃 would be an affine image of the polyhedron 𝑃 with 5 vertices for which the quotient 𝑅𝑟 between the radii of the minimal spherical shell is the smallest possible. It is easy to show that 𝑃¯ would have to be the convex hull of the north and south poles plus the regular ¯ If 𝑃¯ were identical with triangle inscribed in the equator of the circumsphere of 𝑃. the extremal polyhedron 𝑃, then the inscribed sphere and the inscribed ellipsoid of 𝑃¯ would coincide. This is not the case, as the ellipsoid of maximum volume inscribed in 𝑃¯ touches the faces of 𝑃¯ at their centroids, while the inscribed ball does not. The question of whether the insphere 𝑘 and circumsphere 𝐾 of a polyhedron with a prescribed number of vertices or faces, for which the ratio 𝐾𝑘 is the smallest possible, must necessarily be concentric or not, has not been answered yet.

5.7 Isoperimetric Problems for Polyhedra Restricted to convex bodies, the isoperimetric problem in space is as follows: Which bodies of a given surface area have maximum volume? The solution is expressed by the inequality 𝐹 3 − 36𝜋𝑉 2 ≥ 0, (5.7.1) which holds for the surface area 𝐹 and volume 𝑉 of an arbitrary convex body, with equality in the case of a ball only. In contrast to the planar problem, from the validity of the inequality (5.7.1) for convex bodies one cannot immediately conclude the same for nonconvex bodies since taking the convex hull may increase the surface area 𝐹. Still, it was proved that the above isoperimetric inequality holds quite generally for nonconvex bodies as well. Just as in the planar case, a number of problems arise if we search for the best 3 bodies, that is, bodies with the smallest value of the quotient 𝑉𝐹 2 , not in the set of all convex bodies but in a subset defined by certain conditions. Here our primary interest lies in the isoperimetric problem for convex polyhedra. It was Steiner who—following L’Huilier’s explorations—dealt with problems of this kind for the first time in detail. He considered prisms, pyramids and polyhedra of the type of a double pyramid, and he noticed that in these cases the best polyhedron is always circumscribed about a ball, such that its faces are tangent to the ball at their centroid. Inspired by this discovery, Steiner asked whether the best convex

5.7 Isoperimetric Problems for Polyhedra

135

polyhedron of any prescribed type must have this property. Moreover, Steiner conjectured that the regular prism circumscribed about a ball is the best one not only among prisms of the same number of faces, but even among all polyhedra isomorphic to it. According to another crucial conjecture of Steiner, among all polyhedra of the type of a regular polyhedron, the regular polyhedron is the best one. Steiner himself succeeded in proving this conjecture only for the octahedron, besides the simple case of the tetrahedron, already established by L’Huilier. Later, Lindelöf [96] proved that the best polyhedron among those with a given number of faces must be circumscribed about a ball so that its faces are tangent to the ball at their centroids. Lindelöf thought that this theorem answered the above question of Steiner. However, this is not quite so, since the set of all polyhedra isomorphic to a given polyhedron is but a subset (and, except for the tetrahedron, it is in fact a proper subset) of the set of all polyhedra with the same number of faces. Nevertheless, the arguments in Lindelöf’s proof provide the affirmative answer to the question of Steiner for polyhedra with three-valent vertices. Namely, Lindelöf showed that a polyhedron that does not satisfy the Steiner–Lindelöf condition can be improved by an infinitesimal change of one of its faces. Therefore, if there is a best polyhedron at all among the polyhedra of the type of some three-valent polyhedron, then it must satisfy the mentioned condition, otherwise it could be improved by an infinitesimal change of a face. Obviously, such a change preserves the type of a three-valent polyhedron. Lindelöf ignored the question of existence. In contrast, Minkowski [102] proved in a direct way that among all convex polyhedra with prescribed outer normal directions to its faces, the polyhedron circumscribed about a ball is the best one. This is the three-dimensional analogue to L’Huilier’s theorems presented in Section 1.4. The fact that the faces of the best 𝑛-faced polyhedron must touch its inscribed ball at their centroids follows in a simple way. The name of Minkowski is linked with the isoperimetric problem also by other important theorems. We emphasize the following inequalities of Minkowski: 𝐹 2 − 3𝑉 𝑀 ≥ 0,

𝑀 2 − 4𝜋𝐹 ≥ 0 ,

(5.7.2)

that hold for the three fundamental characteristics 𝑉, 𝐹 and 𝑀 of an arbitrary convex body. They immediately imply the inequality (5.7.1). In the notation of Section 1.1, for convex polyhedra Minkowski’s first inequality can be written as follows: 3 ∑︁ 𝐹2 ≥ 𝛼𝑙 . 𝑉 2 But the even stronger inequality ∑︁ 𝛼 𝐹2 ≥3 𝑙 tan 𝑉 2

(5.7.3)

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5 Extremal Properties of Regular Polyhedra

holds as well, where equality occurs only for a polyhedron circumscribed about a ball. An elementary proof of (5.7.3), using the inner parallel domain, was given by Bol [20]. Steinitz [127] devoted the last major work of his life to this kind of problem and settled several questions posed by Steiner. Among others, he turned to the question of existence and he gave an example of a polyhedron in whose type there is no best polyhedron. He showed further that there is a polyhedron that fails to satisfy the Steiner–Lindelöf condition even though it is the best one of its type. Thereby he showed that the answer to the above question of Steiner is, in general, negative. Steinitz also dealt with the conjecture of Steiner concerning the prism and he confirmed it for the number of sides 𝑛 = 3, but disproved it for the cases 𝑛 ≥ 8. The case of 𝑛 = 4, that is, for the cube, was later solved, confirming Steiner’s conjecture under much more general conditions. The cases 𝑛 = 5, 6 and 7 are still unsolved. Let us now quote the view of Steinitz [127, I, p. 134] on Steiner’s conjecture concerning the regular polyhedra: “There is not even a slightest idea for an approach to a proof of this conjecture, and since experience in this area suggests great restraint in pronouncing conjectures, we must regard this question as still completely unresolved, especially so concerning the dodecahedron and the icosahedron.”

Such caution appeared quite justified at that time, as indeed, essentially not a single extremal property of the regular dodecahedron or icosahedron was yet known. However, the various extremal properties of the regular solids that we know already appear to support Steiner’s conjecture. Moreover, it is to be expected that the threevalent regular polyhedra are extremal not only among isomorphic polyhedra, but even among all polyhedra with the same number of faces, and the regular triangular polyhedra—again far beyond Steiner’s conjecture—are best among all polyhedra with the same number of vertices. Our expectations concerning the regular polyhedra with three-valent vertices turn out to be correct. To verify this, we notice that by the Lindelöf–Minkowski theorem, for polyhedra with a prescribed number of faces the problem is reduced to polyhedra circumscribed about the unit ball. However, for such polyhedra, the surface area 3 𝐹 and volume 𝑉 are linked by 3𝑉 = 𝐹, that is, 𝑉𝐹 2 = 9𝐹 = 27𝑉. This way the isoperimetric problem for polyhedra of a prescribed number 𝑛 of faces is reduced to finding the 𝑛-faced polyhedron of minimum volume or surface area circumscribed a ball, and so by (5.4.2), we can state the following remarkable theorem: If a convex polyhedron of 𝑛 faces has surface area 𝐹 and volume 𝑉, then 𝐹3 ≥ 54(𝑛 − 2) tan 𝜔 𝑛 (4 sin2 𝜔 𝑛 − 1) , 𝑉2

(5.7.4)

and equality holds only for the regular three-valent polyhedra. This proves Steiner’s conjecture for the cases of the cube and the dodecahedron, under considerably more general conditions.

5.7 Isoperimetric Problems for Polyhedra

137

In contrast, the case of the icosahedron still remains unclear. However, one can conjecture that for a convex polyhedron with 𝑛 vertices the inequality √ 𝐹3 27 3 ≥ (𝑛 − 2)(3 tan2 𝜔 𝑛 − 1) (5.7.5) 2 𝑉2 holds, with equality only for the triangular regular polyhedra. It is likely that even the more general inequality   2𝜋 𝐹3 2 𝜋 2 𝜋 ≥ 9𝑒 sin tan tan − 1 , (5.7.6) 𝑝 𝑝 𝑞 𝑉2 including both (5.7.4) and (5.7.5), holds, with equality occurring only for the regular polyhedra. Here 𝑒 denotes the number of edges, 𝑝 is the average number of the sides of the faces, and 𝑞 is the average valence of the vertices. A proof of the conjecture (5.7.6) would, in a certain sense, resolve the isoperimetric problem for polyhedra. The proof of the inequality (5.7.4) depends on the Lindelöf–Minkowski theorem containing a necessary condition concerning the best 𝑛-faced polyhedron. We now try to find such a condition for polyhedra with 𝑛 vertices. It turns out that the best polyhedron with a prescribed number of vertices is a proper triangular one. Let 𝐸 𝑖 ≡ 𝐸 be a vertex of a convex polyhedron 𝑉 with 𝑛 vertices, and let 𝐸 1 , . . . , 𝐸 𝑞 be the vertices adjacent to 𝐸, ordered cyclically. In the plane of each triangle Δ𝜈 ≡ 𝐸 𝐸 𝜈 𝐸 𝜈+1 (𝜈 = 1, . . . , 𝑞, 𝐸 𝑞+1 ≡ 𝐸 1 ) draw the vector e𝜈 emanating from 𝐸, perpendicular to 𝐸 𝜈 𝐸 𝜈+1 , pointing into the interior of the triangle, and of magnitude |e𝜈 | = 𝐸 𝜈 𝐸 𝜈+1 . Then, under an infinitesimal translation of 𝐸 by the vector dr, the change of the area of Δ𝜈 is dΔ𝜈 = − 21 e𝜈 dr. Consequently, the change of the surface area 𝐹 of 𝑉 is 1 d𝐹 = − e d r, 2

e = e1 + · · · + e 𝑞 .

Observe that e ≠ 0, as if translation by the vector r moves 𝐸 into the interior of 𝑉 then d𝐹 is negative. Let 𝑉𝐸 be the convex hull of the vertices of 𝑉 different from 𝐸. Consider the locus 𝐿 of those points 𝑃 for which the volume of the convex hull 𝐻 (𝑃) of the union of 𝑉𝐸 and 𝑃 is at most 𝑉. A vertex 𝐸 𝑖 of 𝑉𝐸 is visible from 𝑃 if the segment 𝑃𝐸 𝑖 intersects 𝑉𝐸 only in 𝐸 𝑖 . The boundary of 𝐿, that is the set of points for which 𝐻 = 𝑉, is determined by the sets of vertices of 𝑉𝐸 visible from 𝑃. While 𝑃 sees the same set of vertices of 𝑉𝐸 , it moves on a plane determined by the set of visible vertices. 𝐿 is the convex polyhedron obtained as the intersection of the half-spaces bounded by these planes containing 𝑉𝐸 . The set of visible vertices changes if 𝑃 passes through a face-plane of 𝑉𝐸 , in which case 𝑃 lies on an edge or in a vertex of 𝐿. If among the faces of 𝑉 meeting in 𝐸 there is one with more than three sides, then 𝐸 lies on a face plane of 𝑉𝐸 , thus it is a singular point of 𝐿. Therefore, r can be chosen so that e d r > 0 and the line through 𝐸 parallel to r intersects 𝐿 only in 𝐸. The translation by the vector r sends 𝐸 into the a point 𝑃 lying outside of 𝐿, hence

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5 Extremal Properties of Regular Polyhedra

𝐻 (𝑃) > 𝑉. On the other hand, we have d𝐹 = − 21 e d r < 0, thus 𝐻 (𝑃) has smaller 3 surface area than 𝑉. Therefore the isoperimetric quotient 𝑉𝐹 2 of 𝑉 is not minimal. To obtain the above-mentioned condition that we are trying to find, observe now 3 2 that the best polyhedron must satisfy the equation d 𝑉𝐹 2 = 𝑉𝐹 3 (3𝑉d𝐹 − 2𝐹d𝑉) = 0. We have 1 d 𝑉 = v d r , v = v1 + · · · + v𝑞 , 3 where v𝜈 denotes the outer normal to Δ𝜈 of magnitude |v| = Δ𝜈 . Consequently, (9𝑉e + 4𝐹v)d r = 0 must hold for each d r, which implies the following: At each vertex of a best polyhedron with a prescribed number of vertices, the vectors 9 𝑉 e1 , . . . , 9 𝑉 e𝑞 , 4 𝐹 v1 , . . . , 4 𝐹 v𝑞 are in equilibrium.

5.8 A General Inequality We prove here a general inequality that is useful in various applications. It reads as follows: Let the surface 𝐹 of the unit ball be partitioned into 𝑓 ≥ 4 convex spherical polygons 𝐹1 , . . . , 𝐹 𝑓 by a net 𝑁 with 𝑣 vertices and 𝑒 edges. Let 𝑃1 , . . . , 𝑃 𝑓 be arbitrary points on 𝐹 and let 𝑎(𝑥) be a strictly decreasing function defined for 0 ≤ 𝑥 ≤ 𝜋. Then ∫ 𝑓 ∫ ∑︁ 𝑎 (𝑃𝑖 𝑃) d𝐹 ≤ 4𝑒 𝑎 ( 𝐴𝑃) d𝐹 , (5.8.1) 𝑖=1

𝐹𝑖

Δ

where d𝐹 is the surface element at the variable point 𝑃 and Δ is a right spherical triangle 𝐴𝐵𝐶 with angles 𝛼 = 𝜋2𝑒𝑓 and 𝛽 = 𝜋𝑣 2𝑒 at the vertices 𝐴 and 𝐵, respectively. Equality occurs only if 𝑁 is a regular net with faces-centers 𝑃1 , . . . , 𝑃 𝑓 . The following theorem is a special case of the above: Let 𝑃1 , . . . , 𝑃𝑛 be 𝑛 points on the unit sphere 𝐹, let 𝑑 (𝑃) = min(𝑃𝑃1 , . . . , 𝑃𝑃𝑛 ) denote the spherical distance from the variable point 𝑃 to the nearest one among the 2𝜋 points 𝑃𝑖 , let Δ¯ = 𝑄𝑅𝑆 be an equilateral spherical triangle of area Δ¯ = 𝑛−2 and let ¯ 𝑑 (𝑃) = min(𝑃𝑄, 𝑃𝑅, 𝑃𝑆) be the spherical distance from 𝑃 to the nearest vertex of this triangle. If 𝑎(𝑥) is a strictly decreasing function defined for 0 ≤ 𝑥 ≤ 𝜋, then ∫ ∫
𝑎 (𝑑 (𝑃)) d𝐹 ≤ (4𝑛 − 2) 𝑎 𝑑¯ (𝑃) d𝐹. (5.8.2) 𝐹

Δ¯

Equality occurs only if the points 𝑃1 , . . . , 𝑃𝑛 are vertices of a regular triangular net.

5.8 A General Inequality

139

The inequality (5.8.1) is the spherical analogue of the inequality (3.8.4), and its proof is analogous to the proof thereof as well. We define the function ∫ 𝜔(𝑠) = 𝑎 (𝑂𝑃) d𝐹 , 𝑠

where 𝑠 denotes a segment of a spherical cap 𝐾 centered at 𝑂 cut off by a great circle. Just as in the plane, it can be shown that, on one hand, 𝜔(𝑠) is convex for 0 ≤ 𝑠 ≤ 𝐾2 , and on the other hand, ∫ 𝑎 (𝑂𝑃) d𝐹 ≥ 𝜔(𝑡), 𝑡

where 𝑡 denotes the intersection of 𝐾 with a spherical triangle one of whose vertices is diametrically opposite to 𝑂. ∫ Since, obviously, the integral 𝐹 𝑎 (𝑃𝑃𝑖 ) d𝐹 reaches its maximum at a point inside 𝑖 𝐹𝑖 , we can assume that 𝑃𝑖 lies inside 𝐹𝑖 . Denote the vertices of 𝐹𝑖 by 𝐸 1 , . . . , 𝐸 𝑝 in a cyclic order, let 𝐾𝑖 be the circle of radius 𝐴𝐵 centered at 𝑃𝑖 , and consider the subregions 𝑡1 , . . . , 𝑡 𝑝 of 𝐾𝑖 lying outside 𝐹𝑖 , the first of which is bounded by the great circles 𝑃𝑖 𝐸 1 , 𝐸 1 𝐸 2 , 𝑃𝑖 𝐸 2 , the second one by 𝑃𝑖 𝐸 2 , 𝐸 2 𝐸 3 , 𝑃𝑖 𝐸 3 , and so on. If we omit the integrand 𝑎(𝑃𝑃𝑖 )d𝐹 under the integral sign, then we can write ∫

∫ −

= 𝐹𝑖

𝐾𝑖

𝑝 ∫ ∑︁ 𝜈=1

∫ +

, 𝐹𝑖′

𝑡𝜈

where 𝐹𝑖′ denotes the part of 𝐹𝑖 lying outside 𝐾𝑖 . If we add the corresponding equalities for 𝑖 = 1, . . . , 𝑓 and notice that the number of the regions 𝑡 𝜈 is 2𝑒, then, in view of the above remarks, we get 𝑓 ∫ ∑︁ 𝑖=1

𝐹𝑖

∫ −

= 𝑓 𝐾

2𝑒 ∫ ∑︁ 𝜈=1

𝑡𝜈

+

𝑓 ∫ ∑︁ 𝑖=1

𝐹𝑖′

∫ ≤ 𝑓

− 𝐾

2𝑒 ∑︁ 𝜈=1

𝜔(𝑡 𝜈 ) +

𝑓 ∫ ∑︁ 𝑖=1

𝐹𝑖′

! 𝑓 ∫ 2𝑒 ∑︁ 1 ∑︁ 𝑡𝜈 + , ≤ 𝑓 − 2𝑒 𝜔 ′ 2𝑒 𝜈=1 𝐾 𝑖=1 𝐹𝑖 ∫ where 𝐾 is the circle of radius 𝐴𝐵 centered at 𝐴 and 𝐾 denotes the integral ∫ 𝑎 ( 𝐴𝑃) d𝐹. 𝐾 𝑓 ∑︁ Setting 𝐹𝑖′ = 𝐹 ′, we obviously have ∫

𝑖=1

𝐹 = 𝑓𝐾 −

2𝑒 ∑︁ 𝜈=1

𝑡𝜈 + 𝐹 ′ .

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5 Extremal Properties of Regular Polyhedra

Therefore the inequality can be written as: 𝑓 ∫ ∑︁ 𝐹𝑖

𝑖=1

 ∑︁ 𝑓 ∫ 𝑓 𝐾 − 𝐹 + 𝐹′ ≤ 𝑓 − 2𝑒 𝜔 + ′ 2𝑒 𝐾 𝑖=1 𝐹𝑖   ∫ ∫ 𝑓 ∫ ∑︁ 𝑓𝐾 − 𝐹 = 𝑓 − 2𝑒 𝜔 − 2𝑘𝑒 𝑎 ( 𝐴𝑃) d𝐹 + . ′ 2𝑒 𝐾 𝑡 𝑖=1 𝐹𝑖



∫

to Here 𝑡 denotes that subregion of 𝐾 which completes the segment of area 𝑓 𝐾−𝐹 2𝑒 ′ the segment of area 𝑓 𝐾−𝐹+𝐹 . Due to the monotonicity of the function 𝑎(𝑥), the sum 2𝑒 of the last two terms of the above inequality is non-positive. Hence 𝑓 ∫ ∑︁ 𝑖=1

𝐹𝑖



∫ ≤ 𝑓

− 2𝑒 𝜔 𝐾

    ∫ 𝛼 1 𝑓𝐾 − 𝐹 𝑓𝐾 − 𝐹 = 4𝑒 − 𝜔 . 2𝑒 2𝜋 𝐾 2 2𝑒

However, in view of Δ=𝛼+𝛽−

𝛼 1 𝑓𝐾 − 𝐹 𝜋 = 𝐾− , 2 2𝜋 2 2𝑒

𝑓 𝐾−𝐹 2𝑒

is nothing else but the area of the segment cut off from 𝐾 by 𝐵𝐶. This completes the proof of the inequality (5.8.1). Equality occurs only if the circles 𝐾𝑖 cover the surface 𝐹 completely and the regions 𝑡 𝜈 are congruent sections of circles. Then 𝐹1 , . . . , 𝐹 𝑓 are congruent regular polygons, and 𝑃1 , . . . , 𝑃 𝑓 are their centers. We now give a few applications of the proved theorem that arise by specifying the function 𝑎(𝑥). In some of these applications functions that are not strictly monotonic will occur. This makes a difference only in the case of equality, which, in each specific case is resolved without much trouble. For an increasing function, the inequalities (5.8.1) and (5.8.2) hold with the opposite sign. Project a spherical region 𝐺 from the center of the sphere to the plane tangent to the sphere at the point 𝐴. Clearly, the volume of the resulting cone can be represented ∫ as an integral of the form 𝐺 𝑧( 𝐴𝑃)d𝐹, with an increasing function 𝑧(𝑥), namely 𝑧(𝑥) = 31 sec3 𝑥. Let now 𝐹1 , . . . , 𝐹 𝑓 be the central projections of the faces of a convex polyhedron containing the unit ball, with face-normals 𝑂𝑃1 , . . . , 𝑂𝑃 𝑓 , then (5.8.1) taken with the function 𝑧(𝑥) produces the estimate (5.4.1). If we consider the function ( 1 sec3 𝑥 for 0 ≤ 𝑥 ≤ 𝐴𝐵 𝑧(𝑥) = 31 3 3 sec 𝐴𝐵 for 𝐴𝐵 ≤ 𝑥, then a sharper form of (5.4.1) is produced in which the volume of the polyhedron can be replaced by the volume of its part contained in the concentric ball of radius tan 𝜋𝑝 tan 𝑞𝜋 . This is the spatial analogue of the theorem expressed in (1.3.3).

5.8 A General Inequality

141

We emphasize the following special case of this sharpened theorem: If 𝑆 denotes the intersection of an 𝑛-faced √ polyhedron circumscribed about the unit ball with the concentric ball of radius 3 tan 𝜔 𝑛 , then 𝑆 ≥ (𝑛 − 2) (3 tan2 𝜔 𝑛 − 1) sin 2𝜔 𝑛 .

(5.8.3)

As a quite special, yet interesting application of the inequality (5.8.2), we mention the following extremal property of the icosahedron: The volume of the intersection of twelve congruent balls, each of which contains the unit ball, is minimal when the surfaces of the twelve balls touch the unit ball at the vertices of a regular icosahedron. We now turn to the question of how large a portion of the sphere can be covered by 𝑛 congruent spherical caps of a given size. For convenience of expression, we now introduce the notions of the density 𝑑 and covering measure 𝛿 of a family of domains with respect to a region 𝐺. Denote by Σ the sum of the areas of the intersections of the domains with 𝐺, and let 𝜎 denote the area of the part of 𝐺 that is covered by the 𝜎 Σ and 𝛿 = 𝐺 . Our result domains. Then 𝑑 and 𝛿 are defined as the quotients 𝑑 = 𝐺 reads as follows: Let a family of 𝑛 ≥ 3 congruent spherical caps with density 𝑑 be given on the unit sphere. To estimate the covering measure 𝛿 of the family, we draw an 2𝜋 and at its vertices we place three equilateral spherical triangle Δ of area Δ = 𝑛−2 caps 𝐾1 , 𝐾2 , 𝐾3 , each congruent to those in the family. Then the density of the family {𝐾1 , 𝐾2 , 𝐾3 } with respect to Δ is 𝑑, and 𝛿 satisfies 𝛿 ≤ Δ𝑛 (𝑑) ,

(5.8.4)

where Δ𝑛 (𝑑) denotes the covering measure of {𝐾1 , 𝐾2 , 𝐾3 } with respect to Δ. This theorem corresponds to the theorem expressed in the inequality (3.8.3), and it includes the formulae (5.1.1) and (5.1.2) just as (3.8.3) implies (3.8.1) and (3.8.2). In addition, let us point out the inequality Δ𝑛 (𝑑) ≤ lim Δ𝑛 (𝑑) = Δ(𝑑) :

𝑑 ≥ 0,

𝑛 = 3, 4, . . .

𝑛→∞

which yields the exact estimate 𝛿 ≤ Δ(𝑑) for the covering measure, independent of the number of caps. Δ(𝑑) is identical to the function defined in connection with (3.8.3). It yields the covering measure of an infinite family of circles with density 𝑑, whose centers form the equilateral triangular lattice. For a proof of our theorem, notice that the fact that the density of the family {𝐾1 , 𝐾2 , 𝐾3 } with respect to Δ is 𝑑 is trivial. Namely, if we denote the sum of the 2𝜋 , we get angles of Δ by 𝛼, then in view of Δ = 𝛼 − 𝜋 = 𝑛−2 𝛼𝐾1 𝑛𝐾1 : Δ= =𝑑. 2𝜋 4𝜋

142

5 Extremal Properties of Regular Polyhedra

On the other hand, the essential part of the theorem, that is, the inequality (5.8.4), follows directly from the inequality (5.8.2) applied to the decreasing function ( 1 for 0 ≤ 𝑥 ≤ 𝑟 𝑧(𝑥) = 0 for 𝑟 < 𝑥 ≤ 𝜋 . In this case the left-hand side of the inequality (5.8.2) gives the area of the part of 𝐹 that is covered by the spherical caps of radius 𝑟 centered at the points 𝑃1 , . . . , 𝑃𝑛 . The integral on the right-hand side has a similar meaning. After dividing both sides by 𝐹 = 4𝜋, we get the desired inequality (5.8.4).

5.9 On the Shortest Net Dissecting the Sphere into Convex Parts of Equal Area The total arc length 𝐿 of a curvilinear net that dissects the unit sphere into 𝑛 ≥ 3 equiareal convex parts satisfies   2 𝐿 ≥ 6(𝑛 − 2) arccos √ cos 𝜔 𝑛 . (5.9.1) 3 Equality occurs only for the spherical net of a regular polyhedron with three-valent vertices. This theorem is a corollary of the following more general theorem: The total arc length 𝐿 of a curvilinear net that dissects the unit sphere into equiareal convex parts satisfies cos 𝐿 ≥ 2𝑒 arccos

sin

𝜋 𝑝 𝜋 𝑞

,

(5.9.2)

where 𝑒 is the number of edges of the net, 𝑝 is the average number of sides of the faces and 𝑞 is the average valence of the vertices. Equality occurs only for the spherical net of a regular polyhedron. The first theorem follows from the second one by the fact that a net with a prescribed number 𝑛 of faces can always be considered as a net with three-valent 6𝑛−12 vertices. Therefore we can set 𝑞 = 3, 𝑒 = 3𝑛 − 6 and 𝑝 = 2𝑒 𝑛 = 𝑛 . It should also be noted that the second theorem is valid for the degenerate regular polyhedra {2, 𝑞} and {𝑝, 2} as well. For the proof, we make use of the isoperimetric property of the regular spherical polygons, according to which the regular spherical 𝑝-gon is of minimum perimeter among all 𝑝-sided spherical polygons of the same area. The perimeter of the regular

5.10 On the Total Length of the Edges of a Polyhedron

143

spherical 𝑝-gon of area 𝑡 is 𝜋 𝑝 2 𝜋−𝑡 2𝑝

cos 𝑈 ( 𝑝, 𝑡) = 2𝑝 arccos

cos

.

Consequently, 2𝐿 ≥

𝑓 ∑︁

  4𝜋 , 𝑈 𝑝𝑖 , 𝑓 𝑖=1

where 𝑓 denotes the number of faces of the net and 𝑝 1 , . . . , 𝑝 𝑓 are the numbers of sides of the individual faces. One can show that, with 𝑡 > 0 fixed at an arbitrary value, 𝑈 ( 𝑝, 𝑡) is a convex function of 𝑝 for 𝑝 ≥ 2. Therefore we get   4𝜋 , 2𝐿 ≥ 𝑓 𝑈 𝑝 , 𝑓 which is equivalent to the inequality (5.9.2). The case of equality is verified without difficulty. It seems conceivable that—in certain analogy with theorem on page 79 concerning packing of circles of not too different sizes—the extremal property of the regular spherical mosaics expressed by (5.9.2) also holds under the assumption that the areas of the pieces differ only by a sufficiently small amount. However, this is not the case, as is easily seen for 𝑝 = 4, 𝑞 = 3. Although the assumed convexity of the spherical regions is essential in the above proof, the theorem itself is likely valid even without this assumption. However, in such a general setting, the problem has not been solved, not even for the case of three faces. In contrast, the case of two faces has been resolved by a—non-trivial by any means—theorem of F. Berstein [10]: Among all simple closed curves that halve the surface area of the ball, the great circle has the smallest perimeter.

5.10 On the Total Length of the Edges of a Polyhedron Considerations similar to those in the previous section and in Section 3.9 yield the following theorem: If all faces of a convex 𝑛-faced polyhedron of surface area 𝐹 are of equal area, then its total edge length 𝐿 satisfies the inequality √︁ (5.10.1) 𝐿 ≥ 6(𝑛 − 2)𝐹 tan 𝜔 𝑛 . Equality occurs only for a regular polyhedron with three-valent vertices.

◀ 226

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5 Extremal Properties of Regular Polyhedra

Combining (5.10.1) with the (equivalent to (5.4.2)) inequality 𝐹 ≥ (3𝑛 − 6) (3 tan2 𝜔 𝑛 − 1) sin 2𝜔 𝑛 concerning an 𝑛-faced polyhedron of surface area 𝐹 containing the unit ball, we infer that the total edge length 𝐿 of a convex polyhedron with 𝑛 faces of equal area and containing the unit ball satisfies √︁ 𝐿 ≥ (6𝑛 − 12) sin 𝜔 𝑛 3 tan2 𝜔 𝑛 − 1 . Denoting the bound on the right-hand side by 𝑆 𝑛 , we have the following numerical values: 𝑆4 ≈ 29.4, 𝑆5 ≈ 24.6, 𝑆6 = 24, 𝑆7 ≈ 24.3, 𝑆8 ≈ 24.5. Moreover, since lim 𝑆 𝑛 = ∞, these numerical values suggest that 𝑆 𝑛 attains its minimum at 𝑛 = 6. 𝑛→∞ To verify this claim, notice that, in view of tan 𝜔 𝑛 > tan

and sin 𝜔 𝑛 > sin

𝜋 6

+

𝜋 𝜋 1 > tan + 3𝑛 6 cos2

𝜋 6

𝜋 1 4𝜋 =√ + 3𝑛 3 9𝑛

𝜋 1 = , we get 6 2   √ 2 3𝜋 . 𝑆 𝑛 > 2(𝑛 − 2) 𝑛

Since for 𝑛 ≥ 17 the amount on the right-hand side is greater than 24, all that remains is to check the values of 𝑆 𝑛 for 𝑛 ≤ 16. If instead of the unit ball we consider a ball of a given diameter, then we can express the result obtained above as follows: If a convex polyhedron with faces of equal areas contains a ball of diameter 𝐷, then the total length 𝐿 of its edges satisfies 𝐿 ≥ 12𝐷 and equality occurs only for a cube of edge length 𝐷. 227 ▶

It can be conjectured that this theorem remains valid even without the restriction to polyhedra with faces of equal area. The proof of this general case seems to be difficult. Nevertheless, it is easy to prove the following: The total edge length 𝐿 of a convex polyhedron containing a ball of diameter 𝐷 satisfies 𝐿 > 10𝐷 . For a proof, we set 𝐷 = 2 and we pick one of the polyhedron’s faces, 𝑇, of perimeter 𝑈. If we denote by 𝜏 the central projection of 𝑇 on the ball’s surface, then the area of 𝑇 cannot be smaller than the area of the circle whose plane is tangent to the ball at its center, and whose projection on the ball’s surface is of area 𝜏. Consequently, by the isoperimetric property of the circle, 𝑈 cannot be smaller than

5.10 On the Total Length of the Edges of a Polyhedron

145

the perimeter of this circle. This fact is expressed by the inequality √︁ 𝜏(4𝜋 − 𝜏) 𝑈 > 2𝜋 . 2𝜋 − 𝜏 Naturally, equality cannot occur here, since 𝑇 cannot be a circle. We now use the fact that the curve defined by the equation √︁ 𝑥(4𝜋 − 𝑥) 𝑦 = 2𝜋 , 𝑥>0 2𝜋 − 𝑥 lies above the tangent ray emanating from the origin (Figure 5.5). Through a simple calculation, this implies that 𝑦 ≥ 3.33𝑥. This yields ∑︁ ∑︁ 2𝐿 = 𝑈 > 3.33 𝜏 = 3.33·4𝜋 > 40 , Fig. 5.5 which was to be shown. It is most interesting here that—in contrast to all previously considered theorems— neither the number of vertices nor the number of faces is prescribed, and we search among all convex polyhedra for the extremal one, that is, for the polyhedron whose 𝐿 quotient 𝐷 is as small as possible. In this sense, the cube appears to be a universally best polyhedron. We now turn to an analogous problem and prove the following. The total edge length 𝐿 of a triangular polyhedron containing a ball of diameter 𝐷 satisfies 𝐿 > 14𝐷 .

Uniqueness of the solution to the corresponding problem is not to be expected, since it is likely that both the regular tetrahedron and octahedron are best. The √ 𝐿 comparison of the constant 14 with the common value 6 6 ≈ 14.7 of 𝐷 for the tetrahedron and octahedron shows that our estimate is quite good. The proof is analogous to the previous one. If we vary a triangular face 𝑇 under the constraint that its projection’s area 𝜏 remains constant, and so that its plane does not penetrate the ball, then 𝑇 reaches its minimum when it becomes a regular triangle touching the ball at its centroid. Consequently, by the isoperimetric property of the regular triangle, neither can the perimeter 𝑈 of 𝑇 be smaller than the perimeter of this triangle. Therefore we have     2𝜋 − 𝜏 2 𝑈 ≥ 27 3 cot −1 , 6 from which, by an elementary calculation, we get 𝑈 > 4.46𝜏. Consequently,

◀ 227

146

5 Extremal Properties of Regular Polyhedra

2𝐿 =

∑︁

𝑈 > 4.46

∑︁

𝜏 = 4.46 · 4𝜋 > 56 ,

which was to be shown. We will now estimate from below the total edge length, or more generally, the sum of various powers of the edge lengths, in terms of the minimum of the diameters of the incircles of the polyhedron’s faces. the incircles of the faces of Let 𝑑 denote the diameter of the smallest circle among Í 𝑒 a convex polyhedron with edges 𝑙 1 , . . . , 𝑙 𝑒 . If we set 𝐿 𝛼 = 𝑖=1 𝑙 𝑖𝛼 , then the following inequalities hold: √ 𝐿 1 ≥ 6 3𝑑 , 𝐿 2 ≥ 12𝑑 2 , 𝐿 4.5 ≥ 30 tan4.5 36◦ 𝑑 4.5 . In the first inequality, equality occurs only for the regular tetrahedron, in the second one only for the cube, and in the third one only for the dodecahedron. The proofs of the three inequalities are analogous. It will suffice to prove only the second one. To that end, consider a face of the polyhedron, its incircle’s center 𝑂 and one of the face’s sides 𝑙. If we denote by 2𝛼 the angle at 𝑂 spanned by 𝑙, then 𝑙 ≥ 𝑑 tan 𝛼, that is, 𝑙 2 ≥ 𝑑 2 tan2 𝛼. Altogether there are 2𝑒 such inequalities. If we add them all, then the left-hand side becomes twice the sum 𝐿 2 of squares of the edges. Therefore we have ∑︁ 𝜋𝑛 , 2𝐿 2 ≥ 𝑑 2 tan2 𝛼 ≥ 𝑑 2 2𝑒 tan2 2𝑒 where 𝑛 denotes the number of the polyhedron’s faces. As 𝑒 ≤ 3𝑛 − 6, this yields 𝐿 2 ≥ 𝑑 2 (3𝑛 − 6) tan2 𝜔 𝑛 = 3𝑑 2𝑇 (𝑛) . Now, a numerical calculation shows that for 𝑛 = 4, 5, . . . , 13 we have 𝑇 (𝑛) ≥ 𝑇 (6) = 4. On the other hand, 𝑇 (𝑛) > 12 tan2 𝜋6 = 4 for all 𝑛 ≥ 14. Therefore 𝐿 2 ≥ 12𝑑 2 for an arbitrary number of faces, which completes the proof. In the third case, the reason for considering the non-integer 4.5 for the exponent is the fact that with the fourth power, the corresponding function reaches its minimum at 𝑛 = 11, and with the fifth power, at 𝑛 = 13. Consequently, for example for 𝑛 = 5 our arguments produce just the inequality 𝑑𝐿55 > 33 tan5 𝜔13 = 6.015 . . . , while for the regular dodecahedron we have 𝑑𝐿55 = 30 tan5 𝜔12 = 6.039 . . . . Nevertheless, the inequality 𝑑𝐿55 ≥ 30 tan5 36◦ is probably correct. In any case, we have a good, though not the best possible, estimate 𝐿 5 > 6𝑑 5 . Finally, we mention the inequality (𝑟 1 + · · · + 𝑟 𝑛 ) 2 ≤

𝑛2 𝐹 𝜋𝑛 cot , 2𝑒 2𝑒

where 𝑟 1 , . . . , 𝑟 𝑛 denote the inradii of the faces of an arbitrary 𝑛-faced polyhedron with 𝑒 edges and with surface area 𝐹. Equality holds here only if all the faces are congruent regular polygons, which, naturally, occurs not only for the regular

5.11 The Thinnest Saturated Packing of Spherical Caps

147

polyhedra. For convex polyhedra, this implies the inequality (𝑟 1 + · · · + 𝑟 𝑛 ) 2 ≤

𝑛2 𝐹 cot 𝜔 𝑛 , 6𝑛 − 12

in which equality occurs only for the regular three-valent polyhedra. From the last inequality one can conclude that 𝑛𝐹 (𝑟 1 + · · · + 𝑟 𝑛 ) 2 < √ . 12 This inequality is in exact analogy with (3.7.5); the proof is analogous as well.

5.11 The Thinnest Saturated Packing of Spherical Caps In Section 3.2 we investigated the thinnest saturated family of congruent circles in the plane. We saw that this problem is equivalent to the problem of the thinnest covering of the plane with congruent circles. Indeed, the estimate (3.2.5) stated there is just a re-statement of the inequality (3.2.2). In contrast, the analogous considerations in spherical geometry involve a new moment. For the problem of covering the sphere with two congruent caps we can only state the trivial fact that the density of the covering must be at least 1. On the other hand, the notion of a saturated family of congruent spherical caps lends this very case of two caps special interest. Namely, we have the following theorem: If 𝑑 is the density of an arbitrary saturated family of congruent spherical caps, then √ 2 = 0.29289 . . . 𝑑 ≥ 1− (5.11.1) 2 and equality is attained only by two antipodal spherical caps of spherical radius 𝜋4 . In other words, a saturated family of spherical caps of equal size can leave empty √ = 70.7 . . . % of the sphere’s area. at most 100 2 Since for 𝑛 = 1 and 2 caps the inequality (5.11.1) is obviously true, we restrict ourselves to the case 𝑛 > 2. If we double the radius 𝑟 of the spherical caps, the resulting family of spherical caps covers the sphere. Then, according to (5.1.2), we have 1 cos 2𝑟 = 2 cos2 𝑟 − 1 ≤ √ cot 𝜔 𝑛 . 3 Therefore the density 𝑑 of the original family of disks satisfies (    ) 𝑛 1 1 𝑛2𝜋(1 − cos 𝑟) ≥ 1− 1 + √ cot 𝜔 𝑛 = 𝑑𝑛 . (5.11.2) 𝑑= 4𝜋 2 2 3

5 Extremal Properties of Regular Polyhedra

148

However, it can be shown that the sequence 𝑑3 , 𝑑4 , . . . is monotonically decreasing 𝜋 and converges to the limit lim 𝑑 𝑛 = √ , therefore for 𝑛 ≥ 3 the density 𝑑 satisfies 𝑛→∞ 108 √ 𝜋 2 𝑑> √ > 1− . □ 2 108 It should be noted that the inequality (5.11.2) gives the exact bound for 𝑛 = 1 and, with the convention cot 𝜔2 = 0, for 𝑛 = 2, as well. In Figure 5.6 we show graphically the values of the lower bound (5.11.2) for the density 𝑑, corresponding to a few values of the number 𝑛 of caps. The solid dots represent exact bounds. We remark that our theorem expresses a deeper extremal Fig. 5.6 property of the considered antipodal pair of spherical caps than one would expect from such a simple structure of the extremal configuration. Incidentally, what makes this theorem even more √ 2 𝜋 interesting is that the constant √ is only slightly greater than 1 − 2 . 108 By means of (5.1.1) along with the above observations, the following theorem is obtained: Suppose that some family consisting of 𝑛 ≥ 1 spherical caps of radius 𝑟 is saturated, and that there is another family of 𝑁 mutually non-overlapping caps of the same radius 𝑟. Then 𝑁 ≤ 3𝑛 . (5.11.3) Equality can be reached only if 𝑟 = 𝜋4 and 𝑛 = 2, although the quotient arbitrarily close to 3 for sufficiently small values of 𝑟 and 𝑛1 .

𝑁 𝑛

can be

For the proof, notice that for 𝑛 ≠ 2 the density 𝑑 of a saturated family satisfies 𝑑 > √ 𝜋 . Consequently, the density of a family of at least 3𝑛 equally large caps is 108

𝜋 greater than √3 𝜋 = 12 . Therefore, by (5.1.1), some of the caps of this family must 108 overlap, which proves the sharper inequality√𝑁 < 3𝑛 in this case. For 𝑛 = 2 on the other hand, the density satisfies 𝑑 ≥ 1 − 22 . Therefore the density of 3 · 2 = 6  √  caps of the same size as the original pair must be at least 3 1 − 22 . The bound  √  in (5.1.1) just allows the density 3 1 − 22 = 62 (1 − 12 csc 𝜔6 ) as the density of 6 non-overlapping spherical caps, but excludes the possibility of 7 caps. Inequality (5.11.3) is therefore satisfied in any case. The fact that for 𝑛 = 2 the quotient 𝑁𝑛 can actually reach 3 can be verified directly.

5.12 Approximating a Convex Surface by Polyhedra

149

5.12 Approximating a Convex Surface by Polyhedra In this section we do not claim rigor. We consider a smooth convex surface 𝐹 along with the convex hull 𝑃𝑛 of 𝑛 points of the surface 𝐹. We choose a configuration of the 𝑛 points so that the deviation 𝜂(𝐹, 𝑃𝑛 ) reaches its minimum. In short, we consider the inscribed polyhedron with a prescribed number 𝑛 of vertices that approximates 𝐹 best. If 𝐹 is, for example, a sphere, then 𝑃12 is—as we have seen—the inscribed regular icosahedron. Obviously, as 𝑛 increases, 𝜂(𝐹, 𝑃𝑛 ) approaches zero. More precisely: the order of magnitude of 𝜂(𝐹, 𝑃𝑛 ) is 𝑛1 , so that 𝑛𝜂(𝐹, 𝑃𝑛 ) approaches the limit value 1 = lim 𝑛𝜂(𝐹, 𝑃𝑛 ) 𝐴 𝑛→∞ dependent only on the surface 𝐹. We name 𝐴 the√approximability of 𝐹 by inscribed polyhedra. For the unit ball, this number is 𝐴 = 427 𝜋 . It should be mentioned that we get the same number 𝐴 if we consider circumscribed polyhedra instead of inscribed, or polyhedra with 𝑛 faces instead of 𝑛 vertices. For arbitrary polyhedra, that is, for those that need not be inscribed or circumscribed, the approximability is 2𝐴. Approximability can be defined similarly for elliptically curved pieces of surfaces by polyhedral surfaces. We assert that the approximability 𝐴 of an elliptically curved piece of surface by polyhedral surfaces can be presented as a surface integral of the square root of the Gaussian curvature 𝐾: ∫ √ 1 1 =√ 𝐾 dF . (5.12.1) 𝐴 27 𝐹 It follows that the approximability is invariant under the so-called bendings, or intrinsic isometries. By a bending one understands a mapping from one surface to another that preserves the length of every arc lying on the surface. The claimed invariance of approximability is an immediate consequence of the famous “theorema egregium” of Gauss, that expresses the invariance of 𝐾 under bendings. Furthermore, according to Schwarz’s inequality, the following holds:  ∫ … ∫ 1 𝐹Ω 1 ≤ √ d𝐹 𝐾 d𝐹 = , 𝐴 27 𝐹 𝐹 27 ∫ where Ω = 𝐹 𝐾 d𝐹 denotes the total curvature of the surface. Equality can be reached here only for a surface of constant Gaussian » curvature, that is, for a surface 𝐹 . In particular, among all convex that can be bent into a part of a sphere of radius Ω surfaces of a given surface area, the sphere is worst approximated by polyhedra. To make the formula (5.12.1) plausible, we restrict our attention to a closed convex surface 𝐹 and we consider a small triangle Δ whose vertices lie on 𝐹. Denote by Δ′ the piece of 𝐹 lying near Δ consisting of points at which the normals to 𝐹 intersect Δ, and let Δ vary within a small neighborhood of some point 𝑃 on the surface so

150

5 Extremal Properties of Regular Polyhedra

that while keeping the deviation 𝜂(Δ, Δ′) constant, the area of Δ becomes as large as possible. We push the plane tangent to 𝐹 at 𝑃 by a parallel translation towards the surface by the distance of 𝜂 = 𝜂(Δ, Δ′) and we consider the “Dupin curve” 𝑆 of 𝐹, along which the plane intersects the surface. The desired optimal position of Δ is the triangle of maximum area inscribed in 𝑆. √︁ In√︁its first approximation, 𝑆 can be viewed as an ellipse with semi-axes 2𝜂𝑅1 and 2𝜂𝑅2 , where 𝑅1 and 𝑅2 denote the principal curvatures of 𝐹 at the point 𝑃. Consequently, √ » 27 Δ≤ 4𝜂2 𝑅1 𝑅2 + · · · , 4 where the dots stand for a series whose value is negligible compared to 𝜂. From here we get 2 √ 𝜂(Δ, Δ′) ≥ √ 𝐾Δ + · · · . 27 Thus, if we denote the faces of 𝑃𝑛 by Δ1 , · · · , Δ2𝑛−4 , the corresponding “triangles” ′ on 𝐹 by Δ1′ , · · · , Δ2𝑛−4 and the curvature of 𝐹 at a point picked on each of those “triangles” by 𝐾1 , · · · , 𝐾2𝑛−4 , then, in view of 𝜂(𝑃𝑛 , 𝐹) =

max 𝜂(Δ𝑖 , Δ𝑖′ ) ≥ 𝜂(Δ𝑖 , Δ𝑖′ ),

1≤𝑖 ≤2𝑛−4

𝑖 = 1, . . . , 2𝑛 − 4 ,

we have 2𝑛−4 2𝑛−4 2 ∑︁ √︁ 2 ∑︁ √︁ (2𝑛 − 4)𝜂(𝑃𝑛 , 𝐹) ≥ √ 𝐾 𝑖 Δ𝑖 + · · · = √ 𝐾𝑖 Δ𝑖′ + · · · . 27 𝑖=1 27 𝑖=1

This yields that for an arbitrary sequence 𝑃4 , 𝑃5 , . . . of inscribed polyhedra we have ∫ √ 1 lim inf 𝑛𝜂(𝑃𝑛 , 𝐹) ≥ √ 𝐾 d𝐹 . 27 𝐹 Our observations suggest that the bound on the right-hand side above can be attained by a suitably chosen sequence of polyhedra. For a construction of such ∫ √ a sequence, that is, of polyhedra 𝑃𝑛 with deviation 𝜂(𝐹, 𝑃𝑛 ) ∼ √ 1 𝐹 𝐾 d𝐹, 27𝑛 ∫ √ we cut 𝐹 with a plane at the depth of √ 1 𝐹 𝐾 d𝐹 and we inscribe into the 27𝑛 obtained intersection curve a triangle Δ of maximum area. We then project Δ to the surface 𝐹 along the surface’s normals, and we tile a neighborhood of the resulting triangle Δ′ in 𝐹 by almost congruent replicas of Δ′. This neighborhood is chosen so that the curvature 𝐾 remains essentially constant within it. Then we continue the tiling starting from another triangle Δ until finally the whole surface 𝐹 is tiled. The vertices of the triangles determine an inscribed polyhedron whose deviation ∫ √ from 𝐹 is approximately √ 1 𝐹 𝐾 d𝐹 and whose number of faces, in view of 27𝑛 ∫ √ √ 2 𝐾Δ′ ∼ 𝑛1 𝐹 𝐾 d𝐹, is about 2𝑛. Consequently, the number of vertices of our polyhedra is asymptotically 𝑛.

5.12 Approximating a Convex Surface by Polyhedra

151

It is worth noticing that from the formula (5.12.1) one can derive the exact estimate 𝑁 > √2𝑄 for the number 𝑁 of unit circles that can cover a square 𝑄. Namely, in the 27 case of a sphere, (5.12.1) is equivalent to the limiting case of (5.2.3) as 𝑛 → ∞. This implies (3.2.2), and thereby, via the comment on tiling domains (Section 3.4), the said estimate. Our considerations will now be based on the volume deviation instead of the Hausdorff distance. The approximation problem arising in this context is very closely related to the notion of the affine surface area. The most visual way to present the affine surface area Ω of a convex surface 𝐹 is as follows. We break the surface 𝐹 into surface elements and to each such element we fit a surface element of a unitary ellipsoid. Somewhat more precisely, each surface element of 𝐹 is replaced by an equal area element of the second-order osculating ellipsoid. If we piece together these surface elements of the ellipsoids, via volume-preserving affinities, to form a part 𝑇 of the surface of the unit ball, then Ω can be defined as the ordinary surface area of 𝑇, where, naturally, the multiply covered parts of 𝑇 are to be counted more than once, according to their multiplicity. This explanation allows us to reduce the problem of approximability of a convex surface with respect to volume-deviation to the approximability of the unit ball, in other words, to express its approximability in terms of its affine surface area. We can namely restrict our attention to a surface that is constructed by joining together finitely many, say 𝑚, fragments of unitary ellipsoids, since 𝐹 can be approximated by such a surface with an accuracy of a greater order than by a polyhedron with 𝑚 vertices or faces. From the point of view of approximating by volume, a part of a unitary ellipsoid is equivalent to a part 𝑇 of the unit sphere. The approximability of 𝑇, however, can be easily derived from the approximability of the whole sphere. To see this, one should observe that, say, the vertices of a maximum-volume 𝑛-vertex polyhedron 𝑣 𝑛 inscribed in the unit ball are distributed so that, for large 𝑛, the number 𝜈 of vertices that fall in 𝑇 is asymptotically 4𝑇𝜋 𝑛. Consequently, 𝑇 can be approximated by an inscribed polyhedral surface 𝐹𝜈 with 𝜈 vertices in such away that the volume of the space between 𝐹𝜈 and 𝑇 is asymptotically 4𝑇𝜋 43𝜋 − 𝑣 𝑛 , which, in view of (5.5.2) is asymptotically equal to lead to the following theorem:

√ 𝑇 4 3 𝜋2 4 𝜋 3𝑛

∼

𝑇2 √ 4 3𝜈

. These observations

Let 𝑉 be a convex body with affine surface area Ω. Let 𝑉𝑛 denote the circumscribed 𝑛-faced polyhedron of minimum volume and 𝑣 𝑛 a maximum volume inscribed polyhedron with 𝑛 vertices. Then 14 7 14 lim 𝑛(𝑉𝑛 − 𝑉) = lim 𝑛(𝑉 − 𝑣 𝑛 ) = √ Ω2 . (5.12.2) 5 9 18 3 ∫ √4 In view of the formula Ω = 𝐹 𝐾 d𝐹, by which the affine surface area is usually defined, the equalities in (5.12.2) are in close analogy with (5.12.1). If we combine these equalities with the “isoperimetric inequality” Ω2 ≤ 12𝜋𝑉 discovered by Blaschke [15], in which equality holds for an ellipsoid only, then we get the exact estimates lim 𝑛(𝑉𝑛 − 𝑣 𝑛 ) =

152

14𝜋 lim 𝑛(𝑉𝑛 − 𝑣 𝑛 ) = √ 𝑉, 27

5 Extremal Properties of Regular Polyhedra

5𝜋 lim 𝑛(𝑉𝑛 − 𝑉) = √ 𝑉, 27

9𝜋 lim 𝑛(𝑉 − 𝑣 𝑛 ) = √ 𝑉, 27

stating that among all convex bodies of a given volume, the ellipsoid is the worst approximated by polyhedra. Furthermore, for large values of 𝑛, the vertices of 𝑣 𝑛 , as well as the centroids of the faces of 𝑉𝑛 , are distributed so that the parts of the surface of 𝑉 that have the same affine surface area contain roughly the same number of vertices and centroids of faces, respectively. In the plane, the asymptotic distribution of vertices is uniquely determined by the analogous property. In space, the best approximating polyhedron has another interesting property: The configuration of vertices lying in a small piece of the surface is close to a lattice that is transformed into the equilateral triangular lattice by the affinity that turns the Dupin indicatrix of a point lying on the piece into a circle. Finally, we turn our attention to an unsolved problem. Let 𝐾 be the convex hull of a convex disk 𝐸 and a point lying outside the plane of 𝐸. Let 𝐹 be a piece of surface contained in 𝐾 having common boundary with 𝐸, and let 𝑆 be the convex hull of 𝐹. The problem is to estimate from above the affine area of 𝐹 in terms of the volumes of 𝐾 and 𝑆 only. In other words, we vary 𝑆 and 𝐾 at the same time keeping their volumes constant. We search for the pair 𝑆, 𝐾 such that the affine area of 𝐹 is as large as possible. By means of Steiner symmetrization it can be shown that only such bodies need to be considered that, through some affinity, become bodies of revolution. It is highly probable that the extreme surface 𝐹 is a second-degree algebraic surface.

5.13 Historical Remarks The regular polygons possess an array of long-known extremal properties. It is surprising, on the other hand, that not even a single extremal property of the regular dodecahedron or icosahedron can be found in the older literature. It is even harder to find any kind of a systematic treatment of the extremal properties of regular polyhedra. Perhaps the blame for this lies partially in the fact that all the attention in this respect was focused on the relatively difficult isoperimetric problem. The inequality (5.7.4) was discovered by Goldberg [69]. His proof is based on the inequality (5.4.2), which he obtained by the same conclusions that are presented in Section 5.4 as a first incomplete proof. Thus Goldberg’s proof could be considered valid only after the annoying gap in it has been filled. Independently from Goldberg, the author [39] obtained (5.7.4) in the very same manner. Here, however, the difficulty in the proof is clearly exposed, and the convexity of the involved function is illustrated by a diagram. The rigorous proof based on the considerations of Section 5.8, as well as the second proof given in Section 5.4, date back to the author’s works [49,57]. The additional remarks concerning the isoperimetric problem come from Goldberg’s article [70] and from the article [62] of the author.

5.13 Historical Remarks

153

The remaining results of this chapter are found spread out through the articles [32,33,34,36,39,43,44,45,46,48,49,52,53,54,57,61] of the author. The inequality (5.1.1) was found by Hadwiger at the same time as the author, and later also by Habicht and van der Waerden [73]. The topic treated here is an inexhaustible gold mine of attractive problems only the top layer of which has been touched thus far. The research can continue in two main directions. First, additional extremal problems for polyhedra with a prescribed number of vertices or faces can be considered that would lead to the regular polyhedra with triangular faces or with three-valent vertices. Second, in accordance with the nature of the problem, one can try in some cases to generalize the bounds, valid for polyhedra with 𝑛 vertices or 𝑛 faces, for the case when both the number of vertices and faces is given obtaining exact bounds characterizing all regular polyhedra. We mention for example the following interesting, yet quite hard problem: Given 𝑛 freely moving “material” points on the sphere that are electrically charged by the same amount. Find the most stable equilibrium of the points, that is, a configuration of the points at which the potential energy of the point configuration reaches its absolute minimum (compare Föppl [68]). It is not yet known whether for 𝑛 = 6, 12 and ∞ the solution is {3, 4}, {3, 5} and {3, 6}, respectively. Naturally, it makes no sense to generalize this problem in the second direction mentioned above. However, as a possible generalization of this kind, besides the already mentioned theorems and conjectures, we give another example. Denote by 𝑅1 , . . . , 𝑅𝑣 and 𝑟 1 , . . . , 𝑟 𝑓 the distances from the vertices and faces, respectively, of a convex polyhedron to an interior point 𝑂. Further, let 𝑞 1 , . . . , 𝑞 𝑣 denote the numbers of the edges emanating from the various vertices, and let 𝑝 1 , . . . , 𝑝 𝑓 denote the numbers of sides in the various faces, then it is conjectured that

◀ 258

◀ 257

𝜋 𝜋 𝐴(𝑅; 𝑞) ≥ tan tan . 𝐻 (𝑟; 𝑝) 𝑝 𝑞 This would unify (5.3.5) and (5.3.6) into a more general theorem. The distances from the edges to 𝑂 could also be considered, raising additional problems. Let us now consider the following inequalities for the surface area 𝐹 of a polyhedron, analogous to those mentioned at the beginning of Section 5.6:     2𝜋 2𝜋 2 𝜋 2 𝜋 2 2 𝜋 2 𝜋 tan tan − 1 𝑟 ≤ 𝐹 ≤ 𝑒 sin 1 − cot cot 𝑅2 . 𝑒 sin 𝑝 𝑝 𝑞 𝑝 𝑝 𝑞 The first inequality is equivalent to (5.4.1). In contrast, in our attempt to prove the second one, we encounter difficulties. Here we prove this inequality under the condition that the feet of the perpendiculars drawn from the center of the circumsphere to the planes of the faces and lines of the edges fall on the corresponding faces and edges. The proof is the dual counterpart to the second proof of (5.4.1) given in Section 5.4. The notation will be the same as there with the difference that we let the circumsphere of radius 1 play the role that the insphere played before. We have now 𝐴𝐵 ≤ sin 𝑐, and therefore

◀ 229

154

5 Extremal Properties of Regular Polyhedra

Δ≤

1 1 sin 2𝛼 sin2 𝑐 = sin 2𝛼(1 − cot2 𝛼 cot2 𝛽) 4 4  𝜋 𝜋 − 𝛼, − 𝛽 . = 𝐹¯ (𝛼, 𝛽) = −𝐹 2 2

Since by 2 cot4 𝛼 2 [1 − (cos2 𝛼 + cos2 𝛽)] 2 ≥ 0, 𝐹¯ 𝛼𝛼 𝐹¯𝛽𝛽 − 𝐹¯ 𝛼𝛽 = sin6 𝛽 the function 𝐹¯ (𝛼, 𝛽) is concave for 0 ≤ 𝛼 < 𝐹=

225 ▶

∑︁

𝜋 2,

0≤𝛽
2, in addition to the quantity 𝑎 𝑛 , we introduce the angle 𝛼𝑛 of the equilateral spherical triangle of side length 𝑎 𝑛 . These numbers are linked by the relationship cos 𝑎 𝑛 = cot 𝛼𝑛 cot

𝛼𝑛 cos 𝛼𝑛 = . 2 1 − cos 𝛼𝑛

The bound

𝑛 𝜋 , 𝑛−2 3 which is equivalent to (5.1.1), gives the exact values of 𝛼𝑛 and 𝑎 𝑛 , for 𝑛 = 3, 4, 6 and 12. We now turn to the case 𝑛 = 5 and we show that 𝑎 5 = 𝑎 6 = 90◦ . 𝛼𝑛 ≤

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_6

157

158

6 Irregular Packing on the Sphere

We denote the points by 𝐴, 𝐵, 𝐶, 𝐷 and 𝐸 and we assume that the smallest distance between them is greater than 90◦ . Since the circle of radius 90◦ centered at 𝐸 does not contain any of the points 𝐴, 𝐵, 𝐶, 𝐷, these points lie in an open hemisphere. For the same reason, the points 𝐴, 𝐵, 𝐶 lie in an open quarter of the sphere, and the points 𝐴, 𝐵 in an open octant. But this contradicts the assumption that 𝐴𝐵 > 90◦ . In what follows, we restrict our attention to the case 𝑛 > 6. Then 𝑎 𝑛 ≤ 90◦ . Given 𝑛 points on the unit sphere with the smallest distance 𝑎 ≤ 𝑎 𝑛 between them, we join the pairs of points whose distance is exactly 𝑎 with an arc of a great circle of length 𝑎. A graph is thereby created. The number of joining arcs, called edges of the graph, that meet at a point is called the valence of the point. A point of valence zero is said to be isolated. We treat isolated points as part of the graph. The edges of the graph do not intersect each other except possibly in a common end point. Indeed, if two edges 𝐴𝐵 and 𝐶𝐷 had a point 𝑆 in common which is not a common endpoint of them, then we would have 2𝑎 ≤ 𝐴𝐶 + 𝐵𝐷 < ( 𝐴𝑆 + 𝑆𝐶) + (𝐵𝑆 + 𝑆𝐷) = 𝐴𝐵 + 𝐶𝐷 = 2𝑎, which is a contradiction. We consider a connected subset 𝑇 of a graph with, say, 𝑚 points and 𝑚 edges, in which exactly two edges meet at each point. Since the edges of 𝑇 do not intersect in interior points, 𝑇 is a simple closed polygonal chain that separates the sphere into two regions. If one of the regions does not contain any edge of the graph, then we call the region an 𝑚-gon of the graph. If all edges of the graph that meet at a point 𝑃 are contained in an angle of 180◦ , then by a small displacement of 𝑃 one can achieve that 𝑃 in its new position will be at a distance greater than 𝑎 from all the other points. Namely, if 𝑃 is shifted in a direction perpendicular to the boundary of the 180◦ angle that contains all the edges 𝑃 𝐴, 𝑃𝐵, . . . meeting at 𝑃, moving it outside the angle, then, obviously, all the distances 𝑃 𝐴, 𝑃𝐵, . . . will increase. We call this process a push-away of 𝑃. A graph in which no point can be pushed away is called irreducible. Except for possible isolated points, an irreducible graph contains only points of valence at least three. The polygons of an irreducible graph are convex, since a concave vertex could be pushed away. We now consider a maximal graph, defined as a graph of 𝑛 points with the minimum distance equal to 𝑎 𝑛 . We assert: The maximal graphs of 7 and 8 points are irreducible, contain no isolated points, and contain no polygons other than triangles or quadrangles. If a maximal graph is reducible, then we can change it by repeated push-aways into an irreducible yet still maximal graph. We will show that no maximal graph of 7 or 8 points can contain an isolated point. This will prove that the original graph must be irreducible and free from isolated points. Consider the family of 8 vertices of the Archimedean antiprism (3, 3, 3, 4) inscribed in the unit ball (Figure 6.1). An elementary calculation shows that the smallest spherical distance of this family of points is √ 8−1 ≈ 74◦ 51′31′′ . arccos 7

6.2 The Maximal Configuration for 𝑛 = 7

159

Therefore 𝑎 7 ≥ 𝑎 8 > 74◦ . We now use a famous theorem, stated by Archimedes as an axiom, and valid in spherical geometry as well: If a convex region of perimeter 𝐿 contains another convex region of perimeter 𝑙, then 𝐿 ≥ 𝑙, and equality occurs only if the two regions are identical. Therefore for the perimeter 𝑚𝑎 7 and 𝑚𝑎 8 of an 𝑚-gon in an irreducible graph of 7 and 8 points, respectively, we get 𝑚74◦ < 𝑚𝑎 8 ≤ 𝑚𝑎 7 < 360◦ , from which 𝑚 < 5 follows. Fig. 6.1 Now, an isolated point of an irreducible graph can lie only inside a polygon of the graph. Since in a triangle or quadrangle with equal sides of length 𝑎 the distance from an interior point to one of the vertices must be smaller than 𝑎, there can be no isolated point in an irreducible graph containing only triangles or quadrangles. Thereby our assertion is proved.

6.2 The Maximal Configuration for 𝒏 = 7 Consider an equilateral spherical triangle 𝐴𝐵𝐶 of edge length 𝑎. Let the centroid of the triangle be the sphere’s south pole. To the three sides attach three additional equilateral triangles. The tips 𝑃, 𝑄, 𝑅 of these triangles are equally distant from the north pole 𝑁. If 𝑎 is small, their distance from 𝑁 is greater than 𝑎, but for 𝑎 = 𝑎 4 they all coincide with 𝑁. Therefore there is a value of 𝑎 for which 𝑁 𝑃 = 𝑁𝑄 = 𝑁 𝑅 = 𝑎 (Figure 6.2). Then the quadrilateral 𝐴𝑃𝑁 𝑅 is a spherical rhombus. Therefore its angle at 𝐴 is equal to its angle at 𝑁, namely 120◦ . Moreover, for the Fig. 6.2 angle 𝛼 of an equilateral triangle of side length 𝑎 we ◦ ◦ ◦ get 120 + 3𝛼 = 360 , that is, 𝛼 = 80 . This configuration shows that 𝛼7 ≥ 80◦ . We prove that 𝛼7 = 80◦ by showing that the graph of the considered points 𝐴, 𝐵, 𝐶, 𝑃, 𝑄, 𝑅, 𝑁 is identical to the maximal graph of 7 points. As we have seen, a candidate for a maximal graph of 7 points must be an irreducible graph without isolated points and containing no polygons other than triangles or quadrilaterals. We furthermore assert that in a maximal graph of 7 points only points of valence 3 or 4 can occur. Namely, all angles occurring in the graph must be greater than or equal to 𝛼7 , for otherwise the shortest distance between the points would be smaller than 𝑎 7 . Therefore, in view of 𝛼7 ≥ 80◦ > 15 360◦ , at most 4 edges can meet at a point, just as we asserted.

160

6 Irregular Packing on the Sphere

On the other hand, not every point can be of valence 3, since otherwise the number of edges would be 21 × 3 × 7. Hence the maximal graph has at least one point of valence 4. Let us say 𝐴 is connected to the points 𝐵, 𝐶, 𝑅, 𝑃, in this order. The graph has two more points, 𝑁 and 𝑄. Since every point’s valence is at least 3, each of 𝑁 and 𝑄 is connected with at least two of the points 𝐵, 𝐶, 𝑅, 𝑃. However, each of them can only be connected with a pair of consecutive points 𝐵, 𝐶 or 𝐶, 𝑅 or 𝑅, 𝑃 or 𝑃, 𝐵. For if, say, 𝑁 were connected with 𝐵 and 𝑅, then 𝐴𝐵𝑁 𝑅 would be a quadrilateral chain of edges separating the sphere into two regions. Then each of the regions would contain one of the points 𝑃 and 𝐶, which is obviously impossible. Let then 𝑁 be connected with 𝑃 and 𝑅. Besides 𝑃 and 𝑅, 𝑁 can only be connected with 𝑄. This connection must occur, since the valence of 𝑁 is at least 3. This leaves one of the pairs 𝐵, 𝐶 or 𝐶, 𝑅 or 𝑃, 𝐵 as the only possibilities for being connected with 𝑄. But if 𝑄 were connected with, say, 𝐶 and 𝑅, then 𝐵 would have to be connected with 𝐶 and 𝑃. Then 𝑁𝑄𝐶𝐵𝑃 would form a pentagon without any diagonal connections. The graph would therefore contain a pentagon, contrary to our previous statement. For the same reason 𝑄 cannot be connected with 𝑃 and 𝐵. This leaves the connections of 𝑄 with 𝐵 and 𝐶 as the only possibility. In addition, to avoid creating a pentagon, 𝐵 must be connected with 𝑃, and 𝐶 with 𝑅. The graph still has a degree of freedom since the Fig. 6.3 angles of the two rhombi adjacent to 𝐴 have not yet been fixed. One of these two angles could, in the extreme case, be equal to 𝛼7 , which would create a connection between 𝐵 and 𝐶 or between 𝑃 and 𝑅. Let us think of the segments 𝐴𝐵, 𝐴𝐶, 𝐴𝑅, 𝐴𝑃, 𝑁 𝑃, 𝑃𝐵, 𝐵𝑄, 𝑄𝐶, 𝐶 𝑅, 𝑅𝑁 as rods of length 𝑎 = 𝑎 7 , connected to each other at their common endpoints with hinged joints (Figure 6.3). This rod model is flexible. We claim that the diagonal  𝐴𝑄 = 𝑦 is a concave function of the angle 𝐵𝐴𝐶 = 𝛽. Indeed, since 𝑦 = 2 arctan tan 𝑎 cos 𝛽2 , we have 𝛽 d𝑦 1 tan 𝑎 sin 2 =− , 0 < 𝛽 < 𝜋. d𝛽 2 1 + tan2 𝑎 cos2 𝛽 2

The derivative is a decreasing function of 𝛽, confirming our claim. Likewise, the diagonal 𝐴𝑁 is a concave function of the angle 𝑃 𝐴𝑅. Since that angle is equal to 360◦ − 2𝛼7 − 𝛽, 𝐴𝑁 is a concave function of 𝛽 as well. Therefore the sum 𝑄 𝐴 + 𝐴𝑁 is also a concave function of 𝛽. A concave function attains its minimum at an end of the interval on which it is defined. Now if the angles at 𝐴 in each of the two rhombi were greater than 𝑎 7 , then the sum 𝑄 𝐴 + 𝐴𝑁 could be decreased, which would delete the connection 𝑁𝑄 without which the graph is not maximal. If, say, 𝛽 = 𝛼7 , then a connection between 𝐵 and 𝐶 appears. However, the graph created this way is identical to the graph characterized by 𝛼 = 80◦ considered above.

161

6.3 The Maximal Configuration for 𝑛 = 8 and 9

We express the obtained result in the following theorem: The smallest spherical distance between 7 arbitrary points on the unit sphere is at most arccos(cot 80◦ · cot 40◦ ) ≈ 77◦ 52′10′′. Equality occurs only in a configuration whose graph contains three rhombi adjacent at one point and four triangles.

6.3 The Maximal Configuration for 𝒏 = 8 and 9 We now derive a general inequality for graphs composed of triangles and quadrilaterals. To that end we consider an isosceles spherical triangle 𝐴𝐵𝐶 with legs 𝐴𝐵 = 𝐵𝐶 = 𝑎. Let 𝛽 be the angle at 𝐵. We keep 𝑎 fixed and we consider the area   𝛽 Δ(𝛽) = 𝛽 − 2 arctan cos 𝑎 tan , 0 < 𝛽 < 𝜋, 2 of the triangle as a function of 𝛽. Since the derivative Δ′ (𝛽) = 1 −

cos 𝑎 1 − sin2 𝑎 sin2

𝛽 2

is decreasing, the function Δ(𝛽) is concave. Now, let 𝑉 (𝛽) be a quadrilateral of a graph, with edge length 𝑎 and with an angle 𝛽. If 𝛼 denotes the angle of an equilateral triangle Δ(𝛼) = Δ, then 𝛼 ≤ 𝛽 ≤ 2𝛼. If 𝛽 takes any of the two extreme values 𝛼 and 2𝛼, then 𝑉 (𝛽) splits into two triangles. As a concave function, 𝑉 (𝛽) = 2Δ(𝛽) attains its minimum in this case: 𝑉 (𝛽) ≥ 𝑉 (𝛼) = 𝑉 (2𝛼) = 2Δ. Consider now an irreducible graph without isolated points and containing triangles and quadrilaterals only. Let 𝑑 and 𝑣 be the number of triangles and quadrilaterals, respectively, adjacent to a point 𝑃. Further, if we denote the angles of the quadrilaterals at 𝑃 by 𝛽1 , . . . , 𝛽𝑣 , then the total area of the polygons adjacent to 𝑃 is 𝑆 = 𝑑Δ + 𝑉 (𝛽1 ) + · · · + 𝑉 (𝛽𝑣 ) . If we now keep the sum 𝛽1 + 𝛽2 = 𝑠 fixed, then, besides the function 𝑉 (𝛽1 ), also the function 𝑉 (𝛽2 ) = 𝑉 (𝑠 − 𝛽1 ) is concave, and therefore 𝑉 (𝛽1 ) + 𝑉 (𝛽2 ) is a concave function of 𝛽1 as well. Therefore this function reaches its minimum when 𝛽1 takes on one of the extreme values allowed by the conditions 𝛼 ≤ 𝛽1 , 𝛽2 ≤ 2𝛼; 𝛽1 + 𝛽2 = 𝑠. Then one of the angles, 𝛽1 or 𝛽2 , is equal to 𝛼 or 2𝛼, therefore one of the quadrilaterals, 𝑉 (𝛽1 ) or 𝑉 (𝛽2 ), say 𝑉 (𝛽1 ), splits into two triangles. If we now keep 𝛽2 + 𝛽3 fixed, then, again, at the minimum of 𝑉 (𝛽2 ) +𝑉 (𝛽3 ), one of the quadrilaterals 𝑉 (𝛽2 ) or 𝑉 (𝛽3 ) splits into triangles, and so on. Therefore at the minimum of 𝑆, each of the quadrilaterals, except possibly one, splits into two triangles. If at that minimum we denote by 𝛽0 the quadrilateral’s angle that is not equal to 𝛼 or 2𝛼, then we finally get the inequality

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6 Irregular Packing on the Sphere

𝑆 ≥ (𝑑 + 2𝑣)Δ + 2𝑈 (𝛽0 ) , where 𝑈 (𝛽) = Δ(𝛽) − Δ denotes the surplus of the area of Δ(𝛽) over the area of the equilateral triangle Δ(𝛼) = Δ. The angle 𝛽0 can also be defined in this way: Subtract from 360◦ a multiple of 𝛼 for which the remaining angle 𝛽0 is between 𝛼 and 2𝛼. This makes it clear that 𝛽0 does not depend on the numbers 𝑑 and 𝑣 of the triangles and quadrilaterals meeting at the corresponding point, and therefore the value of 𝛽0 is the same at every vertex of the graph. By adding the above inequalities for all vertices of the graph we get ∑︁ ∑︁ (𝑑 + 2𝑣) + 2𝑛𝑈 = (3 𝑓3 + 8 𝑓4 )Δ + 2𝑛𝑈 , 𝑆 = 3𝑆3 + 4𝑆4 ≥ Δ where 𝑆3 and 𝑓3 denote the total area and the number of the triangles, respectively; 𝑆4 and 𝑓4 are the corresponding quantities for the quadrilaterals, and 𝑛 denotes the number of vertices of the graph. By adding 𝑆3 = 𝑓3 Δ to both sides and then dividing by 4 one gets 1 4𝜋 = 𝑆3 + 𝑆4 ≥ ( 𝑓3 + 2 𝑓4 ) Δ + 𝑛 𝑈 . 2 Now, by Euler’s formula for polyhedra, if 𝑒 denotes the number of edges and 𝑓 the number of faces, we have 𝑛−2= 𝑒− 𝑓 =

1 1 (3 𝑓3 + 4 𝑓4 ) − ( 𝑓3 + 𝑓4 ) = ( 𝑓3 + 2 𝑓4 ) . 2 2

Therefore our last inequality above can be written as 1 2(𝑛 − 2)Δ + 𝑛𝑈 (𝛽0 ) ≤ 4𝜋 . 2

(6.3.1)

Thereby we have produced the inequality promised at the beginning of this section. For graphs consisting of triangles and quadrilaterals only, this inequality strengthens the estimate 2(𝑛 − 2)Δ ≤ 4𝜋, which is equivalent to (5.1.1). Equality in (6.3.1) holds when either only triangles occur in the graph, or exactly one quadrilateral is adjacent to each vertex. In the first case, which occurs only for 𝑛 = 3, 4, 6 or 12, the graph is the net of a triangular regular polyhedron. If quadrilaterals occur too, then all angles of Fig. 6.4 the quadrilaterals are equal to 𝛽0 , and therefore each of them is regular. In that case the graph coincides with the spherical net of an Archimedean polyhedron every vertex of which is surrounded by triangles and one quadrilateral only. There are two such (nondegenerate) polyhedra, namely the earlier considered antiprism (3, 3, 3, 4) with 8 vertices and the polyhedron

163

6.3 The Maximal Configuration for 𝑛 = 8 and 9

of the symbol (3, 3, 3, 3, 4) with 24 vertices (Figure 6.4). Thus equality occurs in (6.3.1), besides for 𝑛 = 3, 4, 6 or 12, only for 𝑛 = 8 and 24. We now consider the maximal graph with 𝑛 = 8 vertices. As we have already seen, the graph cannot have any isolated points or any polygons other than triangles and quadrilaterals. Therefore the inequality (6.3.1) can be applied: 12Δ + 4𝑈 (𝛽0 ) ≤ 4𝜋 , that is, 2Δ + Δ(𝛽0 ) ≤ 𝜋 . We wish to express the left-hand side of this inequality in terms of 𝛼 = 𝛼8 only. Since 72◦ = 𝛼12 < 𝛼8 < 𝛼6 = 90◦ , we have 𝛽0 = 360◦ − 3𝛼. Therefore   2𝜋 − 3𝛼 2 (3 𝛼 − 𝜋) + 2 𝜋 − 3 𝛼 − 2 arctan cos 𝑎 tan ≤ 𝜋; 2 cos 𝑎 = cot 𝛼 cot

𝛼 , 2

that is, 

 𝛼 3𝛼 − 𝜋 3 𝛼 − 2 arctan cot cot 𝛼 cot ≤ 𝜋. 2 2

(6.3.2)

The left-hand side is an increasing function of 𝛼 for 72◦ < 𝛼 < 90◦ . Therefore the inequality (6.3.2) gives an upper bound on 𝛼 that can be reached only by the graph of the 8 vertices of the semiregular solid (3, 3, 3, 4). This shows that the vertices of this solid constitute the best configuration of 8 points: The smallest spherical distance between 8 points on the unit sphere is always √ ◦ ′ ′′ smaller than or equal to arccos 8−1 7 ≈ 74 51 31 . Equality occurs only if the points are the vertices of the semiregular solid bounded by 2 squares and 8 equilateral triangles. Besides the above case, the maximal configuration is known for 𝑛 = 9. It can be described as follows. Let 𝐴𝐵𝐶 be an equilateral triangle inscribed in the equator. On each of the great-circle arcs 𝐴𝐵, 𝐵𝐶, 𝐶 𝐴 we place a rhombus of the same edge length 𝑎 so that the arc is its diagonal. We label the rhombi 𝐴𝐶 ′ 𝐵𝐶 ′′, 𝐵𝐴 ′𝐶 𝐴 ′′, 𝐶 𝐵 ′ 𝐴𝐵 ′′ so that each of the triangles 𝐴 ′ 𝐵 ′𝐶 ′ and 𝐴 ′′ 𝐵 ′′𝐶 ′′ is inscribed in a latitudinal circle (Figure 6.5). If we let 𝑎 continuously increase from 60◦ to 90◦ , then the side length of the triangles 𝐴 ′ 𝐵 ′𝐶 ′ and 𝐴 ′′ 𝐵 ′′𝐶 ′′ continuously decreases from 120◦ to 0◦ . Therefore there is a value

Fig. 6.5

164

6 Irregular Packing on the Sphere

of 𝑎 for which the edge length is exactly 𝑎. At that very moment the triangles 𝐴𝐵𝐶, 𝐴 ′ 𝐵 ′𝐶 ′ and 𝐴 ′′ 𝐵 ′′𝐶 ′′ represent the best possible arrangement of 9 points. The graph of this configuration contains 3 rhombi and 8 triangles. All points are of valence 4. The edge length 𝑎 of the considered graph satisfies the equation cos 𝑎 = 31 . Since this graph is identical to the maximal graph of 9 points, we have: Among any 9 points on the unit sphere a pair of points with distance smaller than or equal to arccos 13 ≈ 70◦ 31′44′′ can always be found, and this constant cannot be replaced by a smaller one. The proof of this theorem is also based on the investigation of the graph associated with the point configuration. For the proof, besides additional lemmata, a series of case distinctions is needed whose number is increased by the fact that the occurrence of pentagons as well as isolated points cannot be excluded as easily as in the case of 7 or 8 points. We omit the proof, which can be found in the article [123] of Schütte and van der Waerden.

6.4 Some Configurations of More Than 9 Points

232 ▶

232 ▶ 233 ▶

Except for 𝑛 = 12, no extremal configuration of 𝑛 > 9 points is known. Therefore, for the time being, we have to be satisfied with conjectures. We present here the likely maximal configurations of 𝑛 points for a few values of 𝑛, as conjectured by Schütte and van der Waerden. In any case, these configurations provide lower bounds for the unknown values of 𝑎 𝑛 . 𝑛 = 10. Attach to both sides of a (spherical) segment 𝐴𝐵 of length 𝑎 a regular quadrilateral, 𝐴𝐵𝐶𝐷 and 𝐴𝐵𝐸 𝐹. Then, to the segments 𝐶𝐷 and 𝐸 𝐹 attach equilateral triangles 𝐶𝐷𝐺 and 𝐸 𝐹 𝐼, and complete the configuration by the two rhombi 𝐴𝐷𝐻𝐹 and 𝐵𝐸 𝐽𝐶 (Figure 6.6). Obviously, for a specific choice of the value of 𝑎, the side length of the rhombus 𝐺𝐻𝐼𝐽 will coincide with 𝑎 itself. Then a graph with 6 triangles and 5 quadrilaterals arises, that is likely the maxFig. 6.6 imal graph of 10 points. 𝑛 = 11 and 12. It is likely that, in analogy with the equality 𝑎 5 = 𝑎 6 , we have 𝑎 11 = 𝑎 12 . 𝑛 = 13. A relatively good arrangement of 13 points is obtained by placing one point at the north pole and arranging the remaining points zonally in quadruples, in a quite symmetric way, and with the greatest possible distances between them. The graph contains one regular quadrilateral, 4 triangles and 8 rhombi, four of which meet at the north pole (Figure 6.7).

165

6.4 Some Configurations of More Than 9 Points

This configuration and (5.1.1) give the estimate 57◦ 8′ < 𝑎 13 < 60◦ 56′ . From a point of view to be discussed later, it is also worth mentioning the result 𝑎 13 < 60◦ proved recently by Schütte and van der Waerden, [124] that strengthens (5.1.1) in this special case. 𝑛 = 14. Let 𝐴𝐵𝐶𝐷𝐸 𝐹 be an equilateral spherical hexagon of side length 𝑎. The points 𝐴, 𝐵, 𝐷, 𝐸 should be at the distance 𝑎 away from the north pole 𝑁, so that the diagonals 𝐴𝐷 and 𝐵𝐸 partition the hexagon into two equilateral triangles 𝑁 𝐴𝐵 and 𝑁 𝐷𝐸 and two rhombi 𝑁 𝐵𝐶𝐷 and 𝑁 𝐸 𝐹 𝐴. We consider a hexagon 𝐴 ′ . . . 𝐹 ′ congruent to 𝐴 . . . 𝐹 centered at the south pole 𝑆, placed so that the Fig. 6.7 triangles 𝐴𝐹 ′ 𝐵, 𝐸 ′𝐶𝐷 ′, 𝐷𝐶 ′ 𝐸, 𝐵 ′ 𝐹 𝐴 ′ are isosceles. At a specific value of 𝑎 each of these triangles is equilateral. Then the vertices of these hexagons together with 𝑁 and 𝑆 make a relatively good arrangement of 14 points (Figure 6.8). The graph contains 8 triangles and 8 rhombi in two groups of 4. All points are of valence 4.

Fig. 6.8

Fig. 6.9

𝑛 = 15. A good arrangement is obtained by grouping the points in 5 zones, 3 points in each zone, so that the points in the same zone form a regular triangle inscribed in a latitudinal circle (Figure 6.9). The minimum distance occurs between the points in the 1st (north most) zone, as well as between those in the 5th (south most) zone. The 2nd and the 4th zones are to be chosen so that each point in them is at a distance 𝑎 away from two points in the 1st zone, respectively the 5th zone. Now, to make the minimum distance as large as possible, for the 3rd zone we do not choose the equator, but another latitudinal circle. The northern points (the 1st and 2nd zones) should be turned about the polar axis with respect to the corresponding southern points (the 4th and 5th zones) to arrive at the position in which the points in the 3rd zone can be placed so that each of the points in the 3rd zone is at the same

◀ 233

166

233 ▶

6 Irregular Packing on the Sphere

minimum distance 𝑎 from one point in each of the 2nd, the 4th and the 5th zone. The corresponding graph contains 12 triangles, 3 quadrilaterals and 3 pentagons. The points in the 5th zone have valence 5, the points in the 3rd zone have valence 3, and all remaining points have valence 4. 𝑛 = 16. A zonal arrangement of 4 points per zone where all points are of valence 4 appears to be the best one. In this configuration there are 8 triangles, 8 congruent rhombi and 2 regular quadrilaterals (Figure 6.10). 𝑛 = 24. A particularly good arrangement is given by the set of vertices of the regular Archimedean solid (3, 3, 3, 3, 4). 𝑛 = 32. In the spherical net of the icosahedron (or the dodecahedron) the vertices along with the centroids of the triangles (or the pentagons) form a quite good arrangement of 32 points. The graph of this configuration of points contains 30 rhombi, 12 points of valence 4 and 20 points of valence 3. The graph is identical with the spherical net of the semiregular body bounded by 30 rhombi of equal area. This polyhedron is the dual one to Fig. 6.10 the semiregular body (3, 5, 3, 5) with congruent vertex figures.

6.5 A Survey Table The table shown here lists the approximate values of 𝑎 𝑛 , or the lower and upper bounds for 𝑎 𝑛 obtained, on one hand, from the graphs discussed above and, on the other hand, from the estimate (5.1.1). The corresponding values in the next column refer to the radius 𝑅 of the smallest sphere on which 𝑛 points can be placed so that the smallest (Euclidean) distance between them is 1, and the last column shows the data for the density 𝐷 𝑛 of the densest packing of 𝑛 congruent spherical caps. The quantities 𝑅𝑛 and 𝐷 𝑛 depend on 𝑎 𝑛 as follows: 1 , 2 − 2 cos 𝑎 𝑛 𝑛 𝑎𝑛  𝐷𝑛 = 1 − cos . 2 2 𝑅𝑛 =

6.5 A Survey Table

𝑛 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 24 32

167

𝑎𝑛 180◦ 120◦ 109◦ 28′16′′ 90◦ 90◦ 77◦ 51′58′′ 74◦ 52′10′′ 70◦ 31′44′′ 66◦ 19′ 69◦ 33′42′′ 63◦ 26′06′′ 66◦ 17′23′′ 63◦ 26′06′′ 57◦ 08′ 60◦ 55′11′′ 55◦ 40′ 58◦ 40′51′′ 53◦ 39′ 56◦ 40′01′′ 52◦ 14′ 54◦ 51′19′′ 43◦ 41′ 44◦ 42′52′′ 37◦ 22′ 38◦ 41′31′′

𝑅𝑛 0.5 0.577 0.612 0.707 0.707 0.795 0.822 0.866 0.916 0.877 0.951 0.914 0.951 1.045 0.986 1.070 1.020 1.097 1.054 1.135 1.086 1.343 1.315 1.560 1.509

𝐷𝑛 1 0.75 0.845 0.732 0.878 0.777 0.823 0.825 0.812 0.893 0.802 0.895 0.896 0.791 0.897 0.810 0.898 0.808 0.898 0.816 0.899 0.861 0.901 0.843 0.903

The lower bounds for 𝑎 17 to 𝑎 23 and 𝑎 25 to 𝑎 31 in Figure 6.11, as well as the bounds for 𝐷 17 to 𝐷 23 and 𝐷 25 to 𝐷 31 in Figure 6.12, come from those circle packings that arise from the conjectured best packings of 24 and 32 circles, respectively by omitting the corresponding number of circles. Therefore, for example, 𝐷 17 is likely to be considerably higher than the presented bound. It is conjectured that for every 𝑛 (= 1, 2, 3 . . .), √ 5 (2 − 2) 4 = 0.732 . . . .

𝐷𝑛 ≥ 𝐷5 =

(6.5.1)

◀ 236

6 Irregular Packing on the Sphere

168

Fig. 6.11

Fig. 6.12

6.6 Historical Remarks

169

6.6 Historical Remarks All the results of this chapter are due to Schütte and van der Waerden. It can be considered as a main result of their article [123] that, apparently for the first time, an extremal property of an Archimedean solid has been recognized. This result, as well as the conjecture concerning the extremal property of the solid (3, 3, 3, 3, 4), supported by the inequality (6.3.1), may help to inspire the discovery of other extremal properties of the semi-regular solids. The proof in Section 6.2 contains a simplification of the argument in the original article [123] of Schütte and van der Waerden, suggested to the author by van der Waerden. The extremal property of the solid (3, 3, 3, 4) was previously expressed as a a relatively good conjecture by Rutishauser [117]. In addition, Rutishauser gave √ 33−3 arrangement of 20 points with the smallest distance arccos 4 ≈ 46◦ 41′. The points are distributed symmetrically with respect to the plane of the equator, namely 3 · 6 points in the corners of three regular hexagons, one inscribed in the equator and two in latitudinal circles, plus one point at each of the poles. The corresponding graph contains 6 rhombi lying in the equatorial zone, 12 triangles, and 2 regular hexagons inside each of which lies one isolated point. However, this second solid of Rutishauser is not extremal. In fact, van der Waerden [136] showed a better arrangement of 20 points, in which the smallest distance is approximately 47◦ 25′. Investigating the problem of the densest packing of balls one encounters the question of how many unit balls can be attached to a unit ball. The question of whether at most 12 or 13 balls can be attached was the subject of a dispute between Newton and Gregory. This question was settled by Schütte and van der Waerden [124] by proving the inequality 𝑎 13 < 60◦ : the number in question is 12. In other words: at most 12 points can be placed on the unit sphere so that the minimum (Euclidean) distance between them is 1. In the determination of the maximal configuration for 𝑛 = 9, as well as in the proof of the inequality 𝑎 13 < 60◦ , the following lemma of Habicht and van der Waerden [73] is used: If an equilateral spherical 𝑛-gon 𝑇𝑛 of side length 𝑎 has the property that any two of its vertices are at a distance of at least 𝑎 from each other, then 𝑇𝑛 ≥ (𝑛 − 2)𝑇3 . For this inequality, whose special case 𝑉 (𝛽) ≥ 2Δ was used in Section 6.3, and from which the inequality (6.3.1) can be derived in a simpler way, Molnár [105] gave a very simple proof. An idea for a proof of the inequality 𝑎 13 < 60◦ can be found in Boerdijk [19]. For additional historical comments concerning this kind of problem, see Whyte [137]. The notion of the graph associated with an arrangement of points was introduced in the article of Habicht and van der Waerden [73]. It can be expected that with a deeper development of packing theory, further applications of graph theory will be found. A textbook presentation of graph theory is given by König [90].

170

234 ▶

6 Irregular Packing on the Sphere

We mention another problem where an application of the notion of a graph can probably be successfully used. The problem is about the thinnest covering of the sphere with congruent circles. We consider a given configuration of points on the unit sphere and we label them black. We then consider those points on the sphere whose distance from the nearest black point reaches the greatest possible value 𝑎. Such a point will be labeled white. By connecting every white point with the nearest black one we get a graph (Figure 6.13). The problem of determining the thinnest circle covering of the sphere is reduced to finding the configuration of the black points for which the edge length 𝑎 of the associated graph is smallest possible. It is easy to prove that the segments of such a graph cannot cross each other. Furthermore, it is obvious that each white point is connected to at least 3 black points not all of which lie on one side of any great circle containing the white point in question. An analogous statement can be made for minimal graphs with respect to the black points: If the edges emanating from a black point 𝑆 are contained in one hemisphere bounded by a great circle containing 𝑆, then Fig. 6.13 𝑆 can be pulled in so that all original edges emanating from 𝑆 become shorter. A graph in which no black point can be pulled in is called irreducible. By pulling in black points repeatedly, every minimal graph can be turned into an irreducible minimal graph. An irreducible graph will then partition the sphere into convex polygons. The vertices of such a polygon are alternatingly black and white. Therefore the number of edges of the polygon is even. Thereby we have created a notion corresponding to the one introduced for the dual problem. It would be desirable to put this notion to use and determine some minimal configurations.

Chapter 7

Packing in Space

First, in Section 7.1 we try to gain a general idea about arrangement problems in space. After that, we turn our attention to the problem of the densest ball packing: How large a part of space can be filled with congruent material (that is, nonoverlapping) balls? Those who encounter this simple-sounding natural question for the first time can hardly believe that it is a difficult, still unsolved problem. Where does the difficulty lie, and how could it be handled? We try to answer these questions in Section 7.2. In Section 7.3, we handle an extremal problem related to space partitions that is also connected to the problem of the densest sphere packing, while in Section 7.4 we concider the spatial analogue of the mean value formula whose planar case was handled in (3.12.1).

7.1 General Remarks The densest circle packing in the plane is (in essence) lattice-like. To get some idea about the analogous problem in space, consider first the densest lattice packing with balls (Figure 7.1). This packing can be described as follows. Let 𝑆0 be a densest layer of balls, that is, a family of equal sized balls whose centers lie in a plane so that each ball is touched by six other balls. On top of 𝑆0 we place a congruent layer 𝑆1 , so that each ball of the layer 𝑆1 touches three balls in the layer 𝑆0 . The translation that sends 𝑆0 to 𝑆1 sends 𝑆1 to 𝑆2 , and so on. By Fig. 7.1 the inverse translation we obtain the layers 𝑆−1 , 𝑆−2 , . . . . The densest lattice packing of balls consists of the layers . . . , 𝑆−1 , 𝑆0 , 𝑆1 , . . . . In it, every ball touches twelve other balls, namely, six in its own layer and three in each of the neighboring layers. In an analogous way as in the plane, to each packing of balls a partition of space into convex polyhedra can be assigned, which we also call cells. In the case of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_7

171

172

7 Packing in Space

lattice packing considered above, each cell is a rhombic dodecahedron. The density of this packing is equal to the ratio between the volume of √ the ball and the volume of the circumscribed rhombic dodecahedron, that is, 43𝜋 : 4 2 = √𝜋 = 0.74048 . . . . 18 The centers of the balls of the densest lattice packing form the so-called face-centered cubic lattice that is obtained from the lattice points of the usual cubic lattice by including also the centers of the cubes’ faces (Figure 7.2). In comparison with the twodimensional case, a difference appears. Namely, there is another regular but not latticelike packing of balls with the same density √𝜋 . Here we say 18 that a family of balls is reguFig. 7.2 lar when—roughly speaking— no single ball in the family can be distinguished from any other, that is, when every two balls can be sent to each other by a symmetry operation of the family. Replace the layer 𝑆2 by the layer 𝑆2′ obtained as the reflection of 𝑆0 in the plane of the ball centers of the layer 𝑆1 . The balls of the layer 𝑆2′ fall in those holes of the layer 𝑆1 that were left unoccupied by the layer 𝑆2 . If we now reflect the layer 𝑆1 in the center-plane of the layer 𝑆2′ , then we get the layer 𝑆3 . The desired ball packing is obtained by repeating such reflections in both directions. In this packing each ball is touched by twelve others as well, but in a different arrangement (Figure 7.3). The cells of this packing can be obtained by the following modification of the rhombic dodecahedron: Cut the rhombic dodecahedron into two parts by a plane passing through six centers of its faces. Each part is a hexagonal column, which—like in a honeycomb—is capped by three rhombi. The cell in question is obtained by rotating one of the parts by 60◦ and then joining the two parts in this new position. The resulting cell is a twelve-faced polyhedron bounded by six rhombi and six Fig. 7.3 isosceles trapezoids, which, along with the rhombic dodecahedron, we will call a double bee-cell (Figure 7.4). Each of these two polyhedra can be considered as a space analogue of the regular hexagon, as they can be defined as those polyhedra circumscribed about a ball whose dihedral angle at every edge measures 120◦ .

173

7.1 General Remarks

Fig. 7.4

As we saw, in building a ball packing layer-by-layer, a new layer can be added to the previous ones in two different ways every time. These two possibilities can be combined in an arbitrary way, resulting in an irregular packing as well, with the same density √𝜋 . In a packing obtained in this way, double bee-cells of each of the 18 two kinds appear. Very likely, the density √𝜋 cannot be ex18 ceeded by the density of any ball packing. This would mean that for this problem, the double bee-cells play in space the same role as regular hexagons do in the plane. But in general, for other problems the same does not happen. The problem of the thinnest space covering by balls already provides a counterexample. Let us place congruent balls with their centers at the points of the face-centered cubic lattice, with radius just large enough so that they cover the whole space. The density of the obtained covering is 23𝜋 = 2.094 . . .. If, on the other hand, we start with the body-centered cubic lattice, Fig. 7.5 consisting of the usual cubic lattice points plus the cubes’ centers (Figure 7.5), then we get a lattice covering of space with balls, √ 5 5𝜋 of a considerably smaller density of 24 = 1.464 . . . . The cells of this point lattice are congruent copies of the Archimedean polyhedron (4, 6, 6), called the truncated octahedron. Let us now consider another problem: Partition space into convex polyhedra of equal volume, say of volume one, such that their average surface area is as small as possible. In view of the Lindelöf–Minkowski theorem and the well-known extremal property of the honeycomb, one would justifiably think first of decomposing space into double bee-cells. But the surface area of a double bee-cell of unit volume is » √ 3 108 2 = 5.345 . . . , while that of the truncated octahedron of equal volume

174

7 Packing in Space

√3 √ amounts to only 43 4(1 + 12) = 5.315 . . .. Therefore building the walls of a cell partition of a very large body into truncated octahedra is about 21 % more economical than building it with double bee-cells. The fact that for different problems the role of the regular hexagon is not taken over by one and the same polyhedron can perhaps be explained by the fact that for some problems it is the polyhedron circumscribed about the ball, while for others the inscribed polyhedron, that is in a certain sense distinguished. We did not see this difference in the plane since the regular hexagon is both inscribed and circumscribed. However, the double bee-cells are circumscribed, but not inscribed polyhedra; similarly, the truncated octahedron is an inscribed polyhedron, but not circumscribed. Insufficient experience prevents us from conjecturing that the solutions of spatial analogs of problems on arrangements that lead to a hexagonal partition of the plane are partitions of space into either double bee-cells or truncated octahedra.

7.2 The Problem of Densest Ball Packing In one of his reviews [Mathematical Reviews 9, 53 (1948)], H.S.M. Coxeter writes: The densest packing of equal circles in the plane is unique: each circle is surrounded by six others. Spheres on these circles as equators form a ‘hexagonal layer.’ It is obvious (though nobody has yet succeeded in proving it) that every densest packing of equal spheres is made up of such hexagonal layers.

According to the remarks of the previous section, the conjecture formulated here can be made precise as follows: If 𝐷 is the density of an arbitrary arrangement of congruent material balls, then 𝜋 𝐷 ≤ √ = 0.74048 . . . , 18 238 ▶

(7.2.1)

and equality occurs only if the arrangement is honeycomb-like. Here, a honeycomblike arrangement is defined in a way quite analogous to the planar definition. In brief, we say that a ball packing is honeycomb-like if almost all cells are close to a double bee-cell circumscribed about a ball, while the total volume of the remaining cells is negligible in comparison with the volume of space. The first step towards the solution of the conjecture (7.2.1) was made by Blichfeldt [18], who proved the inequality 𝐷 < 0.835 in 1929. Rankin [111] has replaced this estimate by a slightly sharper but still quite rough estimate 𝐷 < 0.828. We now give a simple proof of the inequality 𝐷 < 0.835 ,

(7.2.2)

different from the original proof of Blichfeldt. The arguments used in this proof can be easily refined to obtain more accurate estimates. Instead, we will sketch a proof of the estimate

7.2 The Problem of Densest Ball Packing

𝐷
16 sin 2𝜔18 (3 tan2 𝜔18 − 1) = 5.016 . . . . Consequently, the density of the packing is less than 43𝜋 : 5.015 . . . < 0.835 already in every single cell, which proves the inequality (7.2.2). We now turn to the problem of determining the cell of smallest volume among all possible cells. Aside from the fact that this problem is incomparably harder than the corresponding problem in the plane, in space we face another inconvenient circumstance: The smallest possible cell does not tile space. Because of this, the density of a ball packing in the whole space cannot attain the maximum value that it reaches with respect to a single cell. Consequently, the problem of densest packing of balls cannot be solved in this way. We will see that the cell of the smallest possible volume is a regular dodecahedron circumscribed about a ball, which is expressed by the inequality …  √  ◦ 2 ◦ 𝑍 ≥ 10 sin 72 (3 tan 36 − 1) = 10 2 65 − 29 5 = 5.550 . . . . For comparison, we compute the volume of a circumscribed double bee-cell, √ which is 4 2 = 5.656 . . ., indeed greater than the above. The above inequality shows that the packing’s density with respect to every cell 𝑍 satisfies …  √  4𝜋 4𝜋 : 𝑍 ≤ : 10 2 65 − 29 5 . 3 3

◀ 238

176

7 Packing in Space

Since space cannot be tiled with regular dodecahedra, this density cannot be reached with respect to the whole space. Obviously, for the problem of determining the cell of minimum volume, the question of how many congruent material balls can touch another ball of the same size plays an important role. This question is equivalent to: How many points can be placed on the unit sphere so that all distances between them are at least 1? Although the inequality (5.2.1) still allows the possibility of 13 points, and excludes the case of 14 points only, experimental attempts leave no doubt that the number in question is 12. However, we know well that to prove this rigorously is not an easy task on its own, accomplished only recently by Schütte and van der Waerden [124]. Fig. 7.6 To be quite clear about this problem we notice that, while a coin placed on a table can be touched by six same-size coins in one configuration only, twelve material balls can touch another one in various configurations. The two different honeycomb-like configurations are known to us already. However, a ball can be touched by twelve others at the vertices of an inscribed regular icosahedron as well (Figure 7.6). Moreover, since the edges of a regular icosahedron inscribed in a unit ball have length   √︁ 2 2 ◦ 4 − csc 36 = 2 − √ = 1.0515 . . . , 5 hence greater than 1, the twelve outer balls in this configuration have some empty space around them. In addition, the rhombic dodecahedral arrangement has the property that the configuration is globally flexible, although no ball can move alone. However, the six equatorial balls can be moved together, loosening the arrangement and making other balls free to move. Therefore the rhombic dodecahedral arrangement can be turned into the pentagon-dodecahedral (icosahedral) one through a continuous motion during which no outer ball loses its contact with the inner one. One can easily see that the other honeycomb-like configuration is movable as well. The problem of the twelve touching balls raises the Fig. 7.7 next question: How closely can a ball, while being touched by twelve material balls of the same size, be approached by yet another material ball of the same size? Let us assume that a central gravity force keeps the twelve outer balls in contact with the inner one. We start from the icosahedral configuration and we push the thirteenth ball slowly into a hole. Then the hole will gradually enlarge, while the

177

7.2 The Problem of Densest Ball Packing

diametrically opposite hole will contract. The thirteenth ball can move towards the inner one only until the opposite hole shrinks completely, that is, until the three corresponding balls come in contact with each other (Figure 7.7). In this position, the distance from the center of the thirteenth ball to the center of the inner ball is … 64 2 = 2.7534 . . . . 19 3 Figure 7.8 illustrates the central projection on the surface of the inner ball of the twelve balls in this configuration, presented in a stereographic projection in the plane. We now describe an even more favorable configuration, found by Boerdijk [19]. Place one ball at the north pole of the “inner” ball and place five other balls touching the northern one so that the first of the five balls touches the second one, the second one touches the third, and Fig. 7.8 so on. Reflect these six balls in the plane of the equator of the inner ball and get a configuration of twelve balls whose centers are the vertices of a polyhedron bounded by 4 squares, 1 rectangle, 8 equilateral triangles and 2 isosceles triangles (Figure 7.9). If we now insert an additional ball in the hole in the rectangular face, then the distance from the center of this ball to the center of the inner ball is 14 2𝑎 = √ = 2.6943 . . . . 27 Presumably, this is the desired extremal configuration, that is, the thirteenth ball cannot come closer to the inner ball, no matter how the twelve attached balls are arranged. If we now try to push the thirteenth ball closer to the inner one, then the twelve touching balls must move away from it. This operation seems to increase the sum of distances from the thirteen balls to the inner one, that was originally 24 + 2𝑎. It is highly likely that 14 points 𝑃0 , 𝑃1 , . . . , 𝑃13 satisfying the inequalities 𝑃𝑖 𝑃 𝑗 ≥ 1,

𝑖, 𝑗 = 0, 1, . . . , 13,

𝑖≠ 𝑗 Fig. 7.9

also satisfy the inequality 13 ∑︁ 𝑖=1

𝑃0 𝑃𝑖 ≥ 12 + 𝑎 = 13.3471 . . . .

(7.2.4)

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7 Packing in Space

The inequality (7.2.4) would imply that there is not enough room for 13 points with the minimum distance 1 not only on the unit sphere, but not even on a sphere of radius 12+𝑎 13 = 1.0267 . . . . This is consistent with our experience. According to Section 4.4, the radius of the smallest ball for 13 points is probably 1.045 . . . . We should comment here that while, for example, the general inequality (5.2.1) is of some interest from the mathematical point of view, searching for a rigorous proof of the very specialized inequality (7.2.4) is not worth the effort. This inequality would be used only to exclude cells with more than 12 faces, thus, in essence, just for a rough estimate. We will therefore consider the inequality (7.2.4) as a well-grounded empirical fact, whose proof would be arduous and of minor interest in comparison with the results of Section 5.8 on which, essentially, the proof of (7.2.3) is based. In fact, the weaker inequality 13 ∑︁

√ 𝑃0 𝑃𝑖 ≥ 12 + 3 tan 36◦ = 13.258 . . .

(7.2.5)

𝑖=1

will meet our needs already. Let 𝑆 be the spherical shell bounded by the ball 𝐾 and the concentric ball 𝐾12 of radius …  √  √ ◦ 𝑅12 = 3 tan 36 = 3 5 − 2 5 = 1.2584 . . . . If at most twelve of the feet 𝐹𝑖 of the perpendiculars dropped from the center of 𝐾 to the planes of the faces of 𝑍 lie in 𝑆, then (7.2.3) is an immediate consequence of (5.8.3). Let us then assume that, say, 13 feet 𝐹1 , . . . , 𝐹13 lie in 𝑆. Then by (7.2.5) we get 13 ∑︁

𝑂𝐹𝑖 ≥ 12 + 𝑅12 ,

1 ≤ 𝑂𝐹𝑖 ≤ 𝑅12 ,

𝑖 = 1, . . . , 13 .

𝑖=1

It is very easy to see that under these conditions the volume of the portion of 𝑍 that lies in 𝐾12 reaches its minimum in the case 𝑂𝐹1 = · · · = 𝑂𝐹12 = 1, 𝑂𝐹13 = 𝑅12 , by which this case is reduced to the case of 12 feet in the spherical shell 𝑆. The case when more than 13 feet lie in 𝑆 can be handled in a similar way. How could one now continue towards obtaining the exact upper bound for the packing density? Let us turn our attention to a pentagon-dodecahedral cell, where the inner ball is surrounded by twelve other balls in the icosahedral configuration. Every outer ball has five neighbors, with a gap between them of about 0.1. It is easy to see that the volume of the cell of such an outer ball is considerably greater than the volume of a double bee-cell, so that the volume deficit of the regular dodecahedron compared to a double bee-cell is richly compensated by the volume surplus of a neighboring cell. These observations lead us to the idea that instead of just concentrating on a single cell, the neighboring cells must also be considered.

7.3 On an Extremal Space Partition

179

Various possibilities arise for a concrete implementation of this main idea, of which the following appears to be most expedient. Two balls whose centers are less than 24+2𝑎 = 2.0534 . . . away from each other will be called close neighbors. It 13 is easy to see that two close neighbors are close also in the sense that their cells are adjacent along a common face. As a consequence of the conjectured inequality (7.2.4), each ball would have at most twelve close neighbors. We pick one ball. Let 𝑍0 be its cell, while the cells of its close neighbors are denoted by 𝑍1 , . . . , 𝑍 𝑗 (0 ≤ 𝑗 ≤ 12). We consider the average value 𝑍¯ =

𝑍1 + · · · + 𝑍 𝑗 + (12 − 𝑗)𝑍0 12

and we ask under which arrangement of balls the value of 𝑍¯ reaches its minimum. In searching for the best arrangement, one should try to bring the balls as close to each other as possible, so that the cells are kept small. Such an arrangement occurs only in the case 𝑗 = 12, and then when the twelve balls touch the inner one. But even among such arrangements, those in which the balls have much looseness can be excluded. The most advantageous appear to be the two double bee-cells, for there the balls have no looseness at all. Therefore it can be presumed that 𝑍¯ reaches its minimum when 𝑍0 along with all its neighboring cells 𝑍1 , . . . , 𝑍12 are double bee-cells. This means that √ (7.2.6) 𝑍¯ ≥ 4 2 . Thereby the conjecture (7.2.1) would be proved. Namely, if we add the inequalities (7.2.6) for all balls, then each cell on the left-hand side would appear exactly twelve times: once in each of the 𝑗 inequalities related to the close neighbors of the ball in question, and 12 − 𝑗 times in the inequality related to the ball itself. Consequently, on the left-hand side we√get the volume √ of space. On the right-hand side we get the number of balls times 4 2, that is, the 4 2 43𝜋 -fold sum of the volumes of the balls. Thereby the ratio of the total volume of the balls to the volume of space, that is, the density of the packing, comes out less than or equal to √𝜋 . 18 It seems that by the above observations the problem of the maximum density of ball packing in space is reduced to the problem of determining the minimum of a function of finitely many variables. Although an exact treatment of this minimum problem appears to be quite complicated, it cannot be considered hopeless by any means. We have certainly presented a specific, in principle fully implementable program, by which we have come one step closer towards the solution.

7.3 On an Extremal Space Partition The whole space is to be partitioned into convex polyhedra so that almost all polyhedra are close to double bee-cells. This should be understood as follows: Let 𝑛(𝑅) denote the number of those polyhedra that are contained in the ball 𝐾 (𝑅) of radius 𝑅 centered at the origin 𝑂, and let 𝜈(𝑅, 𝜀) denote the number of those polyhedra

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7 Packing in Space

contained in 𝐾 (𝑅) whose deviation from any double bee-cell is greater than 𝜀. Then 𝜈(𝑅, 𝜀) the space partition we have in mind should satisfy the condition lim =0 𝑅→∞ 𝑛(𝑅) for every positive 𝜀. We say that such a partition is honeycomb-like. The main difference between a honeycomb-like partition and the cell partition corresponding to a honeycomb-like ball packing is that in the latter case almost all polyhedra must be close to double bee-cells of equal volume, while in the first case double bee-cells of a quite different volume can occur. After this explanation, we prove the following theorem: Assume the whole space is partitioned into convex polyhedra whose inradii have a positive lower bound and whose circumradii have a finite upper bound. Denote by 𝑉, 𝐹 and 𝑀 the volume, surface area and the edge curvature of a polyhedron, and finally assume that the mean values 𝐹 2 /𝑉 and 𝑀 of the quantities 𝐹 2 /𝑉 and 𝑀 exist. Then √ 𝜋 𝐹 2 /𝑉 ≥ 6 3 𝑀 , (7.3.1) and equality occurs exactly for the honeycomb-like partitions. The inequality (7.3.1) can be viewed as the spatial analogue of (3.5.1). Another analogous task would be to estimate the mean value 𝐹 3 /𝑉 2 from below. For this problem, however, the best partition would not be honeycomb-like, but, presumably, truncated-octahedron-like. For the proof of (7.3.1) we add the inequalities (5.7.3) for all polyhedra contained in 𝐾 (𝑅 + 𝑑), where 𝑑 denotes the upper bound for the diameters of the circumspheres of the polyhedra, and we keep on the right-hand side only those terms that correspond to edges contained in 𝐾 (𝑅). Suppose that 𝜈 ≥ 3 polyhedra meet at some edge and denote the corresponding dihedral angles of the polyhedra at this edge by 𝛽𝑖 (𝑖 = 1, . . . , 𝜈), hence their corresponding edge-angles are 𝛼𝑖 = 𝜋 − 𝛽𝑖 . Consequently, in view of tan

𝛽1 𝜋 √ 𝛼1 𝛼𝜈 𝛽𝜈 + · · · + tan = cot + · · · + cot ≥ 𝜈 cot ≥ 3(𝜈 − 2) , 2 2 2 2 𝜈

we get ∑︁ 𝐹 2 √ ∑︁ >3 3 (𝜈 − 2)𝑙 . 𝑉 𝑅 𝑅+𝑑 Í If we now add the equalities 𝑀 = 21 𝑙𝛼 defining the edge curvature for the polyhedra lying in 𝐾 (𝑅 − 𝑑), then, by 𝛼1 + · · · + 𝛼𝜈 = 𝜋(𝜈 − 2), we easily get ∑︁ 𝜋 ∑︁ (𝜈 − 2)𝑙 . 𝑀< 2 𝑅 𝑅−𝑑 Therefore

√ ∑︁ 𝐹 2 6 3 ∑︁ > 𝑀. 𝑉 𝜋 𝑅−𝑑 𝑅+𝑑

7.3 On an Extremal Space Partition

181

Divide now both sides by 𝑛(𝑅) then take the limit as 𝑅 → ∞, and the desired inequality (7.3.1) follows. Our proof would become even simpler if we could, from the beginning, restrict our attention to space partitions in which exactly three polyhedra meet at each edge. The case of equality remains to be discussed. The fact that in the case of a honeycomb-like partition equality occurs is evident. √ 2 Namely, for a double bee-cell we have 𝜋 𝐹𝑉 = 6 3𝑀, and the functionals 𝑉, 𝐹 and 𝑀 are continuous, therefore for every positive 𝛿 the value of 𝜀 can be chosen so that for every polyhedron whose deviation from a double bee-cell is less than 𝜀, the inequality 2 √ 𝐹 𝜋 𝑉 − 6 3𝑀 < 𝛿 holds. Since in a honeycomb-like space partition this inequality holds for almost all polyhedra, and for the remaining ones—by assumption—the corresponding quantities are uniformly bounded, we have √ 2 𝜋𝐹 /𝑉 − 6 3 𝑀 < 𝛿 , which can be true for every 𝛿 only if equality occurs in (7.3.1). It is somewhat more difficult to prove that equality holds only for honeycomblike space partitions. We show first that for those polyhedra that deviate from every double bee-cell by more than 𝜀 there is a positive number 𝛿 such that either ∑︁ 𝛼 𝐹2 −3 𝑙 tan > 𝛿 𝑉 2

(7.3.2)

or

∑︁ 𝜋 𝑙 𝛼 − > 𝛿 3 holds. Otherwise a sequence of the polyhedra could be chosen for which ∑︁ 𝛼 𝐹2 −3 𝑙 tan → 0 and 𝑉 2

(7.3.3)

∑︁ 𝜋 𝑙 𝛼 − → 0 . 3

If we translate these polyhedra so that they all lie in one ball, then, by the selection theorem, they have an accumulation element 𝐻. By the last limit condition it follows that 𝐻 is a polyhedron. This polyhedron satisfies ∑︁ 𝛼 ∑︁ 𝜋 𝐹2 −3 𝑙 tan = 𝑙 𝛼 − = 0 , 𝑉 2 3 which means that 𝐻 is circumscribed about a ball and each dihedral angle of 𝐻 measures 120◦ . But this is the defining property of a double bee-cell. However, the fact that 𝐻 is a double bee-cell contradicts the assumption that the considered polyhedra deviate from each double bee-cell by more than 𝜀.

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7 Packing in Space

Now, by the reasoning in the above proof of the inequality (7.3.1), it follows that equality can occur only if the number of those polyhedra for which either (7.3.2) or (7.3.3) holds is negligible in comparison with the total number of polyhedra. Thereby the proof is completed. √ It should be noted that the difference 𝜋 𝐹 2 /𝑉 − 6 3 𝑀 can be arbitrarily small also for space partitions unrelated to honeycomb-like partitions. To see this, partition a sphere into a large number of nearly congruent regular spherical hexagons. For a precise construction one could consider, for example, the sphere partition into 𝑛 convex spherical polygons with the smallest sum of perimeters, given in Section 5.9. If we project these polygons from the sphere’s center, then the projection rays partition a cube circumscribed about the sphere into spike-like pyramids. The greater 𝑛 is, the sharper these “spikes” become. The desired space partition is obtained by tiling space with congruent cubes and then partitioning each cube into sufficiently sharp needles. Naturally, other space-tiling polyhedra could be used here instead of cubes, and they do not even need to be congruent. If we partition space so that the spikes become sharper as we move away from the origin, then equality can be reached in (7.3.1) as well. But in this case either the lower bound for the inspheres’ diameters is not positive, or the circumspheres’ diameters are not bounded from above. We now give some applications of our theorem. We denote by 𝐿 the total edge length of a polyhedron, and we consider the mean values 𝐿 and 𝑀. If 𝜈 polyhedra meet around a common edge 𝑙, with edge-angles 𝛼1 , . . . , 𝛼𝜈 , then the coefficients of 𝜈 𝜋 1 ∑︁ 𝛼𝑖 = (𝜈 − 2) and 𝜈, respectively. 𝑙 in the sums of the values of 𝑀 and 𝐿 are 2 𝑖=1 2 𝜋 𝜋 Since 2 (𝜈 − 2) ≥ 6 𝜈 holds for 𝜈 ≥ 3, we have 6𝑀 ≥ 𝜋𝐿.

(7.3.4)

Equality occurs only when the sum of lengths of edges that belong to more than three polyhedra is negligible in comparison with the total edge length. From (7.3.1) and (7.3.4) we get the inequality √ 𝐹 2 /𝑉 ≥ 3 𝐿 . (7.3.5) 2

This inequality is noteworthy because an inequality of the form 𝐹𝑉 > 𝐶 𝐿 with a universal constant 𝐶 > 0 cannot be stated for a single polyhedron. We also emphasise the following special, but not uninteresting consequence of the inequality (7.3.5). Let 𝑐 be a prescribed positive number. For the existence of a space partition into convex polyhedra whose volume 𝑉, surface area 𝐹 and total √edge length 𝐿 satisfy the equality 𝐹 2 = 𝑐𝐿𝑉, it is necessary and sufficient that 𝑐 ≥ 3. The necessity of this condition follows from (7.3.5). That the condition is also sufficient is shown by the following construction.

7.4 The Mean Value Formula in Space

183

We compress the rhombic dodecahedron “telescopically” by shortening its six parallel edges. As soon as these edges completely disappear, the polyhedron becomes a parallelepiped. We continue the telescopic compression by shortening the parallelepiped’s four parallel edges until, in the limit case, the polyhedron collapses into two coincident rhombic faces. The intermediate stages present space-tiling polyhedra √ 𝐹2 whose quotients 𝐿𝑉 run through all values greater than or equal to 3. Let us observe that during a telescopic extension of a double bee-cell, the quotient √ √ 𝐹2 4 𝐿𝑉 , starting from the value of 3, tends monotonically to the limit value of 3 3. Therefore, as we would expect, during the telescopic modification of a double bee𝐹2 cell 𝐿𝑉 attains its minimum value for the double bee-cell itself. Finally, we derive from (7.3.1) the following conclusion concerning the problem of the densest ball packing: If 𝑀 denotes the mean edge curvature of the cells of a family of material unit balls, then the density 𝐷 of the ball packing satisfies 2 𝜋2 . 𝐷 ≤ √ 3𝑀 For any polyhedron containing the unit ball we have 𝑉 ≥

(7.3.6)

1 3

𝐹, that is, 𝑉 ≥

𝐹2 9𝑉 .

Consequently, 𝑉 ≥ 19 𝐹 2 / 𝑉 ≥ √ 2 𝑀 from which, in view of 𝐷 = 43𝜋 : 𝑉, the 3𝜋 desired inequality (7.3.6) follows. Since for every cell the inequality 𝑀 > 4𝜋 holds, we immediately get 𝑀 > 4𝜋, and it is worth noticing that by such a quite rough estimate already, (7.3.6) yields an upper bound smaller than 1 for the packing density. The bound obtained this way, namely √𝜋 , is the density of the densest circle packing in the plane. 12

7.4 The Mean Value Formula in Space In space, the mean value formula corresponding to the formula (3.12.1) is as follows: ! 𝑀𝐹 + 𝑀𝐹 𝑆 = 𝐴 𝑉 +𝑉 + . (7.4.1) 4𝜋 Here 𝑉, 𝐹 and 𝑀 denote the mean value of volume, surface area and mean curvature, respectively, of a fixed family of convex bodies 𝑉1 , 𝑉2 , . . . with point density 𝐴; furthermore, 𝑉, 𝐹 and 𝑀 are the corresponding fundamental characteristics of a rigidly movable convex body 𝑉, and 𝑆 is the average number of bodies 𝑉𝑖 intersected by 𝑉. The mean value 𝑆 can be defined, in a way quite analogous to the planar case, by using the notion of kinematic density in space. The derivation of the formula, just as in the planar case, uses the fundamental formula of kinematics in space.

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7 Packing in Space

As a first application of the formula (7.4.1) we consider the space tiling with √ replicas of the truncated octahedron (4, 6, 6) of circumradius 1. We have 𝑉 = 3225 5 , √ √ √ 6 10 𝜋 1 𝐹 = 12 = 5325 . It should be noticed here that for 5 (1 + 12), 𝑀 = 5 , and 𝐴 = 𝑉

the computation of 𝑀 it is not necessary to know the dihedral angles of the truncated octahedron. Namely, if 𝑙 denotes the edge length of the truncated octahedron, then, by (7.3.4), we get 𝑀 = 𝜋6 𝐿 = 6 𝜋 𝑙. By the above values of 𝑉, 𝐹 and 𝑀, we now have √  √  √ √ 3 5 1 + 12 5 5 15 2 𝑆 = 𝑉+ 𝐹+ 𝑀 +1. 32 64 32 By the same observations as in the plane, this allows us to conclude: Every convex body with the three fundamental characteristics 𝑉, 𝐹 and 𝑀 can always be covered by √  √    √ √  5 5 3 5 1 + 12 15 2   𝑉+ 𝐹+ 𝑀 + 1 (7.4.2)    32 64 32𝜋     unit balls. We now turn to the densest lattice packing with unit balls. Here we have 𝐴 = 3𝑉 = 𝐹 = 𝑀 = 4 𝜋 and consequently √   2 4𝜋 𝑆= 𝑉+𝐹+𝑀+ . 8 3

√ 2 8 ,

If the moving body is the inner parallel body 𝑉−2 at distance 2 from a convex body 𝑉, then, obviously, all balls that intersect 𝑉−2 lie in 𝑉. This implies: For every convex body 𝑉 with inradius at least 2 the number of unit balls that can be packed in 𝑉 is at least √   2 4𝜋 𝑉−2 + 𝐹−2 + 𝑀−2 + , 8 3 where 𝑉−2 , 𝐹−2 and 𝑀−2 denote the three fundamental characteristics of the inner parallel body of 𝑉 at distance 2. For the ball of radius 𝑅 this number is 𝜋 √ (𝑅 − 1) 3 , 18 and for the cube of edge length 𝑎 it is √   2 3 4𝜋 2 𝑎 − 6𝑎 + 6 𝜋 𝑎 + 32 + . 8 3

7.5 Historical Remarks

185

7.5 Historical Remarks The densest lattice packing of space with congruent replicas of an arbitrary convex body was studied by Lord Kelvin [88], motivated by problems in physics. Minkowski [103] considered the same problem quite generally in 𝑛-dimensional space and applied the obtained results in proofs of deep theorems in number theory. The problem of the thinnest lattice covering of 𝑛-dimensional space with balls was investigated by Voronoi [135] and, recently, by Bambah and Davenport [4]. Only a few initial results in this respect have been obtain thus far. As a very interesting introduction to this area, connected with the notion of a regular family of points, we mention Chapter II of the work by Hilbert and Cohn-Vossen [84], compare also the encyclopedia article by Liebisch, Schoenflies, and Mügge [95]. Minkowski used the term “double bee-cell” to denote the convex hull of two congruent, homothetic cubes that share exactly one vertex. This polyhedron is not identical with our double bee-cell. The Archimedean solid (4, 6, 6) is commonly known as the cubo-octahedron. Since this name can be easily confused with the name of cuboctahedron given to the polyhedron (3, 4, 3, 4), the name of truncated octahedron used by us here seems to be more appropriate (see the relevant footnote on p. 29 in Coxeter [23]). Density estimates for non-lattice ball packings and coverings of 𝑛-dimensional space were produced by Blichfeldt [17], Rankin [111], Lekkerkerker [94], and Hlawka [85]. Hlawka’s work also contains other general theorems concerning packing and covering of 𝑛-dimensional space by congruent, homothetic convex bodies. Concerning the problem of the densest ball packing in the usual (three-dimensional) space, see also the articles of Supnick [129], Boerdijk [19], Wise [138] and Hadwiger [80]. The results of Sections 7.2 and 7.3 are found in the articles [35, 38, 63] of the author. The method that led us to the estimate (7.4.2) is due to Hadwiger [75], although originally he used a worse space-tiling polyhedron. Following the author’s suggestion, pointing out to him the truncated octahedron, Hadwiger [80] subsequently obtained the considerably better estimate (7.4.2). Partitions of space into convex polyhedra can be considered as degenerate fourdimensional convex polytopes. What can now in general be expected in connection with the extremal properties of the various four- or higher-dimensional polytopes? The partition of space into rhombic dodecahedra or into truncated octahedra are not regular polyhedra. Thus, to what extent are the regular polytopes related to the solutions of extremum problems? Some extremal properties of the higher-dimensional analogues of the regular tetrahedron, octahedron and hexahedron are easily verified. For example, the 𝑛dimensional regular simplex has the greatest volume among all simplices inscribed in the ball and the smallest volume among simplices circumscribed about the ball. This implies that for the inradius 𝑟 and circumradius 𝑅 of an 𝑛-dimensional simplex the inequality 𝑅 ≥ 𝑛𝑟 holds. It is known that, besides the mentioned “trivial” regular polytopes, only three other non-degenerate regular polytopes exist, and all three in

186

256 ▶

7 Packing in Space

four dimensions. Their Schläfli symbols are {3, 4, 3}, {3, 3, 5} and {5, 3, 3}. Thus far no extremal property of these regular polytopes is known. Consider the configuration of 𝑘 points on the surface of the 𝑛-dimensional ball whose minimum√distance 𝑑 𝑛𝑘 between the points is as large as possible. Then the equality 𝑑 𝑛2𝑛 = 2 holds, which expresses the extremal property of the 𝑛-dimensional analogue of the regular octahedron. We saw that the considered arrangement of 5 points on the surface of the three-dimensional ball is not unique and that 𝑑35 = 𝑑36 . A similar phenomenon occurs in higher-dimensions as well. Namely, as Davenport and √Hajós [24] noticed independently from each other, already the equality 𝑑 𝑛𝑛+2 = 2 holds. Let 𝑁 (𝑛) denote the maximum integer 𝑘 for which 𝑑 𝑛𝑘 ≥ 1, that is, the maximum number of points that can be placed on the surface of the 𝑛-dimensional unit ball so that the smallest distance between them is at least 1. We have 𝑁 (2) = 6 and 𝑁 (3) = 12. According to a not yet published result of C.A. Rogers, the inequalities √︁ √︁ √︁ √ 2 lim sup 𝑛 𝑁 (𝑛) ≤ 2 and lim inf 𝑛 𝑁 (𝑛) ≥ √ hold. Does the limit lim 𝑛 𝑁 (𝑛) 𝑛→∞ 𝑛→∞ 𝑛→∞ 3 exist, and if so, what is its value? In general, problems concerning arrangements in space seem difficult to approach. Therefore, as the next area of research in this respect, one could consider problems of this type in which the allowed arrangements are subject to certain regularity conditions. Lattice arrangements of bodies come to mind in particular. We name here a specific problem. 𝜋 ≤ Let us consider a lattice of circles of a prescribed density 𝐷, where √ 12 2𝜋 𝐷 ≤ √ , that covers as large a portion of the plane as is possible. Here the 27 circles’ centers form an equilateral triangular lattice, therefore one can arrange the circles corresponding to the range of values of 𝐷 so that the circles’ centers remain fixed and only their radii change. However, for the analogous problem in space the circumstances are completely different, since the face-centered cubic lattice must transform into the body-centered cubic lattice. How does this transformation take place? We do not even know whether the transformation in question goes continuously or exhibits jumps.

Part II

Notes and Additional Chapters to the English Edition

Chapter 8

Notes G. Fejes Tóth, L. Fejes Tóth and W. Kuperberg

8.1 Notes on Chapter 1 8.1.1 Notes on Section 1.2 The subject of polarity, in particular the specific problem of the product of the volume of polar convex bodies, which comes up in the inequality (1.2.1), has been intensively investigated in the past decades. For a recent survey of the topic, see the introduction to Böröczky Jr., Makai, Meyer, and Reisner [132] and the paper by Makai [501].

8.1.2 Notes on Section 1.3 The inequality (1.3.3) was “dualized” by L. Fejes Tóth [242]) in the following way. Suppose 𝑃 is an 𝑛-gon inscribed in a circle 𝐾, 𝑃¯ is an inscribed regular 𝑛-gon, 𝑘 is the incircle of 𝑃¯ and 𝐻 is the convex hull of 𝑃 and 𝑘. If, further, 𝐿 and 𝐿¯ denote the ¯ This is a stronger version of the right-hand part ¯ then 𝐿 ≤ 𝐿. perimeters of 𝐻 and 𝑃, of (1.3.2). According to (1.3.2), among the convex 𝑛-gons inscribed in a circle, the regular one has maximum perimeter. In the book Brass, Moser and Pach [139, Section 11.1, Problem 3] Brass dropped the condition of convexity and asked for the maximum perimeter of a simple 𝑛-gon contained in the unit circle. When 𝑛 is even, the solution is obvious: The perimeter of the polygon can be arbitrarily close to 2𝑛 if the polygon is sufficiently close to a diagonal of the circle. Audet, Hansen and Messine [28] solved the problem for odd 𝑛: The supremum of the perimeter is reached by a sequence of polygons approximating an isosceles triangle. An alternative proof was given by Lángi [471], who also extended the result to the hyperbolic plane, and in [472] he determined the supremum of the perimeters of simple 𝑛-gons contained in an arbitrary convex disk in the Euclidean or in the hyperbolic plane. A further © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_8

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proof was given by Dumitrescu [190], who also examined the problem in the case when the simplicity condition is dropped, and obtained an exact formula for the maximum perimeter in this case as well.

8.1.3 Notes on Section 1.4 Answering a question of K. Bezdek, Csikós, Lángi and Naszódi [169] extended the isoperimetric inequality for polygons to circle-polygons, i.e. regions enclosed by a finite number of circular arcs of some given radius 𝑟. Their result is more general: they consider piecewise smooth curves of constant geodesic curvature on a plane of constant curvature. We mention the following variation on the isoperimetric problem raised by L. Fejes Tóth [251]. Place a given finite number of rigidly movable convex disks in the plane so that they enclose a region of maximum area. Such an optimal arrangement of the given disks will be called a Dido arrangement. The set of points lying outside the union of the disks consists of one or more connected components. The region enclosed by the disks is defined as the union of the bounded components (Figure 8.1).

Fig. 8.1

Fig. 8.2

It seems certain that in a Dido arrangement of three or more disks each disk meets exactly two other disks. This conjecture, if true, would imply a stronger statement: In a Dido arrangement of more than three disks, the points common to two disks lie on a circle and each ray emanating from the circle’s center meets the interior of at most one of the disks (Figure 8.2). For three disks this statement does not hold. Remarkably, the proof of this apparently trivial conjecture encountered difficulties even in the case when the disks are reduced to line segments (see L. Fejes Tóth [254]). This particular version of the Dido problem was treated by Pach [555], and was finally solved by Siegel [611]. Independently, the conjecture was also verified by Kertész [446], who also proved it on the hyperbolic plane.

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A. Bezdek and K. Bezdek [67] studied a related problem: Arrange some given segments so that their union is connected and the area of their convex hull is greatest possible. They conjectured that in the optimal arrangement the segments form a polygonal path inscribed in a semicircle. They verified the conjecture in the special case where the segments are edges of a graph embedded in the plane. Siegel [610] proved the conjecture under a weaker assumption. Namely, it is enough Ð to assume that the segments have an ordering 𝑠1 , 𝑠2 , . . . , 𝑠 𝑛 such that 𝑠𝑖+1 ∩ conv 1𝑖 𝑠𝑖 ≠ ∅ for 𝑖 = 2, . . . , 𝑛 − 1. Gori [327] solved the analogous problem in 𝑛 dimensions for 𝑛 and 𝑛 + 1 segments. L. Fejes Tóth and Heppes [270] studied the maximum 𝐴𝑛 of the area enclosed by 𝑛 translates of a convex disk of unit area, taken over all such convex disks, and they proved that 𝐴3 = 1, attained by triangles only, and 𝐴4 = 1, attained by triangles and quadrangles only. For the various improvements on the classical isoperimetric inequality we mention, for example, Pleijel [567]. The ground-breaking book of Pólya and Szegő [568] treats a range of “isoperimetric problems” with respect to various physical quantities. More on this interesting topic can be found in the books and surveys of Payne [560], Demar [180], Bandle [40, 41, 42], Gericke [319], Burago [143], Talenti [636], Blåsjö [89], Osserman [553,554], Martini and Mustafaev [507], and Ros [581]. Variants of the isoperimetric problem, where area and perimeter are replaced by other quantities, such as diameter or width, are discussed in Chapter 10.

8.1.4 Notes on Section 1.5 The inequality (1.5.4) was proposed by Erdős [199] in the “Advanced Problems” section of the American Mathematical Monthly. Trigonometric solutions were given by Mordell and Barrow [534]. The inequality has become known as the Erdős– Mordell inequality. A number of simpler, elementary proofs came about due to D.K. Kazarinoff [439], Bankoff [43], Oppenheim [551], Veldkamp [647], Brabant [138], Avez [29], Komornik [455], Lee [474], Dergiades [181], the most simple being perhaps the visual proof by Alsina and Nelsen [13]. Oppenheim [552] proved the corresponding inequality on the sphere. The Erdős–Mordell inequality and its relatives remained an inspiring source for research through the past decades. At the time of the writing of these notes MathSciNet found 47 papers referring to the Erdős–Mordell inequality in their titles. The literature is too numerous to be listed here. In a non-equilateral triangle consider the line 𝑙 through the incenter 𝐼 and perpendicular to the line joining 𝐼 with the circumcenter 𝐶. Steinig [631] compared the inequalities (1.5.2) and (1.5.4) and found that if 𝑂 lies on the side of 𝑙 containing the vertex opposite to the smallest side of the triangle then (1.5.4), while if it lies on the other side, (1.5.2) is sharper.

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By connecting every other vertex of a regular pentagon with an edge, a so-called pentagram is created, the simplest example of a regular star-polygon. Besides the regular convex polygons, also the regular star-polygons were characterized by their various extremum properties in the papers of Degen and Muny [177] and L. Fejes Tóth [247]. The reader can find a multitude of further inequalities in elementary geometry in the books of N.D. Kazarinoff [441], Bottema, Djordjević, Janić, Mitrinović and Vasić [136] and Mitrinović, Pečarić and Volenec [520].

8.1.5 Notes on Section 1.6 A tiling of the plane with convex polygons is normal if each tile contains a circular disk of radius 𝑟 and is contained in a disk of radius 𝑅. It follows from the considerations in Section 1.6 that, in a normal tiling of the plane, the average number of sides of the tiles is at most 6. This would allow that all tiles have at least six sides and that there are infinitely many tiles with more that six sides. Stehling [625] proved that this is not the case: If in a normal tiling with convex polygons each tile has at least six sides, then all but a finite number of them are hexagons. A simple proof of this result was given by Akopyan [7]. He also gave an upper bound on the number of tiles with more than six sides: If the plane is tiled with convex polygons, which have at least six sides, have area at least 𝐴 and diameter not greater than 𝐷, then the number of tiles with more than six sides is at most 𝜋𝐷 2 /𝐴. Frettlöh, Glazyrin and Lángi [307] proved a counterpart of Stehling’s theorem by showing the existence of normal edge-to-edge tilings in which all tiles have at least six edges and the number of non-hexagon tiles is arbitrarily large. Their construction shows that Akopyan’s bound is asymptotically tight.

8.1.6 Notes on Section 1.7 A polyhedron with regular faces and equivalent vertices is named by Coxeter [159] uniform. The convex uniform polyhedra are the Platonic solids, the Archimedean solids and the semiregular prisms and antiprisms. But if one allows star-polyhedra as well, then in addition to the solids named above, besides the four regular starpolyhedra, a wealth of other polyhedra become included. A list of the uniform star-polyhedra, with beautiful drawings and photographs, is contained in Coxeter, Longuet-Higgins and Miller [166]. The proof that the list is complete was given independently by Sopov [619] and Skilling [614]. Messer [518] presented the main combinatorial and metrical quantities of uniform polyhedra and their duals. The concept of semiregularity of a polyhedron or of a tiling can be generalized in various ways. Johnson [429] and Zalgaller [667,668,669] have, independently of each other, enumerated all convex polyhedra with regular faces (see also Grünbaum

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and Johnson [359]) showing that there are 92 such polyhedra besides the regular ones. Among them there are 5 non-regular polyhedra with congruent faces. These have been enumerated previously by Freudenthal and van der Waerden [308]. All five have triangular faces. In a sense, the Freudenthal–van der Waerden polyhedra and the non-regular isogonal spherical tilings enumerated by Heppes [405] are dual to each other. The latter are characterized as those with equiangular faces and congruent vertex-figures.

8.1.7 Notes on Section 1.8 Alternative proofs, applications and accounts of the history of Lexell’s theorem can be found in the papers by Maehara [496], Maehara and Martini [499], Papadopoulos [558], Simonič [613], Atzema [26], and Papadopoulos and Su [559]. The latter paper also gives the proof for the analogous result for the hyperbolic plane.

8.1.8 Notes on Section 1.11 Regarding the theory of convex bodies, we refer the reader to the textbooks by Yaglom and Boltjanski˘ı [664], Hadwiger [365, 366], Dinghas [182], Hadwiger and Debrunner [368, 370, 371], Eggleston [195], Alexandrov [10], Lyusternik [494], Boltjanski˘ı and Gohberg [108], Burago and Zalgaller [144], Leichtweiß [476, 478] Schneider [593] and Gruber [354]. The papers by Fenchel [272] and Gruber [346] give nice accounts of the history of convexity. As references to tiling and polyhedra we suggest Brøndsted [140], Grünbaum [358], Grünbaum and Shephard [360, 361, 362], Zalgaller [669], Ziegler [670], McMullen and Shephard [510], Böhm and Hertel [103] McMullen and Schulte [511] and Schulte [600]. The Handbook of Convex Geometry edited by Gruber and Wills [355] is an encyclopedic collection of survey articles on various aspects of convexity.

8.2 Notes on Chapter 2 8.2.1 Notes on Section 2.1 In connection with inequalities linking the verious deviations between convex bodies we mention the paper by Groemer [341], which gives explicit inequalities relating the Hausdorff distance to the volume deviation and the surface area deviation.

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8.2.2 Notes on Section 2.2 Maligranda [504] gives a nice survey of the history of the inequalities discussed in Section 2.2. Leuenberger [480] sharpened the inequality (2.2.2) by taking in place of 𝑟 𝑖 the length 𝑤 𝑖 of the bisector of the angle 𝑃𝑖 𝑂𝑃𝑖+1 . With the notation from page 35, the inequality 𝑅1 + · · · + 𝑅𝑛 ≥ (𝑟 1 + · · · + 𝑟 𝑛 ) sec(𝜋/𝑛) holds. This inequality, posed by L. Fejes Tóth [229] as a conjecture, was proved by Florian [280] for 𝑛 = 4, and by Lenhard [479] in general.

8.2.3 Notes on Section 2.3

54 ▶

Modified proofs of Dowker’s theorem are found in L. Fejes Tóth [232, 247]. Analogous theorems can be proved which consider the perimeter instead of the area. The corresponding four theorems also hold true in spherical and hyperbolic geometry (Molnár [524], L. Fejes Tóth [239]). A hyperconvex disk of radius 𝑟 is a planar set with nonempty interior that is the intersection of closed circular disks of radius 𝑟. A convex disk-polygon of radius 𝑟 is a set with nonempty interior that is the intersection of a finite number of closed circular disks of radius 𝑟. G. Fejes Tóth and Fodor [223] considered diskpolygons inscribed in and circumscribed about hyperconvex disks and established the corresponding Dowker-type theorems for planes of constant curvature with the exception of the case concerning the perimeter of circumscribed polygons on the sphere. The questions on page 54 asking whether certain other theorems of Dowkertype hold were answered by Eggleston [194]. He showed that the minimum of the perimeter deviation of a given convex disk from an 𝑛-gon is always attained by a polygon inscribed in the disk, thus this sequence is convex. Further he showed that the sequence 𝜏(𝐸, 𝑃 𝑛 ), the minimum of the area deviation, is also convex. However, none of the Dowker-type theorems hold for the Hausdorff distance. Fodor [305] proved that the above mentioned theorem of Eggleston concerning the minimum of the perimeter deviation of a given convex disk from an 𝑛-gon also holds in the hyperbolic plane, but it fails on the sphere. This leaves open the question whether the sequence of minimum perimeter deviations of 𝑛-gons from a given convex disk on the sphere is convex. G. Fejes Tóth [206] united the three theorems concerning area deviation into a single theorem. Let 𝑝 ≥ 1 be a finite or infinite number. Write 𝑞 = 𝑝/( 𝑝 − 1), where 1/0 = ∞ and ∞/∞ = 1. For two convex disks 𝐴 and 𝐵 the weighted area deviation

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195

𝜏𝑝 ( 𝐴, 𝐵) of 𝐵 from 𝐴 is defined as 𝜏𝑝 ( 𝐴, 𝐵) = 𝑝( 𝐴\𝐵) + 𝑞(𝐵\𝐴). Let 𝑃𝑛 be an 𝑛-gon of minimum weighted area deviation from 𝐸. Then the sequence 𝜏𝑝 (𝐸, 𝑃𝑛 ) is convex. Suppose that 𝐸 has 𝑘-fold rotational symmetry. For all cases when the sequence of deviations of the best approximating polygons from 𝐸 is convex, it follows that among the best approximating 𝑘𝑙-gons there is one with 𝑘-fold rotational symmetry, as G. Fejes Tóth and L. Fejes Tóth [218] noticed. The perimeter deviation, as defined on page 33, also used in Egglestone’s article mentioned above, fails to satisfy the triangle inequality. An alternative definition of perimeter deviation between convex disks 𝐾 and 𝐿 was given by Florian [291], designed to satisfy the triangle inequality: 𝜚 𝑃 (𝐾, 𝐿) = 2𝑝(conv(𝐾 ∪ 𝐿)) − 𝑝(𝐾) − 𝑝(𝐿), where 𝑝(𝑆) denotes the perimeter of 𝑆. Let 𝜚 𝑛 (𝐶) denote the minimum value of 𝜚 𝑃 (𝐶, 𝑃𝑛 ) over all polygons 𝑃𝑛 with at most 𝑛 sides. It is an open question whether the sequence 𝜚 𝑛 (𝐶), 𝑛 = 3, 4, . . ., is convex. In space, theorems analogous to Dowker’s do not hold, respectively are not expected to hold. Indeed, if 𝑣 𝑛 denotes the maximum-volume √ √ polyhedron with 𝑛 vertices inscribed in the unit ball, then 𝑣 4 = 8 3/27, 𝑣 5 = 3/2, 𝑣 6 = 4/3, hence 𝑣 4 + 𝑣 6 > 2𝑣 5 . Other examples can be found in Rodríguez-Arias Fernández [576].

8.2.4 Notes on Section 2.4 Chakerian [148] proved the following theorem. Suppose that 𝑝 is a polygon inscribed in a convex disk 𝑇, and 𝑃 is the polygon circumscribed about 𝑇 so that its sides are parallel to the corresponding sides of 𝑝. Then 𝑝 · 𝑃 ≤ 𝑇 2 . As a corollary, he √ 2𝜋 2𝜋 csc for 𝑛 ≥ 5. The inequalities for proved 𝑈3 ≤ 2𝑇, 𝑈4 ≤ 2𝑇 and 𝑈𝑛 ≤ 𝑇 𝑛 𝑛 𝑈3 and 𝑈4 were previously proved by Chakerian and Lange [149]. As the example of parallelograms show, the inequality for 𝑈3 is the best possible. Concerning the √ inequality for 𝑈4 W. Kuperberg [460] showed the existence of√a constant 𝑐 < 2 such that 𝑈4 ≤ 𝑐𝑇 without producing any explicit value below 2. He conjectured that the extreme convex disk 𝑇 in this respect is the regular pentagon for which the minimum of the area of the circumscribed quadrilateral is √3 𝑇 = 1.34164 . . . 𝑇. 5 For a centrally symmetric convex disk 𝑇, Petty [561] proved that the minimum area parallelogram 𝑃 circumscribed about 𝑇 satisfies 𝑃 ≤ 34 𝑇 and equality holds only if 𝑇 is an affine regular hexagon. For 𝑛 ≥ 5 Ismailescu [423] improved 𝜋 Chakerian’s bound to 𝑈𝑛 ≤ 𝑇 sec . Ismailescu derived his inequality from the 𝑛

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following theorem. For every convex disk 𝑇 the inequality 𝑈𝑛 𝜋 ≤ sec 𝑇2𝑛 𝑛

41 ▶

holds. The problems about the perimeter of inscribed and circumscribed polygons posed on page 41 were solved by Schneider [591, 592]. Namely, he proved the following: If 𝑇 is a convex disk of perimeter 𝐿, 𝐿 𝑛 is the minimum perimeter of a circumscribed 𝑛-gon and 𝑙 𝑛 is the maximum perimeter of an inscribed 𝑛-gon, then the inequalities 𝜋 𝑛 tan , 𝜋 𝑛 𝑛 𝜋 𝑙 𝑛 ≥ 𝐿 sin 𝜋 𝑛

𝐿𝑛 ≤ 𝐿

hold. For the first one, Schneider [592] proved that equality holds for the circle only, but for the second one, he proved the same for a finite number of values of 𝑛 only. Florian and Prachar [299] settled the case of equality in the second inequality for all 𝑛. Thus, in the sense of perimeter deviation, the circle is the worst approximated both by inscribed and circumscribed 𝑛-gons, and in view of Eggleston’s theorem mentioned above, also by general 𝑛-gons. The same holds for approximation in the sense of the perimeter deviation 𝜚 𝑃 . Florian [293, 297] gave two different proofs of the following theorem: If 𝑇 is a convex disk of perimeter 𝐿, then    1 𝜋 2𝑛 arcsin sin 𝜚 𝑛 (𝑇) ≤ 𝐿 1 − , 𝜋 2 𝑛 where equality occurs if 𝑇 is a circle. In order to prove the existence of economic packings and coverings we want to approximate convex disks by polygons that tile the plane. The theorem of Sas implies that every convex disk 𝑇 contains a quadrilateral of area 𝜋2 𝑇. Besicovitch [59] proved a slightly stronger result, namely he showed that 𝑇 contains a centrally symmetric hexagon of area 32 𝑇. By considering a broader class of hexagons, the above area bound can be strengthened. A 𝑝-hexagon, introduced by W. Kuperberg [459], is a convex hexagon, possibly degenerate, with a pair of two parallel opposite sides of equal length. W. Kuperberg [462] proved that for every convex disk 𝑇 there is a pair ℎ and 𝐻 of 𝑝-hexagons inscribed in, and circumscribed about 𝑇, respectively, such that 𝐻ℎ ≤ 43 . The inequality is the best possible, as equality occurs if 𝑇 is an ellipse. It implies that ℎ ≥ 43 𝑇. Later W. Kuperberg [463] showed that 𝑇 contains √ an even larger 𝑝-hexagon of area [(2 3 + 3)/8]𝑇 = (0.808 . . .)𝑇.√Elaborating on Kuperberg’s proof, E.H. Smith [615] proved that the constant (2 3 + 3)/8 is not the best possible, without producing a greater one. Subsequently, Ismailescu [421] proved that every convex disk 𝑇 contains a 𝑝-hexagon of area 0.8142 . . . 𝑇.

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The “linking inequality” 𝐻ℎ ≤ 43 also immediately implies the existence of a 𝑝-hexagon containing 𝑇 with area at most 34 𝑇. This was proved previously by 32 by the W. Kuperberg [459], and the constant 34 was subsequently lowered to 25 same author in [461]. The question of the convex disk that can be best approximated by polygons with at most 𝑛 sides is uninteresting, however, the problem becomes meaningful if we consider only some class of convex disks. Such a class, which plays a particular role in packing and covering problems, is C(𝑎, 𝑝), consisting of convex disks whose area is at most 𝑎 and whose perimeter is at least 𝑝, where 𝑎 and 𝑝 are fixed and satisfy the isoperimetric inequality 𝑝 2 /𝑎 ≥ 4𝜋. We are looking for the convex disks within the class C(𝑎, 𝑝) that are best approximated by polygons with at most 𝑛 sides. For approximation in the sense of area deviation the problem was solved by L. Fejes Tóth [238] for circumscribed, by G. Fejes Tóth and Florian [222] for inscribed, and by Florian [289] for general polygons. Florian [294] settled the problem for general polygons and the perimeter deviation 𝜚 𝑃 and in [292] he extended the results for spherical convex disks.

8.2.5 Notes on Section 2.5 Affine perimeter and affine surface area play an important role in several branches of convexity. The paper of Leichtweiß [477] gives a nice survey of the history and significance of these notions.

8.2.6 Notes on Section 2.7 Modern treatments of integral geometry can be found in the following books and surveys: Santaló [587, 588], Gelfand, Graev and Vilenkin [316], Gelfand, Gindikin and Graev [317], Ren [573], Schneider and Weil [594,595,596] Schneider and Wieacker [597] and Ambartzumian [14].

8.2.7 Notes on Section 2.8 On page 55 it is stated that every convex body 𝐾 of volume 𝑉 contains a convex polyhedron with 𝑛 vertices of volume at least as large as one contained in an ellipsoid of volume 𝑉. The question concerning uniqueness was answered by Bianchi [87] for 𝑛 = 5. The conjecture stated on page 55 about embedding a spherical convex curve in a triangular ring of minimum area was proved by L. Fejes Tóth [236]. As a nice application of spherical polarity, it can be shown that this theorem is equivalent to

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the corresponding theorem for the perimeter. Analogous theorems for the 𝑛-sided rings have not yet been established.

8.3 Notes on Chapter 3 8.3.1 Notes on Sections 3.1–3.2 For brevity and ease of expression, we will use the following notation: Given a convex disk 𝐾, the maximum density of a packing of the plane with congruent copies of 𝐾 is denoted by 𝛿(𝐾) and is called the packing density of 𝐾. With the restriction that only translates of 𝐾 are used, the corresponding maximum density is called the translative packing density of 𝐾 and denoted by 𝛿𝑇 (𝐾). Further restriction to lattice packings of 𝐾 results in the lattice packing density of 𝐾, denoted by 𝛿 𝐿 (𝐾). The corresponding notions of covering density of 𝐾, denoted by 𝜗(𝐾), translative covering density of 𝐾, denoted by 𝜗𝑇 (𝐾), and lattice covering density of 𝐾, denoted by 𝜗𝐿 (𝐾) are defined quite similarly. The use of maximum and minimum in place of supremum and infimum is justified by results of Groemer [335, 337, 338, 339], who proved the existence of arrangements with the corresponding extreme densities. With these notations the theorems of Thue and Kershner state that for the unit circle 𝐵2 we have 𝛿(𝐵2 ) = 𝛿 𝐿 (𝐵2 ) = √𝜋 and 𝜗(𝐵2 ) = 𝜗𝐿 (𝐵2 ) = √2 𝜋 . The basic 27 12 problem in the theory of packing and covering is to determine, or at least give bounds, for the packing and covering densities of different classes of convex bodies. Of course, we have 𝛿(𝐾) = 𝜗(𝐾) = 1 for all disks that tile the plane and 𝛿𝑇 (𝐾) = 𝜗𝑇 (𝐾) = 1 for all disks whose translates tile the plane. Moreover, the conditions 𝛿(𝐾) = 1, 𝜗(𝐾) = 1, and 𝐾 tiles the plane, as well as the conditions 𝛿𝑇 (𝐾) = 1, 𝜗𝑇 (𝐾) = 1, and 𝐾 translatively tiles the plane, are equivalent (see e.g. Groemer [332]). For translative arrangements Groemer [340] proved the stability of these results: A convex disk whose translational packing density or translational covering density is close to 1 is in Hausdorff metric close to a translational tile. Prosanov [570] proved a similar theorem: If the translational packing density of a convex body is close to 1, then so is its translational covering density, and vice versa.

8.3.2 Notes on Section 3.3 Two types of partitions associated with a discrete set of points occur in different parts of the book. The first one associates to each element of a discrete set of points the region consisting of points that are closer than or as close to the point as to any other point of the set. The use of this partition is natural in several branches of mathematics and several authors introduced it without knowing of its earlier occurrence in the literature. It appears that Dirichlet [183] defined it for the first time in the plane,

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and Voronoi [649] for higher dimensions. Still today, the terminology is not unified: Dirichlet cell, Voronoi region and Dirichlet–Voronoi region are the most commonly used terms for the parts of the partition. In these notes we shall use the term Dirichlet cell. The other partition is a triangulation, essentially dual to the first one, where the points from the set are the vertices of the triangles, and the edges connect points whose Dirichlet cells are adjacent. The triangulation is named after Delone who first defined them and described its properties in [179]. Both partitions are defined in Euclidean, in spherical, as well as in hyperbolic space of any dimension. For a finite set of points on the sphere, the two partitions can be described in a particularly nice way: Dirichlet cells are obtained by the radial projection of the faces of the polyhedron bounded by the planes tangent to the sphere at the points of the set, while the cells of the Delone triangulation are the projections of the faces of the convex hull of the set. According to Rogers [580, p. 11], the first article of Thue [641] is all too brief for a complete reconstruction of his proof; and while the second article [642] is convincing, the proof of compactness is missing there, which may be not so easy to fill in. It therefore appears that the first complete proof of the inequality (3.2.1) was given by L. Fejes Tóth in [228]. Segre and Mahler [606] gave another proof: The density of a set of non-overlapping congruent circles contained in a polygon with √ angles at most 120◦ cannot exceed 𝜋/ 12. Analogous theorems for the hyperbolic and spherical planes were proved by Molnár [526]. Recall from Section 8.2.3 that a hyperconvex disk of radius 𝑟 is a planar set with nonempty interior that is the intersection of closed circular disks of radius 𝑟. L. Fejes Tóth [259, 260] gave the following generalizations of the inequalities (3.2.1) and (3.2.2). If hyperconvex disks of radius 𝑟 with average area 𝑎 and average perimeter 𝑝 are packed in the plane then the density 𝑑 of the disks satisfies the inequality 𝑑≤

𝑎 𝑝 𝑎 + 6𝑟 2 tan 12𝑟 −

𝑝𝑟 2

.

If hyperconvex disks of radius 𝑟 with average area 𝑎 and average perimeter 𝑝 cover the plane so that no two disks cross then the density 𝐷 of the disks satisfies the inequality 𝑎 . 𝐷≥ 𝑝𝑟 𝑎 − 2 + 3𝑟 2 sin 6𝑟𝑝 For congruent circular disks these inequalities yield the sharp bounds (3.2.1) and (3.2.2). In the inequality for packings equality occurs only for congruent circles. On the other hand, the inequality for coverings is sharp in many cases. Coverings for which equality is attained can be obtained as follows. Start with a trihedral tiling of the plane by equilateral polygons, e.g. with one of the Archimedean tilings (3, 12, 12), (4, 8, 8) or (4, 6, 12), and replace each of the tiles by the corresponding convex disk-polygon of radius 𝑟. Figure 8.3 shows a covering of the plane with pentagonal, hexagonal, and octagonal disk-polygons obtained this way.

200

8 Notes

Fig. 8.3 √ Inspired by the proof of the inequality 𝑃𝑖 𝑈 ≥ 𝑆 = 12 on page 64, Hajós considered the following problem: Given a pair of concentric circles, find the minimum area polygon among those that contain the smaller circle and whose vertices are outside of the larger circle. The answer given by the so-called Hajós lemma states that the optimal polygon is inscribed in the greater circle so that, with the possible exception of one, all sides are tangent to the smaller circle. The lemma found application in various packing problems. A. Bezdek and Joós [70] proved a generalization of the lemma to non-concentric pairs of circles.

8.3.3 Notes on Section 3.4 As an improvement of inequality (3.4.1), Groemer [329] gave the following upper bound on the number 𝑛 of unit circles that can be packed in a convex disk of a given area 𝐹 and perimeter 𝑈: √ √ √ √ 2− 3 12𝑛 ≤ 𝐹 − 𝑈 + 12 − 𝜋( 3 − 1). 2 Groemer’s proof was simplified by Molnár [532]. The case of equality in the above bound occurs, roughly speaking, if the circles form a “convex hexagonal cluster” (degenerate hexagons allowed) of the densest lattice circle packing as shown in Figure 8.4. After Wegner [655], such arrangements are called Groemer packings. It is conjectured (see L. Fejes Tóth [256]) that a packing of 𝑛 congruent circles that minimizes the area of their convex hull must be a Groemer packing. Wegner [655] proved that the area of the convex hull of 𝑛 non-overlapping unit circular disks

8.3 Notes on Chapter 3

201

Fig. 8.4

is at least

m √ √ l√ 2 3(𝑛 − 1) + (2 − 3) 12𝑛 − 3 − 3 + 𝜋,

which confirms the conjecture for an infinite number of cases. In [654] he studied the unconfirmed cases, the first one being 𝑛 = 121. Böröczky Jr.√and Ruzsa [133] proved that equality occurs in Wegner’s inequality if and only if ⌈ 12𝑛 − 3−2⌉ +3− 12𝑛 ≠ (3𝑘 − 1)9𝑚 for some positive integers 𝑘 and 𝑚, and found that the proportion of these numbers within the set of positive integers is exactly 23/24. They also showed that the bound in Wegner’s theorem is very good even if strict inequality holds: For any 𝑛 ≥ 2, there exists a Groemer packing of 𝑛 unit circular disks whose convex hull has area at most m √ √ l√ 2 3(𝑛 − 1) + (2 − 3) 12𝑛 − 3 − 2 + 𝜋. Kenn [443] gave an alternative characterization of the integers 𝑛 for which equality holds in Wegner’s theorem. A similar problem asks for the minimum diameter of a convex set that can hold 𝑚 mutually nonoverlapping unit circles. Equivalently, what is the minimum ratio 𝑟 (𝑚) of the maximum to the minimum distance between 𝑛 points? Bateman and Erdős [53] established the values of 𝑟 (𝑚) for 𝑚 ≤ 7. A. Bezdek and Fodor [69] determined 𝑟 (8). Audet, Fournier, Hansen and Messine [27] gave upper bounds for 𝑟 (𝑚) for 𝑛 ≤ 30. Schütte [603] considered the same problem in 𝑛 dimensions and solved it for 𝑚 = 𝑛 + 2 points. The optimal set consists of two groups of points, of 𝑛+2 cardinality ⌊ 𝑛+2 2 ⌋ and ⌈ 2 ⌉, lying in orthogonal subspaces and forming the vertices of regular simplices centered at the origin. Alternative proofs of Schütte’s theorem were given by Seidel [607], Dekster and Wilker [178], and Bárány [47]. In relation to the aforementioned result of Wegner, one can formulate a dual problem for coverings: Find the maximum area and maximum perimeter of a convex disk that can be covered by 𝑛 unit circles. The “area” version of this problem is still unsolved, but the “perimeter” version has been settled by G. Fejes Tóth, Gritzmann, and Wills [224]: The maximum perimeter is attained by the arrangement in which √︁ the 𝑛 centers are collinear and equally spaced by the distance of 2 1 − (1/𝑛2 ), see

202

8 Notes

Figure 8.5. The result is even stronger than required by the problem: It only suffices to assume that the boundary of the convex disk is covered by the circles.

Fig. 8.5 We call a region circle-convex of radius 𝜚, in short 𝜚-convex, if to each of its boundary points there exists a support circle (a circle containing the boundary point while its interior is outside the region) of radius 𝜚. Molnár [527] extended (3.4.1) to packings of unit circles in a 𝜚-convex region 𝑇 with 𝜚 ≥ 4.76 . . . and G. Fejes Tóth [211] generalized (3.4.2) to coverings by unit circles of a 𝜌-convex regions with 𝜚 ≥ 16.74 . . .. A further generalization of (3.4.1) is due to Heppes. We say that a region is an 𝑜-region or an 𝑖-region if a unit circle can roll freely along the region’s boundary outside, or inside the region, respectively. In the paper by Heppes and Molnár [413] the following theorem is stated without proof: If 𝑛 𝑖-regions are packed in an 𝑜-region, then the area of the unoccupied portion of the container is at least 2(𝑛 − 1)𝑡, where 𝑡 is the area of the “arc-sided triangle” bounded by three mutually tangent unit circles. Equality is attained here in a variety of cases (see Figure 8.6). The proof of a slightly more general result, valid also for the spherical and hyperbolic plane, was given in Heppes [411].

Fig. 8.6

8.3.4 Notes on Section 3.6 In connection with Section 3.6, we mention the following problem. For a convex disk 𝐶 determine 𝑑 𝑛 (𝐶), the supremum of the densities among all packings of positively-homothetic copies of 𝐶 of at most 𝑛 different sizes. It is conjectured that

8.3 Notes on Chapter 3

203

𝑑 𝑛 (𝐶) = 1 − (𝑑1 (𝐶)) 𝑛 . In these terms, the result mentioned in Section 3.6 confirms the conjecture if 𝐶 is the circle. L. Fejes Tóth [263] confirmed the conjecture for the triangle. The analogous problem for coverings appears to be much more difficult. The only result obtained thus far is by G. Fejes Tóth [211], solving the problem of the thinnest covering with circles of two different radii.

8.3.5 Notes on Section 3.7 The homogeneity of an arrangement of bodies {𝐾𝑖 } is defined as inf 𝐾𝑖 /𝐾 𝑗 . It is pointed out on page 79 that circles that do not differ too much from each other cannot be packed more densely than congruent circles. This remark leads to the following theorems: The density of a circle packing {𝐾𝑖 } with homogeneity 6 tan(𝜋/6) − 7 tan(𝜋/7) = 0.5520 . . . 5 tan(𝜋/6) − 5 tan(𝜋/5) √ is less than or equal to 𝜋/ 12. ℎ≥

The density of a circle covering {𝐾𝑖 } with homogeneity 7 sin(2𝜋/7) − 6 sin(2𝜋/6) = 0.6275 . . . 6 sin(2𝜋/6) − 5 sin(2𝜋/5) √ is greater than or equal to 2𝜋/ 27. ℎ≥

These theorems, along with their complete proofs, were presented by K. Böröczky in a lecture at the Mathematical Institute of the Hungarian Academy of Sciences in 1967. They were rediscovered by G. Blind [92]. The first of the two theorems was proved by Florian [284] under the stronger assumption ℎ ≥ 0.82. G. Blind and R. Blind [100] sharpened this result in the spirit of inequality (3.4.1) by showing that if at least two circles satisfying the inequality 𝐾𝑖 /𝐾 𝑗 ≥ 0.6275 . . . are packed √ in a convex disk, then their density in the disk is at most 𝜋/ 12. Connelly and Pierre [155] constructed a packing {𝐾𝑖 } consisting of circles with three different radii showing that the first theorem does not hold for homogeneity ℎ < 0.43366 . . .. Let 𝑑 (ℎ) be the least upper bound of the densities of packings of circles of homogeneity ℎ and let 𝐷 (ℎ) be the greatest lower bound of the densities of coverings with circles of homogeneity ℎ. Consider three circles with homogeneity at least ℎ so that at least one of them is touched by the other two and none of them contains a segment of the line through the centers of the other two circles. Let 𝑠(ℎ) be the supremum of the density of three such circles with respect to the triangle determined by the circles’ centers. In a similar way we define 𝑆(ℎ) as the infimum of the density of three circles with homogeneity at least ℎ, having a common boundary point but no common interior point, with respect to the triangle spanned by their centers.

◀ 80

204

8 Notes

L. Fejes Tóth and Molnár [271] proved that 𝑑 (ℎ) ≤ 𝑠(ℎ) and 𝐷 (ℎ) ≥ 𝑆(ℎ). They conjectured that ℎ 𝜋ℎ + 2(1 − ℎ) arcsin 1−ℎ 𝑠(ℎ) = √ 2 ℎ + 2ℎ2 and √ 2 𝜋 − 2(1 − ℎ) arctan √1−𝑥 ℎ+𝑥 𝑆(ℎ) = min . √ √ 0≤𝑥 ≤1 2 1 − 𝑥 2 ( ℎ + 𝑥) √ This √ means that 𝑠(ℎ) is attained by three mutually touching circles of radii ℎ, √ ℎ and 1, and 𝑆(ℎ) is attained by the symmetric position of a circle of radius ℎ and two unit circles. The conjecture for packings was confirmed by Florian [281]. After some partial results by Florian [282, 283] and Dorninger [186], the conjecture for coverings was proved by Dorninger [187, 188]. Blind [93, 94] sharpened the bound 𝑠(ℎ) for 0.4536 . . . < ℎ < 0.5520 . . . and the bound 𝑆(ℎ) for 0.4485 . . . < ℎ < 0.6275 . . .. The lower bound for 𝑑 (𝑞) shown in Figure 3.14 was improved for certain values of 𝑞 by Molnár [525] (see also Heppes and Molnár [413] and L. Fejes Tóth [247]). Molnár [525] and Jucovič and Lešo [432] gave upper bounds for 𝐷 (𝑞). The inequalities (3.7.3) and (3.7.4) have been extended by G. Fejes Tóth [209]: the first one for 𝛼 ≤ 0.912 . . . and the second one for 𝛼 ≥ 1.226 . . .. Another pair of bounds involving power means was given by L. Fejes Tóth [257, 258]: ( ) −1 √ 12 − 𝜋 𝑀1/3 (𝐾1 , . . . 𝐾𝑛 ) 𝑑 ≤ 1+ , 𝜋 𝑀1 (𝐾1 , . . . 𝐾𝑛 ) ) −1 √ ! 27 𝑀1/3 (𝐾1 , . . . 𝐾𝑛 ) 1 1− 𝐷 ≥ 1− , 4 4𝜋 𝑀1 (𝐾1 , . . . 𝐾𝑛 ) (

where 𝑑 and 𝐷 are the densities of a packing in, and covering of, a hexagon, respectively, with 𝑛 circles 𝐾1 , . . . 𝐾𝑛 of arbitrary radii. Here, in accordance with the notation on page 30, 𝑀 𝛼 (·) stands for the power mean of degree 𝛼. Consider an infinite sequence of circles 𝐾1 , 𝐾2 , . . . for which lim 𝑛→∞

79 ▶

𝑀1/3 (𝐾1 , . . . 𝐾𝑛 ) ≠ 0. 𝑀1 (𝐾1 , . . . 𝐾𝑛 )

For such a sequence of circles, the inequalities above imply the existence of constants 𝑐 < 1 and 𝐶 > 1 such that for every 𝑛 the density of a packing of the first 𝑛 members of the sequence in a convex hexagon is at most 𝑐 and the density of a covering of a hexagon by the first 𝑛 members is at leat 𝐶. On the other hand, A. Bezdek [65] showed that the plane can be packed,Íand can also be covered with density 1 by any infinite set of circles {𝐾𝑖 } for which 𝐾𝑖 = ∞ and lim 𝐾𝑖 = 0. The conjecture on page 79 concerning the face-incircles of the tiling (4, 8, 8) was proved by Heppes [410].

8.3 Notes on Chapter 3

205

8.3.6 Notes on Section 3.8 Let 𝑂 be a point in a region 𝐺, let 𝑎(𝑥) be a decreasing function defined for 𝑥 ≥ 0 and let d 𝑓 denote the area element at the point 𝑃. The moment of 𝐺 with respect to 𝑂 is defined as the integral ∫ 𝑀 (𝐺, 𝑂) = 𝑎(𝑂𝑃)d 𝑓 . 𝐺

Remarks 1 and 2 on pages 83–83 immediately yield the following moment lemma: Among all convex 𝑛-gons of equal area, the maximum moment is attained by the regular 𝑛-gon centered at 𝑂. Hajós [373] provided an elementary geometric proof of this lemma. The inequality (3.8.4) proved in Section 3.8, also known as the moment-sum theorem, is a valuable tool for various applications. Applications to location theory and point distribution problems are found in Papadimitriou [557], Morgan and Bolton [536], Bollobás [105], Bourne, Peletier and Theil [137], Haimovich and Magnanti [372], Bollobás and Stern [106], Babenko [30], Gersho [320], Gruber [352], and Akimova [5]. Stability of the optimal arrangement showing that if the sum of moments is close to the optimal value then most of the Dirichlet cells are close to the regular hexagon of the given area were established by Gruber [350,351] and G. Fejes Tóth [215]. In a previous paper Gruber [348] proved a similar stability statement for Kershner’s theorem about the thinnest covering by congruent circles. The moment-sum theorem and its generalizations play an important role in the approximation of convex bodies by polytopes, discussed in Chapter 5. Because of the importance of the moment-sum theorem in different areas, the theorem has been independently rediscovered a few times and new proofs have been given. New proofs and generalizations are found in Florian [287], Gruber [349], G. Fejes Tóth [203, 204], and Böröczky Jr. and Csikós [129]. In contrast to dimension 2, in higher dimensions exact bounds for the sum of the moments of Dirichlet cells are not known. Zador [666] proved a bound for the sum of the moments of Dirichlet cells with respect to 𝑎(𝑥) = 𝑥 𝛼 , 𝛼 > 0, in 𝑛-dimensional space. Applications to optimum quantization, approximation of probability measures by discrete measures, and estimation of the error in numerical integration are treated in Gruber [353]. The inequality (3.8.3) remains true for an arbitrary convex disk in place of a polygon with at most six sides, provided that the number of circles is at least two (see G. Fejes Tóth [217]). This is a common generalization of (3.4.1) and (3.4.2). The inequality (3.8.3) is obtained in Section 3.8 as a corollary to (3.8.4). It should be noted, however, that (3.8.3) implies (3.8.4) through approximations by step-functions. Some of the new proofs make use of the equivalence of the two inequalities. The assumption in (3.8.3) that the circles are congruent can be relaxed, as long as their sizes do not differ too much, see G. Fejes Tóth [203].

206

8 Notes

Steinhaus [630] considered the problem of dissecting a convex disk of variable mass distribution into 𝑛 pieces so that the sum of the moments of inertia of the pieces about their respective centroids are as small as possible. It was shown by L. Fejes Tóth [241] that the pieces are asymptotically “nearly” regular hexagons whose moments of inertia are all equal. Heppes and Szüsz [414] proved that if the curves of constant mass density are closed, the dissection will always contain a “line of dislocation” in the crystallographic sense.

8.3.7 Notes on Section 3.9 The inequality (3.9.1) implies that if the Euclidean plane is partitioned into convex cells of unit area, then the average perimeter of the cells is at least as large as the perimeter of a regular hexagon of unit area. In other words, the regular hexagonal tiling of the plane minimizes the average perimeter of the cells. The general conjecture stating the same without the assumption of convexity of the cells (the cells should be measurable, of unit area, and with rectifiable boundaries), became known as the honeycomb conjecture. This old conjecture was solved by Hales [379], who also gave a nice, exhaustive account of the history of the problem (see also the survey by Morgan [535]). Let a given region be packed with a given large number of convex disks of equal areas or of equal perimeters. Under which shape and arrangement of the disks does the sum of their perimeters attain its minimum or does the sum of their areas attain its maximum if the common area 𝑎 or the common perimeter 𝑝 of the disks is prescribed? These two problems lead to the same extreme designs. If we let the value of 𝑎 or 𝑝 increase starting from 0, then at first we have arbitrarily arranged circles, that at a certain value of 𝑎 or 𝑝 turn into the densest packing of circles. Then the disks enlarge themselves into “smooth hexagons” that are obtained from regular hexagons by rounding off their vertices with congruent circular arcs. Finally the disks turn into regular hexagons that fill the region completely. For a precise formulation and proofs see L. Fejes Tóth and Heppes [269] and L. Fejes Tóth [238]. As shown by Heppes [404], the condition of convexity in the problem about disks of equal perimeters can be omitted, while the corresponding generalization for the disks of equal areas appears to be hard. The dual counterpart of the above problem, covering a given region by a large number of convex disks of give perimeter so as to minimize the total area, was considered by G. Fejes Tóth and Florian [222] and solved under the non-crossing assumption. The corresponding results for packing and covering in which perimeter is replaced by affine perimeter were obtained by L. Fejes Tóth and Florian [268] and extended by Florian [290] to the case where the disks have unequal affine perimeters and only the average of their affine perimeters is given. Suppose a convex disk in the plane is dissected into 𝑛 ≥ 2 convex cells 𝑇1 , . . . , 𝑇𝑛 with perimeters 𝐿 1 , . . . , 𝐿 𝑛 . How small can the power mean 𝑀 𝛼 (𝐿 12 /𝑇1 . . . , 𝐿 2𝑛 /𝑇𝑛 ) of the isoperimetric quotients of the cells be? In this notation, the inequality

8.3 Notes on Chapter 3

207

of an arbitrary convex disk can be expressed as (3.5.1) on p. 71 for dissections √ 𝑀1 (𝐿 12 /𝑇1 , . . . , 𝐿 2𝑛 /𝑇𝑛 ) > 8 3, and the inequality (3.9.1) on p. 85 for dissections of √ a convex hexagon as 𝑀1/2 (𝐿 12 /𝑇1 , . . . , 𝐿 2𝑛 /𝑇𝑛 ) ≥ 8 3. G. Blind and Buchta [101] √ improved these results by showing that the inequality 𝑀 𝛼 (𝐿 12 /𝑇1 . . . , 𝐿 2𝑛 /𝑇𝑛 ) ≥ 8 3 holds with 𝛼 ≥ −2 for dissections of all convex disks, and with 𝛼 ≥ −6.7212 . . . for dissections of convex hexagons. Previously, the inequality was established with 𝛼 = 1/2 for dissections of convex disks by G. Blind [95] and with 𝛼 = 0, that is, for the geometric mean, for dissections of convex hexagons by Buchta [142]. Related results were proved by G. Blind [96], G. Blind and R. Blind [99], L. Fejes Tóth [264], G. Fejes Tóth [205], Kertész [445] and Vásárhelyi [644, 645]. Glazyrin and Morić [323] considered the following question: What is the greatest possible sum of perimeters of 𝑛 pairwise disjoint convex disks contained in a given convex disk 𝐷? They conjectured that if the pairwise disjoint convex disks 𝐷 1 , . . . , 𝐷 𝑛 of perimeters 𝐿 1 , . . . , 𝐿 𝑛 are contained in the convex disk 𝐷 of perimeter 𝐿 and diameter 𝑑, then 𝑛 ∑︁ 𝐿 𝑖 ≤ 𝐿 + 2(𝑘 − 1)𝑑. 𝑖=1

The bound is obviously sharp as it can be arbitrarily closely approximated by dividing 𝐷 into 𝑘 subsets by 𝑘 − 1 parallel lines close to the diameter of 𝐷. Glazyrin and Morić proved the conjecture in the case when 𝐷 is either a square, or any triangle, Í𝑛 and established the bound 𝑖=1 𝐿 𝑖 ≤ 1.22195𝐿 + 2(𝑘 − 1)𝑑 for general convex disks 𝐷. Pinchasi [565] confirmed the conjecture in full generality. Further results concerning isoperimetric problems of aggregates of cells have been obtained by McKean, Schreiber and Weiss [509], Bleicher and L. Fejes Tóth [90], and L. Fejes Tóth [240, 243, 245, 248].

8.3.8 Notes on Section 3.10 Concerning the proof of (3.10.2) we mention that the construction of the cells described in the original text contained a flaw. It used a nested family of convex disks similarly as in the construction of the cells for packing, however it was not clear how one could guarantee that the boundary curves of overlapping disks belonging to a given parameter value intersect in at most two points. We replaced that part of the text by the construction given in L. Fejes Tóth. [247, pp. 170–171] A detailed description of an alternative construction of these cells was given by Bambah and Rogers [36]. Crossing pairs of disks in optimal coverings. The condition in (3.10.2) can be stated as a requirement that the disks must not cross each other. Should we wish to remove this condition, it would suffice to show the following: If a hexagon is covered by congruent convex disks, then there is also a crossing-free covering of the hexagon by the same disks. Unexpectedly, this statement is false, being contradicted by the following example due to Heppes. In a square, two diametrically opposite vertices

◀ 89

208

8 Notes

and the midpoints of the sides span a hexagon. The two hexagons so obtained cover the square, but they cross, and there is no other way to cover the square with them. This counterexample shows that the removal of the non-crossing condition from the theorem must be quite difficult. L. Fejes Tóth [255] noticed that in the case of two disks the topological and metric possibilities of a rearrangement are very restricted, hence he asked: Can more than two congruent disks covering a convex region always be rearranged so as to cover the region without crossing each other? This question has been answered in the negative as well, by a series of counterexamples constructed by Wegner [653]. Some partial results towards proving (3.10.2) without the non-crossing restriction use the notion of fatness of convex disks. A convex disk is said to be 𝑟-fat if it is contained in a unit circle and contains a concentric circle of radius 𝑟. Heppes [412] proved that the lowest density among coverings of the plane with congruent, 𝑟-fat ellipses with 𝑟 ≥ 0.8561 can be attained without using crossing pairs, and therefore also in a lattice covering. Subsequently G. Fejes Tóth [216] proved (3.10.2) without the non-crossing assumption for every 𝑟-fat convex disk with 𝑟 > 0.933, and for 𝑟-fat ellipses with 𝑟 > 0.741. Exposing the true nature of the trouble with occurrence of crossings in the thinnest coverings, A. Bezdek and W. Kuperberg [72] presented an example of a convex pentagon 𝑃 𝜀 with the property that in every thinnest covering of the plane with congruent copies of 𝑃 𝜀 crossings must occur. Their construction is as follows.

H P0

Pε

Fig. 8.7 Dissect a regular hexagon 𝐻 into three congruent pentagons by cutting along segments drawn from the center of 𝐻 perpendicularly to three nonadjacent sides of 𝐻, as shown in Figure 8.7. The resulting plane-tiling pentagon 𝑃0 has two right angles and three angles measuring 120◦ each. Keeping the vertices at the 120◦ angles of 𝑃0 fixed, enlarge the pentagon 𝑃0 slightly by moving the vertices at the right angles outside 𝐻 and so that in the modified pentagon the sides emanating from the relocated vertices change their directions by the same small angle 𝜀. Denote the so enlarged pentagon by 𝑃 𝜀 . Obviously, if 𝜀 > 0 is sufficiently small, then 𝑃 𝜀 can no longer tile the plane. If each of the three pentagons into which 𝐻 was partitioned

8.3 Notes on Chapter 3

209

is modified in the same way, then the three congruent copies of 𝑃 𝜀 cover the regular hexagon 𝐻 and they cross each other. The main result is that if 𝜀 is sufficiently small, then in every thinnest covering of the plane with congruent copies of 𝑃 𝜀 crossing pairs must occur. The article includes a brief history of the problem of the thinnest covering with congruent copies of a convex disk. Generalizations of (3.10.1) and (3.10.2). The inequalities (3.10.1) and (3.10.2) have been generalized in various directions. Böröczky, Jr. [128] extended the inequalities to arbitrary convex disks in place of the hexagon, provided that the number of disks is sufficiently high. Böröczky (unpublished) observed that they hold even for affine copies of 𝐾 of total area 𝑛𝐾 whose areas differ only slightly, provided that 𝐾 is not a polygon with at most six sides. G. Fejes Tóth and L. Fejes Tóth [220] gave the following generalization of (3.10.1). Let 𝐾 be a convex disk of unit area in the Euclidean plane. Let 𝑒 𝑘 be the excess over 1 of the area of the 𝑘-gon of minimum area circumscribed about 𝐾. Let 𝐾1 , . . . , 𝐾𝑛 be affine images of 𝐾 such that for any two disks 𝐾𝑖 and 𝐾 𝑗 with 1 ≤ 𝑖, 𝑗 ≤ 𝑛 we have 𝐾𝑖 /𝐾 𝑗 ≤ (𝑒 5 − 𝑒 6 )/(𝑒 6 − 𝑒 7 ). Let 𝐾¯ 𝑖 be the Minkowski sum of 𝐾𝑖 and some other convex disk. Denote the total area of the disks 𝐾1 , . . . , 𝐾𝑛 and 𝐾¯ 1 , . . . , 𝐾¯ 𝑛 by 𝑡 and 𝑡¯, respectively. If congruent copies of 𝐾¯ 1 , . . . , 𝐾¯ 𝑛 are packed in a convex hexagon 𝐻 then 𝐻 ≥ 𝑡¯ + 𝑒 6 𝑡. Observe that (𝑒 5 − 𝑒 6 )/(𝑒 6 − 𝑒 7 ) > 1 unless 𝐾 is a polygon with at most six sides. The generalization of (3.10.1) by Böröczky follows in the special case when 𝐾¯ 𝑖 is congruent to 𝐾𝑖 for 𝑖 = 1, . . . , 𝑛. Based on the inequalities (3.10.1) and (3.10.2), Groemer [331] proved that (a) if the radius of curvature of a convex disk 𝐾 is bounded below by 𝑟, then 𝛿(𝐾) ≤

1  √  1+ 2 3−𝜋

𝑟2 𝐾

and

𝜗𝑇 (𝐾) ≥

1 

1− 𝜋−3

√

3 2



𝑟2 𝐾

and (b) if the radius of curvature of 𝐾 is bounded above by 𝑅, then 𝛿(𝐾) ≤

𝐾 𝐾 6𝑅 2 tan 6𝑅 2

and

𝜗𝑇 (𝐾) ≥

𝐾 . 𝐾 3𝑅 2 sin 3𝑅 2

G. Fejes Tóth [203] gave a common generalization of (3.10.1) and (3.10.2) in the spirit of (3.8.3): Given a convex disk 𝐾, let 𝑓𝐾 (𝑥) denote the maximum area of the intersection of 𝐾 with a convex hexagon of area 𝑥, and let 𝑓¯ denote the smallest concave function such that 𝑓¯ ≥ 𝑓 . Given a convex hexagon 𝑆 and 𝑛 non-crossing congruent copies of 𝐾, let 𝐴 denote the area of the part of 𝐻 covered by the copies of 𝐾. Then   𝑆 ¯ . 𝐴 ≤ 𝑛 𝑓𝐾 𝑛 Moreover, the copies of 𝐾 need not be congruent: it suffices that they are affine copies of 𝐾 whose areas do not differ too much from each other, and whose total area is 𝑛𝐾. This result implies the result of Böröczky stating that (3.10.1) and (3.10.2)

210

8 Notes

hold even for affine copies of 𝐾 of total area 𝑛𝐾 provided that the areas differ only slightly. The inequality 𝐴 ≤ 𝑛 𝑓¯𝐾 (𝑆/𝑛) is sharp for centrally symmetric 𝐾, even if 𝑓¯ ≠ 𝑓 . If 𝑓¯𝐾 (𝑆/𝑛) = 𝑓𝐾 (𝑆/𝑛), then, as 𝑛 → ∞, the bound can be approximated by a block from a lattice arrangement of 𝐾, and if 𝑓¯𝐾 (𝑆/𝑛) ≠ 𝑓𝐾 (𝑆/𝑛), then it can be approximated by a suitably chosen mixture of two lattice arrangements. G. Fejes Tóth [207] constructed examples of centrally symmetric convex disks for which 𝑓¯ ≠ 𝑓 . G. Fejes Tóth and L. Fejes Tóth [221] noted that the phenomenon that by changing the density of the disks the optimal arrangement transforms from one lattice arrangement to another through a mixture of two lattices shows an analogy with the phase transition of crystals. Translational packing and covering. In connection with the theorem of Rogers mentioned on page 96, see Rogers [577] (see also Rogers [579]), we note that his result is more general than (3.10.1) in two ways: It is valid not only for packings in a hexagon, but in arbitrary convex polygons, and the bound is better for non-centrally symmetric convex disks. This is derived from a bound he gave on the area of the polygon 𝑃 determined by the centroids of translates of a strictly convex disk in a finite packing. The disks whose centroids are the vertices are assumed to form a cycle in which the consecutive disks are touching, and the remaining disks in the collection should be contained in the polygon. If the collection fits the optimal lattice packing for the disk, the bound is sharp. Rogers’ result was generalized by Oler [547], who proved the inequality without the restriction that the disks whose centroids are the vertices of 𝑃 form a cycle in which the consecutive disks are touching. In [548] he discussed the case of equality, and in [549] he gave a further generalization where the polygon need not be simple. Additional generalizations of Rogers’ theorem were given by Folkman and Graham [306] in the special case of packing of circles, and by Witsenhausen and Zassenhaus [328] for packing of translates of a convex disk. For coverings, a dual counterpart of Rogers’ theorem was proved by Bambah and Woods [39]. Rogers’ theorem implies that 𝛿𝑇 (𝐾) = 𝛿 𝐿 (𝐾) for every convex disk 𝐾. The same conclusion can be reached by using the theorem from Section 3.10 concerning packing the plane with congruent copies of a centrally symmetric convex disk and a well-known observation of Minkowski: A family of translates of a convex disk 𝐾 forms a packing if and only if the corresponding translates of the centrally symmetric body 21 (𝐾 − 𝐾) forms a packing. L. Fejes Tóth [261] gave an alternative proof going in the reverse way. He gave a proof for a special family of asymmetric convex disks whose centrally symmetric images exhaust the family of centrally symmetric convex disks. In view of Minkowski’s observation, this implies the validity of the theorem for all centrally symmetric disks, and consequently for all convex disks. Hence, for a convex disk 𝐾 the determination of 𝛿𝑇 (𝐾) is reduced to the task of finding the minimum area hexagon circumscribed about 𝐾 − 𝐾. Similarly, in view of (3.10.2) the determination of 𝜗𝑇 (𝐾) for centrally symmetric convex disks 𝐾 is reduced to the task of finding a maximum area hexagon inscribed in 𝐾. For general convex disks these remain difficult tasks, for convex 𝑛-gons, however, Mount and Silverman [538] designed algorithms solving the problems in 𝑂 (𝑛) time.

8.3 Notes on Chapter 3

211

The equality 𝛿𝑇 (𝐾) = 𝛿 𝐿 (𝐾) does not extend over starlike sets, as examples constructed by von Wolf [662] and Groemer [336] show. Bambah, Dumir and Hans-Gill [35] showed that the equality 𝜗𝑇 (𝐶) = 𝜗𝐿 (𝐶) also loses its validity if we replace “centrally symmetric convex disk” by “centrally symmetric starlike set”. A simpler example, composed of nine unit squares forming a cross with arms of length 2, was given by Loomis [493]. For examples in higher dimensions we refer to the survey by Stein [626]. On the other hand, the methods introduced by L. Fejes Tóth in [261] enabled him to extend the equality 𝛿𝑇 (𝐾) = 𝛿 𝐿 (𝐾) over certain non-convex sets. On the boundary of a convex disk 𝑈 let 𝑎 and 𝑏 be two points lying on opposite parallel support lines of 𝑈. Let 𝛼 be one of the two arcs into which 𝑎 and 𝑏 divide the boundary of 𝑈. A semi-convex domain 𝑆 is bounded by the convex arc 𝛼 and an arbitrary Jordan arc 𝛽 connecting 𝑎 with 𝑏 and lying in the convex region 𝑊 bounded by 𝛼 and the two support lines. Choose an arbitrary point 𝑐 on the arc 𝛼 and let 𝛼1 and 𝛼2 denote the sub-arcs 𝑎𝑐 and 𝑐𝑏 of 𝛼, joining 𝑎 with 𝑐 and 𝑏 with → − → 𝑐, respectively. Now, translate 𝛼1 by the vector 𝑐𝑏 and 𝛼2 by − 𝑐𝑎. The original arc 𝛼 and the translates of 𝛼1 and 𝛼2 enclose a region 𝑉. If the Jordan-arc 𝛽 which, along with 𝛼, bounds 𝑆 is restricted to lie in 𝑉 instead of 𝑊 then 𝑆 is called limited semi-convex. Typical examples of semi-convex and limited semi-convex shapes are shown in Figure 8.8. L. Fejes Tóth [265] proved that 𝛿𝑇 (𝐾) = 𝛿 𝐿 (𝐾) for every limited semi-convex disk 𝐾. On the other hand, A. Bezdek and Kertész [71] constructed an example showing that semi-convexity of 𝐾 does not imply 𝛿𝑇 (𝐾) = 𝛿 𝐿 (𝐾). L. Fejes Tóth [266] conjectured that the equality 𝛿𝑇 (𝐾) = 𝛿 𝐿 (𝐾) holds for sets that are the union of two convex disks with a common point and verified the conjecture for the union of two congruent circular disks. Heppes [406] showed that the equality 𝛿𝑇 (𝐾) = 𝛿 𝐿 (𝐾) no longer holds for the union of three convex disks with a common point. Since two translates of a convex disk cannot cross, the density bound (3.10.2) yields 𝐾 𝜗𝑇 (𝐾) ≥ ℎ for the translational covering density of a convex disk 𝐾. Bambah, Rogers and Zassenhaus [37] proved the alternative bound 𝜗𝑇 (𝐾) ≥

𝐾 , 2𝑡

where 𝑡 is a triangle of maximum area contained in 𝐾. For centrally symmetric disks we have 2𝑡 = ℎ, so the two bounds coincide and imply 𝜗𝑇 (𝐾) = 𝜗𝐿 (𝐾). Without central symmetry, sometimes one, sometimes the other bound is better. The conjecture that 𝜗𝑇 (𝐾) = 𝜗𝐿 (𝐾) for every convex disk 𝐾 is equivalent to 𝜗𝑇 (𝐾) = ℎ𝐾∗ , where ℎ∗ denotes the greatest area of a centrally symmetric hexagon inscribed in 𝐾. A small step in the direction of confirming this was made by G. Fejes Tóth [210], who gave the following common generalization of the above two bounds:

212

8 Notes

𝜗𝑇 (𝐾) ≥

𝐾 , ℎ

where ℎ is the greatest area of a 𝑝-hexagon inscribed in 𝐾. For the definition of a 𝑝-hexagon see Section 8.2, page 193. A further proof of the fact that for centrally symmetric convex disks 𝜗𝑇 (𝐾) = 𝜗𝐿 (𝐾) holds was given by Bambah and Woods [38].

a

b

b

a

α1 α

c

α2

Fig. 8.8 For covering, the case of general convex disks cannot be reduced to the symmetric case by an argument analogous to the one used for packing. Groemer [334] proved that there is no transformation assigning to every convex body 𝐾 a centrally symmetric body 𝐾 ∗ with the property that for each convex body 𝐾 a family of translates of 𝐾 forms a covering if and only if the corresponding family of translates of 𝐾 ∗ form a covering. The difficulty in proving the conjecture 𝜗𝑇 (𝐾) = 𝜗𝐿 (𝑇) for all convex disks is shown by the fact that the conjecture for the triangle has only recently been proved by Januszewski [425]. A further small step was made by Sriamorn and Xue [620]. A quarter-convex disk is the affine image of a set of the form {(𝑥, 𝑦) : 0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 𝑓 (𝑥)} for some positive concave function 𝑓 (𝑥) defined for 0 ≤ 𝑥 ≤ 1. The class of quarter-convex disks includes all triangles and convex quadrilaterals. Sriamorn and Xue proved that 𝜗𝑇 (𝐾) = 𝜗𝐿 (𝑇) for every quarter-convex disk. Double-lattice arrangements. For centrally symmetric convex disks 𝐾 the optimal density 𝛿(𝐾) is achieved by a lattice arrangement. For convex disks without a center of symmetry this is not the case. G. Kuperberg and W. Kuperberg [458] introduced the wider class of double-lattice arrangements, and conjectured that certain convex disks, e.g. regular polygons with an odd number of sides, achieve their packing density within this class of arrangements. A double-lattice arrangement is the union of a lattice-like arrangement and its mirror image. G. Kuperberg and W. Kuperberg reduced the problem of finding the densest double-lattice packing

8.3 Notes on Chapter 3

213

of a convex disk to an optimization problem over inscribed parallelograms. They determined in this way the density of the densest double-lattice packing for both the regular pentagon √ and the regular heptagon and found that the corresponding densities are (5− 5)/3 = 0.92131 . . . and 0.8926 . . ., respectively. Based on the idea of G. Kuperberg and W. Kuperberg, Mount [537] constructed an algorithm that determines the densest double-lattice packing of convex polygons with a running time linear in the number of vertices. Kallus and Kusner [437] introduced a technique for proving the local optimality of packing configurations. As an application, they could prove that, under mild assumptions satisfied generically, the optimal doublelattice packing of congruent copies of a convex disk 𝐶 is locally optimal in the full space of packings of copies of 𝐶.

Fig. 8.9 The regular pentagon has attracted special interest. The conjectured densest packing of regular pentagons suggested independently by Henley [401] and G. Kuperberg and W. Kuperberg [458] is shown in Figure 8.9. de Oliveira Filho and Vallentin [550] generalized the linear programming bound for sphere packings to obtain density bounds for other convex bodies as well. They represented the method on the example of the regular pentagon. Their computations yielded the upper bound 0.98103 for the packing density of the regular pentagon. Using completely different methods, Hales and Kusner [395] gave a computer-assisted proof √ confirming the conjecture that the packing density of the regular pentagon is (5 − 5)/3. They point out that their methods are adequate for the solution of other related problems in geometric optimization. The limiting factor is the availability of sufficient computer resources. Given a centrally symmetric convex disk 𝐷, every line through the center of 𝐷 splits 𝐷 into two congruent sets. Either of these two sets is called a semidisk. L. Fejes Tóth asked whether one can obtain from every centrally symmetric convex disk 𝐷 that is not a quadrangle or hexagon a semidisk 𝐷 ′ such that 𝛿(𝐷 ′) > 𝛿(𝐷). Groemer and Heppes [342] answered this question positively by showing that there is such a

214

8 Notes

semidisk 𝐷 ′ whose densest double-lattice packing is denser than the densest lattice packing of 𝐷. Regularity of optimal arrangements. As pointed out on page 87, for every centrally symmetric convex disk the maximum density packing is realized in a lattice arrangement. It is also conjectured that the analogous statement is true for the covering density. However, this property of centrally symmetric convex disks is exceptional. G. Fejes Tóth [212] proved that the family of convex disks 𝐾 for which 𝛿(𝐾) = 𝛿 𝐿 (𝐾) is sparse, namely of first Baire category in the space of all convex disks furnished with the Hausdorff metric. An analogous statement has been obtained for coverings by G. Fejes Tóth and Zamfirescu [227]: The family of convex disks 𝐾 for which 𝜗(𝐾) = 𝜗𝐿 (𝐾) is sparse. This does not mean that packings and coverings realizing the extreme density cannot have a regular structure. In both cases it is only shown that for typical convex disks double-lattice arrangements are more economical than lattice arrangements. Thus, it is natural to ask: Can the packing density and the covering density of every convex disk be realized by an arrangement possessing some regularity? Our knowledge about this problem is very limited. An arrangement of sets is periodic if it is a lattice arrangement of a finite collection of domains. It is clear that the packing density and the covering density of every convex disk can be approximated by the densities of periodic arrangements. However, we do not know whether there exist periodic arrangements for every convex disk 𝐾 with density 𝛿(𝐾) and 𝜗(𝐾). On the other hand, optimal arrangements of congruent non-convex sets can be quite complex: Schmitt [590] constructed a star-shaped set for which no packing of maximum density is periodic. Schmitt’s example is the union of two convex disks with a common point.

8.3.9 Notes on Section 3.13 The theorem of A.W. Goodman and R.E. Goodman about non-separable arrangements of circles has been generalized by K. Bezdek and Lángi [86] to centrally symmetric bodies: Every non-separable family of homothetic copies of a centrally symmetric convex body 𝐾 in 𝐸 𝑛 Í homothety coefficients 𝜆1 , . . . , 𝜆 𝑚 with positive
𝑚 can be covered by a translate of 𝑖=1 𝜆 𝑖 𝐾. They gave an example showing that the theorem does not hold without the assumption of central symmetry. For general convex bodies 𝐾 they showed that a non-separable family of 𝜆
1 𝐾, . . . , 𝜆 𝑚 𝐾, Í𝑚 𝜆𝑖 𝐾. Akopyan, 𝜆𝑖 > 0 for 𝑖 = 1, . . . , 𝑚, can be covered by a translate of 𝑛 𝑖=1 Balitski and Grigorev [8] improved the factor 𝑛 in the last mentioned theorem to (𝑛 + 1)/2. Polyanskii [569] proved the analog of Goodman and Goodman’s theorem in 𝑛-dimensional spherical space. In Figure 3.27 on page 97 the centers of the 19 circles have mutual distances at 𝜋 = 1.93185 . . .. The result of least 1, and lie all in a circle of radius 𝑟 19 = 2 cos 12 Reifenberg, Bateman and Erdős mentioned there implies that no more than 19 points with mutual distances at least 1 can be placed in a circle of radius 𝑟 19 , if

8.3 Notes on Chapter 3

215

one of them lies in the center. Elaborating on the method of Bateman and Erdős, Fodor [301] dropped the condition that one of the points is the center of the circle. He proved that 𝑟 19 is the smallest number 𝑟 such that a circle of radius 𝑟 can accommodate 19 points with mutual distances at least 1. He used similar methods in [302], [303], and [304] for the solution of the analogous problem for 12, 13, and 14 points. The problem of determining the smallest value 𝑟 𝑛 with the property that a circle of radius 𝑟 𝑛 can contain 𝑛 points with mutual distances at least 1 is elementary for 𝑛 ≤ 7 was solved for 𝑛 = 8, 9, and 10 by Pirl [566] and by Melissen [512] for 𝑛 = 11. The sharpened conjecture of Gallai (page 97), stating that all circles of a collection in which every two have a common point can be stabbed by 5 needles, has been proved by Stachó [622]. However, Gallai overestimated the number of needles needed. In [170] Danzer announced that four needles always suffice, but he published a proof only in [173]. The same result was independently proved by Stachó [623]. That four points are indeed needed is shown by an example of 21 pairwise intersecting circles that cannot be stabbed by three points given by Grünbaum [357]. Danzer [173] gave such an example of ten circles. Har-Peled, Kaplan, Mulzer, Roditty, Seiferth, Sharir and Willert [398] gave an algorithm that finds in 𝑂 (𝑛) time five points stabbing a collection of 𝑛 pairwise intersecting circles. A simpler algorithm was given by Biniaz, Bose and Wang [88]. Carmi, Katz and Morin [145] proved that four stabbing points suffice. Their proof also yields a linear time algorithm for finding the four stabbing points. Hadwiger [367] proved that a family of pairwise intersecting unit circles can be stabbed by three points, and showed an example of 9 such circles that cannot be stabbed by two points (see also Hadwiger and Debrunner [368, Problem 44, pp. 18–19]. Dragan, Revenko and Soltan [175] gave an example of 7 pairwise intersecting unit circles that cannot be stabbed by two points, and proved that 7 is the smallest number which has this property. Schopp [599] sharpened Hadwiger’s result by showing that a family of pairwise intersecting unit circles can be partitioned into √ three subfamilies the intersection of each of which contains a circle of radius (2 3 − 3)/3 = 0.1547 . . .. Bose, Carmi and Shermer [135] investigated the problem of piercing pairwise intersecting geodesic discs in a simple polygon and gave the upper bound 14 for the number of points needed to pierce the disks. Abu-Affash, Carmi and Maman [4] showed that five points suffice. The piercing number 𝑝(F ) of a family F of sets is the minimum value 𝑘 such that F can be split into 𝑘 subfamilies, each having a non-empty intersection. Grünbaum [357] proved that the piercing number of a family of pairwise intersecting translates of a centrally symmetric convex disk is at most 3 and a family of pairwise intersecting homothetic copies of a centrally symmetric convex disk can be pieced by 7 needles. He also showed that there are universal upper bounds depending only on the dimension for the piercing number of a family of pairwise intersecting translates or homothetic copies of any convex body in all dimensions. The papers by Kim, Nakprasit, Pelsmajer and Skokan [447] and Dumitrescu and Jiang [191] contain explicit bounds.

◀ 97

◀ 97

216

8 Notes

Gallai’s problem is a special case of a general problem, the so-called ( 𝑝, 𝑞)problem posed by Hadwiger and Debrunner [369]. A family of sets has the ( 𝑝, 𝑞) property if among any 𝑝 members of the family some 𝑞 have a non-empty intersection. Hadwiger and Debrunner conjectured that for every 𝑝 ≥ 𝑞 ≥ 𝑛 + 1 there is a smallest integer 𝑀 ( 𝑝, 𝑞, 𝑛) such that the piercing number of every family of convex bodies in 𝐸 𝑛 that has the (𝑝, 𝑞) property is at most 𝑀 ( 𝑝, 𝑞, 𝑛). They proved that 𝑀 ( 𝑝, 𝑞, 𝑛) = 𝑝 − 𝑞 + 1 if 𝑝(𝑛 − 1) < (𝑞 − 1)𝑛. Their result is sharp and it implies Helly’s theorem in the case 𝑝 = 𝑞 = 𝑛 + 1. After several partial results establishing the case of special classes of convex bodies, the conjecture was finally confirmed by Alon and Kleitman [11, 12]. A survey on the ( 𝑝, 𝑞)-problem can be found in Eckhoff [193]. The papers by Danzer, Grünbaum and Klee [174] and Eckhoff [192] give thorough surveys of the variety of Helly-type theorems. For recent development we refer to Bárány [48], Holmsen and Wenger [417] and Amenta, De Loera and Soberón [16]. Let 𝑇 be a convex disk and let 𝑃1 , . . . , 𝑃𝑛 be 𝑛 points in 𝑇. Let 𝑎 𝑖 be the distance from 𝑃𝑖 to the nearest point 𝑃 𝑗 ( 𝑗 ≠ 𝑖) and let 𝑆 𝑛 be the maximum of 𝑎 1 + · · · + 𝑎 𝑛 taken over all configurations of 𝑃1 , . . . , 𝑃𝑛 in 𝑇. Erdős and L. Fejes Tóth [200] proved that √ lim 𝑆 2𝑛 = 2𝑇/ 3. 𝑛→∞

Let 𝑑 𝑛 be the sum of distances from 𝑃𝑖 to the two nearest points and let 𝑈𝑛 be the maximum of 𝑑1 + · · · + 𝑑 𝑛 taken over all configurations of points 𝑃1 , . . . , 𝑃𝑛 in 𝑇. Confirming a conjecture by Erdős and L. Fejes Tóth, Few [276] proved that √ lim 𝑈𝑛2 = 8𝑇/ 3. 𝑛→∞

Even this theorem is weaker than the conjecture expressed on page 97 with 𝐿 𝑛 instead of 𝑆 𝑛 . A related problem is to estimate the maximum 𝐷 𝑛 of the smallest perimeter of the triangles determined by the points 𝑃1 , . . . , 𝑃𝑛 . Heppes [402] proved that lim sup 𝐷 2𝑛 ≤ 54𝑇/𝜋. 𝑛→∞

This compares quite favorably with the lower bound lim sup 𝐷 2𝑛 ≥ 12𝑇 . 𝑛→∞

proved by Erdős and L. Fejes Tóth [200]. Improving Verblunsky’s inequality (page 97), Few [274] showed by an ingenious construction that 𝑛 points√ in a unit square can always be connected by a polygonal chain of total length 𝐿 𝑛√≤ 2𝑛 + 1.75. An algorithmic approach to the problem yielding a bound of 𝐿 𝑛 ≤ 2𝑛 + 𝑂 (1) was presented by Supowit, Reingold √ and Plaisted [634]. Karloff [438] improved the upper bound to 𝐿 𝑛 ≤ 1.3916 𝑛 + 11. A Steiner minimal tree is the shortest road-net connecting a given finite set of points. In the same article Few [274] also considered the problem of estimating the length of the Steiner minimal tree connecting 𝑛 points in a unit square and showed that it

8.3 Notes on Chapter 3

217

does not exceed 𝑛1/2 + 7/4. Chung and Graham [151] improved Few’s bound to 0.995𝑛1/2 + 𝑂 (1). It is conjectured that the sharp bound is (3/4) 1/4 𝑛1/2 + 𝑂 (1), the worst case being when the points form a regular triangular lattice. Melzak [514] proved that the Steiner minimal tree for any 𝑛 points can be constructed by a sequence of Euclidean constructions. Few [274] considered the analogous problem also in higher dimensions and showed that through 𝑛 points in a 𝑑-dimensional unit cube there exists a path of length at most 𝑑{2(𝑑 − 1)} (1−𝑑)/2𝑑 𝑛 (𝑑−1)/𝑑 + 𝑂 (𝑛1−2/𝑑 ). Few’s bound was improved in low dimensions by Goldstein and Reingold [326] and for large values of 𝑑 by Moran [533], Goddyn [324] and Steele and Snyder [624]. The latter authors proved the existence of constants 𝛽 𝑑 such that the shortest traveling salesman tour through 𝑛 points in the 𝑑-dimensional unit cube has length at most 𝛽 𝑑 𝑛 (𝑑−1)/𝑑 . Several authors investigated the length of the optimal traveling salesmen tour through randomly chosen points. Improving on earlier results by Mahalanobis, [500] Jessen [426] and Ghosh [321], Beardwood, Halton and Hammersley [54] proved that the expected length of the optimal traveling salesman tour through 𝑛 independent identically distributed random points in the 𝑑-dimensional unit cube approaches 𝑐𝑛 (𝑑−1)/𝑑 as 𝑛 → ∞, where 𝑐 is a constant depending on the distribution. They also proved an analogous theorem for the length of the Steiner minimal tree and gave bounds for the constant 𝑐 occurring in their theorem for some special distribution of points. In particular, for the case where the independently chosen points are uniformly distributed in the unit square they proved the bounds 0.625 ≤ 𝑐 ≤ 0.92116 . . .. Recently, Steinerberger [628] decreased the upper bound by (9/16)10−6 . He also claimed to increase the lower bound by 19/5184, however his argument contained a fault. Gaudio and Jaillet [313] corrected the fault and by further refinement of the argument proved 𝑐 ≥ 0.6277. Numerical experiments suggest that 𝑐 ≈ 0.712. The conjecture about the maximum density of the part of the plane consisting of points that are covered exactly once by a family of congruent circles was solved by Balázs [32] in the case when the family forms a lattice, and in full generality by G. Blind and R. Blind [97]. The last two authors also investigated the analogous problem on the sphere [98] and gave an upper bound for the area of the part covered exactly once by 𝑛 congruent circles. In their bound equality occurs for 𝑛 = 3, 4, 6 and 12 when the centres of the circles are the vertices of a regular triangle, tetrahedron, octahedron or icosahedron, respectively, and the intersection points of each circle with its 𝑘 neighbours form the vertices of a regular 2𝑘-gon. Edelsbrunner and Iglesias-Ham proved that among a certain family of lattice arrangements of balls the largest density of the part of the space consisting of points that are covered exactly once is attained by enlarging the balls of the densest lattice packing by a factor of 1.090 . . . . Heilbronn’s conjecture turned out to be false. Namely, Komlós, Pintz and Szemerédi [454] showed that there is a constant 𝑐 > 0 such that for all sufficiently large 𝑛 a set of 𝑛 points exists in the unit circular disk for which every triangle determined by the points is of area greater than 𝑐𝑛−2 ln 𝑛. The original upper bound for the area of the smallest triangle, given by Roth (page 98), was improved first by

◀ 98

◀ 98

218

8 Notes

Schmidt [589] to 500𝑛−1 log 𝑛−1/2√and later, in a series of papers, by Roth [582, 583, 584] to 𝑐𝑛−𝜇 with 𝜇 = (17 − 65) − 𝜀. The currently best result was obtained by Komlós, Pintz and Szemerédi [453] who, refining Roth’s method, showed that for every 𝜇 < 8/7 there is a triangle of area smaller than 𝑐𝑛−𝜇 . Additional material relevant to Chapter 3 can be found in the following surveys and books: Baranovski˘ı [46], Boltjanski˘ı [107], L. Fejes Tóth [262, 267], G. Fejes Tóth [208, 213, 214], G. Fejes Tóth and W. Kuperberg [225, 226], Böröczky Jr. [126], Gruber [354], Pach and Agarwal [556] and Zong [673]. Collections of open problems are presented in Brass, Moser and Pach [139], Croft, Falconer and Guy [168], and Klee and Wagon [450].

8.4 Notes on Chapter 4 8.4.1 Notes on Section 4.1 Courant [158] gave a new proof of Fáry’s inequality (4.1.1). It should be mentioned that this inequality also follows from the theorem of Besicovitch [59] stating that every convex disk 𝑇 contains a centrally symmetric hexagon of area 32 𝑇.

8.4.2 Notes on Section 4.2 With regard to the Reinhardt–Mahler conjecture about the smoothed octagon whose lattice packing density is 0.9024 . . . (Chapter 4, page 106), Ennola [198] showed that every centrally symmetric convex disk admits a lattice packing with density √  √ 1 √ 18 + 3 − 6 = 0.8813 . . . , 𝑑≥ 4 improving upon (4.2.2). By elaborations of Ennola’s method, this lower bound was raised by Tammela [637] to 0.892656 . . . . Nazarov [545] proved that in the space of all centrally symmetric convex disks furnished with the Hausdorff metric, the smoothed octagon is a point at which the packing density attains a local minimum. Hales [386, 390] gave detailed strategies for proving the Reinhardt– Mahler conjecture. He showed that the extreme packing density among centrally symmetric disks with piecewise analytic boundary occurs for a smoothed polygon, that is, for a polygon rounded off at corners by arcs of hyperbolae. The Reinhardt– Mahler conjecture still remains open. The question about the general convex disk with the smallest packing density has received less attention. Kallus [435] proved that the regular heptagon 𝐻 represents a local minimum among double-lattice arrangements and conjectured that 𝛿(𝐻) is a local minimum of 𝛿(𝐾) among all convex disks 𝐾.

8.4 Notes on Chapter 4

219

According to a conjecture of Ulam (see Gardner [312, p. XX]) in threedimensional space every convex body can be packed as densely as the ball. A partial result supporting Ulam’s conjecture was proved by Kallus [434]: The 3-ball represents a local pessimum for 𝛿 𝐿 (𝐾) among centrally symmetric convex bodies. If Ulam’s conjecture is true, then the 3-space is quite exceptional. In all other dimensions where the densest lattice packing of balls is known, that is, for 𝑛 = 4, 5, 6, 7, 8, and 24, there are centrally symmetric convex bodies 𝐾 for which 𝛿 𝐿 (𝐾) < 𝛿 𝐿 (𝐵 𝑛 ). Kallus [436] showed that the ball represents a local pessimum for 𝜗𝐿 (𝐾) among centrally symmetric convex bodies in 3 dimensions but not in 4 and 5 dimensions. One way of proving the existence of dense packings and thin coverings by a convex disk 𝐾 is to find tiling polygons of large area contained in 𝐾 and of small area containing 𝐾. Results in this direction are surveyed in Subsection 8.2.4. The best known results use the 𝑝-hexagons introduced by W. Kuperberg [459]. By this method W. Kuperberg [461] proved that 𝛿(𝐾) ≥ 25/32 = 0.78125 and Ismailescu [421] proved that 𝜗(𝐾) ≤ 1.228 . . . for every convex disk. The method was also used by W. Kuperberg [462] to extend the inequality (4.2.4) over all, not necessarily symmetric, convex disks. Using √ a different method, G. Kuperberg and W. Kuperberg [458] proved that 𝛿(𝐾) ≥ 3/2 = 0.866 . . . , extending the inequality (4.2.2) to all convex disks. By elaborating on their proof, Doheny [184] proved that this bound is not the best possible, without presenting a better one. Ismailescu [422] proved the following inequalities, valid for every centrally symmetric convex disk 𝐾: 𝜗𝑇 (𝐾) − 1 ≥ 1 − 𝛿𝑇 (𝐾) and

5 √︁ 1 − 𝛿𝑇 (𝐾). 4 Also, he showed that these two inequalities are sharp in the following local sense: For every 𝜀 > 0 there exist centrally symmetric convex disks 𝐾 𝜀 and 𝐿 𝜀 with 𝜗𝑇 (𝐾) − 1 ≤

𝜗𝑇 (𝐾 𝜀 ) − 1 < (1 + 𝜀) (1 − 𝛿𝑇 (𝐾 𝜀 )) and 𝜗𝑇 (𝐿 𝜀 ) − 1 > (1 − 𝜀)

5 √︁ 1 − 𝛿𝑇 (𝐿 𝜀 ). 4

Ismailescu and Kim [424] proved that √ 8(13 13 − 35) = 1.172559 . . . 1 ≤ 𝛿𝑇 (𝐾)𝜗𝑇 (𝐾) ≤ 81 for every centrally symmetric convex disk 𝐾. The left inequality is a sharpening of the inequality 𝜗𝑇 (𝐾) − 1 ≥ 1 − 𝛿𝑇 (𝐾), since the geometric mean is smaller than the arithmetic mean. The right inequality yields an improvement of the inequality √︁ 𝜗𝑇 (𝐾) − 1 ≤ 45 1 − 𝛿𝑇 (𝐾) if 𝛿𝑇 (𝐾) < 0.973.

220

8 Notes

C

A

C

P

P

P

α

U

Fig. 8.10

β

Λ

Fig. 8.11

Fig. 8.12

The density bounds for convex disks discussed here can be graphically illustrated as follows. Let Ω be the set of points in the Cartesian plane consisting of all points (𝑥, 𝑦) for which there is a convex disk 𝐾 with 𝛿(𝐾) = 𝑥 and 𝜗(𝐾) = 𝑦, see W. Kuperberg [462]. If only centrally symmetric disks are considered, the corresponding set is denoted by Ω∗ . Similarly, to the pairs of densities 𝛿𝑇 , 𝜗𝑇 and 𝛿 𝐿 , 𝜗𝐿 correspond the sets Ω𝑇 , Ω𝑇∗ and Ω 𝐿 , Ω∗𝐿 . Since 𝛿𝑇 (𝐾) = 𝛿 𝐿 (𝐾) and 𝜗𝑇 (𝐾) = 𝜗𝐿 (𝐾) for every centrally symmetric convex disk 𝐾 (see Section 3.10), the sets Ω𝑇∗ and Ω∗𝐿 coincide. It is conjectured that the sets Ω𝑇 and Ω 𝐿 coincide as well. The white region 𝑃 seen in Figures 8.10, 8.11, and 8.12, determined by the previously mentioned bounds, contains the set Ω𝑇∗ . Ismailescu [422] succeeded in computing the translational packing and covering densities for every given centrally symmetric octagon, which enabled him to describe analytically a convex subset 𝑈 of Ω𝑇∗ consisting of all points that correspond to centrally symmetric octagons, see Figure 8.10. The point 𝐴 marked at the top of 𝑈 corresponds to the regular octagon, while the point (1, 1) (the lower right corner) corresponds to parallelograms and centrally symmetric hexagons. The exact shape of each of the sets Ω, Ω𝑇 , Ω 𝐿 , Ω∗ , and Ω𝑇∗ remains unknown. Due to the affine invariance of the functions 𝛿𝑇 , 𝛿 𝐿 , 𝜗𝑇 and 𝜗𝐿 , and to compactness of the space of affine invariance classes of convex disks (see Macbeath [495]), each of the sets Ω𝑇 , Ω 𝐿 and Ω𝑇∗ = Ω∗𝐿 is compact, but the same is not known about the sets Ω and Ω∗ . To obtain an approximation of these sets from within, W. Kuperberg [464] exhibited a convex region Λ, and proved that it is contained in Ω ∩ Ω∗𝐿 and largely overlapping with 𝑈 (see Figure 8.12, where the point 𝐶 corresponds to the circular disk). The boundary of Λ consists of two smooth arcs 𝛼 and 𝛽 joined at their ends, see Figure 8.11. Points lying on 𝛼 correspond to convex disks obtained as the intersection of a fixed circular disk and a concentric regular hexagon that grows from being

8.5 Notes on Chapter 5

221

inscribed in the circle to being circumscribed about it. Points on 𝛽 correspond to the convex disks obtained as the convex hull of the union of the same circular disk and the growing hexagon. The exact values of packing and covering densities for these disks are easily computed based on Thue’s and Kershner’s classical results, and for each of them are attained by lattice arrangements. The computations yield a parametric description of the loop 𝛼 ∪ 𝛽 which turns out to be a simple closed curve bounding a convex disk Λ. By an argument of a topological nature, the loop is null-homotopic within Ω∗𝐿 , thus Λ must be contained in Ω∗𝐿 . Note that the union 𝑈 ∪ Λ occupies quite a large portion of 𝑃. It seems likely that Ω∗𝐿 differs very little from the convex hull of 𝑈 ∪ Λ by area, and that its shape should resemble a narrow leaf pointed at both ends, very much like each of 𝑈 and Λ. At least, it seems reasonable to conjecture that Ω∗𝐿 is convex.

8.5 Notes on Chapter 5 8.5.1 Notes on Section 5.1 The inequality (5.1.1) was generalized by László [473]: If 𝑢 1 , . . . , 𝑢 𝑛 denote the perimeters of 𝑛 ≥ 3 non-overlapping spherical circles on the unit sphere, then … 1 𝑢 1 + · · · + 𝑢 𝑛 ≤ 2𝜋𝑛 1 − csc2 𝜔 𝑛 . 4 If 𝑟 1 , . . . , 𝑟 𝑛 are the radii of the circles, then it is conjectured that   1 𝑟 1 + · · · + 𝑟 𝑛 ≤ 𝑛 arccos csc 𝜔 𝑛 , 2 which was confirmed by László under the condition that either max 𝑟 𝑖 ≤ 63.69◦ or 𝑛 ≥ 9.

8.5.2 Notes on Section 5.2 The problem of distributing 𝑛 points on the sphere so that the least distance between the points should be as great as possible was first studied by the biologist Tammes [638], who explained the distribution of the orifices on the pollen grains of certain plant by this extremum problem. Therefore, the problem is often referred to as the “Tammes problem”.

222

8 Notes

8.5.3 Notes on Section 5.3 If 𝑅 and 𝑟, respectively, denote the circumradius and the inradius of a convex polyhedron with 𝑒 edges, then, according to Florian [278], 𝑅 𝜋(𝑒 + 2) ≥ tan2 . 𝑟 4𝑒 This inequality is more general and sharper than the inequality (5.3.2) concerning the radii of the minimal spherical shell. Equality is attained only for the regular tetrahedron. A proof of D.K. Kazarinoff’s inequality on tetrahedra (page 122) can be found in N.D. Kazarinoff [440]. The inequality (1.5.7) was generalized to tetrahedra by Berkes [56] and to simplices in 𝑛 dimensions by Schopp [598]: If 𝑅1 , 𝑅2 , . . . , 𝑅𝑛+1 and 𝑟 1 , 𝑟 2 , . . . , 𝑟 𝑛+1 denote the distances to the vertices and to the faces, respectively, from an arbitrary interior point 𝑂 of a simplex in 𝑛 dimensions, then 𝐺 (𝑅1 , 𝑅2 , . . . , 𝑅𝑛+1 ) ≥ 𝑛𝐺 (𝑟 1 , 𝑟 2 , . . . , 𝑟 𝑛+1 ) and equality occurs only if the simplex is regular, centered at 𝑂. It should be noted that Berkes’ proof can also be extended to higher dimensions (see Berkes [57]). The main geometric ingredient of 𝑛+1 ∑︁ 𝑅𝑖 ≥ 𝑛, from which the inequality for the Berkes’ proof is the inequality 𝑅 + 𝑟𝑖 𝑖=1 𝑖 geometric means follows by algebraic operations. Wu and Bencze [663] generalized 𝑛+1 ∑︁ 𝑅𝑖 𝑛(𝑛 + 1) . the above inequality of Berkes as follows: If 𝑘 ≥ 1, then ≥ 𝑅 + 𝑘𝑟 𝑖 𝑛+𝑘 𝑖=1 𝑖 Linhart [488] characterized the 𝑛-dimensional regular simplicial polytopes as those of maximum volume among all simplicial polytopes of the same dimension, the same number of facets, and the same circumradius. From the numerous new extremum properties of the regular simplex we mention the following. Petty and Waterman [562] gave a generalization of inequality (1.5.3), according to which the sum of distances from an arbitrary point to the vertices of a simplex of a given volume reaches its minimum when the simplex is regular and the point coincides with the centroid of the simplex. Sansone [586], and later independently of him Melzak [516], proved that the regular tetrahedron is of maximum volume among all tetrahedra with given sum of lengths of the edges. Weiss [656] proved the following generalization: Among all 𝑛-dimensional simplices of equal sum of measure of their 𝑘-dimensional faces the regular one has extremum sum of measure of their 𝑙-dimensional faces (maximum if 𝑘 < 𝑙, minimum if 𝑘 > 𝑙). Tanner [639] proved that the 𝑛-dimensional regular simplex maximizes the sum of the squared contents of all 𝑖-dimensional faces, for all 𝑖 = 2, . . . , 𝑛, when the sum of the one-dimensional squared contents is fixed. It is conjectured that among all simplices with given sum of lengths of the edges the regular one has the maximum mean width. Böröczky Jr. [124] proved the conjecture for dimensions 3, 4, and 5. The connection of the conjecture with information theory is discussed in the paper by Litvak [492].

8.5 Notes on Chapter 5

223

In [629] Steinhagen considered the following problem: What is the maximum width of a convex body in 𝐸 𝑛 with unit inradius? He showed that the problem can be reduced to the investigation of simplices and proved that among simplices circumscribing the unit ball the regular simplex has the largest width. Steinhagen’s rather complicated proof was simplified by Gericke [318]. Further proofs and generalizations were given by Volenec [648], Lillington [482], Weißbach [657], Juhnke [433], and Betke and Henk [63]. The analogous problem about simplices inscribed in a ball was raised by Sallee (see Guy [363, Problem 8]). Confirming the conjecture of Sallee, Alexander [9] proved that of all simplices inscribed in the unit ball the regular simplex has the largest width. Alternative proofs can be found in Lillington [482], Yang and Zhang [665], Weißbach [657], and Mao and Zuo [505]. We emphasize the significance of a result of Florian [286], namely an analogue to the moment lemma for polyhedra with a given number of edges. The result expresses some general extremum properties of the regular polyhedra with triangular faces. The corresponding problem with a given number of faces appears to be difficult. A result of Böröczky Jr. and Schneider [134] on the minimum mean width of a simplex containing a given convex body characterizes the ball rather than the regular simplex: The ratio between the minimum of the mean width of a simplex containing a convex body 𝐾 and the mean width of 𝐾 is greatest possible when 𝐾 is a ball. An analogous result is proved for circumscribed parallelepipeds. In this case the extreme bodies are those of constant width. For a trihedral polyhedron 𝑃 inscribed in the unit sphere let 𝜂(𝑃) be the quotient of the length of the longest edge and the length of the shortest edge. If 𝑃 is regular then 𝜂(𝑃) = 1, otherwise 𝜂(𝑃) > 1. Thus, 𝜂(𝑃) measures in some sense the regularity of 𝑃 and the question arises: How closely regular can such a polyhedron be in this sense? Let 𝑒 𝑚 denote the infimum of 𝜂(𝑃) for trihedral polyhedra 𝑃 with 𝑚 faces inscribed in the unit sphere. Clinton [152] and Kitrick [448] constructed sequences of polyhedra showing that lim inf 𝑚→∞ 𝑒 𝑚 ≤ 2 sin 36◦ , while, based on an idea of L. Fejes Tóth, Makai [502] proved lim inf 𝑚→∞ 𝑒 𝑚 ≥ 2 sin 36◦ . Makai also gave a detailed description of Clinton and Kitrick’s constructions.

8.5.4 Notes on Section 5.4 By means of some known theorems about convex functions, Florian [277] succeeded in proving the convexity of the function 𝑇 (𝜏, 𝑝) defined on p. 127, thereby completing the proof of inequality (5.4.1), as it was suggested in Section 5.4. The estimate (5.4.1) was generalized by L. Fejes Tóth [235] to non-Euclidean polyhedra as follows:

◀ 127

224

8 Notes

2𝑒 𝑉 ≥ √ 𝜅3

∫

𝜋 2𝑘

0

   √   

𝜅𝑟 − »

© √ arctan ­tan 𝜅𝑟 𝑐2 − sin2 𝜑 « cos 𝜑

»

   𝑐2 − sin2 𝜑 ª  ® d𝜑,  cos 𝜑  ¬

𝜋𝑣 𝜋𝑓 /cos . 2𝑒 2𝑒 Here 𝜅 denotes the curvature of space and 𝑟 denotes the radius of the inscribed ball. In the limiting case, as 𝜅 → 0, this inequality becomes (5.4.1). A corresponding inequality for the surface area was given by Tomor [643]. In 4-dimensional Euclidean space, among the convex polytopes containing the unit ball 𝐵4 and bounded by 5, 16, or 600 tetrahedral cells or by 24 octahedral cells, we are looking for those that have the smallest possible volume. Here by an octahedron we mean a polyhedron topologically isomorphic to the regular octahedron. L. Fejes Tóth [233] proved that the respective solutions of these problems are the regular polytopes {3, 3, 3}, {3, 3, 4}, {3, 3, 5} and {3, 4, 3} circumscribed about 𝐵4 . 𝑐 = sin

8.5.5 Notes on Section 5.5

130 ▶

132 ▶

The inequality (5.5.1) solves the problem of finding the polyhedron of maximum volume with 𝑛 vertices contained in the unit ball for 𝑛 = 4, 6, and 12. We saw that in these cases the extreme bodies are the corresponding triangular regular polyhedra. Berman and Hanes [58] solved the problem for 𝑛 = 5, 7 and 8. By a computer-aided search, Mutoh [544] constructed polyhedra inscribed in the unit ball with up to 30 vertices having conjecturally maximum volume. Á.G. Horváth and Lángi [315] solved this problem for 𝑑-dimensional polytopes with 𝑛 = 𝑑 + 2 vertices in every dimension, and for polytopes with 𝑑 + 3 vertices in odd dimensions. Florian [277, √ 278] proved the inequality (5.5.3) for all polyhedra for which ≤ 3 (see also L. Fejes Tóth [247]). There are only seven cases with tan 𝜋2𝑒𝑓 tan 𝜋𝑣 2𝑒 √ 𝜋𝑓 𝜋𝑣 tan 2𝑒 tan 2𝑒 > 3. These cases were resolved using a stronger version of (5.5.3) by L. Fejes Tóth [244] and H. Florian [282]. It should be noted that, along with the proof of (5.5.3), the inequality (5.6.1) is quite easily verified as well. If, in counting the number of faces and vertices of a star-polyhedron, we include the multiplicity of their occurrences in the faces and edge configurations, then, under certain conditions, inequalities (5.4.1) and (5.5.3) hold for star-polyhedra as well (L. Fejes Tóth [234], Florian [279]). Then, besides the five Platonic solids, equality also occurs for the four Kepler–Poinsot star-polyhedra. Florian [278] investigated the edge curvature 𝑀 of polyhedra inscribed in or circumscribed about a ball. He stated the inequalities 𝑀 ≥ 2𝑒 sin

  1/2   𝜋𝑓 𝜋𝑣 𝜋𝑣 𝜋𝑓 𝜋𝑓 tan2 tan2 −1 arccos cos csc 𝑟 2𝑒 2𝑒 2𝑒 2𝑒 2𝑒

8.5 Notes on Chapter 5

225

and   1/2   𝜋𝑓 𝜋𝑣 𝜋𝑓 2 𝜋𝑓 2 𝜋𝑣 𝑀 ≤ 2𝑒 sin 1 − cot cot arccos cos csc 𝑅 2𝑒 2𝑒 2𝑒 2𝑒 2𝑒 for a polyhedron with 𝑓 faces, 𝑒 edges and 𝑣 vertices containing the unit ball and contained in the unit ball, respectively, but he could prove them only under the foot-point condition for the edges and faces. In [285] he was able to get rid of the foot-point condition for the edges in the first inequality and proved this inequality without any condition in his papers [296] and [298]. Linhart [486] proved the second inequality in full generality. For an 𝑛-dimensional convex polytope with circumradius 𝑅, total edge length 𝐿, and mean width 𝑀 Linhart [485] proved that 𝑀>

4 𝜔 𝑛−1 𝑅 𝑛 𝜔𝑛

and 𝐿 > 2𝑛𝑅, where 𝜔 𝑛 denotes the volume of the 𝑛-dimensional unit ball. The second inequality confirms a conjecture of Lillington [481], who proved it in two special cases, for polytopes of dimension 𝑛 ≤ 3 and for simplicial polytopes of arbitrary dimension. Böröczky [113] and Peyerimhoff [563] proved that in 𝑛-dimensional spherical and hyperbolic space, the regular simplex is of maximum volume among all simplices contained in a given closed ball. Moreover, Peyerimhoff proved that in 𝑛-dimensional hyperbolic space, among all simplices of a given inradius or volume, the regular one is of minimum total edge length.

8.5.6 Notes on Section 5.6 Böröczky and Böröczky Jr. [117, 118] characterized both the regular octahedron and icosahedron as the minimum volume polyhedron among all convex polyhedra containing ball and with vertices lying outside the concentric ball of radius » the unit √ √ 3 or 15 − 6 5, respectively. A further example of an extremum property of the regular polyhedra is linked with the notion of the minimal spherical shell (page 121). If we measure the size of a minimum spherical shell containing a given polyhedron by the quotient 𝑅/𝑟 of its radii, then the following theorem holds (L. Fejes Tóth [234]): Among all convex polyhedra isomorphic to a regular polyhedron, the regular one’s spherical shell is the smallest.

◀ 154

226

8 Notes

8.5.7 Notes on Section 5.7 A simple proof of Steinitz’s result that among 5-faced polyhedra of given volume the regular triangular prism has the smallest surface area was given by Gallagher, Ghang, Hu, Martin, Miller, Perpetua and Waruhiu [311]. The 𝑛-dimensional isoperimetric inequality 𝐹 𝑛 /𝑉 𝑛−1 ≥ 𝑛𝑛 𝜔 𝑛 was strengthened by Hadwiger [364] for polytopes with a given number of facets. For 𝑛 = 2, Hadwiger’s inequality turns into (1.4.1). Böröczky Jr. and G. Fejes Tóth [130] proved stability in the case of equality for (5.7.4), as well as for several other inequalities characterizing regular trihedral polyhedra. In hyperbolic space, the isoperimetric property of the regular tetrahedron was established by L. Fejes Tóth [246]. The corresponding problem in the spherical space, to which Böhm [102] outlined a possible approach, is still unsolved. In the original text the proof of the claim that the isoperimetric quotient of a polyhedron with 𝑛 vertices is reached by a proper triangular polyhedron was based on the assumption that the locus of the points 𝑃 for which the surface area of 𝐻 (𝑃) remains constant is a smooth convex surface. As an example by Akiyama [6] shows, this is not the case. The proof was corrected by Akiyama and also by Böröczky and Böröczky, K. Jr. [116]. We changed the text following the argument by the latter authors, who extended the result to higher dimensions and to all intrinsic volumes. They also solved the isoperimetric problem for polytopes with 𝑛 + 2 vertices in 𝐸 𝑛 . √ 3 In particular, if a polyhedron 𝑃 in 𝐸 3 has at most 5 vertices, then 𝑉𝐹 2 ≥ 243 2 with equality if and only if 𝑃 is a double pyramid over a regular triangle of edge length 𝑎, √ where the other six edges are of length 46 𝑎. For polyhedra 𝑃 with at most 6 vertices √ 3 Böröczky, K. Jr. and Kovács [131] established the inequality 𝑉𝐹 2 ≥ 108 3, where the case of equality characterizes the regular octahedron.

8.5.8 Notes on Section 5.9

143 ▶

In relation to Section 5.9 reference should be made to L. Fejes Tóth [243, 245] and Hirsch, Li, Petti and Xue [416], where besides the spherical tilings, the isoperimetric properties of the three-valent regular Euclidean and hyperbolic tilings are discussed. It was established by Engelstein [197] and Hales [380] that the subdivision of the sphere into 4 and 12 (not necessarily convex) regions of equal area minimizing the total perimeter is the regular tetrahedral and the dodecahedral tiling, respectively. Maurmann, Engelstein, Marcuccio and Pritchard [508] proved that the least total perimeter of a partition of a smooth, compact Riemannian

8.5 Notes on Chapter 5

227

surface into 𝑛 regions of equal area 𝑎 is asymptotically equal to 𝑛/2 times the perimeter of a planar regular hexagon of area 𝑎. A partition of the sphere into regions of given area and minimum total perimeter must satisfy certain conditions. A partition into more than 2 regions must be given by a net whose edges are arcs of circles meeting only in three-valent vertices at equal angles. It follows that, in general, an optimal partition does not consists of convex regions. Heppes [405] enumerated all nets that might provide an optimal partition into convex regions. Besides the four regular trihedral nets dividing the sphere into 3, 4, 6 and 12 cells, he found five irregular ones. In [408] he showed that none of the five irregular convex nets provides the solution of the corresponding general partition problem.

8.5.9 Notes on Section 5.10 The conjecture expressed on page 144, stating that among all convex polyhedra circumscribed about a given ball the cube has the smallest total edge length, was proved by Besicovitch and Eggleston [62]. The corresponding problem for polyhedra with triangular faces (page 145) was solved by Linhart [484]. Coxeter [161] considered the problem in non-Euclidean spaces. Coxeter and L. Fejes Tóth [164] proved the following: In spherical space there are two polyhedra, namely the dihedron {3, 2} and the tetrahedron {3, 3}, that have the same smallest sum of edge lengths among all convex polyhedra with triangular faces and with an inradius of arcsin 41 ; in hyperbolic space, for the inradius 0.364054 . . . the regular octahedron, and for the inradius 0.828375 . . . the regular icosahedron has the smallest sum of edge lengths. In four-dimensional Euclidean space L. Fejes Tóth [250] established the following result. Let 𝐹 be the total area of the 2-dimensional faces of a 4-dimensional convex polytope containing a unit sphere; then 𝐹 > 67.5. If the polytope has tetrahedral cells only, then 𝐹 > 110.4; and if it is bounded by octahedral cells only, then 𝐹 > 81.6. Think of the edges of a convex polyhedron as thin rods made of a perfectly rigid material and rigidly connected at the vertices. We say that the rods form a cage. In his review of the paper by Besicovitch and Eggleston [62], Coxeter [160] asked to find a cage of minimum sum of edges holding a ball and not permitting to slide it out. Coxeter conjectured √ that the optimal cage is a right triangular prism all of whose edges have length 3. This conjecture was refuted√by Besicovitch [61], who constructed for every 𝜀 > 0 a cage of total length 38 𝜋 + 2 3 + 𝜀 around the unit ball from which the ball cannot √ escape. Aberth [1] proved that no cage of length less than or equal to 38 𝜋 + 2 3 will do. Fruchard and Zamfirescu [309] investigated cages for different convex bodies. Among other things, they proved that the minimum total length of edges for a cage of a regular tetrahedron of unit edge length is 3. In Besicovitch’s example not all edges of the cage touch the ball. So, Coxeter [162, 163] posed a modified problem: He asked for the minimum total edge length of a crate around the unit ball, that is, a cage that holds the unit ball and such that every edge is tangent to it. He conjectured that the regular triangular prism of total

◀ 144

◀ 145

228

8 Notes

√ edge length 9 3 = 15.5884 . . . might be the optimal crate for the unit ball. However, this conjecture turned out to be false, as well. Shephard [609] constructed a crate of total edge length arbitrarily close to 6 + 3𝜋 = 15.4247 . . . . We wish to knit a net around the unit ball, using pieces of some inextensible string, so that the ball cannot slip out from the net. Besicovitch [60] showed that the total length of string in such a net must be greater than 3𝜋. The constant 3𝜋 cannot be replaced by a smaller one. Alternative proofs were given by Croft [167] and Stein [627]. Results on extremum problems for polyhedra are surveyed by Florian [288,295, 296].

8.5.10 Notes on Section 5.12 In Section 5.12 several asymptotic formulae for approximation of convex bodies by convex polyhedra were given without rigorous proofs. They provided, however, the direction and motivation for future development. The following quote from Gruber [344, p. 326] underscores the importance of such influence of this section: “In his fundamental treatise Lagerungen in der Ebene, auf der Kugel und im Raum L. Fejes Tóth stated for 𝑑 = 2, 3 many asymptotic formulae ... and in some cases indicated proofs. Through the effort of L. Fejes Tóth, McClure and Vitale, Gleason, Schneider, Enomoto, Gruber and Ludwig most of these formulae have been proved, some new ones were added and several formulae were extended to general dimensions.”

Approximation of convex bodies by convex polyhedra rapidly developed into an important part of convexity theory that is still growing at a fast rate. Mentioning even the most important new results of the field would explode the frame of these notes. We refer the reader to the surveys by Gruber [343,344,347] and Bronshte˘ın [141]. Concerning formula (5.12.1) the original text contained the following note: “For hyperbolically curved surfaces formula (5.12.1) does not hold anymore. In that case already the above definition of approximability fails. Let for example 𝑇 be the part of a one-sheeted hyperboloid of revolution bounded by two congruent circles 𝐴 and 𝐵. Inscribe regular 𝑛-sided polygons 𝐴1 . . . 𝐴𝑛 and 𝐵1 . . . 𝐵𝑛 in the circles 𝐴 and 𝐵, respectively, so that the lines 𝐴1 𝐵1 , . . . , 𝐴𝑛 𝐵𝑛 lie on the hyperboloid, and let 𝐹2𝑛 be the polyhedral surface with faces 𝐴1 𝐴2 𝐵1 , . . . , 𝐴𝑛 𝐴1 𝐵𝑛 , 𝐵1 𝐵2 𝐴2 , . . . , 𝐵𝑛 𝐵1 𝐴1 . The deviation 𝜂 (𝐹2𝑛 , 𝑇) coincides with the deviation of the 𝑛-gon 𝐴1 . . . 𝐴𝑛 from the circle 𝐴; hence its order of magnitude is 𝑛12 and not 𝑛1 .”

We omitted this paragraph from the text of the book, since, as it was pointed out by Wintraecken [661] (see also Atariah, Rote and Wintraecken [24]) the distance between the hyperboloid and the approximating polyhedron is not of order 𝑛12 but 1 𝑛 . Moreover, Vegter and Wintraecken [646] showed that for analytic negatively curved surfaces with boundary that contain no straight line segments approximability can be defined in the same way as for convex surfaces.

8.5 Notes on Chapter 5

229

8.5.11 Notes on Section 5.13 The inequality 𝐹 ≤ 𝑒 sin

  2𝜋 𝜋 𝜋 1 − cot2 cot2 𝑅2 𝑝 𝑝 𝑆𝑞

is proved on page 153 under the foot-point condition for the faces and edges. Linhart [487] showed that it suffices to assume that the foot-point condition holds for the edges. As a corollary Linhart derived the inequality √   𝜋 𝑣  27 1 (𝑣 − 2) 1 − cot2 𝑅2 , 𝐹≤ 2𝑅 3 6 𝑣−2 valid for all polyhedra with 𝑣 vertices, with equality in the cases of regular polyhedra with triangular faces. The special case 𝑣 = 4 had been proved earlier by Krammer [457] and Heppes [403]. Besides the here mentioned cases of 𝑣 = 4, 6 and 20, the maximum surface area of a polyhedron with 𝑣 vertices inscribed in a sphere was determined for 𝑣 = 5 by Donahue, Hoehner and Li [185]. The problems posed on page 154 about the sum of the lengths of the edges of a polyhedron with given volume or surface area were treated by Aberth. In the proof of his theorem about the smallest cage containing a ball, Aberth [1] used the following inequality as a lemma. If 𝐹 is the surface area and 𝐿 is the sum of the lengths of edges of a convex polyhedron, then 𝐿 2 /𝐹 > 6𝜋 = 18.849 [452] constructed  √. . . . Kömhoff 8𝜋 2 a sequence of polyhedra for which 𝐿 /𝐹 ≤ 2 3 + 3 = 20.219 . . . , leaving the question open whether polyhedron exists for which 𝐿 2 /𝐹 equals, or  √ a convex 8𝜋 possibly is less than 2 3 + 3 . For polyhedra with triangular faces Kömhoff [451] √ proved that 𝐿 2 /𝐹 ≥ 12 3, with equality only for the regular tetrahedron. For the sum of the lengths of edges of a convex polyhedron of volume 𝑉, Aberth [2] proved that 𝐿 3 /𝑉 √> 432𝜋. According to a conjecture of Melzak [515, problem 13], 𝐿 3 /𝑉 ≥ 972 3, where equality is attained only by a right triangular prism with height equal to a side of the base. Analogous problems in higher dimensions are treated in the works of Eggleston, Grünbaum and Klee [196], Klee [449], Berger [55] and Scott [605]. It is known (see e.g. Abrosimov, Makai, Mednyik, Nikonorov and Rote [3]) that for a given sequence of numbers 𝑠 𝑚 ≥ 𝑠 𝑚−1 ≥ . . . ≥ 𝑠1 , there exists a, possibly degenerate, 𝑚-faced 𝑛-dimensional polytope with facet areas 𝑠1 , . . . , 𝑠 𝑚 if and only if 𝑚 > 𝑛 and 𝑠 𝑚 ≥ 𝑠 𝑚−1 + . . . + 𝑠1 . Generally, these conditions do not determine a unique polytope. So the question arises: What is the greatest, and the smallest volume of such a polytope? The question about the largest volume was answered long ago for 𝑚 = 𝑛 + 1 by Lagrange [469] for the three-dimensional case and by Borchardt [110] for all dimensions. The solution is a unique orthocentric simplex, that is, a simplex whose altitudes concur at its orthocenter. For 𝑚 ≥ 𝑛 + 1 the problem remains open.

◀ 153

230

155 ▶

8 Notes

The problem concerning the minimum volume in the case 𝑛 = 2 was solved independently by Böröczky, Kertész and Makai [119] and Nikonorova [546]. The solution is a well-defined triangle whose shape depends on whether we search among all simple polygons or just among convex ones. The situation changes in higher dimensions: Abrosimov, Makai, Mednyik, Nikonorov and Rote [3] showed that, in dimensions greater than 2, among the considered polytopes there are convex polytopes with arbitrary small volume. A zonotope is the Minkowski sum of a finite number of line segments. A zonotope in 𝐸 3 is regular if its faces are congruent rhombi and its vertex figures are regular polygons. There are three 3-dimensional regular zonotopes: the cube, the rhombic dodecahedron and the rhombic triacontahedron generated by the segments connecting antipodal vertices of the regular octahedron, cube, and icosahedron, respectively. Linhart [489] characterized these by showing that among all zonotopes generated by 3, 4, or 6 unit segments the regular zonotopes have maximum surface area, as well as maximum inradius. Linhart [490] gave an upper bound for all intrinsic volumes of a zonotope generated by 𝑘 segments in 𝑛 dimensions. K. Bezdek [80] showed that the intrinsic 𝑖-volume of a zonotope generated by 𝑛 + 1 line segments and containing the unit ball in 𝑛 dimensions is at least as large as the intrinsic 𝑖-volume of a zonotope with unit inradius generated by 𝑛 + 1 line segments connecting the center of a regular simplex with its vertices. A zone of width 𝑤 is the parallel domain at distance 𝑤/2 of a great circle. Jiang and Polyanskii [427] proved that if a centrally symmetric convex domain on the sphere is covered by zones of total width 𝑤, then it can be covered by one zone of width 𝑤. Observe that a digon of area 𝑎 can be covered by a zone of width 𝑎/2 and if a convex region on the sphere is covered by a zone of width 𝑤, then it can also be covered by a digon of area 2𝑤. Thus, for centrally symmetric regions the answer to the problem about covering a convex region by digons posed on page 155 is affirmative. The result of Molnár that if at least three congruent circles are packed in a convex spherical region 𝐺, or if 𝐺 is covered by at least three congruent circles, then the density of the packing is smaller than √𝜋 and the density of the covering 12

is greater than √2 𝜋 does not extend to the hyperbolic plane. On the other hand, 27 L. Fejes Tóth conjectured that, in the hyperbolic plane, the density of a packing of at least two congruent circles in a circle cannot exceed √𝜋 , and the density of at 12

leat two congruent circles covering a circle cannot be less than √2 𝜋 . K. Bezdek [75] 27 confirmed the conjecture for packing. Moreover, in [76], he showed that the density of a finite packing of circles of radius 𝑟 in the parallel domain at distance 𝑟 from the convex hull of the circles’ centers is at most √𝜋 , and in [77] he extended this 12 result to the sphere. An analogous result for coverings in the hyperbolic plane was proved by Böröczky Jr. [127] a corollary of which confirms the above conjecture of L. Fejes Tóth about covering a circle by circles. Böröczky Jr. [125] also proved an inequality for the number of circles of radius 𝑟 packed in a convex domain of given area and perimeter.

231

8.5 Notes on Chapter 5

The result mentioned on page 155 on circle packing in the hyperbolic plane states that for the density 𝑑 of the circles relative to every Delone triangle the inequality 𝑑≤

3 csc 𝜋𝑘 − 6 ; 𝑘 −6

csc

𝜋 𝑟 = 2 cos 𝑘 𝑅

holds. L. Fejes Tóth [230]) proved a similar bound for coverings: Given a covering of a sphere, with a real or imaginary radius 𝑅, with at least three congruent circles of radius 𝑟 the density 𝐷 of the circles relative to every Delone triangle satisfies √ 12 cot 𝜋𝑘 − 6 𝜋 √ 𝑟 𝐷≥ ; cot = 3 cos . 𝑘 −6 𝑘 𝑅 √ As 𝑘 → ∞ this bound converges decreasingly to 12/𝜋 (see Krammer [456]). √ Consequently we have 𝐷 > 12/𝜋. L. Fejes Tóth [231, 237] extended the investigation to arrangements of horocycles √ (circles of infinite radius) and proved the bound 𝑑 ≤ 3/𝜋 for packings and 𝐷 ≥ 12/𝜋 for coverings.

Fig. 8.13

The density bounds given on page 155 are illustrated by the upper curve in Figure 8.13. Molnár [527, 528, 529] succeeded in improving upon this bound (the lower curve in Figure 8.13). Let 𝑐 be a circle of radius 𝑟, and let 𝐶 be a concentric circle of radius 𝑅, where 𝑅 is the radius of the circle circumscribed about the equilateral triangle of side 2𝑟. Further, let 𝑃 be a polygon inscribed in 𝐶 all of whose sides, with possibly one exception, touch 𝑐. Then 𝑐/𝑃 is Molnár’s bound. It should be stressed, however, that here the density relative to the Dirichlet cells was considered. Molnár [531] studied the densest circle packing and thinnest circle covering in spaces of constant curvature under the condition that the radii of the circles lie in a given interval. According to well-grounded conjectures of Molnár, several interesting optimal configurations are to be expected.

232

8 Notes

Research on packing and covering in hyperbolic space started at the time when the first edition of Lagerungen was written. We survey new developments of this topic in Chapter 10.

8.6 Notes on Chapter 6 8.6.1 Notes on Sections 6.1–6.3 The problems of the densest spherical circle packing and the thinnest spherical circle covering with 𝑛 congruent circles can be interpreted as follows (see Meschkowski [517, p. 1], G. Fejes Tóth [202]): A planet is ruled by 𝑛 mutually hostile dictators. How should the residences of these “gentlemen” be distributed so that the shortest among all distances between them is as large as possible? A planet is ruled by 𝑛 allied dictators. How should the residences of these “gentlemen” be distributed so that the entire planet is easiest to control, in the sense that the maximum distance from a point on the planet to the nearest dictator’s residence is as small as possible? These problems, along with the general problems solved by means of inequality (5.8.2) (page 138) for 𝑛 = 3, 4, 6 and 12, as well as the corresponding problems in higher dimensions, appear to play a role in biology (Tammes [638], van der Waerden [650], Şerban and Stroilaˇ [608], Goldberg [325], L. Fejes Tóth [247]), in information theory (van der Waerden [650], Landau and Slepian [470]) and also in stereochemistry (Atiyah and Sutcliffe [25], Melnyk, Knop and Smith [513]).

8.6.2 Notes on Sections 6.4–6.5 164 ▶

The cases of 10 and 11 hostile dictators were solved by Danzer in his Habilitationsschrift [171] but the result was published only in [172]. In the meantime, independent solutions were given for the case of 10 dictators by Hárs [399] and for the case of 11 by Böröczky [112]. The case of 13 hostile dictators attracted special attention. This case is related to the problem of 13 spheres—the subject disputed by Newton and Gregory— asking whether 13 congruent material balls can touch a 14-th ball of the same size. A proof of the inequality 𝑎 13 < 60◦ deciding the dispute in favor of Newton was first given by Schütte and van der Waerden [604]. Alternative proofs are due to Leech [475], Böröczky [115], Anstreicher [22], Musin [541], and Maehara [497,498], and Glazyrin [322]. The bound 𝑎 13 < 60◦ was sharpened by Böröczky and Szabó [121] to 𝑎 13 < 58.869187◦ and by Bachoc and Vallentin [31] to 𝑎 13
𝑎 24 . They also found arrangements of 47, 59 and 119 points with minimum mutual distances greater than the conjectured values for 48, 60, and 120 points. This supports the conjecture that 𝑎 𝑛−1 = 𝑎 𝑛 only if 𝑛 = 6 and 𝑛 = 12. This conjecture seems to be very difficult to resolve. We do not even know whether there exists a natural number 𝑘 such that 𝑎 𝑛+𝑘 < 𝑎 𝑛 for all 𝑛. Let 𝑎 𝑛𝑘 denote the supremum of the 𝑘-th minimal distance between 𝑛 distinct points on the unit sphere. A. Bezdek and K. Bezdek [68] investigated 𝑎 2𝑛 . They proved that 𝑎 92 = 2𝜋/3 and gave an upper bound on 𝑎 2𝑛 for 𝑛 > 9. They conjectured that 𝑎 2𝑛 = 𝑎 ⌈𝑛/3⌉ for 𝑛 > 12. Füredi [310] determined all values of 𝑎 2𝑛 for 𝑛 ≤ 12 and verified the above conjecture for sufficiently large values of 𝑛. He also proved that 𝑎 3𝑛 = 𝑎 ⌈𝑛/5⌉ for sufficiently large 𝑛 and gave an upper bound for 𝑎 𝑛𝑘 for all 𝑛 and 𝑘.

◀ 164 ◀ 165

◀ 166

234

8 Notes

Böröczky [114] investigated a related problem. He considered 𝑛 points on the sphere and called two points of the set neighbors if their Dirichlet cells have a common boundary point. He looked for the arrangement of the points for which the greatest distance between neighboring points attains its smallest possible value 𝑏 𝑛 . He determined 𝑏 𝑛 for 𝑛 ≤ 10 and 𝑛 = 12. For 𝑛 = 4, 6 and 12 the points in an optimal arrangement are vertices of a regular tetrahedron, octahedron and icosahedron. It is conjectured that 𝑏 𝑛 is decreasing, but it is not strictly decreasing. We have 𝑏 5 = 𝑏 4 and 𝑏 8 = 𝑏 7 = 𝑏 6 .

8.6.3 Notes on Section 6.6

170 ▶

The problem of 𝑛 allied dictators was solved for 𝑛 = 5 and 7 by Schütte [602], for 𝑛 = 8 by Wimmer [660] and for 𝑛 = 10 and 14 by G. Fejes Tóth [202]. Wimmer also showed that the combinatorial structure of the best arrangement of 9 points coincides with the structure of the set of points that Jucovič [431] conjectured to be optimal. The solution of the cases 𝑛 = 5, 7 and 𝑛 = 10, 14 points can be grouped together with the earlier settled cases for 6 and 12 in the following theorem: Consider the residences of 𝑛 allied dictators in their best possible distribution. If 𝑛 = 5, 6 or 7, then the points lie at the vertices of a regular bipyramid. For 𝑛 = 10, 12 or 14 the points lie at the vertices of a regular antiprismatic bipyramid. An antiprismatic bipyramid arises when two pyramids are attached to an antiprism: one to its floor and one to its ceiling. If the solid is inscribed in a sphere and all of its faces are congruent, then it is regular (see Figure 8.14). We know of some configurations of points that solve both problems, of the hostile and of the allied dictators alike, namely the vertices of the tilings {2, 3}, {3, 2}, {3, 3}, {3, 4}, {3, 5} and the vertices and face-centers of {2, 3}. Are there any other configurations of this kind? This question has not been answered yet, but G. Fejes Tóth and L. Fejes Tóth [219] proved that the number of such configurations is finite.

Fig. 8.14

235

8.6 Notes on Chapter 6

We consider certain configurations of points on the sphere, subject to a stronger condition. We place a circle of radius 𝑟 centered at each of the points and require that the portion of the sphere covered by the circles be of maximum area in comparison with all other locations of the points. If this happens for every value of 𝑟 (0 < 𝑟 ≤ 𝜋/2), then we call it a perfect configuration. L. Fejes Tóth [253] proved that the only perfect configurations are those that consist of the vertices of the tilings {2, 3}, {3, 2}, {3, 3}, {3, 4} and {3, 5}. An interesting area of research is the study of the two problems of dictators on the elliptic plane, that is, on the sphere with the additional requirement that the configurations of points be symmetric with respect to the sphere’s center. For 𝑛 ≤ 6, the problem of 𝑛 hostile dictators on the elliptic plane was solved by L. Fejes Tóth [249] and independently also by Rosenfeld [585]. The solution is given by the antipodal pairs of vertices of the tiling {4, 2} for 𝑛 = 2, {3, 4} for 𝑛 = 3, {4, 3} for 𝑛 = 4, and {3, 5} for 𝑛 = 6. Each of these
spherical configurations has the property that the distance between all of the 𝑛2 pairs of points is the same. For 𝑛 = 4 there are two such equiangular line configurations: the diagonals of the cube and four of the diagonals of the regular icosahedron. The greater angle occurs between the diagonals of the cube. The study of equiangular lines was initiated by Haantjes [397] and van Lint and Seidel [491]. It became an intensively investigated area of combinatorial geometry with recent breakthrough works by Balla, Dräxler, Keevash and Sudakov [33] and by Jiang, Tidor, Yao, Zhang and Zhao [428]. The connection of equiangular lines in 𝑛-dimensional complex spaces with quantum theory is discussed in the book by Stacey [621].

Fig. 8.15

In the notes to the second edition of this book the conjecture was formulated that the set of face-centers of the cuboctahedral tiling (Figure 8.15) is the best distribution of 7 points for each of the two dictator problems on the elliptic plane. The part about the hostile dictators was confirmed by Conway, Hardin and Sloane [156], the

236

167 ▶

8 Notes

uniqueness follows from a more general theorem of Cohn and Woo [154]. Conway, Hardin and Sloane [156] also gave putatively optimal packings for all 𝑛 ≤ 50. Their conjecture for 𝑛 = 8 was recently confirmed by Mixon and Parshall [521]. Heppes [409] solved the problem of 4 allied dictators on the elliptic plane. Molnár [523] proved the inequality 𝐷 𝑛 ≥ 𝐷 5 stated on page 167 as a conjecture. Alternative proofs and generalizations of the inequality 𝑇𝑛 ≥ (𝑛−2)𝑇3 of Habicht and van der Waerden (page 169) were given by Molnár [522] and Bollobás [104]. A generalization for the 3-dimensional case was given by Molnár [530]. Let 𝑃1 , 𝑃2 , . . . be a given sequence of arbitrary points on the unit sphere and let 𝑏 𝑛 denote the minimum distance between distinct points among 𝑃1 , . . . , 𝑃𝑛 . Then—as Groemer [330] proved—the inequality √ 2 lim inf 𝑏 𝑛 𝑛 ≤ √ = 3.22 . . . ln 4 − 1 holds.

8.7 Notes on Chapter 7 8.7.1 Notes on Sections 7.1–7.2 The sphere packing problem. The old essay of Kepler [444] includes a description of the structure of the densest lattice packing with congruent balls, the familiar face-centered cubic lattice arrangement (see Figure 8.16). Kepler noticed that the arrangement can be viewed as a stack of hexagonal layers or a stack of square layers at the same time (see Figure 8.17, copied fromKepler’s essay). The beauty of the arrangement alone convinced Kepler of its optimality, which he proclaimed as “the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container”, with no hint of a proof. It is doubtful that Kepler meant thereby to state a conjecture as some of the modern articles ascribe it to him. Nevertheless, Kepler’s belief was shared by many for centuries to come, long before the problem was√resolved. That the density of no packing of congruent spheres could ever exceed 𝜋/ 18 (the density of the face-centered cubic lattice packing) seemed quite natural, and was, in the words of C.A. Rogers [578, p. 610], what “many mathematicians believe, and all physicists know”. The optimality of the face-centered cubic lattice sphere packing among all lattice packings was first established by Gauss [314] in the context of ternary quadratic forms and elementary number theory, but in general the sphere packing conjecture remained unsolved for centuries. Hilbert [415] popularized the conjecture by including it in his Problem 18, part 3. Before the conjecture was resolved, a series of upper bounds for the density of a sphere packing improving the bounds by Blichfeldt [91] and Rankin [572] mentioned on page 174 were produced by:

237

8.7 Notes on Chapter 7

Fig. 8.16

Fig. 8.17

Rogers [578], Baranovski˘ı [45], Lindsey [483], and Muder [539, 540]. The last author gave a bound of 0.773 . . . , still above the dodecahedral bound. Answering a question posed by Böröczky [111] A. Bezdek, W. Kuperberg and E. Makai [74] strengthened Gauss’ result by proving that if a packing consists of a collection of √ parallel strings of balls, then its density cannot exceed 𝜋/ 18. A string of balls is a collection of non-overlapping congruent balls whose centers are collinear and each of the balls is tangent to two of the other balls. Dauenhauer and Zassenhaus [176] proved that the face-centered cubic lattice sphere packing, restricted to a large tetrahedron properly embedded in the lattice, is locally optimal under sufficiently small perturbations.

238

174 ▶

175 ▶

8 Notes

In the early 1990s, two authors independently announced progress towards a solution of the sphere packing conjecture. The first of them, W.-Y. Hsiang, actually claimed to have already proved it, and, following a series of corrections prompted by critical remarks, published an article [419]. His approach, making use of Dirichlet cells assigned to the spheres, is a strategy similar to that proposed by the author of this book in Section 7.3, though much more elaborate. His nearly 100-page-long article, followed by a book [420], describe the strategy in great detail. Hales [376] pointed out the shortcomings of Hsiang’s work. Although for insufficient rigor and incompleteness in some crucial steps Hsiang’s claim has not been accepted as a confirmation of the conjecture, his highly detailed strategy may still serve as a solid basis for a proof. The other author, T.C. Hales, at first used Delone cells associated with the spheres [374,375], which he later modified in [377,378]. Subsequently, in collaboration with his student S. Ferguson, he changed the strategy more significantly, introducing a partition of space that combines modified Dirichlet and Delone cells. The theoretical foundation of the proof was published in the Annals of Mathematics (Hales [381]), and the details, along with historical notes, were given in a series of articles in a special issue of Discrete and Computational Geometry (Hales [382, 383, 384, 385], Ferguson [273] and Hales and Ferguson [392]). The implementation of the strategy required, however, extremely extensive case analysis, impossible to accomplish by a human. The description of the theoretical approach to the problem and of the results of computer work occupy nearly 300 pages in the articles cited above. At the computational stage of the proof, some 5000 computer-generated cases were examined, each of them requiring the solution of optimization problems with non-linear constraints and a large number of variables. The same method of the solution of the sphere packing problem, including computational techniques and employed software, was used by Hales and McLaughlin [396] to confirm the dodecahedral conjecture as well. Anstreicher [22] describes an attempt to prove the dodecahedral bound along the lines indicated in Section 7.2. In 2003 Hales launched a project named flyspeck—a near acronym for “Formal Proof of Kepler”—designed for an automatic (computerized) formal verification of his proof. The project involved a number of experts in formal languages. Yet another article on the topic of the sphere packing conjecture, revising the originally published proofs appeared (Hales et al. [394]), written by the flyspeck team as a byproduct of this monumental project. Hales’ book [387], based on the formal verification of his proof, shows in detail how geometric ideas and elements of proof are arranged and processed in preparation for the formal proof-checking scrutiny, see also the review of the book by Lagarias [468]. Marchal [506] proposed an alternative strategy for a computer-aided solution of the ball packing problem. The subdivision associated with the packing appears to be simpler and his strategy less complex in comparison with those of Hales. Lagarias’ article [466] describes and explains the common ideas in the various specific strategies to the solution of the ball packing problem, including the one

8.7 Notes on Chapter 7

239

proposed in Section 7.3, the one outlined by Hsiang and the one implemented by Hales. The strategy of Marchal also fits the same pattern. On August 10, 2014 the team of the Flyspeck project announced the successful completion of the project, see Flyspeck [300], where they made a remarkable comment: “The formal proof takes the same general approach as the original proof, with modifications in the geometric partition of space that have been suggested by Marchal.” Thereby the correctness of the proof was confirmed. The official published account of the completed Flyspeck project appeared in Hales et al. [391]. It is worth noting that the statement Kepler formulated for finite packings of balls in a container is false. Schürmann [601] proved that if 𝐾 is a smooth convex body in 𝑛-dimensional space (𝑛 ≥ 2), then there exists a natural number 𝑛0 , depending on 𝐾, such that the densest packing of 𝑛 ≥ 𝑛0 congruent balls in 𝐾 cannot be part of a lattice arrangement. The papers containing the detailed proof of the Kepler conjectures, together with the two papers indicating the original approach of Hales are reprinted in the book [393] by Hales and Ferguson. The book contains an introductory chapter by Lagarias [467] where a brief history of the Kepler problem is given. The reader will find popular expositions of the long story of the ball packing problem in the papers by Joswig [430], Henk and Ziegler [400] Pfender and Ziegler [564], Casselman [146], and in the book of Szpiro [635]. Investigations after the solution of the sphere packing problem. It is likely that, with the possible exception of Hales, nobody has checked all the details of the proof of the Kepler conjecture, and after the completed computer verification probably no one will sacrifice the time and energy to check the proof. So, it is natural that investigations about the sphere-packing problem in three dimensions continued. These investigations offer broader insight into the problem and might lead to a more digestible proof. K. Bezdek [79] formulated the Strong Dodecahedral Conjecture, according to which in three-dimensional space the surface area of every Dirichlet cell in a packing of unit balls is at least as large as that of a regular dodecahedron circumscribed about the ball, that is, 16.6508 . . . . He showed that the surface area of such a cell cannot be smaller than 16.1182 . . . . The bound was improved successively to 16.1445 . . . by K. Bezdek and Daróczy-Kiss [85], then to 16.1977 . . . by Ambrus and Fodor [15]. The conjecture was finally confirmed by Hales [388, 389]. The third fundamental characteristic, the mean width, of Dirichlet cells in packstudied by K. Bezdek [78]. He gave the lower bound ings √ of unit balls√was √ 𝜋/(6 6 arcsin (1/ 3) − 6𝜋) = 2.3264 . . . for the mean width of a Dirichlet cell in a unit ball packing. It is conjectured that the minimum mean width of such a cell is 2.3736 . . . achieved for the regular dodecahedron circumscribed about the unit ball. Instead of Dirichlet cells of packings of unit balls one can consider tilings of the Euclidean 3-space by general convex cells, each containing a unit ball. Of course, then an individual cell can be arbitrarily close to the unit ball. On the other hand, for normal tilings in which the cell diameters have a finite upper bound, we can seek lower bounds for the average volume, surface area, and mean width of the cells.

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8 Notes

√ √ From 𝛿(𝐵3 ) = 𝜋/ 18 it follows that the average volume of the cells is at least 4 2. K. Bezdek√[83] proved that the average surface area in such a tiling is greater than or equal to 8 3. The√average surface area of the Dirichlet cells in the densest packing of unit balls is 12 2. It is an open question whether the average surface area in a partition of space into √ uniformly bounded convex cells, each containing a unit ball, can be less than 12 2. For a polyhedron 𝑃 let 𝐺 (𝑃) = tan 𝛼2 , where 𝑙 is the length of an edge, 𝛼 is the angle between the outer normals of the faces meeting at this edge and the summation extends over all edges of 𝑃. This quantity is analogous to the edgeÍ curvature 𝑀 (𝑃) = 12 𝑙𝛼 introduced by Steiner (see Section 1.1). K. Bezdek [84] proved that the average of 𝐺 (𝐶) of the cells 𝐶 in a partition of√space into uniformly bounded convex cells, each containing a unit ball, is at least 8 3. Since the surface area of a polyhedron 𝑃 containing the unit ball is at least 𝐺 (𝑃), this is a sharpening of the result about the average of the surface area of the cells. Böröczky, Heppes and Makai [120] determined the densest packing of translates √ of strings of balls whose centres lie equidistantly with distance 2𝑑 ∈ [2, 2 2] on a line. They also showed that in 4 dimensions the density of translates of a 2dimensional layer of balls whose centers form either a square lattice or a regular triangular lattice of edge length 2 is at most 𝜋 2 /16, with equality for the densest lattice packing of unit balls in 𝐸 4 . An arrangement of unit balls is an 𝜀-quasi-twelve-neighbor packing if no two balls of the packing touch each other but for each ball 𝐵 of the packing there are twelve other balls in the packing with the property that the distance from the centre of each of these twelve balls to the centre of 𝐵 is less than 2 + 𝜀. Böröczky and Szabó [123] constructed for every positive √ number 𝜀 an 𝜀-quasi-twelve-neighbour packing of balls in 𝐸 3 whose density is 𝜋/ 18 and the local density of a positive percentage of the balls relative to the corresponding Dirichlet cell is greater than √ 𝜋/ 18. This shows the extreme difficulty of the sphere packing problem. Covering with balls. The dual counterpart to Gauss’ result, the determination of the thinnest lattice covering with balls, was achieved by Bambah [34]. Simpler proofs are due to Barnes [52], Few [275] and Baranovski˘ı [44]. As was expected (page 173), in the thinnest lattice covering, the centers√of the balls form the body-centered cubic lattice. The density of this covering is 5𝜋 5/24, and is conjectured to be the minimum density over all space coverings with congruent balls. For the density of a general covering of space with congruent balls, Coxeter, Few and Rogers [165] √ proved the upper bound 9 2 3 (arccos 13 − 𝜋3 ) = 1.431 . . . . This bound differs only a bit more than 2% from the conjectured value. Besides that, virtually no new results have been obtained about covering with balls in 3 dimensions. Bounds for the packing and covering density of convex bodies. Prior to the confirmation of the sphere packing conjecture, the first non-trivial (i.e., not space tiling) example of a convex 3-dimensional body with its packing density determined was given by A. Bezdek [66], namely a rhombic dodecahedron slightly truncated at one of its 3-valent vertices. We should remark that the same example can now be obtained as a simple corollary of the sphere packing theorem of Hales. Generally, if 𝐾 is a

8.7 Notes on Chapter 7

241

convex body containing the unit ball 𝐵 and has volume smaller than the volume of the rhombic dodecahedron 𝐷 circumscribed about 𝐵, then 𝛿(𝐾) = 𝐾/𝐷. As we mentioned above, the densest lattice packing of balls was already known to Gauss. For computing the lattice packing density of other convex bodies, Minkowski [519] reduced the problem to centrally symmetric bodies and described an algorithmic method which he used to find the lattice packing density of the regular octahedron. Minkowski’s method was implemented by Whitworth [658] for the slab of cube of width 𝑤 (the intersection of the unit cube with a slab of space determined by a pair of planes perpendicular to the cube’s main diagonal and concentric with the cube), by Chalk [150] for a slab of ball (defined similarly as the slab of cube), and by Whitworth [659] for the double cone. The problem of densest lattice packing for the tetrahedron was already of interest to Minkowski. Groemer [333] found a lattice packing of tetrahedra with density 18/49, and he conjectured that this density is the tetrahedron’s lattice packing density, which was confirmed by Hoylman [418]. Minkowski’s method often requires tedious computations with a large number of cases to analyze, which, if done “by hand”, for some convex bodies becomes prohibitively complex. With the emergence of computer technology, however, it became possible to accomplish such tasks. A fairly fast computer algorithm designed by Betke and Henk [64] finds the lattice packing density of any reasonably simple convex polyhedron in a reasonably short time. To show the algorithm’s efficiency, the cited article lists the lattice packing density of each of the regular and Archimedean polytopes, many of which would be practically impossible to handle without computers. Dostert and Vallentin [189] used the method of Minkowski to find new lattice packings of superballs in three dimensions. E.H. Smith [617] showed that 𝛿 𝐿 (𝐾) ≥ 0.538 . . . for every 3-dimensional centrally symmetric convex body 𝐾. Makeev [503] proved the slightly weaker bound 𝛿 𝐿 (𝐾) ≥ 0.527 . . . and proved that for convex bodies symmetric in a plane 𝛿 𝐿 (𝐾) ≥ 8/27 = 0.296 . . . . As a corollary of a more general theorem, Zong [672] proved that 𝜗𝐿 (𝐾)/𝛿 𝐿 (𝐾) ≤ 7/43 for every three-dimensional convex body 𝐾. It follows that 𝛿 𝐿 (𝐾) ≥ (4/7) 3 = 0.186 . . . and 𝜗𝐿 (𝐾) ≤ (7/4) 3 = 5.359 . . . . For centrally symmetric convex bodies 𝐾, E.H. Smith [616] showed that 𝜗𝐿 (𝐾)/𝛿 𝐿 (𝐾) ≤ 4, which he improved to 𝜗𝐿 (𝐾)/𝛿 𝐿 (𝐾) ≤ 3 in [618]. It follows that 𝜗𝐿 (𝐾) ≤ 3 for such bodies.

8.7.2 Notes on Section 7.3 Concerning the isoperimetric problem for space partitions into cells of equal volume, in contrast to the planar case, where the unique optimal solution naturally produces convex cells (see Section 8.3), in the three-dimensional case no cell in an optimal space partition is convex. By a modification of the tiling of space with truncated octahedra Kelvin [442] constructed a space partition into congruent non-convex cells that he conjectured to be optimal. The conjecture, generally believed to be true for

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8 Notes

more than a century, was disproved by Weaire and Phelan [652]. Sullivan [632] described a family of cell partitions that seem useful for generating good equalvolume cells, including the counterexample found by Weaire and Phelan. Gallagher, Ghang, Hu, Martin, Miller, Perpetua, and Waruhiu [311] investigated the following problem: Among 𝑛-faced convex polyhedra of given volume, congruent copies of which form a face-to-face tiling of space, find the one with minimum surface area. They found the best tetrahedron among tetrahedra admitting an orientation preserving tiling. It is a suitably scaled similar replica of a tetrahe√ dron 𝑇0 bonded by four isosceles triangles with two sides of 3 and one side of 2. They also gave conjectures for 7 ≤ 𝑛 ≤ 14. Bongiovanni, Diaz, Kakkar, and Sothanaphan [109] proved that the tetrahedron 𝑇0 is optimal among all tetrahedra admitting a face-to-face tiling. The best 5-faced polyhedron is the regular triangular prism and the best 6-faced polyhedron is the cube, since these are optimal even without the condition of admitting a tiling. Heppes [407] considered partitions of a given convex polyhedron into a given number of cells of equal volume and of minimum total surface area. He classified all cases in which every cell is convex. The result of Heppes implies that in the generic case the cells of the optimal partition are not convex. An article of L. Fejes Tóth [248] discusses the question of optimality of the honeycomb cells, that is, the cells of a partition of the slab of space enclosed between two parallel planes, requiring that each cell has a face on exactly one of the two planes. The author points out that the cells that bees build, of the shape of a hexagonal tube capped off by three congruent rhombi on one end, are not optimal, as replacing the three rhombi by two rhombi and two hexagons makes the cells more economical. According to Heppes’ classification, the economy of these cells can still be improved by making them non-convex. For more information on the Kelvin conjecture see Weaire [651], R. Kusner and Sullivan [465] and Sullivan and Morgan [633]. Additional results related to the subject of Chapter 7 can be found in Aste and Weaire [23] K. Bezdek [81,82], A. Bezdek and W. Kuperberg [73], Böröczky, Jr. [126], Cohn [153], Conway and Sloane [157], G. Fejes Tóth and W. Kuperberg [225], Gruber [354], and Zong [671]. Number-theoretical connections with packing problems are discussed in the books of Cassels [147], Gruber and Lekkerkerker [356], Erdős, Gruber and Hammer [201] and in the survey by Gruber [345].

Chapter 9

Finite Variations on the Isoperimetric Problem G. Fejes Tóth

Let 𝑎, 𝑝, 𝑤, and 𝑑 denote the area, perimeter, width and diameter of a convex disk. Fixing one of these four quantities, what is the infimum and the supremum of another one of them? Of course, fixing one quantity and asking for the supremum of another one is equivalent to the problem of fixing the second quantity and asking for the infimum of the first one. The solution of one half of the twelve problems arising in this way is obvious: The answer is either zero or infinity. In the case of the six meaningful problems, we can ask for minima and maxima and for the convex disks attaining the optimum. The isoperimetric problem asks for the convex disk of maximum area with given perimeter. Its solution is the circle, which was known already in Ancient Greece, although a mathematically rigorous proof was obtained only in the 19th century. The solution of the problem of the maximum width for a given diameter is obvious: The optimal sets are convex disks of constant width. Convex disks of constant width were also characterized by Blaschke [18] as those among domains of a given width that have minimum perimeter, and by Rosenthal and Szász [41] as those among with a given diameter, that have maximum perimeter. Bieberbach [12] proved that among all domains of a fixed diameter the one of maximum area is the circular disk, and Pál [39] proved that the minimum area of a convex disk with given width is attained by the regular triangle. An interesting area of research is to consider these optimum problems restricted to polygons with at most a given number of sides. The isoperimetric problem for 𝑛-gons was solved centuries ago by Zenodorus (see [19]): Implicitly assuming the existence of a solution, he proved that a regular 𝑛-gon has greater area than all other 𝑛-gons with the same perimeter. This is expressed in the inequality 𝑝 2 ≥ 4𝑛𝑎 tan

𝜋 . 𝑛

Pál’s theorem solves the problem of minimum area for a given width. Concerning the remaining four problems only partial results are known.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_9

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9 Finite Variations on the Isoperimetric Problem

Reinhardt [40] considered the problem of maximizing the perimeter of a convex 𝑛-gon with a given diameter and proved that the perimeter 𝑝 of an 𝑛-gon of diameter 𝑑 satisfies the inequality 𝜋 𝑝 ≤ 2𝑛 sin 𝑑. 2𝑛 Equality is attained here if and only if 𝑛 is not a power of 2. This result, together with the characterization of the case of equality, was rediscovered by Larman and Tamvakis [33], Datta [20] and A. Bezdek and Fodor [11]. To describe the polygons for which equality is attained, we start with a convex polygon with an odd number of sides such that each vertex is at distance 𝑑 from the endpoints of the opposite side. Replacing each side by a circular arc of radius 𝑑 centered at the opposite vertex we obtain a Reuleaux polygon. If 𝑛 is not a power of 2, the 𝑛-gons of diameter 𝑑 with perimeter 2𝑛 sin(𝜋/2𝑛) are inscribed in a Reuleaux polygon in such a way that every vertex of the Reuleaux polygon is a vertex of the polygon, and all sides of the polygon are of equal length. Such polygons are called by Audet, Hansen and Messine [5] clipped Reuleaux polygons, while Mossinghoff [38] uses the term Reinhardt polygons for them.

Fig. 9.1 Consider a Reuleaux polygon with 𝑚 vertices. Its diagonals form an 𝑚-gon which for 𝑚 > 3 is a star polygon. The sum of the angles of this polygon is 𝜋, so in order that it can accommodate a clipped Reuleaux polygon its angles must be integer multiples of 𝜋/𝑛. The clipped Reuleaux polygons were studied by Gashkov [23,24], Mossinghoff [38] and Hare and Mossinghoff [27, 28]. For a given 𝑛, there are clipped Reuleaux 𝑛-gons with 𝑘-fold rotational symmetry for some divisor 𝑘 of 𝑛. Besides these, called periodic by Mossinghoff, there may be some others, called sporadic. The latter name is misleading, since it turned out that the sporadic clipped Reuleaux polygons outnumber the periodic ones for almost all 𝑛. Figure 9.1 shows a regular and a sporadic clipped Reuleaux polygon with 30 vertices. Finding the maximum perimeter of an 𝑛-gon of given diameter when 𝑛 is a power of 2 is difficult. Only the cases of the quadrangle and octagon are solved. The best quadrangle was determined by Tamvakis [43] and rediscovered by Datta [20]. The octagon’s case was settled by Audet, Hansen and Messine [2]. Tamvakis

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245

described a sequence of unit-diameter 𝑛-gons for 𝑛 = 2 𝑘 whose perimeter exceeds 𝜋 that of the regular 𝑛-gon, and differs from the upper bound 2𝑛 sin 2𝑛 by 𝑂 (𝑛−4 ). By improved constructions the difference from the upper bound was reduced to 𝑂 (𝑛−5 ) by Mossinghoff [35] and lately to 𝑂 (𝑛−6 ) by Bingane [14]. 𝜋 𝑑 with the isoperimetric inequality 𝑝 2 ≥ Combining the inequality 𝑝 ≤ 2𝑛 sin 2𝑛 𝜋 4𝑛 tan 𝑛 𝑎, Reinhardt [40] obtained the inequality 𝑎≤

𝜋 𝜋 𝑛 cos tan 𝑑 2 2 𝑛 2𝑛

with equality only for odd 𝑛 and regular 𝑛-gons. Thus, for odd 𝑛, among all 𝑛-gons of a given diameter the regular one has maximum area. Alternative proofs were given by Lenz [34], Griffiths and Culpin [26], and Gashkov [22]. Reinhardt proved that for even 𝑛 ≥ 6, the optimal 𝑛-gon is never regular. Alternative proofs were given by Schäffer [42], Audet, Hansen and Messine [4] and Mossinghoff [35]. The latter author constructed 𝑛-gons with unit diameter for all even 𝑛 whose area exceeds the area of the unit-diameter regular 𝑛-gon by 𝑂 (𝑛−2 ), and whose area differs from the maximum area of such 𝑛-gons by a term of at most 𝑂 (𝑛−3 ). By a different construction Bingane [15] improved Mossighoff’s lower bound for the maximum are by a term of order 𝑂 (𝑛−5 ).

Fig. 9.2 The maximum area of a quadrangle of diameter 𝑑 is 𝑑 2 /2. The diagonals of the optimal quadrangles are perpendicular and have length 𝑑. The case of the hexagon was solved by Graham [25]. He confirmed the conjecture of Bieri [13] that the non-regular hexagon shown in Figure 9.2 is the unique optimal solution. He also formulated a conjecture for all even 𝑛 ≥ 6, stating that for such 𝑛, every optimal 𝑛-gon’s diameter graph consists of an (𝑛 − 1)-cycle with one additional edge emanating from one of the cycle’s vertices. Note that the conjecture leaves the geometric realization of the best polygon undetermined, subject to an optimization problem. Graham’s conjecture was confirmed for 𝑛 = 8 by Audet, Hansen, Messine, and Xiong [9], who also solved the corresponding optimization problem, thus determining the best octagon. Subsequently, Graham’s conjecture was confirmed in general by Foster

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9 Finite Variations on the Isoperimetric Problem

and Szabo [21]. The corresponding optimal polygons for 𝑛 = 10 and 𝑛 = 12 were determined by Henrion and Messine [29]. Graham [25] also asked for the solution of the higher-dimensional analogue of the problem: Which convex 𝑑-polytope with 𝑛 vertices and unit diameter has the largest volume? For 𝑛 = 𝑑 + 1 the solution is the regular simplex. Kind and Kleinschmidt [30] solved the problem for 𝑛 = 𝑑 + 2 and described all the extremal polytopes. The case 𝑛 = 𝑑 + 3 was attacked by Klein and Wessler [31], however their proof turned out to be incomplete (cf. Klein and Wessler [32]), thus, this case is still open. In the Russian journal for high school students Quant, Gashkov [22] wrote a small article about the isoperimetric problem and its relatives. There he gave a proof of Reinhardt’s theorem based on central symmetrization. He used the following facts about a convex 𝑛-gon 𝑃 and its central symmetric image 𝑃∗ = 21 (𝑃 − 𝑃): 𝑃∗ is a convex polygon with at most 𝑚 ≤ 2𝑛 sides and the same width 𝑤, diameter 𝑑 and perimeter 𝑝 as 𝑃. The inradius of 𝑃∗ is at least 𝑤 and the circumradius of 𝑃∗ is at least 𝑑. It follows that 2𝑚 sin

𝜋 𝜋 𝑑 ≥ 𝑝 ≥ 𝑚 tan 𝑤. 2𝑚 𝑚

Since 𝑚 ≤ 2𝑛, using the monotonicity of the functions 𝑥 sin 1𝑥 and 𝑥 tan 1𝑥 , we get, on one hand, Reinhardt’s inequality for the maximum perimeter of an 𝑛-gon with given diameter, and on the other hand, the new inequality 𝑝 ≥ 2𝑛 tan

𝜋 𝑤. 2𝑛

The combination of these inequalities yields the inequality 𝑤 ≤ cos

𝜋 𝑑 2𝑛

between the width and diameter of a convex 𝑛-gon. Equality is attained in these inequalities by a clipped Reuleaux polygon for every 𝑛 ≥ 3 that has an odd factor. Gashkov’s article remained unnoticed. The last inequality was rediscovered by A. Bezdek and Fodor [11], and the inequality between perimeter and width by Audet, Hansen and Messine [5]. These authors also solved the case of the quadrangle for both problems. The octagon of a given diameter with maximum width was determined by Audet, Hansen, Messine and Ninin [7]. Motivated by a question of Erdős, Vincze [44] studied the problem of finding the maximum perimeter of an equilateral 𝑛-gon with given diameter. He solved the problem if 𝑛 is not a power of 2. Of course, this case is an immediate consequence of Reinhardt’s theorem. However, Vincze’s argument works without the assumption that the sides have equal length, so it yields an alternative proof of Reinhardt’s theorem. The case when 𝑛 is a power of 2 is of similar difficulty as the problem for general 𝑛-gons. The only case solved is the one for the octagon settled by Audet, Hansen, Messine and Perron [8]. Mossinghoff [37] constructed a sequence of equilateral 𝑛-gons with unit diameter for 𝑛 = 2 𝑘 , 𝑘 ≥ 4, and proved that their

9 Finite Variations on the Isoperimetric Problem

247

perimeter differs from the maximum perimeter of such 𝑛-gons by a term of at most 𝑂 (𝑛−4 ). By constructing a differen sequence of polygons Bingane and Audet [16] further improved the lower bound for the optimum perimeter. The question about the maximum area of an equilateral 𝑛-gon with given diameter 2 𝜋 𝑑 is solved for all 𝑛: It is 𝑑2𝑛 cos 𝑛𝜋 tan 2𝑛 , attained only for a regular 𝑛-gon. This follows from Reinhardt’s theorem for odd 𝑛 and was proved by Audet [1] for even 𝑛. Bingane and Audet [17] determined the equilateral octagon of unit diameter with maximum width. They also provided a family of equilateral 𝑛-gons of unit diameter, for 𝑛 = 2𝑠 with 𝑠 ≥ 4, whose widths are within 𝑂 (𝑛−4 ) of the maximum width. It appears that the question about the maximum width of an equilateral polygon with 𝑛 = 2 𝑘 sides and a given perimeter has not been studied so far. By restricting the class of competing polygons to equilateral polygons some problems with obvious solutions become interesting. The area, perimeter, and diameter of a general unit-width convex 𝑛-gon can be arbitrarily large. This is still the case for an equilateral polygon with an even number of sides. However, these quantities are bounded for equilateral convex 𝑛-gons when the number of sides is odd. Audet and Ninin [10] determined the maximum perimeter, diameter and area of an equilateral unit-width convex 𝑛-gon for every odd 𝑛 ≥ 3. The optimal polygon is the same for all three problems: For 𝑛 = 3 it is an equilateral triangle of side length √2 , and for 3

𝑛 = 2𝑘 + 1 ≥ 5 a trapezoid whose non-parallel sides have length equal to 𝑚 √2 3

1) √2 . 3

√2 , 3

and

and (𝑚 − the parallel ones have length The papers by Mossinghoff [36] and Audet, Hansen and Messine [3, 5, 6] contain nice surveys about variations of the isoperimetric problem for polygons.

Chapter 10

Higher Dimensions G. Fejes Tóth

10.1 Existence of Economic Packings and Coverings The celebrated Minkowski–Hlawka theorem (Hlawka [73]) gives the lower bound 𝛿 𝐿 (𝐾) ≥

𝜁 (𝑛) 2𝑛−1

for an arbitrary 𝑛-dimensional centrally symmetric convex body. For sufficiently √ large 𝑛, Schmidt [129, 130] improved the bound to 𝑐𝑛2−𝑛 , provided 𝑐 < ln 2. By Minkowski’s observation concerning translates of an arbitrary convex body 𝐾 and 1 2 (𝐾 − 𝐾) (see the notes to Chapter 3 in Section 8.3, p. 198) and the inequality   vol(𝐾 − 𝐾) 2𝑛 ≤ vol(𝐾) 𝑛 of Rogers and Shephard [115], it follows that, for sufficiently large 𝑛, √ 𝛿 𝐿 (𝐾) ≥ 𝜋𝑐𝑛3/2 /4−𝑛 , √ where 𝑐 < ln 2. Slightly better lower bounds were proved for the packing density of the ball by Ball [8], Vance [144], and Venkatesh [146]. Venkatesh proved that for any constant 𝑐 > sinh2 (𝜋𝑒)/𝜋 2 𝑒 3 = 65963.8 . . . there is a number 𝑛(𝑐) such that for 𝑛 > 𝑛(𝑐) we have 𝛿 𝐿 (𝐵 𝑛 ) ≥ 𝑐𝑛2−𝑛 . Moreover, there are infinitely many dimensions 𝑛 for which 𝛿 𝐿 (𝐵 𝑛 ) ≥ 2−𝑛−1 𝑛 ln ln 𝑛. Leech [86,87] constructed lattice ball packings in dimensions 𝑛 ≤ 24 and 𝑛 = 2𝑚 . He found a remarkable packing in 𝐸 24 . This lattice, called after its inventor the Leech lattice, has density 𝜋 12 /12! = 0.001929 . . ., and each ball in it is touched by 196560 others. As it was shown (see later on page 256), no other packing in 𝐸 24 has higher density and no ball can be touched by more than 196560 other balls of the same size. All the densest known lattice packings in dimensions less than 24 can be obtained as sections of the Leech lattice. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_10

249

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All packings constructed by Leech were extensions of error-correcting codes, that is packings where the centers are vertices of the unit cube. The book [140] by Thomson gives a nice account of the development of ideas from the birth of error correcting codes through sphere packings to the field of simple groups. A systematic study of constructions of ball packings based on error-correcting codes was initiated by Leech and Sloane [88]. Their work inspired further research, which led to more constructive lower bounds for the packing density of special classes of convex bodies, approaching, and in some cases also improving the Minkowski–Hlawka bound. Let 𝐵 𝑛𝑝 denote the 𝑛-dimensional unit superball, that is, the ball for the 𝑙 𝑝
1/ 𝑝 Í𝑛 norm 𝑖=1 |𝑥𝑖 | 𝑝 . Rush and Sloane [123] adopted the constructions by Leech and Sloane for packings of superballs. They obtained an improvement of the Minkowski–Hlawka bound for all integers 𝑝 > 2. As examples let us mention that 𝛿 𝐿 (𝐵3𝑛 ) ≥ 2−0.8226𝑛+𝑜(𝑛) and 𝛿 𝐿 (𝐵4𝑛 ) ≥ 2−0.6742𝑛+𝑜(𝑛) . With further elaboration of the method Rush [117] constructed lattice packings with density 2−𝑛+𝑜(𝑛) for every convex body which is symmetric through each of the coordinate hyperplanes. Moreover, Elkies, Odlyzko and Rush [54] were also able to use the method for packings of centrally symmetric convex bodies. This made it possible to construct dense packings of generalized superballs defined as 𝑓 (𝑥1 , . . . , 𝑥 𝑘 ) 𝜎 + 𝑓 (𝑥 𝑘+1 , . . . , 𝑥 2𝑘 ) 𝜎 + . . . , + 𝑓 (𝑥 𝑛−𝑘+1 , . . . , 𝑥 𝑛 ) 𝜎 ≤ 1

𝜎 ≥ 1,

where 𝑓 (𝑥1 , . . . , 𝑥 𝑘 ) is a distance function, that is, 𝑓 (0) = 0, 𝑓 (𝑥) > 0 for 𝑥 ≠ 0, and 𝑓 (𝑡𝑥) = 𝑡 𝑓 (𝑥) for 𝑡 > 0. With a further generalization of the notion of superball Rush [120] could also handle bodies like (|𝑥1 | + |𝑥2 | 2 ) 3 and max(|𝑥1 |, |𝑥2 |, |𝑥3 | 5 ). More results based on the construction of packings through error correcting codes can be found in Rush [118, 121, 122] and Liu and Xing [94]. Concerning coverings, Rogers [110] gave the bound 𝜗𝑇 (𝐾) ≤ 𝑛 ln 𝑛 + 𝑛 ln ln 𝑛 + 5𝑛 , for an arbitrary 𝑛-dimensional convex body 𝐾. Rogers’ proof uses a mean value argument combined with saturation. There are several alternative arguments yielding an upper bound for 𝜗𝑇 (𝐾) of the same order of magnitude. Naszódi [101] gave a proof of the inequality 𝜗𝑇 (𝐾) ≤ 𝑛 ln 𝑛 + 𝑛 ln ln 𝑛 + 5𝑛 that relies on an algorithmic result of Lovász [95]. Erdős and Rogers [55] showed that every 𝑛-dimensional convex body 𝐾 admits a covering by translates of 𝐾 with density 𝑛 ln 𝑛 + 𝑛 ln ln 𝑛 + 4𝑛 so that no point is covered more than 𝑒(𝑛 ln 𝑛 + 𝑛 ln ln 𝑛 + 4𝑛) times. Füredi and Kang [67] used the Lovász local lemma to give a simple proof of a slightly weaker result: There exists a covering by translates of 𝐾 such that every point is covered at most 10𝑛 ln 𝑛 times. G. Fejes Tóth [57] observed that with a modification of a proof of Rogers [112] one can show for every 𝑛-dimensional convex body 𝐾 the existence of a lattice arrangement of 𝐾 such that 𝑂 (ln 𝑛) translates of this arrangement form a covering of space with density not exceeding 𝑛 ln 𝑛 + 𝑛 ln ln 𝑛 + 𝑛 + 𝑜(𝑛). Thus, a low density covering can be achieved with an arrangement of relatively simple structure. Rolfes and Vallentin [116] suggest a greedy approach to constructing

10.1 Existence of Economic Packings and Coverings

251

coverings of compact metric spaces by metric balls. Balls are iteratively chosen to cover the maximum measure of yet uncovered space. Their method is an extension of the argument of Chvátal [27] for the finite set cover problem to the setting of compact metric spaces. Rogers [112] proved that 𝜗𝐿 (𝐾) ≤ 𝑛log2 ln 𝑛+𝑐 for some suitable constant 𝑐 and all 𝑛-dimensional convex bodies 𝐾. The same bound was proved by Butler [25] in a different way, providing a covering by translates of 𝐾 with the above density such that the corresponding translates of 𝜆𝐾, 𝜆 = vol 𝐾/vol(𝐾 − 𝐾), form a packing. For the covering density of balls Rogers [112] proved the bound 1

𝜗𝐿 (𝐵 𝑛 ) ≤ 𝑐𝑛(ln 𝑛) 2 log2 2 𝜋𝑒 . This result was generalized by Gritzmann [68], who showed that 𝜗𝐿 (𝐾) ≤ 𝑐𝑛(ln 𝑛) 1+log2 𝑒 for every 𝑛-dimensional convex body 𝐾 that has an affine image symmetric through at least log2 ln 𝑛 + 4 coordinate hyperplanes. Using covering codes instead of error correcting codes, Rush [119] adapted a construction by Leech and Sloane for the construction of thin lattice coverings by star-shaped bodies. All the above-mentioned results about lattice coverings have been superseded by Ordentlich, Regev and Weiss [104], who proved that there is a constant 𝑐 such that for any 𝑛-dimensional convex body 𝐾, 𝜗𝐿 (𝐾) ≤ 𝑐𝑛2 . Chabauty [26], Shannon [137], and Wyner [153] investigated the problem of packing balls in spherical space. Independently of each other, they proved the lower bound √ cos 𝜑 𝑀 (𝑛, 𝜑) ≥ (1 + 𝑜(1)) 2𝜋𝑛 𝑛−1 , sin 𝜑 for the maximum number 𝑀 (𝑛, 𝜑) of balls of angular diameter 𝜑 that can be packed on the sphere. Their bound was improved by Jenssen, Joos and Perkins [76] by a linear factor. Rogers [113], Böröczky and Wintsche [22], Verger-Gaugry [148], and Dumer [52] showed the existence of reasonably thin coverings of the 𝑛-dimensional sphere by congruent spherical caps and an 𝑛-dimensional ball by congruent balls. Dumer’s result implies 𝜗(𝐵 𝑛 ) ≤ ( 21 + 𝑜(1))𝑛 ln 𝑛 as 𝑛 → ∞, improving the above bound of Rogers in the case of a ball by a factor 1/2. We note that the published version of Dumer’s paper contained a small error which he corrected in [53].

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Naszódi [101] established the existence of thin coverings of the 𝑛-dimensional sphere by congruent copies of a spherical convex body. Bacher [6] considered the problem of finding universal covering sets. Let F be a family of convex bodies all having the same area. A set 𝑈 is a universal covering set for F if the translates of any member of F by the vectors of 𝑈 cover the space. He showed the existence of such sets in all dimensions. In 𝑛-dimensional space he constructed a universal covering set 𝑈 for convex bodies of unit volume such that 𝑟 𝐵 𝑛 contains at most ln(𝑟) 𝑛−1 𝑟 𝑛𝑉 (𝐵 𝑛 ) points of 𝑈.

10.2 Upper Bounds for 𝜹(𝑩 𝒏 ) and Lower Bounds for 𝝑(𝑩 𝒏 ) 10.2.1 Blichfeldt’s bound The first upper bound for the packing density of the 𝑛-dimensional ball was proved by Blichfeldt [16]. Blichfeldt’s idea was the following. Consider √ a packing of unit balls in 𝐸 𝑛 . Replace each ball 𝑆 by a concentric ball of radius 2 such that the density at distance 𝑑 from the center of the ball is 2− 𝑑 2 . Of course, the enlarged balls may overlap, however, it can be shown that their total density is at most 2 at every point of space. Comparing the volume of the unit ball to the mass of an enlarged ball, we get the bound 𝑛 + 2 −𝑛/2 2 𝛿(𝐵 𝑛 ) ≤ 2 for the packing density of the 𝑛-dimensional ball. With a modified density function Blichfeldt obtained a slightly better bound, and a further improvement was given by Rankin [106]. Blichfeldt’s method was used by Rankin [109] and Bloh [18] to derive bounds for ball packings in 𝑛-dimensional spherical space.

10.2.2 The simplex bound In an 𝑛-dimensional space of constant curvature let 𝑑 𝑛 (𝑟) be the density of 𝑛 + 1 mutually touching balls of radius 𝑟 with respect to the simplex spanned by the centers of the balls. Coxeter [41] and L. Fejes Tóth [61] conjectured that in an 𝑛-dimensional space of constant curvature the density of a packing of balls of radius 𝑟 cannot exceed the simplex bound 𝑑 𝑛 (𝑟). The corresponding simplex bound for coverings, formulated by L. Fejes Tóth [62, 63], reads as follows: In an 𝑛dimensional space of constant curvature the density of a covering by balls of radius 𝑟 cannot be less than the density 𝐷 𝑛 (𝑟) of 𝑛 + 1 balls of radius 𝑟 centered at the vertices of a regular simplex of circumradius 𝑟 relative to the simplex. In Euclidean space the bounds do not depend on 𝑟, so in this case we write them simply as 𝑑 𝑛 and

10.2 Upper Bounds for 𝛿 ( 𝐵𝑛 ) and Lower Bounds for 𝜗 (𝐵𝑛 )

253

𝐷 𝑛 . Concerning the notion of density and the interpretation of the simplex bound in hyperbolic space, see Chapter 11. For Euclidean space, the conjecture about packings was verified by Rogers [111] and independently by Baranovski˘ı [10], and the conjecture about coverings was proved by Coxeter, Few and Rogers [42]. It can be seen that the simplex bound for 𝛿(𝐵 𝑛 ) is sharper than Blichfeldt’s bound. However, the improvement is only a constant factor, namely we have 𝛿(𝐵 𝑛 ) ≤ 𝑑 𝑛 ≈

𝑛 −𝑛/2 2 , 𝑒

𝑛 → ∞.

The simplex bound 𝑛 𝜗(𝐵 𝑛 ) ≥ 𝐷 𝑛 ≈ √ , 𝑒 𝑒

𝑛 → ∞,

for the covering density of the ball compares quite favorably with the bound 𝜗(𝐵 𝑛 ) ≤ ( 21 + 𝑜(1))𝑛 log 𝑛. The three-dimensional case of the conjecture about packing was settled by Böröczky and Florian [21] and, finally, Böröczky [19] proved the conjecture in full generality. More precisely, Böröczky proved that the density of each ball in its Dirichlet cell is at most 𝑑 𝑛 (𝑟). The conjecture about covering for spherical and hyperbolic space is still open. Consider the balls inscribed in the cells of the 3-dimensional spherical tilings {2, 3, 3}, {3, 3, 3}, {4, 3, 3} or {5, 3, 3}. The corresponding radii are 21 arccos(− 31 ), 1 1 2 arccos(− 4 ), 𝜋/4, and 𝜋/10, and the density of the balls in the Dirichlet cells is 0.61613 . . ., 0.68057 . . ., 0.72676 . . ., and 0.77412 . . ., respectively. These densities agree with the corresponding tetrahedral density bound. Böröczky, Böröczky Jr., Glazyrin and Kovács [20] proved the stability of the simplex bound in the cases mentioned here. Recall the conclusion on page 116 that the ordinary sphere cannot be packed as densely as the Euclidean plane by at least two congruent circles, nor can it be covered as thinly. Remarkably, in three dimension, the analogous statement does not hold for packings, and probably does not hold for coverings either. Namely, the density 𝜋 60  𝜋 − sin = 0.77412 . . . 𝜋 5 5 √ of the 120 balls inscribed in the cells of the tiling {5, 3, 3} is greater than 𝜋/ 18 = 𝛿(𝐵3 ). Similarly, the 120 balls circumscribed about the same cells form a covering of the spherical space, with density 1 60 2𝜋 (𝜔 − sin 𝜔) = 1.44480 . . . , 𝜔 = − arccos , 𝜋 3 4 √ which is smaller than 5 5𝜋/24 = 1.46350 . . ., the conjectured value of 𝜗(𝐵3 ). It is conjectured (see L. Fejes Tóth [62]) that the inspheres and the circumspheres of {5, 3, 3} form the densest packing and the thinnest covering, respectively with at least 4 spheres in 𝑆 3 . Florian [65]) proved that 𝑑3 (𝑟) is a strictly decreasing function for

254

10 Higher Dimensions

√ 0 < 𝑟 ≤ arctan 2. It√follows that the density of a packing of at least 4 equal spheres in 𝑆 3 is at most lim 18(arctan 13 − 𝜋3 ) = 0.77963 . . .. This bound is rather close to 𝑟→0

the conjectured minimum density. Molnár [99] defined a Segre–Mahler polytope as a convex 𝑛-dimensional polytope in a space of constant curvature, every dihedral angle of which is at most 120◦ . He conjectured that, when equal spheres of radius 𝑟 are packed in such a region of 𝑛space, the density cannot exceed the simplex bound 𝑑 𝑛 (𝑟). He verified the conjecture for the 3-dimensional case. G. Fejes Tóth [56] observed that if a packing of 𝑛-dimensional balls arises as the intersection of a higher-dimensional packing of congruent balls with an 𝑛dimensional subspace, then the simplex bound 𝑑 𝑛 for the density of the packing still holds. Obviously, no similar conclusion can be drawn for an arbitrary covering with balls. However, A. Bezdek [13] proved that if a circle covering of the plane arises from a planar section of a lattice covering√ with balls in three dimensions, then the covering’s density cannot be less than 2𝜋/ 27+0.017 . . .. Moreover, equality occurs for exactly one lattice and only for certain cutting planes. It should be mentioned that in general there is no connection between the density of an arrangement of bodies and the densities of the sections of the arrangement. Groemer [71] gave examples of packings P1 and P2 with density 0 and 1, respectively, such that in each plane parallel to a given plane the density of the intersection with the sets of P1 is 1, while in each of these planes the density of the intersection with the sets of P2 is 0. He gave similar examples for coverings. Florian [66] gave a nice survey on the simplex bound for packings of balls in spaces of constant curvature with emphasis on dimension 2 and 3.

10.2.3 The linear programming bound It took more than 40 years until an improvement in exponential order was achieved for Blichfeldt’s bound of 𝛿(𝐵 𝑛 ). The first step was made by Sidelńikov [138,139], who proved 𝛿(𝐵 𝑛 ) ≤ 2−0.509619𝑛+𝑜(𝑛) . Subsequently, Leven˘ste˘ın [89] improved the bound to 𝛿(𝐵 𝑛 ) ≤ 2−0.5237𝑛+𝑜(𝑛) , and Kabatjanski˘ı and Levenšte˘in [78] proved 𝛿(𝐵 𝑛 ) ≤ 2−(0.599+𝑜 (1))𝑛

(as 𝑛 → ∞),

which remains the best known asymptotic upper bound for 𝛿(𝐵 𝑛 ). The gap between this bound and the lower bound by Venkatesh [146] remains considerable. It is ln 𝛿(𝐵 𝑛 ) ln 𝛿 𝐿 (𝐵 𝑛 ) and lim exist, and if so, whether even unknown whether lim 𝑛→∞ 𝑛→∞ 𝑛 𝑛 they are equal. All these improvements of Blichfeldt’s bound were obtained as a corollary of a bound for ball packings in spherical space. A spherical code in dimension 𝑛 with minimum angle 𝜑 is a set of points on the unit sphere in 𝐸 𝑛 with given minimum angular distance 𝜑 among them. Let 𝑀 (𝑛, 𝜑) denote the greatest size of such a

10.2 Upper Bounds for 𝛿 ( 𝐵𝑛 ) and Lower Bounds for 𝜗 (𝐵𝑛 )

255

spherical code. Observe that this notation is consistent with the previous definition of 𝑀 (𝑛, 𝜑) in Section 10.1 as the maximum number of balls of angular diameter 𝜑 that can be packed on the sphere. Delsarte, Goethals and Seidel [46] and independently Kabatjanski˘ı and Levenšte˘in [78] adopted the linear programming bound developed by Delsarte [45] for bounding the cardinality of error-correcting codes of given minimum distance to spherical codes. In the linear programming bound a basic role is played by a sequence of polynomials 𝑃 𝑛𝑘 , 𝑘 = 0, 1, . . ., called ultraspherical polynomials. They form a complete orthogonal system on the interval [−1, 1] with respect to the measure (1−𝑡 2 ) (𝑛−3)/2 d𝑡. In other words, for 𝑖 ≠ 𝑗, ∫

1

𝑃𝑖𝑛 (𝑡)𝑃 𝑛𝑗 (𝑡) (1 − 𝑡 2 ) (𝑛−3)/2 d𝑡 = 0.

−1

For the purpose of the linear programming bound, the normalization is irrelevant; the sign should be chosen so that 𝑃 𝑛𝑘 (1) > 0. With different normalizations, they are special Jacobi polynomials or Gegenbauer polynomials. The property of ultraspherical polynomials that is crucial for the application is that they are positive-definite kernels, that
is, for any 𝑁 and any points 𝑥1 , . . . , 𝑥 𝑁 ∈ 𝑆 𝑛−1 , the 𝑁 × 𝑁 matrix 𝑃 𝑛𝑘 (𝑥𝑖 · 𝑥 𝑗 ) 1≤𝑖, 𝑗 ≤𝑁 is positive semidefinite. Schoenberg [131] proved that every continuous positive-definite kernel on 𝑆 𝑛−1 has an ultraspherical expansion with nonnegative coefficients which converges absolutely and uniformly. Since the sum of the entries of a positive-semidefinite matrix is nonnegative, it follows that ∑︁ 𝑃 𝑛𝑘 (𝑥𝑖 · 𝑥 𝑗 ) ≥ 0 1≤𝑖, 𝑗 ≤ 𝑁

for any points 𝑥 1 , . . . , 𝑥 𝑁 ∈ 𝑆 𝑛−1 . Now suppose that 𝑥1 , . . . , 𝑥 𝑁 is a spherical code in dimension 𝑛 with minimum Í 𝑛 angle 𝜑 and let 𝑃(𝑡) = 𝑚 𝑘=0 𝑎 𝑘 𝑃 𝑘 (𝑡) be a real polynomial. Then ∑︁ ∑︁ 𝑃(𝑥𝑖 · 𝑥 𝑗 ) 𝑃(𝑥𝑖 · 𝑥 𝑗 ) = 𝑁 𝑃(1) + 1≤𝑖, 𝑗 ≤ 𝑁

1≤𝑖, 𝑗 ≤ 𝑁 , 𝑖≠ 𝑗

= 𝑁 2 𝑎0 +

𝑚 ∑︁ 𝑘=1

∑︁

𝑎𝑘

𝑃 𝑛𝑘 (𝑥𝑖 · 𝑥 𝑗 ).

1≤𝑖, 𝑗 ≤ 𝑁

Suppose that 𝑃(𝑡) ≤ 0 for −1 ≤ 𝑡 ≤ cos 𝜑, 𝑎 0 > 0, and 𝑎 𝑘 ≥ 0 for 𝑖 = 1, 2, . . . , 𝑘. 𝑚 ∑︁ ∑︁ ∑︁ Then 𝑃 𝑛𝑘 (𝑥𝑖 · 𝑥 𝑗 ) ≥ 0. Hence we get the 𝑃(𝑥𝑖 · 𝑥 𝑗 ) ≤ 0 and 𝑎𝑘 1≤𝑖, 𝑗 ≤ 𝑁 , 𝑖≠ 𝑗

𝑘=1

1≤𝑖, 𝑗 ≤ 𝑁

linear programming bound for 𝑀 (𝑛, 𝜑): 𝑚 ∑︁ 𝑎 𝑘 𝑃 𝑛𝑘 (𝑡) be a real polynomial such that 𝑃(𝑡) ≤ 0 for −1 ≤ 𝑡 ≤ Let 𝑃(𝑡) = 𝑘=0

cos 𝜑, 𝑎 0 > 0, and 𝑎 𝑘 ≥ 0 for 𝑖 = 1, 2, . . . , 𝑚. Then

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10 Higher Dimensions

𝑀 (𝑛, 𝜑) ≤ 𝑃(1)/𝑎 0 . 𝑛 denote the largest root of 𝑃 𝑛 (𝑡). With the choice of appropriate polynoLet 𝑡1,𝑘 𝑘 𝑛 then mials Kabatjanski˘ı and Levenšte˘in [78] proved that if cos 𝜑 ≤ 𝑡 1,𝑘



  −1 𝑘 +𝑛−2  𝑛 𝑀 (𝑛, 𝜑) ≤ 4 . 1 − 𝑡1,𝑘+1 𝑘 From this they derived the asymptotic bound 𝑀 (𝑛, 𝜑) ≤ (1 − cos 𝜑) −𝑛/2 2−𝑛(0.099+𝑜(1))

(as 𝑛 → ∞)

for all 𝜑 ≤ 𝜑∗ = 62◦ . . .. Improvements by a constant factor, thus not affecting the exponent in the asymptotic bound, were given by Levenšte˘in [93] and Sardari and Zargar [127]. The asymptotic upper bound for 𝑀 (𝑛, 𝜑) yields a similar bound for 𝛿(𝐵 𝑛 ). Kabatjanski˘ı and Levenšte˘in used the inequality 𝛿(𝐵 𝑛 ) ≤ sin𝑛 (𝜑/2)𝑀 (𝑛 + 1, 𝜑).

186 ▶

However, there is a slightly better inequality using Blichfeldt’s method. For, it is easily seen that a ball of radius 𝑟 = 1/sin(𝜑/2) ≤ 2 can contain at most 𝑀 (𝑛, 𝜑) centers of a packing of unit balls. Thus the concentric balls of radius 1/sin(𝜑/2) form an 𝑀 (𝑛, 𝜑)-fold packing, yielding 𝛿(𝐵 𝑛 ) ≤ sin𝑛 (𝜑/2)𝑀 (𝑛, 𝜑). This argument was well known in the community approaching problems of packing by geometric methods. It was first published by Levenšte˘in [91, p. 108], then rediscovered by Cohn and Zhao [39], who also used it to derive an asymptotic improvement of the simplex bound in hyperbolic space. The linear programming bound for spherical codes was used by Odlyzko and Sloane [102] and Levenšte˘in [90] to solve the problem of densest packing of 𝑁 balls in 𝑆 𝑛 for some special values of 𝑛 and 𝑁. The arrangements that were characterized as optimal spherical codes in this way are listed, with the exception of the set of 120 vertices of the 600-cell {3, 3, 5}, in Levenšte˘ın [92, p. 72] and [93, p. 621]. The optimality of the vertices of {3, 3, 5} follows from the simplex bound, and was proved using the linear programming bound by Andreev [4]. Elaborations of the linear programming bound by Musin [100], Boyvalenkov [23], Pfender [105], Bachoc and Vallentin [7] and Mittelmann and Vallentin [98] yielded improved upper bounds for 𝑀 (𝑛, 𝜑) in some low dimensions. The surveys by Boyvalenkov, Dodunekov and Musin [24], Cohn [29, 31], and Viazovska [150] describe these methods in detail. Cohn and Elkies [33] and Cohn [28] modified the linear programming method, obtaining bounds for 𝛿(𝐵 𝑛 ) directly. Although they did not improve on the asymptotic bound given above, their method proved to be exceptionally efficient for 𝑛 = 8 and 𝑛 = 24. The method enabled Cohn and Kumar [34, 36] to show that the Leech lattice is the unique densest lattice in 24 dimensions and to give an alternative proof of Blichfeldt’s result [17] about the densest packing of balls in 8 dimensions.

10.2 Upper Bounds for 𝛿 ( 𝐵𝑛 ) and Lower Bounds for 𝜗 (𝐵𝑛 )

257

Furthermore they proved that no packing of congruent balls in 24 dimensions can exceed the Leech lattice’s density by a factor of more than 1 + 1.65 × 10−30 . A breakthrough was achieved by Viazovska [149], who succeeded in proving that 𝛿(𝐵8 ) = 𝛿 𝐿 (𝐵8 ) = 𝜋 4 /384. Some days later Cohn, Kumar, Miller, Radchenko and Viazovska [38] showed that 𝛿(𝐵) 24 = 𝛿 𝐿 (𝐵24 ) = 𝜋 12 /12!. The paper by Cohn [32] explains the main ideas leading to this landmark achievement. The survey and interview by de Laat and Vallentin [43] is also interesting reading on this subject. See also the paper [151] by Viazovska and the Fields Medal laudatio of Maryna Viazovska by Henry Cohn [30].

10.2.4 Arrangements of points with minimum potential energy Given a decreasing potential function 𝑓 defined on (0, 2] and an integer 𝑁 > 1, we wish to place 𝑁 distinct points {𝑥1Í, 𝑥2 , . . . , 𝑥 𝑁 } on the unit sphere in 𝑛-dimensional 𝑓 (|𝑥𝑖 − 𝑥 𝑗 |) is as small as possible. Yudin [154] space so that the potential energy 𝑖≠ 𝑗

extended the linear programming bound to obtain an lower bound for the potential energy. His result contains the linear programming bound for 𝑀 (𝑛, 𝜑) as a corollary. Using Yudin’s result the minimum potential energy of 240 points on 𝑆 7 , of 196560 points on 𝑆 23 , and of 120 points on 𝑆 3 with the potential of 𝑓 (𝑥) = 𝑥 𝑛−1 on 𝑆 𝑛 was determined by Kolushov and Yudin [79] and Andreev [3, 5]. The corresponding optimal arrangements are the minimal vectors in the 𝐸 8 root lattice, the minimal vectors in the Leech lattice and the vertices of the 600-cell {3, 3, 5}. Cohn and Kumar [35] succeeded in proving optimality of certain arrangements not just for a specific, but for a whole class of potential functions. A real function 𝑓 (𝑥) is completely monotonic if it is decreasing, infinitely-differentiable and satisfy the inequalities (−1) 𝑙 𝑓 (𝑙) (𝑥) ≥ 0. Typical examples are the inverse power functions 𝑓 (𝑥) = 1/𝑥 𝑠 with 𝑠 > 0 and exponential functions 𝑓 (𝑥) = 𝑒 −𝑐𝑥 with 𝑐 > 0. In Cohn and Kumar’s investigation the argument of potential functions is the square of the distance, rather that the distance itself. They define an arrangement of points as universally optimal if it is optimal under every potential function 𝑓 (|𝑥 − 𝑦| 2 ) where 𝑓 (𝑥 2 ) is completely monotonic. Note, that 𝑓 (𝑥 2 ) is completely monotonic on an interval (𝑎, 𝑏), 𝑎 > 0, then 𝑓 (𝑥) is completely monotonic on (𝑎 2 , 𝑏 2 ), but not vice versa. Cohn and Kumar [35] proved universal optimality of the sets of vertices of all regular simplicial polytopes in every dimension, as well as of several other arrangements in dimensions 2–8 and 21–24. These arrangements are listed in Table 1 of their paper. The list coincides with the list of optimal spherical codes mentioned above. For 𝑓 (𝑥) = 𝑥 −𝑡 with 𝑡 large, the energy is asymptotically determined by the minimal distance, thus minimizing energy requires maximizing the minimal distance. Hence, universal optimality of an arrangement implies that it is an optimal spherical code. With the special choice of the potential function 𝑓 (𝑥) = 2 − 𝑥 1/2 and 𝑓 (𝑥) = ln(4/𝑥), respectively, it also follows that these arrangements maximize

◀ 153

258

153 ▶

10 Higher Dimensions

the sum, as well as the product, of the distances between pairs of points. Earlier, Andreev [2, 3] proved that the vertices of the regular icosahedron maximize the product of the distances between 12 points on 𝑆 2 and the minimal vectors of the Leech lattice maximize the sum of the distances between 196560 points on 𝑆 23 . Also, Kolushov and Yudin [80] proved that the vertices of the regular simplex, the regular cross-polytope and the minimal vectors of the 𝐸 8 root lattice maximize both the sum and the product of the distances between the corresponding number of points lying on the respective sphere. On 𝑆 2 , the only universal optima are a single point, two antipodal points, an equilateral triangle on the equator, and the vertices of a regular tetrahedron, octahedron, or icosahedron. That this list is complete follows from a result of Leech [85] who enumerated all those configurations on 𝑆 2 that are in a (stable or unstable) equilibrium under any force that depends on the distance only. He showed  that the only configu rations of this kind are the vertices of the tilings {𝑝, 𝑞} 𝑝, 𝑞 ≥ 2, 𝑝1 + 𝑞1 > 12 or the face-centers of a tiling {2, 𝑞} (𝑞 ≥ 2). Our knowledge in higher dimensions is limited. It appears that universally optimal arrangements of points are rare. Through a computer search, Ballinger, Blekherman, Cohn, Giansiracusa, Kelly and Schürmann [9] found two arrangements of points, one consisting of 40 points on 𝑆 9 , and another of 64 points on 𝑆 13 , which they conjecture to be universally optimal. Besides the arrangements of points proved by Cohn and Kumar to be universal optimal, exact solutions are only known for a few special cases. Dragnev, Legg and Townsend [50] investigated the problem of minimizing the energy under the logarithmic potential ln(1/𝑥). This is the same problem as finding the maximum of the product of the distances between 𝑘 points on 𝑆 𝑛 . They solved the problem for 𝑘 = 5 points on 𝑆 2 . The optimum is attained by the opposite poles of the sphere and three points distributed evenly on the equator, that is, by the vertices of a bipyramid. Dragnev [49] investigated the problem in higher dimensions. He made the conjectured that the product of the distances between 𝑛 + 3 points on 𝑆 𝑛 attains its maximum for an arrangement of points that is the union of two mutually orthogonal 𝑛+2 regular simplices, one of dimension ⌊ 𝑛+1 2 ⌋, the other of dimension ⌊ 2 ⌋. He proved the conjecture for 𝑛 = 3 and 𝑛 = 4. Dragnev and Musin [51] verified the conjecture for all 𝑛 by enumerating all stationary configurations of 𝑛 + 2 points on 𝑆 𝑛 for the logarithmic potential. Due to applications in stereochemistry, the potential sign (𝑠)/𝑥 𝑠 received special attention. Through a computer search, Melnyk, Knop and Smith [97] found conjecturally optimal solutions for up to 16 points for different values of 𝑠. For the case of 5 points they conjectured that there exists a phase transition constant 𝑠0 = 15.04808 so that for 𝑠 < 𝑠0 the triangular bipyramid, and for 𝑠 > 𝑠0 a quadrilateral pyramid is optimal. The conjecture was confirmed in three special cases. The triangular bipyramid was proved to be optimal by Hou and Shao [74] if 𝑠 = −1 and by Schwartz [134] if 𝑠 = 1 or 𝑠 = 2. Subsequently, Schwartz [135, 136] determined the exact value of 𝑠0 = 15.0480773927797 . . . , proved that the triangular bipyramid is optimal for −2 < 𝑠 < 𝑠0 and the quadrilateral pyramid is optimal for 𝑠0 < 𝑠 ≤ 15 + 25/512 = 15.048828125 = 𝑠1 . For 𝑠 = 𝑠0 both the triangular bipyra-

10.3 Bounds for the Packing and Covering Density of Convex Bodies

259

mid and the quadrilateral pyramid minimize the energy. For 𝑠 > 𝑠1 the problem is unsolved. A further characterization of the triangular bipyramid is due to Tumanov [143] who proved that this arrangement of points on the sphere constitutes the unique minimizer position under a potential of the form 𝑓 (𝑟) = 𝑎𝑟 4 − 𝑏𝑟 2 + 𝑐, 𝑎 > 0, 𝑏 > 8𝑎. Remarkably, the solution of the sphere packing problem in 8 and 24 dimensions was not the end of the story: Cohn, Kumar, Miller, Radchenko and Viazovska [37] proved that the 𝐸 8 root lattice and the Leech lattice are universally optimal among point arrangements in Euclidean spaces of dimensions 8 and 24, respectively. In other words, they have minimum energy among all arrangements of points with given density for every potential function that is a completely monotonic function of the squared distance.

10.2.5 Lattice arrangements of balls Besides dimensions 3, 8, and 24, where the maximum density of general packings of congruent balls is known, the value of 𝛿 𝐿 (𝐵 𝑛 ) was determined for 𝑛 = 4 and 𝑛 = 5 by Korkine and Zolotareff [81, 82] and for 𝑛 = 6, 7 and 8 by Blichfeldt [17]. The lattice covering density of the ball is known only in dimensions 4 and 5. The case 𝑛 = 4 has been established by Delone and Ryškov [44] and the case 𝑛 = 5 by Ryškov and Baranovski˘ı [124, 125]. Schürmann and Vallentin [133] designed an algorithm for approximating the values of the lattice covering density of the ball along with the critical lattice, with arbitrary accuracy. Implementing the algorithm in dimensions 6, 7, and 8 enabled them to find the best known lattices for thin sphere covering in these dimensions. An 𝑚-periodic arrangement is the union of 𝑚 translates of a lattice arrangement. Andreanov and Kallus [1] presented an algorithm to enumerate all locally optimal 2-periodic sphere packings in any dimension, provided there are finitely many. They implemented the algorithm in 3, 4, and 5 dimensions and showed that no 2-periodic packing of balls surpasses the density of the optimal lattices in these dimensions.

10.3 Bounds for the Packing and Covering Density of Convex Bodies Finding the packing density 𝛿(𝐾) for a given convex body 𝐾, even finding a meaningful upper bound for it, is generally a very difficult task. Let 𝑑 (𝐾) denote the density 𝑛) of the insphere of an 𝑛-dimensional body 𝐾 in 𝐾. Then, obviously, 𝛿(𝐾) ≤ 𝛿𝑑(𝐵 (𝐾) , which yields a non-trivial upper bound for 𝛿(𝐾) if 𝑑 (𝐾) ≥ 𝛿(𝐵 𝑛 ). Using this bound Torquato and Jiao [141,142] gave non-trivial upper bounds for the packing density of the icosahedron and dodecahedron, as well as of several Archimedean solids and

260

10 Higher Dimensions

superballs. The bound obtained in this way for the octahedron is greater than 1, however we get a nontrivial bound for the packing density of the regular cross-polytope in dimensions greater than 23. Moreover, we get that the packing density of the 𝑛-dimensional regular cross-polytope approaches zero exponentially as 𝑛 tends to infinity. Blichfeldt’s method was used to obtain upper bounds for the translational packing density of the superball 𝐵 𝑛𝑝

   

𝑛

= (𝑥1 , . . . , 𝑥 𝑛 ) ∈ 𝐸 |   

𝑛 ∑︁ 𝑖=1

! 1/ 𝑝 |𝑥 𝑖 |

𝑝

≤1

      

by van der Corput and Schaake [145], Hua [75] and Rankin [107, 108] (see also Zong [155, Section 6.3]). For 𝑝 ≥ 2 their bound was recently improved by Sah, Sawhney, Stoner and Zhao [126] based on the Kabatjanski˘ı–Levenšte˘in bound for spherical codes. With an extension of Blichfeldt’s method, G. Fejes Tóth and W. Kuperberg [59] were able to give non-trivial upper bounds for the packing density of other, suitable convex bodies, e.g., for “longish” bodies such as sufficiently long cylinders 𝐵 𝑛−1 × [0, ℎ] and “sausage-like” solids 𝐵 𝑛 + [0, ℎ] in 𝑅 𝑛 . Applying this method G. Fejes Tóth, Fodor and Vígh [58] gave a non-trivial upper bound for the packing density of the regular cross-polytope in all dimensions greater than 6. Elaborating on the method used by A. Bezdek and W. Kuperberg [14], W. Kusner [83] obtained another bound for the √ packing density of finite-length circular cylinders, namely 𝛿(𝐵2 × [0, ℎ]) ≤ 𝜋/ 12 + 10/ℎ. Although this bound is meaningful (smaller than 1) only if ℎ > 100, and it improves upon the bound given in G. Fejes Tóth and W. Kuperberg only for ℎ greater than about 250, the advantage of Kusner’s bound is that it approaches the packing density of the circle as ℎ → ∞. In a further paper W. Kusner [84] extended the result of A. Bezdek√and W. Kuperberg by showing that the packing density of the set 𝐵2 × 𝑅 𝑛 is also 𝜋/ 12. We can get upper bounds for the translational packing density of convex bodies by Minkowski’s observation that a family of translates of a convex body 𝐾 forms a packing if and only if the corresponding translates of the centrally symmetric body 1 2𝑛−1 2 (𝐾 − 𝐾) form a packing. Groemer [69] proved in this way that 𝛿𝑇 (𝐶) ≤ 2𝑛 −1 for every convex
cone 𝐶. For the 𝑛 dimensional simplex 𝑆 𝑛 this argument yields −𝑛 𝛿𝑇 (𝑆 𝑛 ) ≤ 2𝑛 2𝑛 . In particular, we have 𝛿𝑇 (𝑇) ≤ 0.4 for a tetrahedron 𝑇. 𝑛 Zong [156] proposed a method based on the shadow regions introduced by L. Fejes Tóth [64] to give upper bounds for the translative packing density of three-dimensional convex bodies.√Applying √ the method for the tetrahedron 𝑇, he established the bound 𝛿𝑇 (𝑇) ≤ 36 10/(95 10 − 4) = 0.3840610 . . . . de Oliveira Filho and Vellentin [103] extended the linear programming method to estimate the packing density of congruent copies of a convex body. Dostert, Guzmán, de Oliveira Filho and Vallentin [48] exploited this method to obtain upper bounds for the translative packing density of some three-dimensional convex bodies with tetrahedral symmetry, such as superballs and of Platonic and

10.4 The Structure of Optimal Arrangements

261

Archimedean solids. They improved Zong’s upper bound for the translative packing density of the tetrahedron to 0.3745. In some cases it can be proved that for a body 𝐾 we have 𝛿(𝐾) < 1 or 𝜗(𝐾) > 1 without establishing a concrete bound. Hlawka conjectured that the packing density of circular tori cannot be 1. Motivated by this conjecture Schmidt [128] proved a general theorem which implies as a corollary, besides the positive answer to Hlawka’s conjecture, 𝛿(𝐾) < 1 and 𝜗(𝐾) > 1 for every smooth convex body. Schmidt’s theorem does not apply for packings of cones. W. Kuperberg proved that a packing consisting of translates of a cone 𝐶 and its images −𝐶 in 𝐸 3 cannot have density 1. Bárány and Matoušek [11] succeeded in proving an explicit bound smaller than 1 for the density of such a packing.

10.4 The Structure of Optimal Arrangements In higher dimensions the occurrence of less organized arrangements among the optimal ones seems to be more frequent. It is likely that the equality 𝛿 𝐿 (𝐾) = 𝛿𝑇 (𝐾) fails in dimensions greater than 2, although no convex body is known for which 𝛿 𝐿 (𝐾) < 𝛿𝑇 (𝐾). Rogers [114, page 15] conjectured that 𝛿 𝐿 (𝐵 𝑛 ) < 𝛿𝑇 (𝐵 𝑛 ) for all sufficiently large 𝑛. Best [12] constructed non-lattice ball packings in dimensions 10, 11, and 13 that are denser than the densest known lattice packings. There is a special class of convex bodies for which the equality 𝛿 𝐿 = 𝛿𝑇 holds: Venkov [147] and independently McMullen [96] proved that parallelohedra, that is, those polytopes whose translates tile space also admit a lattice tiling. We note, that according a theorem of Groemer [70] parallelohedra are also characterized by the property that space can be tiled by positive homothetic copies of them. The equality 𝛿(𝐾) = 𝛿𝑇 (𝐾), which holds in the plane for all centrally symmetric disks, fails already in dimension 3. There exist centrally symmetric convex bodies whose congruent copies can pack space perfectly (tile it without gaps), but whose maximum density attained in a packing of translates is smaller than 1. One such body is the right double-pyramid erected over and under the unit square, with height 1/2. Moreover, A. Bezdek and W. Kuperberg [15] observed that for 𝑛 ≥ 3, there exist ellipsoids in 𝐸 𝑛 for which packing with congruent copies can exceed the maximum density by translates, that is, the ball’s packing density. In their construction A. Bezdek and W. Kuperberg used the theorem of Heppes [72] that one can place infinite circular cylinders in the void of every lattice packing of balls. Packing in the cylinders long ellipsoids of the same volume as the balls we get a mixed packing of balls and ellipsoids. With a suitable affinity, the balls and ellipsoids are then transformed into congruent ellipsoids. Refining this construction, Wills [152] produced a packing of congruent ellipsoids with density 0.7549 . . . and Schürmann [132] constructed dense ellipsoid packings in dimensions up to 8. The packing of congruent ellipsoids constructed by Schürmann in 𝐸 8 exceed the density of the densest packing of balls by more than 42.9%.

262

10 Higher Dimensions

Motivated by the problem of understanding the structure of certain materials like crystals and glasses Donev, Stillinger, Chaikin and Torquato [47] constructed in 𝐸 3 packings of congruent copies of ellipsoids close to the ball with density grater √ than 𝜋/√ 18 = 0.74048 . . . . If 𝑎 ≤ 𝑏 ≤ 𝑐 are the semiaxes of √the ellipsoid and 𝑐/𝑎 ≥ 3, then the construction has density 0.7707, exceeding 𝜋/ 18 considerably. In the case 1.365 ≤ 𝑎/𝑏 ≤ 1.5625 a packing constructed by Jin, Jiao, Liu, Yuan and Li [77] has higher density. The idea of the combination of a lattice arrangement with an arrangement in infinite cylinders was used by G. Fejes Tóth and W. Kuperberg [60] for coverings. They proved that for every 𝑛-dimensional (𝑛 ≥ 3) strictly convex body 𝐾 there is an affine-equivalent body 𝐿 whose congruent copies can cover space more thinly than any lattice covering. The assumption of strict convexity is essential: There exist convex polyhedra, e.g. a cube, which tile space in a lattice-like manner. In 𝐸 3 they presented an ellipsoid 𝐸 for which 𝜗(𝐸) ≤√ 1.394. Since by the simplex bound of Coxeter, Few and Rogers [42] 𝜗(𝐵3 ) ≥ 3 2 3 (3 arccos 31 − 𝜋) = 1.431 . . ., it follows that 𝜗(𝐸) < 𝜗(𝐵3 ) = 𝜗𝑇 (𝐸). The excellent book of Rogers [114] gives an exhaustive account of packing and covering in high dimensions. The book of Conway and Sloane [40] is an encyclopedic source of information about sphere packing.

Chapter 11

Ball Packings in Hyperbolic Space G. Fejes Tóth, L. Fejes Tóth and W. Kuperberg

It is natural to extend the study of packing and covering problems to the hyperbolic plane, as well as hyperbolic spaces of higher dimension. Research in that direction began essentially only after the original publication of this book. Research in dimensions higher than 2 was restricted to packings of balls. Difficulties with defining a “reasonable” notion of global density in the hyperbolic plane were already mentioned on page 155. Böröczky [5] exposed that these difficulties lie much deeper than one would expect. Namely, he constructed an arrangement of congruent circles along with two different decompositions 𝑍1 and 𝑍2 of the hyperbolic plane into congruent cells, each containing one circle, with the property that the circles’ density in each cell of 𝑍𝑖 presents the same value 𝑑𝑖 (𝑖 = 1, 2), and yet 𝑑1 ≠ 𝑑2 . Two ways were considered to overcome the aforementioned difficulties with defining global density in hyperbolic geometry. One way was to consider local density relative to Dirichlet cells; the other was to come up with suitable notions alternate to arrangements of optimal density.

11.1 The Simplex Bound Recall the simplex bound of Böröczky [6] mentioned in Section 10.2.2: For any packing of balls of radius 𝑟 in 𝑛-dimensional hyperbolic space the density of a ball in its Dirichlet cell cannot exceed the density 𝑑 𝑛 (𝑟) of 𝑛 + 1 mutually touching balls of radius 𝑟 with respect to the simplex spanned by the centers of the balls. Böröczky stated this for balls of finite radius, however the proof can be extended to packings of horoballs (balls of infinite radius). As in the case of balls of finite radius, the bound refers to the density of the horoballs in their Dirichlet cell in this case as well. It should be mentioned that the volume of the horoballs, as well as their Dirichlet cells, is infinite. However, as the hyperbolic metric on a horosphere is Euclidean, the density of a horoball in its Dirichlet cell can be defined by a limit of the density in growing sectors of the horoball. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_11

263

264

11 Ball Packings in Hyperbolic Space

The definition of the simplex bound needs further explanation in the case of horoball packings. Let 𝑆 be a totally asymptotic regular simplex in 𝑛-dimensional hyperbolic space. Take 𝑛 + 1 mutually tangent horoballs centered at the vertices of 𝑆 and consider their density relative to 𝑆. In contrast to balls of finite radius, the condition that the horoballs are mutually tangent does not determine the configuration. We get the simplex bound 𝑑 𝑛 (∞) occurring in Böröczky’s theorem if the symmetry group of the arrangement of the horoballs coincides with that of 𝑆. Szirmai [49,50] investigated the density of mutually tangent horoballs centered at the vertices of a totally asymptotic regular simplex 𝑆 in the case when the symmetry group of the arrangement of the horoballs does not coincide with that of 𝑆. It turned out that for 𝑛 ≥ 3 there are arrangement of horoballs that produce local density in 𝑆 higher than 𝑑 𝑛 (∞). Of course, these configurations are only locally optimal and cannot be extended to the entire hyperbolic space. But their existence shows that there might also be arrangements of mutually tangent balls of finite radius 𝑟 centered at the vertices of a simplex with density in the simplex higher than 𝑑 𝑛 (𝑟). We recall that for a packing of circles of radius 𝑟 in a plane of constant curvature the density of each circle in its Dirichlet cell, as well the density of the circles in each Delone cell, is at most 𝑑2 (𝑟). It appears that in higher-dimensional hyperbolic space 𝑑 𝑛 (𝑟) is an upper bound only for the local density in the Dirichlet cells and not for the density in the Delone simplices, although no explicit example is known. Also, it is an interesting open question whether, for some 𝑛 ≥ 3, in 𝑛-dimensional Euclidean or spherical space the density of a packing of balls of radius 𝑟 in the Delone simplices can exceed 𝑑 𝑛 (𝑟). In 3-dimensional hyperbolic space there is a remarkable tiling {6, 3, 3} whose cells are degenerate Euclidean polyhedra {6, 3}, circumscribed about horoballs. The horoballs inscribed in the cells of this tiling form a packing while the horoballs circumscribed about the cells form a covering. Coxeter [12] calculated the density of the horoballs in the cells and found that it is   −1 1 1 1 1 1 𝑑3 (∞) = 1 + 2 − 2 − 2 + 2 + 2 − . . . ≈ 0.853 2 4 5 7 8 in the case of packing, and 

1 1 1 1 1 𝐷 3 (∞) = 1 − 2 + 2 − 2 + 2 − 2 + . . . 2 4 5 7 8

 −1 ≈ 1.280

in the case of covering (see also Zeitler [56] for the case of covering). Of course, these densities are the same as the density of the respective horoballs in the asymptotic tetrahedral cells of the dual tiling {3, 3, 6}, that is, the bound proved by Böröczky for packings and the still conjectured corresponding tetrahedral density bound for coverings. Also, we have lim 𝑑3 (𝑟) = 𝑑3 (∞) and lim 𝐷 3 (𝑟) = 𝐷 3 (∞). Florian 𝑟→∞ 𝑟→∞ (see Böröczky and Florian [7]) showed that in hyperbolic space the tetrahedral density bound 𝑑3 = 𝑑3 (𝑟) is a strictly increasing function of 𝑟. Therefore, for an arbitrary packing of congruent balls of finite or infinite radius in hyperbolic space,

11.2 Hyperspheres

265

the density of each ball in its Dirichlet cell is at most 𝑑3 (∞). We can therefore say that in 3-dimensional hyperbolic space the packing of the balls inscribed in the cells of the tiling {6, 3, 3} is the densest among all packings with congruent balls. It is conjectured that 𝑑 𝑛 (𝑟) is a strictly increasing function of 𝑟 for all 𝑛. Marshall [39] gave a partial verification of this conjecture by proving that it holds for sufficiently large values of 𝑛. Kellerhals [33] proved that 𝑑 𝑛 (𝑟) > 𝑑 𝑛+1 (𝑟) for all 𝑟 > 0. Szirmai [47, 48, 51] and Kozma and Szirmai [34, 35, 36, 37, 38] determined the minimum density of certain classes of periodic packings of horoballs in low dimensions.

11.2 Hyperspheres Besides spheres of finite radius and horospheres, there is a third type of “sphere” in hyperbolic space, namely hyperspheres. In 𝑛-dimensional hyperbolic space a hypersphere is the set of points of the space that are at the same distance 𝑟 from an (𝑛 − 1)-dimensional hyperplane. A hypersphere consists of two disjoint surfaces that bound a connected component of the space called the hyperball of radius 𝑟. The study of packing and covering by hyperballs was initiated by Vermes. In [53] he investigated packings of congruent hypercircles of radius 𝑟, and gave an upper bound for the local density of the hypercircles in the cells of the dual subdivision of the plane into Dirichlet cells. His bound is sharp for all values of 𝑟, increases monotonously, and its limit at infinity is 3/𝜋, the density of the densest packing of horocircles. Unaware of Vermes’s result, Marshall and Martin [40] discovered the same bound. Przeworski [46] proved an upper bound for the density of packings of congruent hyperballs. His bound is an analogue of the simplex bound for ball packings. In the proof he uses Delone cells for the base-hyperplanes of the hyperballs. These cells, which he introduced in his paper [45], are truncated ultraideal simplices. The truncation faces are the base-hyperplanes of the hyperballs, and they are perpendicular to all non-truncation faces that intersect them. The maximum density of the hyperballs in the truncated simplices occurs for a regular simplex. Hence he gets the following: In 𝐻 𝑛 , let 𝑑 𝑛 (𝑟) denote the density of 𝑛 + 1 pairwise touching hyperballs in the truncated simplex bounded by the base hyperplanes of the hyperballs and the hyperplanes orthogonal to the base planes though the touching points of the hyperballs. Then the density of any packing of hyperballs of radius 𝑟 within every Delone cell, as well as in every Dirichlet cell, is at most 𝑑 𝑛 (𝑟). The same result was proved by Miyamoto [42] in another context. For 3-dimensional space an alternative decomposition into truncated tetrahedra was suggested by Szirmai [52]. Vermes [54] described for every 𝑟 > 0 a class of regular packings of hypercircles of radius 𝑟, calculated the density of each of them and determined the minimum density for all values of 𝑟. It turned out√ that the minimum density Θ(𝑟) is an increasing function of 𝑟 with limit value 12/𝜋 as 𝑟 → ∞. In [55] he considered general coverings of hypercircles of radius 𝑟 subject to the condition that there exists

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11 Ball Packings in Hyperbolic Space

a number 0 < 𝑟 < 𝑟 such that the hypercircles of radius 𝑟 around the same base lines form a packing. This condition guarantees that no point of the plane is covered by infinitely many members of the covering. For the density of coverings of the plane by hypercircles of radius 𝑟 satisfying the above property, Vermes [55] gave a lower bound that is sharp for the thinnest √ regular coverings. The infimum of the density of coverings by hyperspheres is 12/𝜋, attained by the thinnest covering of horospheres. A hyperball is the parallel body of an (𝑛 − 1)-dimensional hyperplane. It is natural to consider packings of parallel bodies of lower dimensional planes as well. Marshall and Martin [41] and Przeworski [43, 44] proved upper bounds for the density of packings of tubes, that is, of parallel bodies of lines in 3-dimensional hyperbolic space.

11.3 Solid Arrangements As one of the possible substitutes for the notions of densest packing and thinnest covering, L. Fejes Tóth [15] defined solidity of packings and coverings. We say that a circle packing (covering) is solid if no finite subset of the set of circles can be rearranged so that the resulting packing (covering) is not congruent to the original one. Roughly speaking, if we remove any finite number of arbitrarily chosen circles from a solid circle packing (covering), and we wish to place them again so that we still obtain a packing (covering), then they have to be returned to their original places. It follows from a theorem of Imre [32] that the circles inscribed in the faces of the tiling {𝑘, 3} (𝑘 = 2, 3, . . .) form a solid packing, and the circles circumscribed about the faces of the tiling {𝑘, 3} (𝑘 = 3, 4, . . .) form a solid covering. Moreover, L. Fejes Tóth [15] conjectured that the circles inscribed in, and the circles circumscribed about, the faces of every three-valent Archimedean tiling form a solid packing and a solid covering, respectively. On the other hand, he conjectured that the faceincircles and the face-circumcircles of a more than trihedral uniform tiling never form a solid arrangement. In many cases this has been confirmed, see L. Fejes Tóth [15], Heppes [26], Heppes and Kertész [30], Florian [17, 18, 19, 20, 21], Florian and Heppes [23] and G. Fejes Tóth [13]. In particular, both conjectures were confirmed for the face-incircles of all spherical and Euclidean uniform tilings. As a particularly appealing example we mention the “football”, i.e., the tiling of the truncated icosahedron {5, 6, 6}, whose face-incircles form a solid packing. In some cases, e.g., for the incircles of the tilings (8, 8, 4) and (4, 4, 𝑛) for 𝑛 ≥ 6, in addition to the solidity of the packing it is shown that the arrangement has the greatest possible density among all packings with any collection of circles of the given radii (see Florian [19], Florian and Heppes [22], Heppes [28,29] and Heppes and Kertész [30]).

11.3 Solid Arrangements

267

We emphasize the following special case of the above conjecture: Augmenting the incircles of a regular (spherical, Euclidean or hyperbolic) tiling by circles inscribed in the holes results in a solid circle packing. This indeed occurs in the case of the Euclidean tilings {6, 3} and {3, 6}, since the inscribed circles in {6, 3} form a solid packing by themselves (Figure 3.13, page 79), and augmenting the incircles of {3, 6} results in the incircles of {6, 3}. For the case of the third regular Euclidean tiling, namely {4, 4}, the solidity of the corresponding packing (Figure 3.12, page 79) was proved by Heppes [26] using the idea of weighted density. The solidity of a packing P follows if we can show that in every packing of circles with the given radii the local density in the Delone triangles cannot exceed the density of the circles of P in the Delone triangles. For the face incircles of the tiling (4, 4, 8) this is not true. Heppes observed that the solidity of P follows if some weights can be assigned to the circles so that the above statement holds for the weighted density. Using appropriate weights he could prove, besides the incircles of the tiling (4, 4, 8), the solidity of several other packings. The work of Heppes inspired further research. Weighted density of packings has also been studied for its own sake (see Hárs [24, 25]). A notion weaker than solidity was introduced and investigated by A. Bezdek, K. Bezdek and Connelly [3, 4]. A packing or a covering is uniformly stable if there is an 𝜀 > 0 such that no finite subset of the arrangement can be rearranged so that each member is moved by a distance less than 𝜀 and the rearranged members, together with the rest, form a packing or covering, respectively, different from the original arrangement. Using techniques from rigidity theory they proved uniform stability of certain packings. A solid circle packing is strongly solid if it remains solid even after any one of the circles is removed. The incircles of the faces of the tilings {4, 3} and {5, 3} are strongly solid. L. Fejes Tóth conjectured that the incircles of the faces of the tiling {𝑝, 3} for 𝑝 ≥ 6 form a strongly solid packing as well. A. Bezdek [2] confirmed the conjecture for all 𝑝 ≥ 8. The two remaining cases, 𝑝 = 6 (in the Euclidean plane) and 𝑝 = 7 (in the hyperbolic plane) remain unsolved. A partial result supporting the case 𝑝 = 6 of the conjecture is due to Bárány and Dolbilin [1]. They proved that the packing obtained by removing one circle from the densest lattice packing of unit circles is uniformly stable with 𝜀 = 1/40. Another result supporting the conjecture was given by Heppes [27]. He proved that the hexagonal tiling {6, 3} is strongly translationally solid. This means that in the packing of hexagons arising by omitting one tile from the hexagonal tiling, every rearrangement of a finite number of hexagons by translations results in a packing congruent to the original one. L. Fejes Tóth [16] strengthened Bezdek’s result in the following sense. For any packing of congruent circles, a packing obtained from it by removing a finite number 𝑘 of circles is called a 𝑘-truncation of the packing. We say that a circle packing is solid of order 𝑘 if every 𝑘-truncation has the property that when finitely many additional circles are removed from it and then placed back in any place where there is room for them, the resulting packing is again some 𝑘-truncation of the original packing. The grade of saturation of a packing of congruent circles is the maximum number 𝑔 such that every circle congruent to, but not identical with a circle of the

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11 Ball Packings in Hyperbolic Space

packing intersects at least 𝑔 circles of the packing. L. Fejes Tóth [16] proved that for 𝑝 ≥ 8, the order of solidity of the face-incircles of {𝑝, 3} equals the grade of saturation minus 2. In particular, it follows that the order of solidity of these packings becomes arbitrarily large as 𝑝 → ∞.

11.4 Completely Saturated Packings and Completely Reduced Coverings G. Fejes Tóth, G. Kuperberg and W. Kuperberg [14] introduced the notion of a 𝑘-saturated packing (𝑘 = 1, 2, . . .) as a packing with congruent copies of a set 𝐾, such that deleting 𝑘 − 1 members of the packing never creates a void large enough to pack in it 𝑘 copies of 𝐾. Similarly, a covering with congruent copies of 𝐾 is 𝑘-reduced if deleting 𝑘 members of the covering always creates a void too large to be covered by 𝑘 − 1 copies of 𝐾. A packing that is 𝑘-saturated for every 𝑘 is completely saturated. Completely reduced coverings are defined similarly. Obviously, in Euclidean spaces a completely saturated packing with congruent copies of a convex body 𝐾 must have density 𝛿(𝐾); similarly, a completely reduced covering with congruent copies of 𝐾 must be of density 𝜗(𝐾). This suggested that the notions of completely saturated packings and completely reduced coverings could serve in hyperbolic spaces as a substitute for packings and coverings of maximum and minimum density, respectively. The question of existence of completely saturated packings and completely reduced coverings for every convex body turned out to be non-trivial by any means, and it was answered affirmatively for Euclidean spaces in G. Fejes Tóth, G. Kuperberg and W. Kuperberg [14] and for hyperbolic spaces by Bowen [8]. The notion of completely reduced coverings found applications in approximation theory, see Hinrichs and Richter [31].

11.5 A Probabilistic Approach to Optimal Arrangements and their Density Bowen and Radin [10, 11] proposed a probabilistic approach to define optimal arrangements and their density, especially useful in hyperbolic geometry. Their main idea is that with a properly defined probability measure on the set of all packings with copies of a body 𝐾, the density of a specific packing is the same as the probability that its randomly chosen congruent copy contains the origin. Below we sketch some details. Let S denote an 𝑛-dimensional space of constant curvature, namely the Euclidean 𝑛-space, the 𝑛-sphere, or the 𝑛-dimensional hyperbolic space. Instead of studying individual packings in S, Bowen and Radin consider the space Σ𝐾 consisting of all packings of S by congruent copies of a body 𝐾. A suitable topology derived from the Hausdorff metric on Σ𝐾 is introduced which makes Σ𝐾 compact and makes the

11.5 A Probabilistic Approach to Optimal Arrangements and their Density

269

natural action of the group G of rigid motions of S on Σ𝐾 continuous. We consider Borel probability measures on Σ𝐾 invariant under G. For such an invariant measure 𝜇 the density of 𝜇, 𝑑 (𝜇), is defined as 𝑑 (𝜇) = 𝜇( 𝐴), where 𝐴 is the set of packings P ∈ Σ𝐾 for which the origin of S is contained in some member of P. It follows easily from the invariance of 𝜇 that this definition is independent of the choice of the origin. A measure 𝜇 is ergodic if it cannot be expressed as the positive linear combination of two other invariant measures. The relationship between density of measures and density of packings is established by the following theorem. Suppose that 𝜇 is an ergodic invariant Borel probability measure on Σ𝐾 . If a packing 𝑃 is chosen 𝜇-randomly, then with probability 1, for every 𝑝 ∈ S, 𝑑 (𝜇) = lim 𝜆→∞

1 𝑉 (𝐵𝜆 ( 𝑝))

∑︁

𝑉 (𝑃 ∩ (𝐵𝜆 ( 𝑝)).

𝑃∈P

The packing density 𝛿(𝐾) of 𝐾 can then be defined as the supremum of 𝑑 (𝜇) for all ergodic invariant measures on Σ𝐾 . A packing P ∈ Σ𝐾 is optimally dense if the closure of its orbit under G is the support of an ergodic invariant measure whose density reaches this supremum. It is shown in Bowen and Radin [10, 11] that, for every 𝐾, an ergodic invariant measure 𝜇 with 𝑑 (𝜇) = 𝛿(𝐾) exists whose support contains a set of full 𝜇-measure of optimally dense packings. The existence of completely saturated packings with copies of 𝐾 was proved in the same way as for optimally dense packings. In fact, with probability 1, a 𝜇-randomly chosen optimally dense packing is completely saturated. Bowen and Radin [10,11] proved several statements justifying that they proposed a workable notion of optimal density and optimally dense packings. In particular, in the Euclidean case, that is S = E𝑛 , the Bowen–Radin notion of 𝛿(𝐾) coincides with the corresponding traditional notion of the packing density of 𝐾. The probabilistic approach of Bowen–Radin can be naturally applied to coverings, or more generally, to locally finite arrangements of congruent copies of 𝐾. The definition of density in such a setting, however, requires a modification: the “measure” 𝜇( 𝐴) should be replaced by another quantity that takes into account the multiplicity with which portions of the areas of the union of 𝐴 are covered. More precisely, 𝜇( 𝐴) should be replaced with a combination of weighted measures, assigning the weight of 𝑤 to the regions in S consisting of points covered exactly 𝑤 times by the given arrangement of copies of 𝐾. The advantage of this probabilistic approach is that it focuses on periodic packings and neglects pathological packings such as in Böröczky’s example. Another important feature of this approach is that we get bounds for the packing density defined in this framework by considering the local density in the Dirichlet or the Delone partitions. Concerning packings of balls in hyperbolic 𝑛-space, it was shown by Bowen and Radin [10] that there are only countably many radii admitting an optimally dense periodic packing of balls. Thus, for most radii 𝑟, no periodic packing is densest.

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11 Ball Packings in Hyperbolic Space

For the hyperbolic plane, Bowen [9] proved that the packing density of circles of radius 𝑟 is a continuous function of 𝑟, and it is the supremum of densities of periodic packings.

Chapter 12

Multiple Arrangements G. Fejes Tóth

We say that a family of sets is a 𝑘-fold packing if every point of the space belongs to the interior of at most 𝑘 sets. Quite analogously, we say that a family of sets forms a 𝑘-fold covering if every point of the space belongs to the closure of at least 𝑘 sets. Let 𝛿 𝑘 (𝐾) and 𝜗 𝑘 (𝐾), respectively, denote the densities of the densest 𝑘-fold packing and the thinnest 𝑘-fold covering of the space with congruent copies of the convex body 𝐾. Similarly we use the notation 𝛿𝑇𝑘 (𝐾), 𝜗𝑇𝑘 (𝐾), 𝛿 𝐿𝑘 (𝐾) and 𝜗𝐿𝑘 (𝐾) for the optimum densities of the corresponding 𝑘-fold translative and lattice arrangements (compare the corresponding definitions of 𝛿, 𝛿𝑇 , 𝛿 𝐿 , 𝜗, 𝜗𝑇 and 𝜗𝐿 in Section 8.3.1).

12.1 Multiple Arrangements on the Plane The literature on multiple packing and covering is relatively extensive, and it mostly deals with arrangements of congruent copies of the circular disk 𝐵2 . The values of 𝛿 𝑘 (𝐵2 ) and 𝜗 𝑘 (𝐵2 ) are not known for any 𝑘 > 1. We know that 𝛿2 (𝐵2 ) > 2𝛿1 (𝐵2 ) and 𝜗2 (𝐵2 ) < 2𝜗1 (𝐵2 ), as was shown by Heppes [36] and Danzer [14], respectively. G. Fejes Tóth [17] established the bounds 𝛿 𝑘 (𝐵2 ) ≤ 𝑘

𝜋 𝜋 cot 6 6𝑘

and

𝜗 𝑘 (𝐵2 ) ≥ 𝑘

𝜋 𝜋 csc . 3 3𝑘

𝜋 𝜋 Observe that 𝜋6 cot 6𝑘 and 𝜋3 csc 3𝑘 are equal to the density of a disk with respect to the circumscribed and inscribed regular 6𝑘-gon. For 𝑘 = 1 these inequalities are sharp as they coincide with Thue’s and Kershner’s theorems. Bolle [6] proved that there are positive constants 𝑐 𝑖 such that 2

1

1

2

𝑘 − 𝑐 1 𝑘 5 ≤ 𝛿 𝐿𝑘 (𝐵2 ) ≤ 𝑘 − 𝑐 2 𝑘 4 and

𝑘 + 𝑐 3 𝑘 4 ≤ 𝜗𝐿𝑘 (𝐵2 ) ≤ 𝑘 + 𝑐 4 𝑘 5

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_12

271

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12 Multiple Arrangements

and showed in [9] that the exponent 41 is best possible in these inequalities. In [10] he proved that for convex disks 𝐾 with piecewise twice differentiable boundary there are positive constants 𝑐(𝐾) and 𝐶 (𝐾) such that 2

𝛿 𝐿𝑘 (𝐾) ≥ 𝑘 − 𝑐(𝐾)𝑘 5 and

2

𝜗𝐿𝑘 (𝐾) ≤ 𝑘 + 𝐶 (𝐾)𝑘 5 . Moreover, for a polygon 𝑃 the stronger inequalities 1

𝛿 𝐿𝑘 (𝑃) ≥ 𝑘 − 𝑐(𝑃)𝑘 3 and

1

𝜗𝐿𝑘 (𝑃) ≤ 𝑘 + 𝐶 (𝑃)𝑘 𝑟3 hold. The exact values of 𝛿 𝐿𝑘 (𝐵2 ) have been found for 𝑘 ≤ 10 (see Heppes [37], Blundon [5], Bolle [6], Yakovlev [68], Temesvári [64] and Temesvári and Végh [67]). The values of 𝜗𝐿𝑘 (𝐵2 ) are known for 𝑘 ≤ 8 (see Blundon [4], Haas [33], Subak [56] and Temesvári [58, 61, 61, 62, 63]. Linhart [41] described an algorithmic approach for approximating the values of 𝛿 𝐿𝑘 (𝐵2 ) with arbitrarily high accuracy. Elaborating on results by Yakovlev [69] Temesvári, Horváth and Yakovlev [66] described a method for finding the densest 𝑘-fold lattice packing with circles. They reduced this task to a finite number of optimization problems, each over an explicitly given compact domain. A similar method for the determination of the thinnest 𝑘-fold lattice covering with circles was given by Temesvári [60]. For a triangle 𝑇, Sriamorn [53] determined 𝛿 𝐿𝑘 (𝑇) and 𝜗𝐿𝑘 (𝑇) for all 𝑘. Moreover, Sriamorn [54] showed that 𝛿𝑇𝑘 (𝑇) = 𝛿 𝐿𝑘 (𝑇) =

2𝑘 2 2𝑘 + 1

and Sriamorn and Wetayawanich [55] showed that 𝜗𝑇𝑘 (𝑇) = 𝜗𝐿𝑘 (𝑇) =

2𝑘 + 1 2

for all 𝑘. It is worth mentioning that 𝛿 𝐿𝑘 (𝐵2 ) = 𝑘𝛿1𝐿 (𝐵2 ) for 𝑘 = 2, 3 and 4, and also 2 𝜗𝐿 (𝐵2 ) = 2𝜗𝐿1 (𝐵2 ). These equalities for the very same multiplicities have been extended to an arbitrary centrally symmetric convex disk in place of the circle by Dumir and Hans-Gill [15, 16] and G. Fejes Tóth [19]. The equality 𝜗𝐿2 (𝐵2 ) = 2𝜗𝐿1 (𝐵2 ) was further generalized by Temesvári, who proved in [59] that the density of a 2-periodic double covering by circles is at most 2𝜗𝐿1 (𝐵2 ), and in [65] proved the analogous result for 2-periodic double coverings by translates of a centrally symmetric convex disk. Recall from Section 10.2.5 that an 𝑚-periodic arrangement is the union of 𝑚 translates of a lattice arrangement.

12.3 Multiple Arrangements in Space

273

12.2 Decomposition of Multiple Arrangements The equalities 𝛿3𝐿 (𝐾) = 3𝛿1𝐿 (𝐾) and 𝛿4𝐿 (𝐾) = 4𝛿1𝐿 (𝐾) for centrally symmetric disks 𝐾 were derived by noticing that every 3-fold lattice packing by such a disk is the union of three simple (1-fold) packings, and every such 4-fold packing is the union of two 2-fold packings. This observation belongs to the topic concerning decompositions of multiple arrangements into simple ones, problems and results that focus on the combinatorial structure of such arrangements. Research in this direction was initiated by Pach [45]. He proved, among other things, that every 2-fold packing with positively homothetic copies of a convex disk can be decomposed into four (simple) packings. For coverings, he made the conjecture that for every convex disk 𝐾 there exists a minimal natural number 𝑚(𝐾) such that every 𝑚(𝐾)-fold covering of the plane by translates of 𝐾 can be decomposed into two coverings. In [46] he proved this conjecture for centrally symmetric polygons. New interest arose in the topic after Tardos and Tóth [57] proved the conjecture for triangles. Soon after, Pálvölgyi and Tóth [50] proved the conjecture for every convex polygon 𝑃. Unfortunately, the number 𝑚(𝑃) increases with the number of sides of 𝑃, thus the attempt to extend the result to all convex disks through polygonal approximation fails. Still, it came as a surprise when Pálvölgyi [49] (see also Pach and Pálvölgyi [47]) disproved Pach’s conjecture by showing that it does not hold for the circle. For subsequent developments on decomposition of multiple arrangements we refer the reader to the survey article of Pach, Pálvölgyi and Tóth [48].

12.3 Multiple Arrangements in Space The densest 2-fold lattice packing and the thinnest 2-fold lattice covering of balls in three dimensions were determined by Few and Kanagasabapathy [25] and Few [22], respectively. Purdy [51] constructed a threefold lattice packing of balls which he conjectured to be of maximum density. He supported the conjecture by proving that it provides a local maximum of the density among threefold lattice packings of balls. Adapting Blichfeldt’s idea, Few [21] gave the following upper bound for the 𝑘-fold packing density of the 𝑛-dimensional ball: 𝛿 𝑘 (𝐵 𝑛 ) ≤ (1 + 𝑛−1 ) [(𝑛 + 1) 𝑘 − 1] [𝑘/(𝑘 + 1)] 𝑛/2 . This is better than the trivial bound 𝑘 only for large values of 𝑛 compared to 𝑘. By a further elaboration on the same idea for 𝑘 = 2, Few [23] obtained the stronger inequality   𝑛/2 2 4 . 𝛿2 (𝐵 𝑛 ) ≤ (𝑛 + 2) 3 3

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G. Fejes Tóth [18] gave a non-trivial upper bound for 𝛿 𝑘 (𝐵 𝑛 ), as well as a non-trivial lower bound for 𝜗 𝑘 (𝐵 𝑛 ) for every 𝑛 and 𝑘. For multiple lattice arrangements of balls, Bolle [7, 8] established sharper estimates. He proved that there are positive constants 𝑐 𝑛 and 𝐶𝑛 such that 𝛿 𝐿𝑘 (𝐵 𝑛 ) 1 ≤ 1 − 𝑐 𝑛 𝑘 𝑛+ 𝑛 𝑘

and

𝜗𝐿𝑘 (𝐵 𝑛 ) 𝑛+1 ≥ 1 + 𝑐 𝑛 𝑘 2𝑛 𝑘

and

𝜗𝐿𝑘 (𝐵 𝑛 ) 𝑛+3 ≥ 1 + 𝑐 𝑛 𝑘 2𝑛 𝑘

when 𝑛 . 1 (mod 4) and 𝛿 𝐿𝑘 (𝐵 𝑛 ) 𝑛+3 ≤ 1 − 𝑐 𝑛 𝑘 2𝑛 𝑘

when 𝑛 ≡ 1 (mod 4). Extending the result of Schmidt [52] to multiple arrangements, Florian [26] proved that 𝛿 𝑘 (𝐾) < 𝑘 and 𝜗 𝑘 (𝐾) > 𝑘 for every smooth convex body 𝐾, without establishing a concrete bound. By a Blichfeldt-type argument Few [21] proved  𝛿 𝑘 (𝐵 𝑛 ) ≥ 𝛿(𝐵 𝑛 )

2𝑘 𝑘 +1

 𝑛/2 .

In [24] Few studied the multiplicity of partial coverings of space, and, as an application of a general theorem, obtained a better lower bound for 𝛿 𝑘 (𝐵 𝑛 ) for large values of 𝑘 and 𝑛. Groemer [31] proved lower bounds for the 𝑘-fold lattice packing density of a convex body 𝐾 involving the intrinsic volumes of 𝐾. From his results follows the existence of positive constants 𝑐 𝑛 such that 𝛿 𝐿𝑘 (𝐾) ≥ 𝑘 − 𝑐 𝑛 𝑘 (𝑛−1)/𝑛 for every convex body 𝐾 ∈ 𝐸 𝑛 . Cohn [13] proved that 𝜗𝐿𝑘 (𝐾) < [(𝑘 + 1) 1/𝑛 + 8𝑛] 𝑛 = 𝑘 (1 + 𝑂 (𝑛2 𝑘 −1/𝑛 ))

as 𝑘 → ∞

for every 𝑛-dimensional convex body 𝐾. The best known upper bound for 𝜗 𝑘 (𝐾) is due to Naszódi and Polyanskii [44] who, improving slightly on an earlier result by Frankl, Nagy and Naszódi [27], proved that 𝜗 𝑘 (𝐾) ≤ 3.153(1 + 𝑜(1)) max{𝑛 ln 𝑛, 𝑘 } for every convex body 𝐾 ∈ 𝐸 𝑛 . Blachman and Few [1] gave bounds for the density of multiple packings of spherical caps. Also L. Fejes Tóth [20], Galiev [28], Blinovsky [2] and Blinovsky and Litsyn [3] investigated multiple ball packings in spherical spaces.

12.4 Multiple Tiling

275

12.4 Multiple Tiling A system of bodies forms a 𝑘-fold tiling if each point of the space is covered exactly 𝑘 times, except perhaps the boundary points of the bodies. There are centrally symmetric polygons that admit a translational 𝑘-fold tiling, but no simple tiling. The simplest example is perhaps the regular octagon of side-length 1, whose translates by the unit square lattice form a 7-fold tiling. Bolle [11] proved that a convex polygon that admits a 𝑘-fold tiling of the plane by translations is centrally symmetric. He also gave a characterization of those convex polygons that admit a 𝑘-fold lattice-tiling. Kolountzakis [39] gave an algorithm which decides for a centrally symmetric convex polygon if it can tile the plane by translations at some level. His algorithm runs in polynomial time in the number of sides of the polygon. Yang and Zong [71, 72] characterized those convex polygons that admit two-, three-, four- or five-fold translational tiling. Only parallelograms and centrally symmetric hexagons admit a two-, three- or four-fold translational tiling. There are two more classes of polygons admitting five-fold tilings: the affine images of a special octagon and of a decagon. Gravin, Robins and Shiryaev [30] proved that if translates of a convex polytope form a 𝑘-fold tiling of 𝐸 𝑛 , then it is centrally symmetric and its facets are centrally symmetric as well. This generalizes a theorem of Minkowski [43] concerning simple tilings. For the three-dimensional case this means that only zonotopes admit a translational 𝑘-fold tiling. For rational polytopes Gravin, Robins and Shiryaev also proved the converse of their above-mentioned theorem: Every rational polytope in 𝐸 𝑛 that is centrally symmetric and has centrally symmetric facets admits a 𝑘-fold lattice tiling for some positive integer 𝑘. For zonotopes, this was proved earlier by Groemer [32]. A quasi-periodic set is a finite union of translated lattices, not necessarily of the same lattice. Kolountzakis [38] proved that a 𝑘-fold tiling by translates of a convex polygon other than a parallelogram is quasi-periodic. Gravin, Kolountzakis, Robins and Shiryaev [29] proved an analogous theorem for the three-dimensional case: A 𝑘-fold tiling by translates of a polytope that is not a two-flat zonotope is quasi-periodic. A two-flat zonotope is the Minkowski sum of two 2-dimensional symmetric polygons one of which may degenerate into a single line segment. Gravin, Robins and Shiryaev [30] raised the problem of whether the following generalization of the Venkov–McMullen theorem holds: If translates of a polytope 𝑃 form a 𝑘-fold tiling of 𝐸 𝑛 , then 𝑃 also admits an 𝑚-fold lattice tiling for some, possibly different, multiplicity. The two-dimensional case of this conjecture was confirmed independently by Liu [42] and Yang [70]. A further step in the direction of proving the conjecture was made by Chan [12], who proved it for certain quasiperiodic 𝑘-fold tilings. Lev and Liu [40] gave a characterization of those polytopes in 𝐸 𝑛 that tile with some multiplicity 𝑘 by translations along a given lattice. In three dimensions for 𝑘 = 2, 3 and 4 Han, Yang, Sriamorn and Zong [34, 35] proved the following strong version of the conjecture. If a convex body in 𝐸 3 admits a two, three or fourfold tiling by translations then it is a parallelohedron, that is, it admits a lattice tiling.

Chapter 13

Neighbors G. Fejes Tóth, L. Fejes Tóth and W. Kuperberg

Two members of a packing are neighbors if they have a common boundary point. In this chapter we survey results connected with the number of neighbors in a packing.

13.1 The Newton Number of Convex Disks Let 𝑁 (𝑆) denote the maximum of the number of neighbors that one member can have in a packing of congruent copies of 𝑆. After L. Fejes Tóth [53] we call 𝑁 (𝑆) the Newton number of 𝑆. The name recalls the dispute between Newton and Gregory about the maximum number of congruent balls that can touch another one of the same size without overlapping with each other. A rigorous proof settling the dispute in favor of Newton was given by Schütte and van der Waerden [159]. An alternative name for 𝑁 (𝑆) is the kissing number of 𝑆. Some experiments with regular polygonal disks point to the conjecture that the Newton number of a regular 𝑛-gon is: 12 for 𝑛 = 3; 8 for 𝑛 = 4; and 6 for 𝑛 ≥ 5. Indeed, with the exception of 𝑛 = 5, this was proved by Böröczky [28] (see also Youngs [187] for the case of a square, Klamkin, Lewis and Liu [103] for the cases 𝑛 = 3, 4 and 6, and Zhao [188] for the case 𝑛 > 6). The Newton number of the regular pentagon was proved to be 6 by Linhart [122] and independently by Pankov and Dolmatov [146, 147] and Zhao and Xu [189]. Schopp [157] proved that the Newton number of any disk of constant width is at most 7. The Newton number of the Reuleaux triangle is equal to 7 (Figure 13.1). Kemnitz and Möller [98] determined the Newton number of all rectangles. L. Fejes Tóth [51] proved the inequality 𝑁 (𝑆) ≤ (4 + 2𝜋)

𝑤 𝐷 +2+ 𝑤 𝐷

for a convex disk 𝑆 with diameter 𝐷 and width 𝑤. This estimate is exact in many 𝑤 𝜋 cases. For example, for the isosceles triangle Δ with 𝐷 = sin 19 we get 𝑁 (Δ) ≤ 64. But a simple construction (Figure 13.2) shows that 𝑁 (Δ) ≥ 64, hence 𝑁 (Δ) = 64. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_13

277

278

13 Neighbors

Fig. 13.1

Fig. 13.2

Other bounds for the Newton number of a convex disk involving different parameters were given by Hortobágyi [93, 94, 95] and Wegner [185]. Each of them considered a generalization of the Newton number produced by counting the number of congruent copies of a convex disk 𝐾 that can touch another convex disk 𝐶. Wegner’s result includes the computation of the Newton number of the 30◦ -30◦ -120◦ triangle, which, as conjectured, turns out to be 21. Further problems about Newton numbers of convex discs are treated in Harborth, Koch and Szabó [90], Kemnitz, Möller and Wojzischke [99], Kemnitz and Szabó [100] and Kemnitz, Szabó and Ujváry-Menyhárt [101].

13.2 The Hadwiger Number of Convex Disks The Hadwiger number is defined in a similar way as the Newton number, but considering only translated copies of 𝑆 instead of allowing all congruent ones. The notion was so named by L. Fejes Tóth [55] because of the following result of Hadwiger [86]: If each of 𝑘 mutually non-overlapping translates of an 𝑛-dimensional convex body 𝐾 touches 𝐾, then 𝑘 ≤ 3𝑛 −1. The inequality is sharp, as equality occurs for the parallelotope, and, as Groemer [79] proved, only for the parallelotope (see also Grünbaum [83] for the planar case). The Hadwiger number of a convex body 𝐾, denoted by 𝐻 (𝐾), is alternately called the translative kissing number of 𝐾. In the plane, how large can the Hadwiger number be for a non-convex Jordan region? While for each such region the number is finite, Cheong and Lee [37] showed that, surprisingly, there exist Jordan regions with arbitrarily large Hadwiger

13.4 The Number of Touching Pairs in Finite Packings

279

number. For starlike regions, however, A. Bezdek [12] showed that 75 is an upper bound and Lángi [115] lowered Bezdek’s bound to 35. For the Hadwiger number of centrally symmetric starlike regions Lángi [114] established the stronger upper bound of 12. It is still unknown if the Hadwiger number of a starlike region can be greater than 8. Lángi [115] determined the Hadwiger number of a special class of regions. According to Lángi a pocket of 𝑅 is a connected component of conv 𝑅 \ 𝑅. In their examples, Cheong and Lee achieve an arbitrarily large Hadwiger number only for a sequence of polygons whose number of pockets is not bounded above. Lángi determined the Hadwiger number of a region with one pocket; depending on certain properties of the region, the number is either 6 or 8. Boju and Funar [26,27] investigated the generalization of the Hadwiger number in which mutually non-overlapping, 𝜆-homothetic copies of 𝐾 for some 𝜆 > 0 touch 𝐾, and they gave corresponding bounds.

13.3 Translates of a Jordan Disk with a Common Point An interesting related question was addressed by A. Bezdek, K. Kuperberg and W. Kuperberg [16]: What is the maximum number of non-overlapping translates of a Jordan disk that can have a common point? They proved that the number is at most 4, and they characterized the disks for which 4 non-overlapping translates can have a common point. Earlier, K. Kuperberg and W. Kuperberg [109] proved the same bound for star-like disks. In contrast, in space the corresponding number can be arbitrarily large, even for star-like solids.

13.4 The Number of Touching Pairs in Finite Packings A problem of Erdős [45] asks for the maximum number of occurrences of the minimum distance between 𝑛 points in the plane. Erdős proved the upper bound 3𝑛 − 6 and pointed out that the example √of the regular triangular lattice shows that the minimum distance can occur 3𝑛 − 𝑐 𝑛 times. Unaware of the work of Erdős, Reuter [153] restated the problem as a conjecture in the language of circle packings: The maximum number √ of touching pairs among 𝑛 unit circles forming a packing in the plane is ⌊3𝑛 − 12𝑛 − 3⌋. The conjecture was verified by Harborth [89]. Kupitz [112] gave a complete description of the extremal packings; they are all subsets of the triangular lattice. Brass [34] extended Harborth’s result to packings consisting of translates of a convex disk different from a parallelogram and showed √ that the corresponding number for parallelograms is ⌊4𝑛 − 28𝑛 − 12⌋. The papers by K. Bezdek [17, 18] and K. Bezdek and Reid [24] give bounds for the number of touching pairs in packings of congruent balls. The latter paper investigates the number of mutually touching triples and quadruples as well. Bowen [31] studied contact numbers in circle packings in the hyperbolic plane. K. Bezdek, B. Szalkai and

280

13 Neighbors

I. Szalkai [25], K. Bezdek and Naszódi [23], K. Bezdek, Khan and Oliwa [22], and Naszódi and Swanepoel [139] investigated the number of touching pairs in totally separable packings (see Section 16.16 for the definition of totally separable packings). The paper [21] by K. Bezdek and Khan gives a survey on contact numbers in packings.

13.5 𝒏-Neighbor Packings If every member of a packing has exactly 𝑛, or at least 𝑛 neighbors, we call it an 𝑛-neighbor or an 𝑛+ -neighbor packing, respectively. An easy construction shows that there exists a zero-density 5-neighbor packing of the plane with translates of a parallelogram. It turned out that this property characterizes parallelograms (see L. Fejes Tóth [56]). For a convex disk 𝐾 other than a parallelogram, Makai [130] proved that every 5+ -neighbor packing with translates of 𝐾 is of density greater than or equal to 3/7. Equality can occur only if 𝐾 is a triangle. For centrally symmetric convex disks the corresponding lower bound is 9/14, attained only for affine regular hexagons. In [130] Makai only sketched the proof for this last statement; details are given in Makai [131].

Fig. 13.3 Concerning 6+ -neighbor packings, the following is known: L. Fejes Tóth [56] showed that the density of a 6+ -neighbor packing with translates of a convex disk is at least 1/2, and Makai [130] obtained the corresponding lower bound of 3/4 for centrally symmetric disks. Each of these bounds is sharp, and, again, the extreme values are produced only by triangles and affine regular hexagons, respectively. Chvátal [38] proved that the density of a 6+ -neighbor packing with translated parallelograms is at least 11/15. + According to L. Fejes Tóth [56] √ the density of a 5 +-neighbour packing of congruent circular disks is at least 3𝜋/7. Concerning 5 neighbor packing of noncongruent circles, G. Fejes Tóth and L. Fejes Tóth [49] proved that there is a constant ℎ0 = 0.53329 . . . such that any 5+ neighbor packing of circular disks whose homogeneity exceeds ℎ0 has positive density. The constant ℎ0 is the unique real root of the equation 8ℎ3 + 3ℎ2 − 2ℎ − 1 = 0. It cannot be replaced by a smaller one: A 5+ -neighbor packing of circles of homogeneity ℎ0 and density 0 is shown in Figure 13.3.

13.5 𝑛-Neighbor Packings

281

L. Fejes Tóth [58] stated the following interesting conjecture: The homogeneity of a 6+ neighbor packing of circles is either 1 or 0. The conjecture was confirmed by Bárány, Füredi and Pach [11]. They proved the somewhat stronger statement that in a 6+ -neighbor packing of circular disks either all disks are congruent or arbitrarily small disks occur. Their proof combines a geometric idea with a combinatorial one, each of interest on its own. The angle at the boundary point 𝑎 of a convex disk 𝐾 is the measure of the smallest angular region with apex 𝑎 containing 𝐾. The minimum of the angle taken for all boundary points of 𝐾 is called the minimal angle of 𝐾. A further problem due to L. Fejes Tóth [54] asked for the numbers 𝑛 for which there is an 𝑛-neighbor packing consisting of convex disks with given minimal angle 𝜋/ℎ. Linhart [124] proved that for such a packing 𝑛 ≤ max{5, 2⌊ℎ⌋ − 1} and showed that this bound is sharp for ⌊ℎ⌋ ≤ 6.

Fig. 13.4 Let 𝑡 (𝐾) denote the largest number 𝑛 for which there is a finite 𝑛+ -neighbor packing of translates of a convex disk 𝐾, and let 𝑚(𝐾) denote the minimum cardinality of such a packing. Talata [179] proved that 𝑡 (𝐾) = 4 and 𝑚(𝐾) = 12 if 𝐾 is a parallelogram, otherwise 𝑡 (𝐾) = 3 and 𝑚(𝐾) = 7. Wegner [184] solved two problems posed by L. Fejes Tóth [54] concerning the existence of certain packings in which every member has the same number of neighbors. On one hand, he constructed for every 𝑛 ≥ 3 a stable 𝑛-neighbor packing of congruent convex disks, and, on the other hand, he presented a 5-neighbor packing consisting of 32 congruent smooth convex disks (see Figure 13.4). Independently from Wegner, Linhart [123] also found a finite 5-neighbor packing of congruent smooth convex disks using a very similar idea.

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13.6 Maximal Packings A packing of congruent copies of a convex disks is said to be a maximal packing if the number of neighbors of every disk equals its Newton number. The faces of each of the tilings {6, 3}, {4, 4} and {3, 6} form such a maximal packing.

Fig. 13.5 Gács [73] proved that there is an absolute constant 𝑁, such that the number of neighbors in a maximal packing cannot exceed 𝑁. In the tiling obtained by dissecting each face of {3, 6} into three congruent triangles (Figure 13.5) every cell has 21 neighbors. In view of the result of Wegner [185] mentioned above, it follows that this tiling is a maximal packing. On the other hand, Linhart [126] showed that in no maximal packing of convex disks can the number of neighbors exceed 21. Böröczky [28] proved that the faces of every regular spherical tiling form a maximal packing. However at most 48 regular hyperbolic tilings are maximal (see Florian [67] and A. Florian and H. Florian [68,69]). Linhart [125] showed that the tiling {7, 3} is maximal.

13.7 Higher-Order Neighbors We say that in a packing of convex disks each disk is the “zero-th neighbor” of itself. A disk 𝐵 is the 𝑘-th neighbor of the disk 𝐴 if it is a neighbor of a (𝑘 − 1)th neighbor of 𝐴 other than a 𝑗-th neighbor of 𝐴 with 0 ≤ 𝑗 < 𝑘 − 1. The 𝑘-th Newton number 𝑁 𝑘 (𝐷) of a convex disk 𝐷 is the maximum of the total number of its 𝑗-th neighbors for 𝑗 = 1, . . . , 𝑘. The densest lattice packing of circles yields the lower bound 𝑁 𝑘 (𝐵2 ) ≥ 3𝑘 (𝑘 + 1). One could conjecture that, for small values of 𝑘, say for 𝑘 ≤ 12, 𝑁 𝑘 (𝐵2 ) = 3𝑘 (𝑘 + 1) holds. This is obvious for 𝑘 = 1 and was confirmed by L. Fejes Tóth and Heppes [60] for 𝑘 = 2 and by Golovanov [77] for 𝑘 = 3. But L. Fejes Tóth [52] showed that 𝑁14 (𝐵2 ) ≥ 636, which is greater than

13.8 The Newton Number of Balls

283

3 · 14 · 15 = 630. He also determined the asymptotic behavior of 𝑁 𝑘 (𝐵2 ) for large 𝑘, namely √ lim 𝑁 𝑘 (𝐵2 )/𝑘 2 = 2𝜋/ 3 . 𝑘→∞

Higher-order Hadwiger numbers are defined and treated in L. Fejes Tóth [57] and L. Fejes Tóth and Heppes [61].

13.8 The Newton Number of Balls Alternatives to Schütte and van der Waerden’s proof of the equality 𝑁 (𝐵3 ) = 12 were given by Leech [119], Böröczky [29], Anstreicher [5], Musin [134], Glazyrin [76] and Maehara [129], the last one being perhaps the most elementary among them. Flatley, Tarasov, Taylor and Theil [66] proved that the maximum number of tangent pairs among twelve non-overlapping unit balls tangent to a thirteenth unit ball is 24, attained only in the case when the centers of the twelve balls are the vertices either of a cuboctahedron or of a twisted cuboctahedron. A twisted cuboctahedron is obtained by cutting a cuboctahedron into two parts by a plane containing 6 edges forming a regular hexagon, and rotating one part by an angle of 𝜋/3 around the axis through the center of the hexagon and perpendicular to its plane. R. Kusner, W. Kusner, Lagarias and Shlosman [113] described the configuration space of 12 non-overlapping equal spheres of radius 𝑟 touching a central unit sphere. They also gave a nice survey of the history of the twelve spheres problem and the Tammes problem. The value of 𝑁 (𝐵4 ) was determined by Musin [133, 136], while 𝑁 (𝐵8 ) and 𝑁 (𝐵24 ) were determined by Odlyzko and Sloane [140] and, independently, by Levenšte˘ın [121]. The cases 𝑛 = 8 and 𝑛 = 24 were resolved by means of the linear programming method, and the case 𝑛 = 4 by its modification. The corresponding values are: 𝑁 (𝐵4 ) = 24,

𝑁 (𝐵8 ) = 240,

and

𝑁 (𝐵24 ) = 196560.

Each of these Newton numbers is realized in the unique densest lattice packing of balls in the corresponding dimension. Moreover, as shown by Bannai and Sloane [10], each of the arrangements of balls that realize the Newton number for 𝑛 = 8 and 𝑛 = 24 is unique, which for 𝑛 = 4 is still only conjectured. The Kabatjanski˘ı–Levenšte˘ın bound yields 𝑁 (𝐵 𝑛 ) ≤ 20.4041𝑑+𝑜(𝑑) , while the best known √︁ lower bound √ due to √Jenssen, Joos and Perkins [96] is 𝑁 (𝐵 𝑛 ) ≥ (1 + 𝑜(1)) 3𝜋/8 ln(3 2/4)𝑛3/2 (2 3/2) 𝑛 . Dostert and Kolpakov [41] gave upper bounds for the Newton number of balls in spherical and hyperbolic space. Let 𝑁 𝑘𝑛 be the 𝑘-th Newton number of the 𝑛-dimensional ball. We have 𝑁21 = 4 and, as expected, 𝑁22 = 18. L. Fejes Tóth and Heppes [60] showed that 56 ≤ 𝑁23 ≤ 63 and 168 ≤ 𝑁24 ≤ 232. The lower bound 𝑁23 ≥ 56 comes from the enumeration of balls

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in the first and second neighborhood in the trapezo-rhombic dodecahedral packing. The outcome is 12 + 44 = 56. Interestingly enough, the corresponding outcome for the rhombic dodecahedral packing amounts to just 12 + 42 = 54. This could be of some significance to the still unanswered question of why certain metals form the trapezo-rhombic dodecahedral structure in their crystals, while others form a rhombic dodecahedral structure. In the densest lattice packing of the 4-dimensional ball, found by Korkine and Zolotareff [107], the corresponding number is 24 + 144 = 168.

13.9 𝒏-Neighbor Packing of Congruent Balls What is the chromatic number of a finite packing of congruent balls? For which numbers 𝑛 does there exist a finite 𝑛-neighbor packing of congruent balls? Two consecutive layers from the densest lattice packing of balls is a nine-neighbor packing of density 0. Does there exist a ten-neighbor packing of zero density? Motivated by these questions L. Fejes Tóth and Sachs [62] stated the following conjectures. “Conjecture A. The maximum number of points which can be placed on an open unit hemisphere with at least unit distance from one another is equal to eight. Conjecture B. The maximum number of points which can be placed on a closed unit hemisphere with at least unit distance from one another is equal to nine. Conjecture AB. If nine points lie on a closed unit hemisphere with at least unit distance from one another then six of them are on the boundary of the hemisphere.”

Since the condition that six points lie on the boundary uniquely determines the configuration, Conjecture AB implies the other two. It follows from Conjecture A that the chromatic number of a finite packing of unit balls is at most 9, while the highest known chromatic number of a finite packing of unit balls is 5 (see Maehara [128]). It also follows that no finite nine-neighbor packing of congruent balls exists. Conjecture B implies that the density of a ten-neighbor packing is positive. Conjecture B was confirmed by G. Fejes Tóth [48]. Alternative proofs based on the difficult result that the Newton number of the 3-ball is 12 were given by Sachs [155] and by A. Bezdek and K. Bezdek [15]. Finally, Kertész [102] proved Conjecture AB. The analogous problem was investigated in higher dimensions as well. Let 𝐵(𝑛) be the maximum number of points which can be placed on a closed unit hemisphere in 𝐸 𝑛 with at least unit distance from one another. Equivalently, 𝐵(𝑛) is the maximum number of non-overlapping unit balls that can touch another unit ball at points of a closed hemisphere. Corresponding to this, 𝐵(𝑛) is called the one-sided kissing number of 𝐵 𝑛 . Szabó [173] proved that 𝐵(4) ≤ 20. Based on an extension of Delsarte’s method Musin [135] lowered this bound to the sharp value 18. Bachoc and Vallentin [8] proved that 𝐵(8) = 183 and Dostert, de Laat and Moustrou [43] proved uniqueness of the arrangement. Clearly, a finite 𝑘 + -packing of congruent balls in 𝐸 𝑛 exists only for 𝑘 < 𝐵(𝑛). It appears, however, that the maximum number

13.9 𝑛-Neighbor Packing of Congruent Balls

285

𝑘 for which there exists a finite 𝑘 + -packing of congruent balls in 𝐸√ 𝑛 is considerably less than 𝐵(𝑛). Alon [1] gave an explicit construction of a finite 2 𝑛 -packing of unit balls.

Fig. 13.6 Unit balls centered at the vertices of a triangle, tetrahedron, octahedron, and dodecahedron form a two-, three-, four-, and five-neighbor packing, respectively. It is easy to see that for 𝑛 ≤ 4 these are the arrangements of minimal cardinality. G. Fejes Tóth and Harborth [50] showed that a five-neighbor packing of congruent balls has at least 12 members, and reported a six-neighbor packing consisting of 240 congruent balls constructed by Gerd Wegner (see Figure 13.6). It is an open question whether a finite seven-neighbor or eight-neighbor packing of congruent balls exists. It is natural to ask for the minimum density of an 𝑛-neighbor packing of congruent balls for 𝑛 = 10, 11 and 12. For 𝑛 = 10 and 11 our knowledge is limited. Even the question of whether an 11-neighbor packing of congruent balls exists is open. The case 𝑛 = 12 is of special interest. L. Fejes Tóth [53, 59] conjectured that if in a packing with congruent balls each ball is touched by exactly twelve other balls, then the packing must consist of parallel hexagonal layers. This long-standing conjecture was verified by Hales [87,88]. Böröczky and Szabó [30] gave an alternative proof based on the result of Musin and Tarasov [137] about the densest packing of 13 spherical caps. Harborth, Szabó and Ujváry-Menyhárt [91] dropped the condition that the balls are congruent, and constructed finite 𝑛-neighbor packings of balls for all 𝑛 ≤ 12 except for 11. The question of whether a finite 11-neighbor packing of balls exists remains open. Since the smallest ball in a finite ball-packing has at most twelve neighbors, there is no finite 𝑛-neighbor packing for 𝑛 ≥ 12. On the other hand, the average number of neighbors in a finite packing of balls can be greater than 12. G. Kuperberg and Schramm [108] constructed a finite packing of balls in which the average number

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of neighbors is 666/53 = 12.566 . . .. A packing of balls with a slightly greater average number of neighbors of 7656/607 = 12.612 . . . was given by Eppstein, G. Kuperberg and √ Ziegler [44]. G. Kuperberg and Schramm [108] proved the upper bound 8 + 4 3 = 14.928 . . . for the average number of neighbors in a packing of balls, which was improved to 13.955 by Glazyrin [75] and to 13.606 by Dostert, Kolpakov and Oliveira Filho [42]. The latter two papers also give upper bounds for the average number of neighbors in ball packings in higher dimensions. K. Bezdek, Connelly and Kertész [20] investigated packings of circles of radius 𝑟 on the sphere and proved that there are positive numbers 𝜀 and 𝑟 0 such that for 𝑟 ≤ 𝑟 0 the average number of neighbors in such packings is at most 5 − 𝜀.

13.10 Results About Convex Bodies Talata [179] investigated packings of translates of cylinders in 𝐸 3 and showed that the maximum 𝑛 for which there exists a finite 𝑛+ -neighbor packing of translates of a cylinder 𝐶 is 10 if 𝐶 is not a parallelepiped and 13 if 𝐶 is a parallelepiped. He constructed a 10+ -neighbor packing of 172 translates of a cylinder if the base is not a parallelogram and a 13+ -neighbor packing of 382 translates of a parallelepiped. L. Fejes Tóth and Sauer [63] proved that if in a packing of translates of an 𝑛-dimensional cube, for each cube the total number of its 𝑗-th neighbors for 0 ≤ 𝑗 ≤ 𝑘 is more than (𝑘 + 1) (2𝑘 + 1) 𝑛−1 , then the packing has positive density. The result is sharp, as shown by the example of two consecutive layers in the grid of cubes in which every cube has a total number of (𝑘 + 1) (2𝑘 + 1) 𝑛−1 𝑗-th neighbors with 0 ≤ 𝑗 ≤ 𝑘. K. Bezdek and Brass [19] generalized the case 𝑘 = 1 of this result for packings by translates of an arbitrary 𝑛-dimensional convex body. Zong [191] constructed a lattice packing of tetrahedra in which every tetrahedron touches 18 others, conjectured that for a tetrahedron 𝑇 𝐻 (𝑇) ≤ 18, and proved that 𝐻 (𝑇) ≤ 19. Since in a lattice packing each member has an even number of neighbors, it follows that the number of neighbors of a member in a lattice packing of tetrahedra cannot exceed 18. Talata [175] gave a simple alternative proof for this. Later, Talata [176] succeeded in proving Zong’s conjecture about the Hadwiger number of tetrahedra. Moreover, he showed that the packing of 18 translates of a tetrahedron touching a nineteenth one is unique. He also gave a description of all possible packings of 17 translates of a tetrahedron touching an eighteenth one. He applied this result for the determination of the minimum and maximum densities of 17+ -neighbor translative packings of tetrahedra. The Hadwiger number of the octahedron is 18. This was proved independently by Robins and Salowe [154], Talata [178] and Larman and Zong [117]. The latter authors showed that the Hadwiger number of the rhombic dodecahedron is also 18. Grünbaum [83] proved that the Hadwiger number of a convex disk is always attained in a lattice packing. Zong [190] showed that this statement does not hold in any dimension 𝑛 ≥ 3, namely it fails for a cube truncated at some of its vertices.

13.11 Mutually Touching Translates of a Convex Body

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In the same article, Grünbaum conjectured that the Hadwiger number of every convex body is always even. Disproving the conjecture, Joós [97] constructed a 3-dimensional convex body whose Hadwiger number is 15. Talata [174] and Bourgain (see Füredi and Loeb [72]) proved that there is an absolute constant 𝑐 > 0 such that 𝐻 (𝐾) ≥ 2𝑐𝑛 for every 𝑛-dimensional convex body 𝐾. In [177] Talata gave the bound 𝐻 (𝑆 𝑛 ) ≥ 1.13488 (1−𝑜 (1))𝑛 for the 𝑛dimensional simplex 𝑆 𝑛 , and in [180] he showed the existence√of strictly convex 16 (𝑛−1)/2 bodies 𝐾 in 𝐸 𝑛 with 𝐻 (𝐾) ≥ 35 7 . We have 𝐻 (𝐵 𝑛 ) ≥ ( 3/2) 𝑛−𝑜(𝑛) and it was shown by Arias-de-Reyna, Ball and Villa [6] that the same lower bound holds for the Hadwiger number of every convex body in 𝐸 𝑛 . Robins and Salowe [154], Swanepoel [166], Larman and Zong [117] and Xu [186] gave lower bounds for the Hadwiger number of superballs. Zong [192] studied the Hadwiger number of Cartesian products of convex bodies 𝐾 and 𝐿, and he proved that if dim𝐿 ≤ 2, then 𝐻 (𝐾 × 𝐿) = (𝐻 (𝐾) + 1) (𝐻 (𝐿) + 1) − 1. Talata [180] showed that if the dimension of 𝐿 is higher than 2 the equality does not always hold. It is clear that the density of a saturated packing of congruent copies of a convex body with a large average number of neighbors cannot be arbitrarily small. Groemer [80] proved that the density of a saturated packing of translates of a convex body in 𝐸 𝑛 with average number of neighbors 𝜇 is at least 3𝑛1−𝜇 .

13.11 Mutually Touching Translates of a Convex Body A problem of Erdős [46] asked for the greatest cardinality of a set of points in 𝐸 𝑛 with the property that no angle determined by three points is greater than 90◦ . Another problem posed by Klee asked for the largest antipodal set in 𝐸 𝑛 . Two points of a set are antipodal if there are two parallel supporting hyperplanes of the set, each containing one of the points while the whole set is contained in the closed slab bounded by the supporting planes. The two points are strictly antipodal, if except the two points no other points lie on the supporting planes. A set is antipodal or strictly antipodal if every pair of its points are antipodal or strictly antipodal. Danzer and Grünbaum [39] proved that the two problems are equivalent and confirmed the conjecture of Erdős that the answer is 2𝑛 . Moreover, they showed that these problems are also equivalent to the problem of finding the largest family of mutually touching translates of a convex body in 𝐸 𝑛 . The touching number 𝑡 (𝐾) of a convex body 𝐾 in 𝐸 𝑛 is the maximum number of pairwise touching translates of 𝐾. Thus, the theorem of Danzer and Grünbaum states that 𝑡 (𝐾) ≤ 2𝑛 with equality attained only for a parallelotope.

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Gerencsér and Harangi [74] constructed for every 𝑛 a set of 2𝑛−1 + 1 points in 𝐸 𝑛 with the property that any three points form an acute triangle. These sets are strictly antipodal, so they can be augmented to construct strictly convex sets with touching number 2𝑛−1 + 1. Erickson and Kim [47] constructed, for any natural number 𝑛, a family of 𝑛 congruent convex polyhedra in 𝐸 3 , such that every pair intersects in a common facet. This shows that there is no upper bound for the cardinality of packings of pairwise touching congruent convex bodies. On the other hand, Károly Bezdek and János Pach (see [35, p. 98, Conjecture 13]) conjectured that if 𝐶 is a centrally symmetric convex body, then even a family of pairwise touching homothetic copies of 𝐶 has at most 2𝑛 members. Naszódi [138] proved the upper bound 2𝑛+1 without assuming symmetry of 𝐶 and replacing “homothetic” with “positively-homothetic”. For centrally symmetric bodies Naszódi’s bound was lowered by Lángi and Naszódi [116] to 3 × 2𝑛−1 . Földvári [70] proved that the maximum number of pairwise touching positive homothetic copies of a convex disk is 4. There is a fourth equivalent formulation of the problem of Erdős and Klee. A subset 𝑆 of a metric space 𝑀 is said to be equilateral provided each pair of points in 𝑆 have the same distance. The maximum number of elements in an equilateral set in 𝑀 is denoted by 𝑒(𝑀). If 𝐾 is the unit ball of a Minkowski space 𝑀, then the set of mutually touching translates of 𝐾 corresponds to a set of equilateral points in 𝑀, and vice versa. Thus, for an 𝑛-dimensional Minkowski space 𝑀, 𝑒(𝑀) ≤ 2𝑛 , as was noted by Petty [148] and P.S. Soltan [164]. 𝑛 For 1 ≤ 𝑝 ≤ ∞ and 𝑛 ≥ 1 let 𝑙 𝑛𝑝 and 𝑙 ∞ 𝑝 denote 𝑅 endowed with the norm
Í𝑛 1/ 𝑝 ∥ (𝑥1 , . . . , 𝑥 𝑛 ∥ = 𝑖=1 |𝑥𝑖 | 𝑝 and ∥ (𝑥 1 , . . . , 𝑥 𝑛 ∥ = max1≤𝑖 ≤𝑛 |𝑥 𝑖 |, respectively. In 𝑛 𝑙 1 the standard basis vectors and their negatives form an equilateral set of 2𝑛 points, and the set of standard basis vectors together with an appropriate multiple of the all 1 vectors shows that 𝑒(𝑙 𝑛𝑝 ) ≥ 𝑛 + 1. R.B. Kusner (see Guy [85]) conjectured that both examples are extremal, that is 𝑒(𝑙1𝑛 ) = 2𝑛 and 𝑒(𝑙 𝑛𝑝 ) = 𝑛 + 1 for 1 < 𝑝 < ∞. The conjecture concerning 𝑙 1𝑛 was confirmed for 𝑛 = 3 by Bandelt, Chepoi and Laurent [9] and for 𝑛 = 4 by Koolen, Laurent and Schrijver [106]. Thus, the touching number of an octahedron is 6, and the touching number of a crosspolytope in 𝐸 4 is 8. Besides the Euclidean case 𝑙 2𝑛 settled by Smyth [162], Kusner’s conjecture was confirmed for 𝑙4𝑛 by Swanepoel [167] who also proved that the conjecture is false for all 1 < 𝑝 < 2 and sufficiently large 𝑛, depending on 𝑝. However, it follows by continuity that, for fixed 𝑛, if 𝑝 is close to 2 or 4 then 𝑒(𝑙 𝑛𝑝 ) = 𝑛 + 1. Smith [162] and Swanepoel [169] gave explicit bounds for 𝑝: If or | 𝑝 − 4| < 4 ln(1+2/𝑛) then 𝑒(𝑙 𝑛𝑝 ) = 𝑛 + 1. Smyth [163] | 𝑝 − 2| < 2 ln(1+2/𝑛) ln(𝑛+2) ln(𝑛+2) proved that there exists a constant 𝑐 𝑝 such that 𝑒(𝑙 𝑛𝑝 ) ≤ 𝑐 𝑝 𝑛 ( 𝑝+1)/( 𝑝−1) . Extending Smyth’s method, Alon and Pudlák [3] proved 𝑒(𝑙 𝑛𝑝 ) ≤ 𝑐 𝑝 𝑛 (2 𝑝+1)/(2 𝑝−1) , and for odd integers 𝑝 ≥ 1 established the bound 𝑒(𝑙 𝑛𝑝 ) ≤ 𝑐 𝑝 𝑛 ln 𝑛. Petty conjectured that 𝑒(𝑀) ≥ 𝑛 + 1 for every 𝑛 dimensional Minkowski space. In other terms the conjecture states that every convex body 𝐾 in 𝐸 𝑛 admits a packing of 𝑛 + 1 mutually touching translates of 𝐾. Petty proved the bound 𝑒(𝑀) ≥ min{4, 𝑛 + 1}, confirming the conjecture for 𝑛 = 3. Alternative proofs for the 3-

13.12 Mutually Touching Cylinders

289

dimensional case were given by Kobos [104] and Väisälä [181]. This case is well understood: Grünbaum [84] proved that 𝑒(𝑀) ≤ 5 if the unit ball of 𝑀 is strictly convex, and Schürmann and Swanepoel [158] proved that 𝑒(𝑀) ≤ 6 if the unit ball is smooth. The latter authors gave an example of a smooth space 𝑀 with 𝑒(𝑀) = 6 and also characterized the 3-dimensional Minkowski spaces that admit equilateral sets of 6 and 7 points. The case 𝑛 = 4 of Petty’s conjecture was confirmed by Makeev [132]. For 𝑛 ≥ 5 the conjecture is still open, except for special classes of spaces. The Banach–Mazur distance between two 𝑛-dimensional Minkowski spaces is defined as 𝑑 (𝑋, 𝑌 ) = inf ∥𝑇 ∥ ∥𝑇 −1 ∥, where the infimum is taken over all linear, invertible operators 𝑇 from 𝑋 to 𝑌 . Let 𝑀 be an 𝑛-dimensional Minkowski space. Brass [33] and Dekster [40] proved that if 𝑑 (𝑀, 𝑙2𝑛 ) ≤ 1 + 1/𝑛, then 𝑒(𝑀) ≥ 𝑛 + 1. Swanepoel and Villa [171] verified Petty’s conjecture also for the case when 𝑛 ) ≤ 3/2, and Averkov [7] proved that even 𝑑 (𝑀, 𝑙 𝑛 ) ≤ 2 guarantees 𝑑 (𝑀, 𝑙∞ ∞ these results Swanepoel and Villa [171] derived the lower 𝑒(𝑀) ≥ 𝑛 + 1. From √ bound 𝑒(𝑀) ≥ 𝑒 𝑐 ln 𝑛 using the theorem of Alon and Milman [2] stating that for every 𝜀 > 0 there exists a constant 𝑐(𝜀) such that any √ 𝑛-dimensional Minkowski 𝑐 ( 𝜀) log 𝑛 whose Banach–Mazur space contains a subspace of dimension at least 𝑒 𝑛 is at most 𝜀. distance to either 𝑙 2𝑛 or 𝑙∞ González Merino [78] verified Petty’s conjecture for spaces whose unit ball satisfies certain intersection properties and Kobos [105] proved it if the unit ball is symmetric in each of the hyperplanes 𝑥𝑖 = 𝑥 𝑗 . Kobos also proved that the conjecture 𝑛 . Frankl [71] gave the following holds for any (𝑛 − 1)-dimensional subspace of 𝑙∞ extension of this result: To every integer 𝑘 ≥ 2 there is a bound 𝑁 (𝑘) such that 𝑛 contains a set of 𝑘 + 1 equidistant for 𝑛 > 𝑁 (𝑘) any 𝑘-dimensional subspace of 𝑙 ∞ points. As a corollary she obtained that if a centrally symmetric polytope in 𝐸 𝑛 has √ 1+ 8𝑛+9 4 opposite pairs of facets, then there are 𝑛 + 1 mutually translates at most 3 𝑛 − 6 of it. Unfortunately, the difference body of the simplex has more faces, so this does not give a lower bound for the touching number of simplices. Koolen, Laurent and Schrijver [106] gave examples of 𝑛 + 2 mutually touching 𝑛-dimensional simplices. Moreover, Lemmens and Parsons [120] proved that for 𝑛 ≥ 5 and 𝑛 ≡ 1 (mod 4) the touching number of the 𝑛-dimensional simplex is at least 𝑛 + 3. Lawlor and Morgan [118] applied equidistant sets for area-minimization problems in soap films. Further reading about equilateral sets can be found in the papers by Swanepoel [168, 170].

13.12 Mutually Touching Cylinders Littlewood ( [127, Problem 7 on p. 20]) asked the following question: “Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others?” There are several examples of six cylinders mutually touching each other.

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All known examples are flexible with one degree of freedom. Bozóki, Lee and Rónyai [32] answered Littlewood’s question in the affirmative. Fixing the angle of two cylinders the position of the remaining five cylinders can be described by 20 parameters satisfying 20 multivariate polynomial equations. Fixing the position of the initial cylinders perpendicularly two essentially different approximate solutions were found by numerical methods. Having found the approximate solutions, it was proved by Smale’s 𝛼-theory (see Smale [160]), as well as by interval-arithmetic computations, that the system of equations does indeed have a real solution in the neighborhood of the approximate solutions. Further numerical investigation by Bozóki indicated that by fixing the angle of the initial two cylinders between any angle 0 < 𝜙 < 𝜋 gives a one-parametric class of solutions (see the demonstration by Scherer [156]). The two arrangements found by fixing the angle perpendicularly are special elements of this class. Figure 13.7 shows seven mutually touching cylinders. Unaware of Littlewood’s problem, Pikhitsa [149] studied the same question motivated by applications in physics. He also described a configuration of seven mutually touching infinite cylinders numerically, without giving a mathematically rigorous proof for its existence. Pikhitsa and Choi [150] gave numerical evidence for the existence of nine mutually touching incongruent infinite cylinders. W. Kuperberg presented a seemingly convincing physical model of eight mutually touching congruent cylinders and asked whether the cylinders really are mutually touching. Ambrus and A. Bezdek [4] showed that in this model there are two cylinders that do not touch. The question of whether there are eight mutually touching congruent cylinders remains unanswered.

Fig. 13.7 A. Bezdek [13, 14] gave two different proofs of the statement that the number of mutually touching congruent infinite cylinders is bounded. The first proof uses Ramsey’s theorem, and the argument given there shows that the number of cylinders is bounded even if we allow incongruent cylinders. In the second paper he proved

13.14 Neighbors in Lattice Packings

291

that no more than 24 congruent infinite cylinders can mutually touch. Pikhitsa and Pikhitsa [151] also proved that the number of mutually touching infinite cylinders of arbitrary base is bounded. Moreover, in [152] they claimed that no more than 10 infinite cylinders can mutually touch, and gave numerical evidence for the existence of 10 mutually touching elliptic cylinders.

13.13 Cylinders Touching a Ball W. Kuperberg [110, 111] asked for the maximum number of unit-radius infinite cylinders touching a unit-radius ball. He conjectured that the number in question is six, which can be realized in several different ways. Heppes and Szabó [92] gave two different proofs of the upper bound 8 on the number of cylinders. They also discussed the same problem for higher dimensions, and for other radii of the touching cylinders. Brass and Wenk [36] computed the portion of area √ cut out by a cylinder touching the unit ball from a concentric sphere of radius 4.7, which came out to be greater than 1/8, showing that the number of touching cylinders is at most 7. While the question of whether 7 mutually disjoint infinite cylinders of unit radius can touch the unit ball remains open, it turned out that 6 cylinders of radius 𝑟 > 1 can touch it. The first such example with radius 𝑟 = 1.049659 was given by Firsching [65] (see also Firsching [64]) by a numerical exploration of the corresponding 18-dimensional configuration manifold. Ogievetsky and Shlosman [141, 142, 143, 144, 145] devoted a series of papers to the study of the configuration space of cylinders√touching a ball. They found a packing of 6 infinite cylinders of radius 𝑟 = 81 (3 + 33) ≈ 1.093070331 touching the unit ball, and believe that this value of the radius is the maximum. Starostin [165] investigated tubes touching a ball or another tube.

13.14 Neighbors in Lattice Packings Concerning the number of neighbors in densest lattice packings Swinnerton-Dyer [172] proved that for every convex body 𝐾 in 𝐸 𝑛 there is a lattice packing of 𝐾 in which every member touches at least 𝑛(𝑛 + 1) others. M.J. Smith [161] extended Swinnerton-Dyer’s result to compact sets 𝑆 for which 𝑆 − 𝑆 has non-empty interior. On the other hand, Gruber [82] proved that, in the sense of Baire category, typical convex bodies have at most 2𝑛2 neighbors in their densest lattice packings. The ball is not typical: Vlăduţ [182] constructed for a sequence of dimensions 𝑛𝑖 lattice ball packings in 𝐸 𝑛𝑖 in which the balls have 20.0338𝑛𝑖 +𝑜(𝑛𝑖 ) neighbors, and in [183] he constructed sequences of lattice packings of superballs in 𝐸 𝑛 with an exponential number of neighbors. The difference between the Hadwiger number and the maximum number 𝐻 𝐿 (𝐾) of neighbors of 𝐾 in a lattice packing of 𝐾 can be

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large. Talata [175] proved that for every 𝑛 ≥ 3 there exists an 𝑛-dimensional convex body 𝐾 such that 𝐻 (𝐾) − 𝐻 𝐿 (𝐾) ≥ 2𝑛−1 . Groemer [81] studied the number of neighbors in connected lattice packings and proved that in a thinnest connected lattice packing of an 𝑛-dimensional convex body, each body has at least 2𝑛 and at most 2(2𝑛 − 1) neighbors.

Chapter 14

Packing and Covering Properties of Sequences of Convex Bodies G. Fejes Tóth

Consider the following problem: Given a convex set 𝐾 and a sequence {𝐶𝑖 } of convex bodies in 𝐸 𝑛 , is it possible to pack the sequence of bodies in 𝐾 or cover 𝐾 with the bodies? More specifically, is it possible to find for each 𝑖 a congruent copy 𝐶 𝑖 of each 𝐶𝑖 , so that the bodies {𝐶 𝑖 } form a packing in, or a covering of, 𝐾? If so, then we say that the sequence {𝐶𝑖 } permits an isometric packing in 𝐾, or an isometric covering of 𝐾, respectively. If translates of the bodies 𝐶𝑖 are used, then we say that {𝐶𝑖 } permits a translative packing in 𝐾, or a translative covering of 𝐾. Problem 10.1 from the Scottish Book (see Mauldin [78, p. 74]) reads as follows: “PROBLEM 10.1: MAZUR, AUERBACH, ULAM, BANACH Theorem. If {𝐾𝑛 }∞ 𝑛=1 is a sequence of convex bodies, each of diameter ≤ 𝑎 and the sum of their volumes is ≤ 𝑏, then there exists a cube with the diameter 𝑐 = 𝑓 (𝑎, 𝑏) such that one can put all the given bodies in it disjointly. Corollary. One kilogram of potatoes can be put into a finite sack. Determine the function 𝑐 = 𝑓 (𝑎, 𝑏).”

Because of the corollary, packing problems of this type are sometimes called potato-sack problems. The Scottish Book does not give a proof of the theorem. A proof was found later by Kosiński [64], who presented an explicit bound on 𝑓 (𝑎, 𝑏). The main idea is to enclose each 𝐶𝑖 in a box, that is, in a rectangular parallelepiped whose volume is greater than that of 𝐶𝑖 at most by a constant factor independent of 𝐶𝑖 , and whose diameter is not too much larger than the diameter of 𝐶𝑖 , thereby reducing the problem to packing a sequence of boxes in a box. Then it is shown that every sequence of 𝑛-dimensional boxes of edges at most 𝐷 and total volume at most 𝑉 can be packed in a box whose 𝑛−1 edges are of length 3𝐷 and the 𝑛-th one is of length (𝑉 + 𝐷 𝑛 )/𝐷 𝑛−1 . Moon and Moser [80] improved the above bound. They proved that such a family of boxes can be packed in a box of sides 2𝐷, 2𝐷, . . . , 2𝐷, 2(𝑉 + 𝐷 𝑛 )/𝐷 𝑛−1 .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_14

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Moon and Moser also addressed the problem of covering and proved that a family of 𝑛-dimensional boxes of edges at most 𝐷 and total volume 𝑉 can cover 2·4·8···2𝑛 a cube of side 𝐷 if 𝑉 ≥ 𝑐 𝑛 (2𝐷) 𝑛 , where 𝑐 𝑛 = 1·3·7··· (2𝑛 −1) < 2.463. Meir and Moser [79] improved this result slightly. In particular, they showed that any family of 𝑛-dimensional cubes of total volume 𝑉 can transitively cover a cube of volume 𝑉/(2𝑛 − 1). Concerning packing cubes in a cube they proved that a family of cubes of total volume 𝑉 can be transitively packed in a cube of√volume 2𝑛−1𝑉 provided that the side-lengths of the cubes do not exceed 2 (𝑛−1)/𝑛 𝑛 𝑉. The bound 2𝑛−1𝑉 is the best possible, since two cubes of volume 𝑉/2 cannot be transitively packed in a cube of volume smaller than 2𝑛−1𝑉. Let F be a family of convex bodies. For a given convex body 𝐶 we define 𝑝 𝑖 (𝐶 |F ) as the greatest number such that every sequence of members of F whose diameter does not exceed the diameter of 𝐶 and whose total volume is at most 𝑝 𝑖 (𝐶 |F ) vol(𝐶), permits an isometric packing in 𝐶. Further, we define 𝑐 𝑖 (𝐶 |F ) as the smallest number such that every sequence of members of F with total volume at least 𝑐 𝑖 (𝐶 |F ) vol(𝐶) permits an isometric covering of 𝐶. We define similarly 𝑝 𝑡 (𝐶 |F ) and 𝑐 𝑡 (𝐶 |F ), where the extreme values of the total volume are taken for translative packings and translative coverings, respectively. In most of the investigated cases F consists of homothetic or similar copies of a convex body. For a given convex body 𝐶 let Cℎ and C𝑠 denote the family of (positive and negative) homothetic copies of 𝐶 and the family of similar copies of 𝐶, respectively. Further, let Cℎ+ and Cℎ− be the families of positive homothetic copies and negative homothetic copies of 𝐶, respectively.

14.1 Packing and Covering Cubes and Boxes Let 𝐼 𝑛 (𝑠) denote an 𝑛-dimensional cube of edge length 𝑠. If the size is irrelevant, we simply write 𝐼 𝑛 . Then the theorems of Meir and Moser [79] mentioned above state that 𝑝 𝑡 (𝐼 𝑛 |Iℎ𝑛 ) = 2(1/2) 𝑛 and 𝑐 𝑡 (𝐼 𝑛 |Iℎ𝑛 ) = 2𝑛 − 1. The latter equality was also independently proved by A. Bezdek and K. Bezdek [3]. Better bounds have been obtained under the assumption that the sets used for packing and covering are uniformly bounded and the container is large. Let B 𝑛 denote the family of 𝑛-dimensional boxes of edge length at most 1. Groemer [27] proved that for 𝑠 ≥ 3 𝑠−1 ((𝑠 − 1) 𝑛−2 − 1) and 𝑠−2 𝑐 𝑖 (𝐼 𝑛 (𝑠)|B 𝑛 ) ≤ (𝑠 + 1) 𝑛 − 1.

𝑝 𝑖 (𝐼 𝑛 (𝑠)|B 𝑛 ) ≥ (𝑠 − 1) 𝑛 −

14.2 Results for General Convex Bodies

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14.2 Results for General Convex Bodies Macbeath [76] proved that to every 𝑛-dimensional convex body 𝐶 there exist two boxes, 𝑄 1 and 𝑄 2 , with 𝑛𝑛𝑉 (𝑄 1 ) ≥ 𝑉 (𝐶) ≥

1 𝑉 (𝐶), 𝑛!

such that 𝑄 1 ⊂ 𝐶 ⊂ 𝑄 2 . Let F 𝑛 denote the family of 𝑛-dimensional convex bodies whose diameters are at most 1. The combination of Macbeath’s theorem with the above mentioned theorems of Groemer yields 𝑝 𝑖 (𝐼 𝑛 (𝑠)|F 𝑛 ) ≥

1 𝑠−1 ((𝑠 − 1) 𝑛 − ((𝑠 − 1) 𝑛−2 − 1)) 𝑛! 𝑠−2

and 𝑐 𝑖 (𝐼 𝑛 (𝑠)|F 𝑛 ) ≤ 𝑛𝑛 ((𝑠 + 1) 𝑛 − 1). Macbeath’s theorem was also proved by Hadwiger [32]. The part about the box containing 𝐶 was also proved by Kosiński [64] and the part about the box contained in 𝐶 by Chakerian [6]. Lassak [68] improved the bound for the volume of the box 𝑄 1 by a factor of 2. In particular, every convex disk 𝐶 of area 𝑎 in the plane is contained in a rectangle of area 2𝑎. This was previously proved also by Radziszewski [83] and Süss [89]. Since two copies of ( 21 + 𝜀)𝐶 cannot be packed transitively in 𝐶 we have 𝑝 𝑡 (𝐶 |Cℎ+ ) ≤ 21 . Soifer [85] and Novotny [82] conjectured that 𝑝 𝑡 (𝐶 |Cℎ+ ) = 21 for every convex disk 𝐶. We are far from being able to prove or disprove this conjecture. The best known lower bound for general convex disks is 𝑝 𝑡 (𝐶 |Cℎ+ ) ≥ 41 due to Januszewski [46]. Concerning packing positive and negative homothetic copies 7 . The in a convex disk Januszewski [48] established the inequality 𝑝 𝑡 (𝐶 |Cℎ ) ≥ 40 1 stronger conjecture of Soifer [85] that we have even 𝑝 𝑖 (𝐶 |Cℎ ) = 2 for every convex disk 𝐶 was refuted by Novotny √︁ [82] by showing that for the rectangle 𝑅 with sides 31/4 and 21/4 , 𝑝 𝑖 (𝑅|R ℎ ) = 3/8. L. Fejes Tóth conjectured (see Brass, Moser and Pach [4, p. 131]) that 𝐶𝑡 (𝐶 |Cℎ+ ) ≤ 3 for every convex disk 𝐶. The first upper bound of 𝐶𝑡 (𝐶 |Cℎ+ ) ≤ 12 for general convex disks 𝐶 is due to A. Bezdek and K. Bezdek [3]. Bálint, Bálintová, Branická, Grešák, Hrinko, Novotný and Stacho [2] lowered this bound √ to 9(9 − 17)/4. Further improvements were achieved by Januszewski, [35, 38, 42] bringing down the upper bound to 6.5. In [35] Januszewski considered the 𝑛dimensional case as well, and showed that 𝑐 𝑡 (𝐶 |Cℎ+ ) ≤ (𝑛 + 1) 𝑛 − 1 for every convex body 𝐶 in 𝐸 𝑛 . Naszódi [81] improved this bound to 6𝑛 for general convex bodies and to 3𝑛 if 𝐶 is centrally symmetric. A further improvement was achieved recently by Livshyts and Tikhomirov [74]: 𝑐 𝑡 (𝐶 |Cℎ+ ) ≤ 2𝑛 𝑛 ln 𝑛(1 + 𝑜(𝑛)) if 𝐶 is centrally symmetric, and 𝑐 𝑡 (𝐶 |Cℎ+ ) ≤ √ 1𝜋𝑛 4𝑛 𝑛 ln 𝑛(1 + 𝑜(𝑛)) otherwise.

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Soltan (see Brass, Moser and Pach [4, Sect. 3.2, Conjecture 2]) formulated a convex body in 𝐸 𝑛 and 𝐶 ⊂ ∪𝜆𝑖 𝐶𝑖 for some the following conjecture: If 𝐶 is Í positive coefficients 𝜆𝑖 < 1 then 𝜆𝑖 ≥ 𝑛. Soltan and Vásárhelyi [87] proved the conjecture for 𝑛 = 2 and showed that equality characterizes parallelograms. The special case of the conjecture when 𝐶 is a triangle or a parallelogram was also proved by Dumitrescu and Jiang [10]. Soltan and Vásárhelyi settled the conjecture also for the case when the number of copies covering 𝐶 is at most 𝑛 + 1. Naszódi [81] proved the following asymptotic version of the conjecture: For any 𝜈 < 1 there is an 𝑛0 such that if 𝑛 > 𝑛0 then for every 𝑛-dimensional convex body 𝐶, if some Í𝑚 𝜆 ≥ 𝜈𝑛. homothetic copies of 𝐶 of ratios 0 < 𝜆1 , 𝜆1 , . . . , 𝜆 𝑚 < 1 cover 𝐶 then 𝑖=1 Glazyrin [21] proved the conjecture for balls. Ambrus [1] proved that if an 𝑛-dimensional simplex is covered by its negative homothetic copies then the sum of the absolute values of the coefficients is at least 𝑛. Suggested by this result we raise the question: Is it true that if 𝐶 is a convex body Í in 𝐸 𝑛 and 𝐶 ⊂ ∪𝜆𝑖 𝐶𝑖 for some coefficients −1 < 𝜆 𝑖 < 0 then |𝜆𝑖 | ≥ 𝑛. Since, in general, it is more difficult to cover a convex body by its negative than by its positive homothetic copies, we risk the conjecture that the answer is affirmative. Maybe, the same conclusion holds if we only suppose that the absolute value of the coefficients is less than 1.

14.3 On-Line Packing and Covering The problem of packing a container with a sequence of convex bodies has an algorithmic version: The bodies of the sequence are given one at a time, as on a conveyor belt, and the algorithm is to decide on the placement of the arriving body before the next body is revealed; once placed, the body cannot be moved. We call this an on-line packing problem. On-line covering problems are defined similarly. Research in this direction was initiated by Lassak and Zhang [73] for packing and by W. Kuperberg [66] for covering. Analogously to the quantities 𝑝 𝑖 (𝐶 |F ), 𝑐 𝑖 (𝐶 |F ), 𝑝 𝑡 (𝐶 |F ) and 𝑐 𝑡 (𝐶 |F ) we define 𝑝 ∗𝑖 (𝐶 |F ), 𝑐∗𝑖 (𝐶 |F ), 𝑝 ∗𝑡 (𝐶 |F ) and 𝑐∗𝑡 (𝐶 |F ), where the extreme values are taken for on-line arrangements only. Recall that 𝑝 𝑡 (𝐼 𝑛 |Iℎ𝑛 ) = 2( 21 ) 𝑛 . Improving on an earlier result by Lassak [71], Januszewski and Lassak [57] proved that for 𝑛 ≥ 5 an equally efficient on-line algorithm exists: 𝑝 ∗𝑡 (𝐼 𝑛 |Iℎ𝑛 ) = 2(1/2) 𝑛 . For 𝑛 = 3 and 𝑛 = 4 they proved the somewhat weaker result 𝑝 ∗𝑡 (𝐼 𝑛 |Iℎ𝑛 ) ≥ 23 ( 12 ) 𝑛 . The 4-dimensional case was settled recently by Januszewski and Zielonka [61]: 𝑝 ∗𝑡 (𝐼 4 |Iℎ4 ) = 1/8. The question of whether 𝑝 ∗𝑡 (𝐼 𝑛 |Iℎ𝑛 ) = 2( 21 ) 𝑛 for 𝑛 = 2 and 𝑛 = 3 remains open.

14.4 Special Convex Disks

297

Concerning packing boxes in a cube, Lassak [71] proved that √ 𝑝 ∗𝑡 (𝐼 𝑛 |B𝑛 ℎ ) ≥ (1 − 3/2) 𝑛−1 , which was improved by Januszewski and Zielonka [63] to √ 𝑝 ∗𝑡 (𝐼 𝑛 |B𝑛 ℎ ) ≥ (3 − 2 2)3−𝑛 . It is an open question whether 𝑝 ∗𝑡 (𝐼 𝑛 |B𝑛 ℎ ) = 𝑝 𝑡 (𝐼 𝑛 |B𝑛 ℎ ) = 2( 21 ) 𝑛 for 𝑛 ≥ 2. The algorithm by W. Kuperberg [66] yields 𝑐∗𝑡 (𝐼 𝑛 |Iℎ𝑛 ) ≤ 4𝑛 . Better algorithms by Januszewski and Lassak [55] and by Lassak [69] provide the bound 𝑐∗𝑡 (𝐼 𝑛 |Iℎ𝑛 ) ≤ 2𝑛 (1 + 𝑜(𝑛)). The breakthrough was achieved by Januszewski, Lassak, Rote and Woeginger [58], who constructed an on-line algorithm showing that 𝑛+2 𝑐∗𝑡 (𝐼 𝑛 |Iℎ𝑛 ) ≤ 2𝑛 + 3 − 222𝑛 +2−8 − 2 . This bound comes astoundingly close to the best value for off-line coverings. Lassak [72] further improved this bound to 𝑐∗𝑡 (𝐼 𝑛 |Iℎ𝑛 ) ≤ 2𝑛 +

5 (1 + 2−𝑛 ). 3

For the 3-dimensional case this yields 𝑐∗𝑡 (𝐼 3 |Iℎ3 ) ≤ 8 + 15/8 = 9.875. The main tool used in the two articles cited above is the 𝑞-adic on-line algorithm for covering the interval [0, 1] with a sequence of segments 𝑆𝑖 of length 𝛿𝑖 (𝑖 =  1, 2, . . .), where 𝑞 ≥ 2 is an integer, 𝛿𝑖 ∈ 𝑞 −1 , 𝑞 −2 , . . . , and each 𝑆𝑖 must be placed on one of the intervals of the form [𝑘𝛿𝑖 , (𝑘 + 1)𝛿𝑖 ] ⊂ [0, 1], for some integer 𝑘. This approach was earlier suggested by W. Kuperberg [67], explicitly for 𝑞 = 2 and implicitly for all 𝑞 ≥ 2. The suggestion was put in the form of a question asking for the existence of a winning algorithm in a 2-adic interval covering game between two players. The solution of the problem appeared in Januszewski, Lassak, Rote and Woeginger [59].

14.4 Special Convex Disks A considerable amount of research has been devoted to packing and covering problems involving sequences of special convex disks, such as squares, rectangles, or triangles. For the special case of a square 𝑆 the theorem of Meir and Moser [79] mentioned above states that 𝑝 𝑡 (𝑆|Sℎ ) = 21 and 𝑐 𝑡 (𝑆|Sℎ ) = 3. Januszewski [39] proved that 𝑐 𝑖 (𝑆|Sℎ ) = 2 and in his papers [47, 49] he proved that 𝑐 𝑡 (𝑆|S𝑠 ) = 3.

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Let S ′ be the family of squares with diagonals parallel to the sides of 𝑆. Januszewski [40, 53] proved that 𝑝 𝑡 (𝑆|S ′) = 4/9 and 𝑐 𝑡 (𝑆|S ′) = 2.5. Concerning packing rectangles into rectangles Yuan, Xu and Ding [92] proved the following: If R 𝑎 denotes the family of rectangles of side lengths not greater than 𝑎, and 𝑅 𝑎𝑏 denotes a rectangle with sides 𝑎 and 𝑏 ≥ 𝑎, then 𝑝 𝑖 (𝑅 𝑎𝑏 |R 𝑎 ) =

𝑎𝑏 . 2

The case 𝑎 = 𝑏 was proved earlier by Januszewski [37]. For on-line packing squares into a square Januszewski and Lassak [57] established the bound 5 . 𝑝 ∗𝑡 (𝑆|Sℎ ) ≥ 16 The lower bound on 𝑝 ∗𝑡 (𝑆|Sℎ ) was subsequently improved to 31 by Han, Iwama and 11 by Fekete and Hoffmann [14] and, finally, to 52 by Brubach Zhang [33], to 32 [5]. Concerning on-line packing rectangles in a square, Lassak [71] proved the 5 , which was improved by Januszewski and Zielonka [62] to bound 𝑝 ∗𝑖 (𝑆|R) ≥ 36 ∗ 𝑝 𝑖 (𝑆|R) ≥ 0.28378. For on-line√covering a square with squares Januszewski and Lassak [56] proved ∗ 𝑐∗𝑡 (𝑆|Sℎ ) ≤ 47 39+ 13 8 ≈ 5.265. This was improved to 𝑐 𝑡 (𝑆|Sℎ ) ≤ 4 by Januszewski [50]. Concerning on-line covering a square by rectangles Januszewski [34] proved 𝑐∗𝑖 (𝑆|R) ≤ 15 2 . Richardson [84] proved that 𝑝 𝑖 (𝑇 |T𝑠 ) ≥ 21 for every triangle 𝑇. In fact, his algorithm used only positive and negative homothetic copies of 𝑇 and he conjectured that the packing is possible by using positive homothetic copies only. This conjecture is equivalent to the statement that 𝑝 𝑡 (𝑇 |Tℎ+ ) = 21 . In [41] Januszewski established 19 and later in [51] he refined the method and confirmed the bound 𝑝 𝑡 (𝑇 |Tℎ+ ) ≥ 56 Richardson’s conjecture. On the other hand, Januszewski disproved Soifer’s conjecture [86] that 𝑝 𝑖 (𝑇 |T𝑠 ) = 21 for every triangle 𝑇. In [43] he showed that 𝑝 𝑖 (𝑇 |T𝑠 ) = 21 if and only if 𝑇 is equilateral. Moreover, in [44] he proved that if 𝑇 is an √ √ isosceles right triangle then 0.511 ≤ 31 (5 − 1 3) ≤ 𝑝 𝑖 (𝑇 |Tℎ ) ≤ 27 − 2 2 ≤ 0.6715. For translative packing of positive and negative homothetic copies of a triangle Januszewski [45] proved 2 𝑝 𝑡 (𝑇 |Tℎ ) = . 9 Concerning covering a triangle 𝑇 with homothetic copies Vásárhelyi [90] proved that 𝑐 𝑡 (𝑇 |Tℎ− ) = 4 and Füredi [19] proved that

14.4 Special Convex Disks

299

𝑐 𝑡 (𝑇 |Tℎ+ ) = 2. Januszewski [36] strengthened Vásárhelyi’s result by showing that 𝑐 𝑡 (𝑇 |Tℎ ) = 4. For a right isosceles triangle 𝑇 Füredi [20] established the equality √ 1+ 2 𝑐 𝑖 (𝑇 |Tℎ ) = . 2 Denote by 𝑇𝜑 the triangle obtained from 𝑇 by a rotation through the angle 𝜑. Vásárhelyi [91] proved that 𝑐 𝑡 (𝑇30◦ |T30◦ +ℎ ) = 4 and 𝑐 𝑡 (𝑇𝜑 |T𝜑 +ℎ ) < 4 for every triangle that is not regular. Let 𝑇 (ℎ) be a triangle with base length 1 and with height ℎ such that the angles at the base are at most 𝜋/2. Januszewski, Liu, Su and Zielonka [60] proved that    4ℎ 4ℎ 2ℎ , max , 𝑝 𝑡 (𝑇ℎ |Sℎ ) = min . (ℎ + 1) 2 (2ℎ + 1) 2 (ℎ + 1) 2 Previously, the case of an equilateral triangle was proved by Januszewski [52] and the cases 𝑇√2/2 and 𝑇√2/3 were proved by Su, Lu and Liu [88]. For the equilateral triangle 𝑇√3/2 Januszewski [52] established √ 𝑐 𝑡 (𝑇√3/2 |Sℎ ) = 2 3. His result was extended by Lu and Su [75] to covering an isosceles triangle 𝑇 (ℎ). They showed that √  2 2  if ≤ ℎ ≤ 1,  ℎ √2     4ℎ if 2 ≤ ℎ ≤ 1, 2 𝑐 𝑡 (𝑇 (ℎ)|Sℎ ) = 4 √  if 1 ≤ ℎ < 2,  ℎ  √   2 if 2 ≤ ℎ.  √ For a right triangle 𝑇0 with legs 1 and 2 and a square 𝑆 with sides parallel√ to the 2 legs of 𝑇0 Fu, Lian and Zhang [18] proved the inequality 𝑝 𝑡 (𝑇0 |Sℎ ) ≥ 16−6 23 . Let 𝑇𝑟 denote the tetrahedron with three mutually perpendicular edges of lengths 1, 1, √ and 2. Fu, Lian and Zhang [18] also investigate the problem of covering 𝑇𝑟 by homothetic copies of a cube 𝐶 with sides parallel to the edges of 𝑇𝑟 and prove that 𝑐 𝑡 (𝑇𝑟 |Cℎ ) ≤ 18 + √3 . 2 Concerning packing circles in a circle Fekete, Keldenich and Scheffer [16] proved that 1 𝑝 𝑡 (𝐵2 |Bℎ2 ) = . 2

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14 Packing and Covering Properties of Sequences of Convex Bodies

For the on-line case Januszewski [54] established the bound 𝑝 ∗𝑡 (𝐵2 |Bℎ2 ) > 0.197. For the corresponding covering problem Dumitrescu and Jiang [11] proved that 𝑐 𝑡 (𝐵2 |Bℎ2 ) ≤ 2.97 thereby confirming the conjecture of L. Fejes Tóth for the first convex disk that is not a polygon. They also considered the on-line version of the problem and proved that 𝑐∗𝑡 (𝐵2 |Bℎ2 ) ≤ 9.763. Januszewski [54] improved the latter bound to 𝑐∗𝑡 (𝐵2 |Bℎ2 ) < 6.488. Fekete, Morr and Scheffer [17] investigated the problem of packing sequences of circles in a square or triangle. They proved that 𝑝 𝑡 (𝑆|Bℎ2 ) =

𝜋 √ . 3+2 2

Further, if 𝑇 is a non-acute triangle with an incircle of area 𝑎 then 𝑝 𝑡 (𝑇 |Bℎ2 ) = 𝑎. For on-line packing circles in a square Fekete, von Höveling and Scheffer [15] proved the inequality 𝑝 ∗𝑡 (𝑆|Bℎ2 ) ≤ 0.350389. For packing squares in a circle Fekete, Gurunathan, Juneja, Keldenich, Kleist and Scheffer [13] established the equality 8 . 𝑝 𝑡 (𝐵2 |Sℎ ) = 5𝜋 The problem of covering the square by a sequence of circular disks was solved by Fekete, Gupta, Keldenich, Scheffer and Shah [12]. They proved that 𝑐 𝑡 (𝑆|Bℎ2 ) =

195𝜋 . 256

More generally, they gave an algorithm for covering the rectangle 𝑅(1, »√𝜆) with sides 7/2 − 1/4 = 1 and 𝜆 ≥ 1, and showed that there is a threshold value 𝜆0 = 1.035797 . . ., such that for 𝜆 < 𝜆 0   2 5 9 𝜆 2 + + , 𝑐 𝑡 (𝑅(1, 𝜆)|Bℎ ) = 3𝜋 16 32 256𝜆2 and for 𝜆 ≥ 𝜆0 𝑐 𝑡 (𝑅(1, 𝜆)|Bℎ2 ) =

(𝜆2 + 2)𝜋 . 4

14.5 Packing in and Covering of the Whole Space The investigation of covering the whole space by sequences of convex bodies was initiated by Chakerian [6]. Clearly, for a sequence {𝐶𝑖 } of convex bodies to permit a covering of space it is necessary that the sum of the volumes 𝑉 (𝐶𝑖 ), as well as the sum of the widths 𝑤(𝐶𝑖 ), be divergent. These conditions are not sufficient. Chakerian

14.6 Covering with Slabs

301

and Groemer [7] gave necessary and sufficient conditions for a sequence of convex disks to permit a covering of the plane. A sequence {𝐶𝑖 } of convex disks permits a covering of the plane if and only if either the total area of the subsequence with diameter at most 1 is infinite or the total width of the subsequence with diameter greater than 1 is infinite. The sequence {𝐶𝑖 } of convex bodies is bounded if the set of the diameters of the bodies is bounded. In particular, it follows that a bounded sequence of convex disks permits a covering of the plane if and only if the total area of the disks is infinite. The analogous statement for 𝐸 𝑛 , 𝑛 ≥ 3, was proved by Groemer [22]. Chakerian and Groemer [8] gave necessary and sufficient conditions for a sequence of convex bodies to permit a covering of 𝐸 𝑛 . Groemer [23] proved that for a sequence of convex bodies to permit a covering of 𝐸 𝑛 it is necessary and sufficient that the sequence permits a translative covering of almost all points of 𝐸 𝑛 . Groemer [24, 28] investigated coverings of space by finite sequences of unbounded convex sets and in [29] he gave an upper bound for the total volume of a sequence of convex bodies permitting a covering of the 𝑛-dimensional sphere. As consequences of the results of Groemer [27] mentioned in Section 14.1, we note the following. If {𝐶𝑖 } is a bounded sequence of 𝑛-dimensional convex bodies Í such that 𝑉 (𝐶𝑖 ) = ∞, then it permits an isometric covering of the 𝑛-dimensional 1 space with density 21 𝑛𝑛 and an isometric packing with density 𝑛! . Moreover, if all the sets 𝐶𝑖 are boxes, then {𝐶𝑖 } permits a translative covering of space and a translative packing in space with density 1. In the plane, any bounded sequence {𝐶𝑖 } of convex disks with infinite total area permits even a translative packing and covering with density 21 and 2, respectively. It is an open problem whether for 𝑛 > 2 every bounded sequence {𝐶𝑖 } of 𝑛-dimensional convex bodies of infinite total volume permits a translative covering of 𝐸 𝑛 . Groemer [31] investigated the question under which conditions a sequence of convex bodies {𝐶𝑖 } in 𝐸 𝑛 permits an isometric packing or covering of density 1. Í He showed that the conditions 𝑉 (𝐶𝑖 ) and lim𝑖→∞ 𝑑 (𝐶𝑖 ) = 0 on the volume and diameter of the sets 𝐶𝑖 are sufficient for constructing such packings and coverings.

14.6 Covering with Slabs Concerning the problem of covering space with a sequence of slabs, Groemer Í [25,26] proved that every sequence of slabs of widths 𝑤 𝑖 in 𝐸 𝑛 for which 𝑤 𝑖(𝑛+1)/2 = ∞ permits a translative covering. Makai and Pach [77] conjectured that a sequence of slabs in 𝐸 𝑛 permits a translative covering if and only if the sum of their widths is infinite. They verified the conjecture for the case of the plane. In higher dimensions the conjecture is still unresolved. Kupavskii and Pach [65] proved that if 𝑤 1 ≥ 𝑤 2 ≥ . . . is an infinite sequence of positive numbers such that lim sup 𝑖→∞

𝑤1 + 𝑤2 + . . . 𝑤𝑖 > 0, ln(1/𝑤 𝑖 )

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14 Packing and Covering Properties of Sequences of Convex Bodies

then every sequence of slabs of width 𝑤 𝑖 (𝑖 = 1, 2, . . .) permits a translative covering of 𝐸 𝑛 . With this result they got rather close to the proof of the conjecture: For example, it implies that a sequence of slabs of width 𝑤 𝑖 = 1/𝑖 (𝑖 = 1, 2, . . .) permits a translative covering, while this is false for the sequence of widths 𝑤 𝑖 = 1/𝑖 1+𝜀 for any 𝜀 > 0. A detailed account on the topic of packing and covering properties of sequences of convex bodies is found in the survey by Groemer [30]. For surveys on on-line algorithms see Lassak [70] and Csirik and Woeginger [9].

Chapter 15

Four Classic Problems G. Fejes Tóth and W. Kuperberg

15.1 The Borsuk Problem Borsuk [48] asked the question whether every bounded set in 𝑛-dimensional space can be partitioned into 𝑛 + 1 subsets of smaller diameter. Although Borsuk did not suggest a positive solution of the problem, for a long while there was general belief that the answer is yes, so, the problem became known as Borsuk’s conjecture. The truth of the conjecture for 𝑛 = 2 follows easily by the theorem of Pál [138] stating that √ every set of unit diameter can be covered by a regular hexagon of side length 1/ √ 3. Dissecting the hexagon into three pentagons yields the sharp upper bound 3/2 for the diameter of the pieces. The same upper bound was obtained by Gale [76], who obtained it by dissecting a suitable truncation of an equilateral √ triangle of side-length 3. For sets of constant width some stronger theorems were proved. For a convex disk 𝐾, let 𝑑 (𝐾) denote the smallest number with the property that 𝐾 can be covered by three sets of diameter 𝑑 (𝐾). Lenz √ √ [120] proved that for a set 𝐾 of constant width 1 the inequality 3 − 1 ≤ 𝑑 (𝐾) ≤ 3/2 holds, where equality is reached in the upper bound only for the circle, and in the lower bound only for the Reuleaux triangle. Let 𝑙 (𝐾) denote the side-length of the largest equilateral triangle inscribed in 𝐾 and let 𝐿 (𝐾) denote the side-length of the smallest equilateral triangle circumscribed about 𝐾. Melzak [128] proved that a set√ 𝐾 of constant width 1 can be covered by three sets of diameter at most min{𝑙 (𝐾), 3 − 𝑙 (𝐾)} and Schopp [155] proved that it can √ be covered by three √ circular disks of diameter 𝐿(𝐾)/2. Schopp also proved that 12 − 2 ≤ 𝐿(𝐾) ≤ 3 for every set of constant width 1. In a theorem of Chakerian and Sallee [52] the roles of disks is changed to the opposite: They proved that every convex disk of unit diameter can be covered by three copies of any set of constant width 0.9101. Eggleston [69] settled the three-dimensional case of Borsuk’s conjecture by a rather complicated analytic argument. Simpler proofs were given by Grünbaum [84], and Heppes [93]. Both Grünbaum and Heppes started with the result of Gale [76] that every set of diameter 1 can be imbedded in a regular octahedron the distance © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_15

303

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between whose opposite faces is 1. Then they observed that suitably truncating the octahedron at three vertices, the resulting polyhedron still can contain every body of diameter 1, and can be partitioned into four sets of diameter less than 1. For the diameter of the pieces Heppes proved the bound 0.9977 . . . , while with a more detailed analysis Grünbaum got the bound 0.9885 . . . . The presently best known bound, 0.98, is due to Makeev [124], who obtained it by proving that every convex body of diameter 1 is contained in a rhombic dodecahedron with parallel faces at distance 1 apart. This last statement was also proved by Hausel, Makai and Szűcs [91] and by G. Kuperberg [111]. Katzarowa-Karanowa [107] proved that every set of diameter 2 in three-dimensional space can be covered by four unit balls and she stated that her method can be refined to lower the radii of the balls to 0.999983. For finite sets of points in three dimensions a simple proof was given by Heppes and Révész [94]. The proof of this case follows by induction from the following conjecture of Vázsonyi (see Erdős [70]): Among a set of 𝑛 ≥ 4 points in 𝐸 3 there are at most 2𝑛−2 pairs realizing the diameter of the set. Proofs of Vázsonyi’s conjecture were given by Grünbaum [83], Heppes [92], Straszewicz [161], and Dol’nikov [68]. These proofs use the ball polytope obtained by taking the intersection of the balls centred at the points of the set with radius equal to the diameter of the set. Proofs avoiding the use of ball polytopes were given by Swanepoel [163] and Perlstein and Pinchasi [140]. Rissling [150] proved that every set in three-dimensional hyperbolic space can be divided into four parts of smaller diameter and the same is true for the spherical space for sets whose diameter is smaller than 𝜋/3. The theorem that Borsuk’s conjecture holds for smooth convex bodies is generally credited to Hadwiger [86]. Hadwiger only proved that Borsuk’s conjecture holds for a smooth body of constant width. From this, the validity of Borsuk’s conjecture for general smooth convex bodies follows, if we know that every smooth convex body can be enclosed in a smooth body of constant width of the same diameter. However, this was proved only later by Falconer [72] and Schulte [157]. Direct proofs of the Borsuk conjecture for smooth convex bodies were given by Lenz [120] and Melzak [129]. The condition of smoothness was weakened by Dekster [64] who proved that the conjecture holds for every convex body for which there exists a direction in which every line tangent to the body contains at least one point of the body’s boundary at which the tangent hyperplane is unique. Consider a convex body 𝐾 in 𝐸 𝑛 for which to any boundary point 𝑥 of 𝐾 there is a ball of radius 𝑟 contained in 𝐾 and containing the point 𝑥. This means that a 𝑟 can ball of radius  freely roll in 𝐾. Hadwiger [87] proved the upper bound 𝑑 − » 1 2𝑟 1 − 1 − 𝑛2 for the diameter of the pieces in an optimal partition of such bodies   √ 1− 1−4/(𝑛+3) into 𝑛+1 parts. Dekster [63] proved a similar bound, namely 𝑑 2 − 𝑟 2 √ 1+ 1+4/(𝑛+3) » 1 2 2 for odd 𝑛, and 𝑑 − 𝑟 𝑛+1 for even 𝑛.

15.1 The Borsuk Problem

305

Borsuk’s conjecture is known to be true for some further special cases: Rissling [150] proved it for centrally symmetric sets, Rogers [151, 152] for sets whose symmetry group contains that of the regular simplex, and Kołodziejczyk [110] for sets of revolution. The result of Kołodziejczyk was obtained independently by Dekster [66], who proved the same also for hyperbolic and spherical spaces. The obvious simplicia1 decomposition shows that 𝐵 𝑛 can be decomposed into    1/2   1/2  1/2 𝑛+1 1 𝑛−1 1 𝑛 + 1 subsets of diameter 𝑛+2 , if 𝑛 is even, and 2 + 2 𝑛+3 if 𝑛 is odd. It is conjectured that this is the lower bound for the diameter 𝑑 of the pieces. Hadwiger    1/2  1/2 1 𝑛−1 1 [88] confirmed this for 𝑛 ≤ 3 and proved the lower bound 𝑑 ≥ 2 + 2 2𝑛 for 𝑛 ≥ 4. Larman and Tamvakis [114] improved Hadwiger’s bound to 𝑑 ≥ 3 ln 𝑛 + 𝑂 ( 𝑛1 ). 1 − 2𝑛 Let 𝑏(𝑛) denote the 𝑛-th Borsuk number, that is, the smallest integer such that every bounded set in 𝑛-dimensional space can be partitioned into 𝑏(𝑛) subsets of smaller diameter. Lassak [115] established the upper bound 𝑏(𝑛) ≤ 2𝑛−1 + 1. Lassak’s result was improved significantly by Schramm [156] and Bourgain and Lindenstrauss [49] to √︁ 𝑏(𝑛) ≤ ( 3/2 + 𝑜(1)) 𝑛 = (1.2247 . . . + 𝑜(1)) 𝑛 , which is the best presently known bound. Despite the fact that some doubt in the truth of the conjecture was announced by Erdős [71], Larman [113], and Rogers [151], it came as a surprise when Kahn and Kalai [105] proved that the conjecture fails in all dimensions 𝑛 ≥ 2015. Moreover, √ they proved that 𝑏(𝑛) ≥ (1.2) 𝑛 for sufficiently large 𝑛. The best lower bound for 𝑏(𝑛) presently known is  𝑏(𝑛) ≥

2 √ 3

! √𝑛

 √2 + 𝑜(1)

√

= (1.2255 . . . + 𝑜(1))

𝑛

,

due to Ra˘ıgorodski˘ı [146]. Kahn and Kalai [105] claimed without giving any details that Borsuk’s conjecture fails in dimension 𝑛 = 1325. Weissbach [168] pointed out that this statement does not follow from the argument of Kahn and Kalai (see also Jenrich [101]). The lower bound for the dimension 𝑛 in which Borsuk’s conjecture fails was lowered by Nilli [135] to 946, by Grey and Weissbach [81] to 903, by Ra˘ıgorodski˘ı, [145] to 561, by Weissbach [168] to 560, by Hinrichs [95] to 323, by Pikhurko [141] to 321, by Hinrichs and Richter [96] to 298, and by Bondarenko [46] to 65. The last step thus far was done by Jenrich and Brouwer [102], who found a 64-dimensional subset of 352 points of the set constructed by Bondarenko that cannot be divided into fewer than 71 parts of smaller diameter. The notion of the 𝑘-fold Borsuk number of a set was introduced by Hujter and Lángi [98] as follows. Let 𝑆 be a set of diameter 𝑑 > 0. The smallest positive integer 𝑚 such that there is a 𝑘-fold covering of 𝑆 with 𝑚 sets of diameters strictly smaller

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than 𝑑, is called the 𝑘-fold Borsuk number of 𝑆. Besides presenting a few other results concerning this notion, Hujter and Lángi determined the 𝑘-fold Borsuk number for every bounded planar set. Lángi and Naszódi [112] investigated multiple Borsuk numbers in normed spaces. For comprehensive surveys of the topic see Grünbaum [85], Ra˘ıgorodski˘ı [147, 148, 149], Kalai [106], Zong [173] and the corresponding chapters of the books Boltjanski˘ı, Martini and Soltan, [44] and Martini, Montejano and Oliveros [126].

15.2 Tarski’s Plank Problem In [166] Tarski raised the following problem: Is it true that if a convex body 𝐶 of width 𝑤 is covered by parallel slabs with the widths 𝑤 1 , . . . , 𝑤 𝑙 then 𝑤 1 +. . .+𝑤 𝑙 ≥ 𝑤? If 𝐶 is a circle the solution was given by Moese [130]. Straszewicz [160] solved the problem in the plane for two strips. An affirmative answer to the question for general convex bodies was given by Bang [8, 9]. Variations of Bang’s proof were given by Fenchel [74] and Bognár [37]. The width of a slab relative to a convex body 𝐶 is the width of the slab divided by the width of 𝐶 in the direction perpendicular to the slab. Bang [8] asked whether the following generalization of his theorem is true. If some slabs cover a convex body 𝐶 then the sum of the widths of the slabs relative to 𝐶 is at least 1. For centrally symmetric bodies Bang’s question was answered in the affirmative by Ball [7]. As a corollary Ball proved the following theorem. Given a centrally symmetric convex body 𝐶 and 𝑛 hyperplanes in 𝑛-dimensional Euclidean space, then there is 1 a translate of 𝑛+1 𝐶 inside 𝐶 whose interior does not meet any of the hyperplanes. The result is obviously sharp for every 𝑛 and 𝐶 and is a generalization of a result by Davenport [62] who considered the special case when 𝐶 is a cube. For nonsymmetric sets 𝐶 Bang’s problem was solved for coverings of 𝐶 by two slabs (see Bang [10], Moser [131], Alexander [2], and Hunter [99]). Akopyan, Karasev and Petrov [1] proved the conjecture for an arbitrary number of planks whose normals have only two distinct directions. Related to the above corollary is Conway’s fried potato problem, phrased by Croft, Falconer and Guy [53, Problem C1, p. 80] as follows. In order to fry it as expeditiously as possible Conway wishes to slice a given convex potato into 𝑛 pieces by 𝑛 − 1 successive plane cuts (just one piece being divided by each cut) so as to minimize the greatest inradius of the pieces.

This problem was solved by A. Bezdek and K. Bezdek [14]. In A. Bezdek and K. Bezdek [15] the problem is generalized and solved for the case in which the role of the inradius is played by the maximum positive coefficient of homothety of a given convex body contained in the slices.

15.2 Tarski’s Plank Problem

307

Ohmann [136] proved the following generalization of the planar case of Bang’s theorem: If a convex disk is covered by a finite family of convex disks, then the sum of the inradii of the covering disks is at least as large as the inradius of the covered domain. Theorem 1 in A. Bezdek [13] directly implies the same result. Kadets [104] extended this result to 𝑛 dimensions. Instead of the inradius Akopyan and Karasev [4] measured a convex body 𝐵 by the size 𝑟 𝐾 (𝐵) = sup{ℎ ≥ 0 : ℎ 𝐾 + 𝑡 ⊂ 𝐵} of the greatest positively homothetic copy of a given convex body 𝐾 contained in it. They proved that if in the plane 𝐶1 , . . . , 𝐶 𝑘 form a convex partition of the convex Í𝑘 𝑟 𝐾 (𝐶𝑖 ) ≥ 1. In higher dimensions they proved the analogous disk 𝐾, then 𝑖=1 statement for special partitions only. It should be mentioned that not all coverings can be reduced to a partition. The question of whether an analogous result holds for coverings remains open. A. Bezdek [12] made the following conjecture: For every convex disk there exists an 𝜀 > 0 such that the minimum total width of planks needed to cover the annulus obtained by removing from the disk its 𝜀-homothetic copy contained in its interior is the same√as for the whole disk. He verified the conjecture for the case of a square with 𝜀 = 1 − 2/2. He further supported the conjecture by proving that it is true for every polygon whose incircle is tangent to two of its parallel sides. It turned out that the conjecture stated in such generality is false, as White and Wisewell [169] noticed. In fact, they characterized all convex polygons for which Bezdek’s conjecture holds as the polygons with no minimum-width chord that meets a vertex and divides the angle at that vertex into two acute angles. Zhang and Ding [170] showed, with a very short proof, that the equilateral triangle with an arbitrarily small hole placed anywhere in its interior is a counterexample to Bezdek’s conjecture. Also, they gave a positive result for parallelograms. Smurov, Bogataya and Bogaty˘ı [158] proved that Bezdek’s conjecture holds for the cube of dimension 𝑛 ≥ 2, even with infinitely many cubical holes, each homothetic to the covered cube, if the hole’s total edge length is sufficiently small. L. Fejes Tóth [73] considered the following problem: Place 𝑘 great circles on a sphere so that the maximum inradius of the regions into which the circles partition the sphere is as small as possible. He conjectured that in the optimal arrangement the great circles dissect the sphere into the regular tiling {2, 2𝑘 } with congruent digonal faces. Rosta [154] confirmed the conjecture for 𝑘 = 3 and Linhart [122] proved it for 𝑘 = 4. A great circle of the unit sphere and a number 𝑟 > 0 define a zone of width 2𝑟 on the sphere, consisting of points that are at a distance at most 𝑟 from the great circle. A different way to state the same problem is: Find the smallest number 𝑟 𝑘 such that the sphere can be covered by 𝑘 zones of width 2𝑟 𝑘 . A zone of width 2𝑟 𝑘 has area sin 𝑟 𝑘 , thus 𝑤 𝑘 ≤ arcsin 1/𝑘. Fodor, Vígh and Zarnócz [75] gave an improvement of this trivial bound. This reformulation of the problem gives rise to the following more general conjecture: If the sphere is covered by a finite number of zones, then the total width of the zones is at least 𝜋. Jiang and Polyanskii [103] gave a short, elegant proof of this conjecture valid in all dimensions. As a corollary of their theorem Jiang and Polyanskii also proved that if a centrally symmetric convex body on the sphere is covered by zones of total width 𝑤, then it can be covered by one zone of width

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𝑤. Ortega-Moreno [137] gave an alternative proof for the special case of covering the sphere by zones of the same width. A short simplification of Ortega-Moreno’s proof was given by Zhao [171]. Another spherical version of Tarski’s plank problem, on covering the 𝑛dimensional spherical ball with a family of spherical convex bodies, was considered by K. Bezdek and Schneider [36]. In their theorem, the inradius of a set used for the covering plays the role of the width of a plank: If on the 𝑛-dimensional sphere the spherically convex bodies 𝐶1 , . . . , 𝐶𝑚 cover a spherical ball 𝐵 of radius 𝑟 (𝐵) ≥ 𝜋/2, then the sum of their inradii is greater than or equal to 𝑟 (𝐵). Steinerberger [159] gave lower bounds for the sum of the 𝑠-th power of the areas of pairwise intersections of 𝑛 congruent zones on 𝑆 2 . His bound is asymptotically sharp for 0 < 𝑠 < 1. A. Bezdek, Fodor, Vígh and Zarnócz [16] investigated the multiplicity of points covered by zones. They showed that it is possible to arrange 𝑛 congruent zones of suitable width on 𝑆 𝑑−1 such that no point belongs to more than a constant number of zones, where the constant depends only on the dimension and the width of the zones. They also proved that it is possible to cover 𝑆 𝑑−1 by 𝑛 equal zones such that each point of the sphere belongs to at most 𝑐 𝑑 ln 𝑛 zones. Concerning Tarski’s plank problem and its generalizations, we refer the reader to the book and survey by K. Bezdek [21, 24].

15.3 The Kneser–Poulsen Problem The following attractive problem was stated independently by Poulsen [142] and Kneser [109]: If in 𝑛-dimensional Euclidean space finitely many balls are rearranged so that no distance between their centers increases, then the volume of their union does not increase. The problem turned out to be more difficult than it appears, even in the plane. The first result supporting the conjecture was obtained by Habicht (see Kneser [109, p. 388]) and Bollobás [38], who proved it for congruent circular disks under the assumption that the rearrangement is the result of a continuous motion during which all distances between the disks’ centers change monotonically. Csikós [54] and, independently, Bern and Sahai [11] extended this result to arbitrary circular disks. Soon after he published this result, Csikós [55] generalized it to balls in every dimensions. Under the assumption of a continuous motion of the ball’s centers the monotonicity of the volume of the intersection also holds in spherical and hyperbolic space (see Csikós [56]). However, Csikós and Moussong [61] showed that in the 𝑛-dimensional elliptical (real projective) space the conjecture is false. Yet, in spite of the counterexample, a configuration of 𝑛 + 1 balls reaches maximum volume of their union if the distances between their centers become equal to 𝜋/2, the diameter of the space. In his proof, Csikós represents the moving configuration of 𝑁 balls of radii 𝑟 1 , . . . 𝑟 𝑁 , centered at 𝑃1 (𝑡), . . . , 𝑃 𝑁 (𝑡) in 𝐸 𝑛 (0 ≤ 𝑡 ≤ 1), by a single moving point 𝑃(𝑡) = (𝑃1 (𝑡), . . . , 𝑃 𝑁 (𝑡)) in the configuration space 𝑅 𝑛𝑁 and he assumes that,

15.3 The Kneser–Poulsen Problem

309

while the distances between 𝑃𝑖 (𝑡) and 𝑃 𝑗 (𝑡) do not increase for 𝑖, 𝑗 = 1, . . . , 𝑁, the function 𝑃(𝑡) is analytic. Then he derives a formula expressing the derivative of the volume of the union of the balls as a linear combination of the derivatives of the distances between their centers with nonnegative coefficients. This yields directly that the volume of the union does not increase. K. Bezdek and Connelly [28], suitably modifying Lemma 1 of Alexander [3, p. 664] and using the volume formula derived by Csikós [55], succeeded in confirming the planar case of the Kneser–Poulsen conjecture. The conjecture that under a contraction of the centers the volume of the intersection of a set of balls does not decrease was stated by Gromov [82] and Klee and Wagon [108, Problem 3.1.]. The special case of the conjecture for congruent circular disks and continuous motion was established by Capoyleas [50] before K. Bezdek and Connelly [28] proved it in full generality for the plane. For a compact set 𝑀 in a space of constant curvature, consider those balls 𝐵 ⊂ 𝑀 (of possibly zero radius) that are not a proper subset of any ball 𝐵 ′ ⊂ 𝑀. The set of centers of these balls is called the center set of 𝑀. Gorbovickis [80] invented a method dealing with Kneser–Poulsen-type problems based on the investigation of the properties of central sets. He proved that if on a plane of constant curvature the union of a finite set of (not necessarily congruent) closed circular disks has a simply connected interior, then the area of the union of these disks cannot increase after any contractive rearrangement. We emphasise the following corollary of the spherical case of this theorem: (i) If a finite set of circular disks on the sphere with radii not smaller than 𝜋/2 is rearranged so that the distance between each pair of centers does not increase, then the area of the union of the disks does not increase. (ii) If a finite set of disks on the sphere with radii not greater than 𝜋/2 is rearranged so that the distance between each pair of centers does not increase, then the area of the intersection of the disks does not decrease. For two dimensions, this generalizes the result of K. Bezdek and Connelly [29] in which they proved the analogous statement for 𝑛 dimensions but only for hemispheres. The limiting case of the application of claim (ii) when the radii approach zero yield an alternative proof of the corresponding theorem in the Euclidean plane. With a suitable adaptation of Gorbovickis’ method Csikós and Horváth [59] also proved the monotonicity of the area of the intersection of circular disks on the hyperbolic plane. Alexander [3] conjectured that, under an arbitrary contraction of the centers of finitely many congruent circles, the perimeter of the intersection of the circles does not decrease. The proof of this conjecture does not seem to lie within reach. K. Bezdek, Connelly and Csikós [31] settled some special cases of the conjecture. Among other cases, they proved it for four circles. The weaker result concerning the perimeter of the convex hull of the circles was proved by Sudakov [162], rediscovered by Alexander [3] and extended to the hyperbolic plane and to the hemisphere by Csikós and Horváth [59]. For the Euclidean case, a simpler proof

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was given by Capoyleas and Pach [51], who also established a similar result in the case where the Euclidean norm is replaced by the maximum norm. In dimensions higher than 2, for non-continuous contractions, there are only a few partial results concerning the Kneser–Poulsen conjecture. In 𝑛 dimensions, the conjecture was verified for 𝑛+1 balls by Gromov [82]. K. Bezdek and Connelly [28] extended Grovov’s result to at most 𝑛 + 3 balls. The conjecture is proved for special arrangements of balls, e.g. for a small number of intersections or large radii by [35] proved that none of the intrinsic Gorbovickis [78,79]. K. Bezdek and Naszódi √ volumes of the intersection of 𝑘 ≥ (1 + 2) 𝑑 congruent balls in 𝐸 𝑛 decreases under uniform contractions of the centers, that is under contractions where all the pairwise distances in the first set of points are larger than all the pairwise distances in the second set of points. K. Bezdek [25] gave an alternative proof of a slightly stronger theorem. Moreover, in [26] he proved that in a Minkowski space, under uniform contraction of the centers, the volume of both the intersection and the union of the balls changes monotonously. The Kneser-Poulsen conjecture makes sense in any connected Riemannian manifold. However, Csikós and Kunszenti-Kovács [60] proved that if the conjecture is true in a Riemannian manifold, then the manifold must be of constant curvature. Moreover, Csikós and Horváth [58] showed that the same consequence holds even for balls of the same radii. Further results and generalizations on this topic can be found in Csikós [56] and K. Bezdek and Connelly [30]. The articles by K. Bezdek [20] and Csikós [57] contain surveys on the subject of the Kneser–Poulsen conjecture.

15.4 Covering a Convex Body by Smaller Homothetic Copies Hadwiger [89], Levi [121] and Gohberg and Markus [77], independently of each other, asked for the smallest integer ℎ(𝐾) such that a given 𝑛-dimensional convex body 𝐾 can be covered by ℎ(𝐾) smaller positively homothetic copies of 𝐾. A boundary point 𝑥 of the convex body 𝐾 is illuminated from the direction of a unit vector 𝑢 if the ray issuing from 𝑥 in the direction of 𝑢 intersects the interior of 𝐾. Let 𝑖(𝐾) denote the minimum number of directions from which the boundary of 𝐾 can be illuminated. The problem of finding the maximum value of 𝑖(𝐾) was raised by Boltjanski˘ı [39] and in a slightly different, but for compact sets equivalent form, by Hadwiger [90]. Boltjanski˘ı observed that for convex bodies ℎ(𝐾) = 𝑖(𝐾). Both Boltjanski˘ı and Hadwiger conjectured that 𝑖(𝐾) ≤ 2𝑛 for every 𝑛-dimensional convex body 𝐾 and that equality holds only for parallelotopes.

15.4 Covering a Convex Body by Smaller Homothetic Copies

311

Both Levi [121] and Gohberg and Markus [77] verified the conjecture for the plane. Lassak [117] proved the sharp√result that every convex disk can be covered by four homothetic copies with ratio 2/2. An extreme example is the circle. The conjecture remains open for 𝑛 ≥ 3. In three dimensions it was proved for centrally symmetric convex bodies by Lassak [116], for convex bodies symmetric in a plane by Dekster [67], and for convex polyhedra with an arbitrary affine symmetry by K. Bezdek [17]. Zong [172] confirmed the conjecture for the unit ball of the 𝑙 𝑝 norm (1 ≤ 𝑝 ≤ ∞) and for cones. There is a great variety of results confirming the conjecture for special classes of bodies in 𝐸 𝑛 by establishing upper bounds for ℎ(𝐾) or 𝑖(𝐾) smaller than 2𝑛 . Schramm [156] proved that if 𝐾 is a set of constant width then 𝑖(𝐾) <   𝑛/2 √ . Schramm’s bound is less than 2𝑛 for 𝑛 ≥ 16 the cases 5𝑛 𝑛(4 + ln 𝑛) 32 of lower dimension were settled by K. Bezdek and Kiss [32] and Bondarenko, Prymak and Radchenko [47]. K. Bezdek [22, 23] extended Schramm’s inequality to a wider class of convex bodies, namely for those convex bodies 𝐾 that are the intersection of congruent balls with centers in 𝐾. Martini [125] proved the bound ℎ(𝐾) ≤ 3 · 2𝑛−2 for every zonotope other than a parallelotope. The bound ℎ(𝐾) ≤ 3 · 2𝑛−2 , 𝐾 not a parallelotope, was also verified for zonoids by Boltjanski˘ı and P.S. Soltan [45], and Boltjanski˘ı [40] extended it to an even wider class of convex bodies, the so-called belt bodies. Boltjanski˘ı and Martini [43] characterized those belt bodies 𝐾 for which ℎ(𝐾) = 3 · 2𝑛−2 . K. Bezdek and Bisztriczky [27] proved the conjecture for dual cyclic polytopes. For the dual 𝑃 of an 𝑛-dimensional cyclic polytope Talata [165] proved the inequality (𝑛+1) (𝑛+3)/4 ≤ ℎ(𝑃) ≤ (𝑛 + 1) 2 /2 and, for the case when 𝑛 is even, he gave the sharp bound ℎ(𝑃) ≤ (𝑛/2 + 1) 2 . Tikhomirov [167] considered convex bodies 𝐾 in 𝐸 𝑛 with the property that for any point (𝑥1 , . . . , 𝑥 𝑛 ) ∈ 𝐾, any choice of signs 𝜀1 , . . . , 𝜀 𝑛 ∈ {−1, 1} and any permutation 𝜎 on 𝑛 elements, 𝜀1 𝑥 𝜎1 , . . . , 𝜀 1 𝑥 𝜎𝑛 ∈ 𝐾. He proved that for sufficiently large 𝑛, we have 𝑖(𝐾) < 2𝑛 for every such convex body 𝐾 different from a cube. Smooth convex bodies in 𝐸 𝑛 can be illuminated by 𝑛 + 1 directions, see e.g. Boltjanski˘ı and Gohberg [41, Theorem 9, p. 61]. Dekster [65] extended this result by replacing the assumption of smoothness of the body by a certain smoothness of just a single belt of the body. Livshyts and Tikhomirov [123] proved that the cube represents a strict local maximum for these problems: If a convex body, that is not a parallelotope, is close to the cube in the Banach–Mazur metric, then it can be covered by 2𝑛 − 1 smaller homothetic copies of itself, and 2𝑛 −1 light sources suffice to illuminate its boundary. A simple consequence of the upper bound of Rogers for the translational covering density combined with the Rogers–Shephard inequality for the volume of
the difference body is that ℎ(𝐾) ≤ 2𝑛 (𝑛 ln 𝑛 + ln 𝑛 + 5𝑛) for every convex body 𝑛 and ℎ(𝐾) ≤ 2𝑛 (𝑛 ln 𝑛 + ln 𝑛 + 5𝑛) for centrally symmetric convex bodies in 𝐸 𝑛 (see Rogers and Zong [153]). An improvement by a sub-exponential factor was recently obtained by Huang, Slomka, Tkocz Vritsiou [97]. For general convex bod√
−𝑐and 𝑛 for some universal constant 𝑐 > 0. For low ies their bound is of the order of 2𝑛 𝑒 𝑛

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dimensions better bounds were given by Lassak [118] and Prymak and Shepelska [144]. In three dimensions Lassak [119] proved ℎ(𝐾) ≤ 20, which was improved by Papadoperakis [139] to ℎ(𝐾) ≤ 16 and subsequently by Prymak [143] to 14. A cap body of a ball is the convex hull of a closed ball 𝐵 and a countable set {𝑝 𝑖 } of points outside the ball such that for any pair of distinct points 𝑝 𝑖 , 𝑝 𝑗 the line segment 𝑝 𝑖 𝑝 𝑗 intersects 𝐵. The difficulty of the illumination problem is exposed by the following example of Naszódi [133]. Clearly, we have 𝑖(𝐵 𝑑 ) = 𝑑 + 1. On the other hand, for any 𝜀 > 0 there is a centrally symmetric cap body 𝐾 of 𝐵 𝑑 and a positive constant 𝑐 = 𝑐(𝜀) such that 𝐾 is 𝜀 close to 𝐵 𝑑 and 𝑖(𝐾) ≥ 𝑐 𝑑 . Ivanov and Strachan [100] studied the illumination number of cap bodies in 3 and 4 dimensions and proved that 𝑖(𝐾) ≤ 6 for centrally symmetric cap bodies of a ball in 𝐸 3 , and 𝑖(𝐾) ≤ 8 for unconditionally symmetric cap bodies of a ball in 𝐸 4 . Weighted illumination was introduced by Naszódi [132] and weighted covering was introduced by Artstein-Avidan and Raz [5] and Artstein-Avidan and Slomka [6]. A collection of weighted light sources illuminates a convex body 𝐾 if for every boundary point 𝑥 of 𝐾 the total weight of light sources illuminating 𝑥 is at least 1. Similarly, a collection of weighted bodies covers 𝐾 if for every point 𝑥 of 𝐾 the total weight of bodies containing 𝑥 is at least 1. The weighted or fractional illumination number 𝑖 ∗ (𝐾) of 𝐾 is the infimum of the total weight of light sources illuminating 𝐾. The weighted covering number ℎ∗ (𝐾) of 𝐾 is the infimum of the total weight of smaller weighted homothetic copies of 𝐾 covering 𝐾. Naszódi [132] conjectured that 𝑖 ∗ (𝐾) ≤ 2𝑛 for every 𝑛-dimensional convex body 𝐾. He proved the conjecture
for centrally symmetric bodies and established the inequality 𝑖 ∗ (𝐾) ≤ 2𝑛 𝑛 for general convex bodies. Artstein-Avidan and Slomka [6] proved that ℎ∗ (𝐾) ≤ 2𝑛 for convex bodies 𝐾 ⊂ 𝐸 𝑛 with equality only for a parallelotope centrally symmetric 2𝑛
∗ and ℎ (𝐾) ≤ 𝑛 for general convex bodies. They pointed out that the proof of the equivalence between the illumination problem and the Levi–Hadwiger covering problem carries over to the weighted setting. This way they gave an alternative proof of Naszódi’s result. K. Bezdek and Lángi [34] considered the illumination problem on the sphere. A boundary point 𝑞 of a convex body 𝐾 in 𝑆 𝑑 is illuminated from a point 𝑝 ∈ 𝑆 𝑛 \ 𝐾 if it is not antipodal to 𝑝, the spherical segment with endpoints 𝑝 and 𝑞 does not intersect the interior of 𝐾, but the great-circle through 𝑝 and 𝑞 does. The illumination number of 𝐾 is the smallest cardinality of a set that illuminates each boundary point of 𝐾 and lies on an (𝑛 − 1)-dimensional great-sphere of 𝑆 𝑛 which is disjoint from 𝐾. Bezdek and Lángi proved that the illumination number of every convex polytope in 𝑆 𝑛 is 𝑛 + 1 and raised the question whether there is a convex body in 𝑆 𝑛 whose illumination number is greater than 𝑛 + 1. The paper by Naszódi [134] surveys different problems about covering, among others the Hadwiger–Levi problem. For literature and further results concerning the illumination problem we refer to the surveys by K. Bezdek [18, 19], K. Bezdek and Khan [33], Boltjanski˘ı and Gohberg [42], Szabó [164], Martini and Soltan [127] and to the book by Boltjanski˘ı, Martini and P.S. Soltan [44].

Chapter 16

Miscellaneous Problems About Packing and Covering G. Fejes Tóth, L. Fejes Tóth and W. Kuperberg

16.1 Arranging Houses Suppose a large area is designated for a housing project in which the minimum distance between the congruent rectangular outlines of the houses is prescribed. Which arrangement of the rectangles allows for the greatest number of houses in the area? By the inequality 3.10.1, the problem is reduced to the determination of the densest lattice packing of the parallel domain of the rectangle. This problem was solved completely by L. Fejes Tóth [90] and Florian [113]. Let 𝑎 denote the length of the shorter side of the rectangle and 𝑑 the prescribed minimum distance between the houses. √ There are√three types of optimal depending on whether √ arrangement √ 𝑎/𝑑 ≤ 4 − 12, 4 − 12 < 𝑎/𝑑 < 2 − 2, or 2 − 2 ≤ 𝑎/𝑑. Let 𝐻 denote the minimum area√ hexagon circumscribed about the parallel domain of the rectangle. If 𝑎/𝑑 ≤ 4 − 12 then 𝐻 has bilateral symmetry √ about a line parallel to the longer side of the rectangle (see Figure 16.1). If 2 − 2 ≤ 𝑎/𝑑 then, besides a pair of sides parallel to the longer sides of the rectangle, 𝐻 also has parallel to √ a pair of sides √ the shorter sides of the rectangle (Figure 16.3). If 4 − 12 < 𝑎/𝑑 < 2 − 2 then 𝐻 has neither bilateral symmetry nor sides parallel to the shorter sides of the rectangle (Figure 16.2). 1010 000 111 10 000 111 000 111 000 111 000 111 000 111 000 111 1010 10101010 101010 101010 10 000 11110 000 111 000 111 000 111 000 111 000 111 000 111

0000 11111010 10110010 10101010 101010 10 000 11110 000 111 000 111 000 111 000 111 000 111 000 111 1100110 1010010100110 1011001011001001 10101011001010 10101 010 0000 1111

1010 000 11110 111 000 111 000 111 000 111 000 000 111 1010 101010 10101010 101010 1010 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 11001 100

0000 1111 110011100 100101101010 10010101001 1010101010101010 10110 010 0000 1111 1111 0000 1111 0000 1111 0000 1111 0000 10 0000 1111 10101010 101010101010 010 10101101010101010 10101010 10101010 0000 1111

10 10 0000 1111 1111 0000 1111 0000 1111 0000 1111 0000 0000 1111 1 1001 100101 101001 1001010 101 0 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 1010 10

000 111 10101 10010 1010110 1001010 101 0 0000 1111 1111 0000 1111 0000 1111 0000 1111 0000 10 0000 1111 10101010 1010101010 010 10101101010101010 10101010 10101010 000 111

1010 0000 111110 1111 0000 1111 0000 1111 0000 1111 0000 1010 0000 1111 101100 110011001 1001010 1010 0000 1111 1111 0000 1111 0000 1111 0000 1111 0000 0000 11111010 1100

Fig. 16.1

000 1111010 110010 1010101 01010 1100 0000 111110 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 1010 1010 01 000 111

000 111 000 111 000 111 000 111 000 111 000 111 000 111 1010 100110 10101010 101001 010 1 000 111 000 111 000 111 000 111 000 111 000 111 000 111 11001 100

101 0010 11 101 111111 000000 1010 00000 11111 0 0010 11 111111 000000 00000 11111 10 00 1001 111111 000000 1010 00000 11 11111 11111 00000 0 001100 11 111111 000000 00000 11111 11111 00000 11010 101 00 1001 10 00000 11 11111 11111 00000 010 00 11 010 111111 000000 1 11111 0 00000 1010 00 11 1011111 111111 000000 10111111 0000010 11111 000000 00 11 1100101000000 1010 00000 000000 11111 111111 1111111 00000 00 11111 00000 00000 10 10000000 11111 11111 000000 111111 00000 11111 10101010 111111 11001010 000000 1 100110011111 00000010 00000 111111 1010 000000 0 1 0 1 1 10101010 111111 0 1 000000 1010 111111 11111 00000 11001100 00000 11001100 11111 000000 111111 11111 00000 00000 11111 10 000000 1010 11111 00000 11001100 00000 101100 111111 11111 000000 10101010 111111 1010 11111 00000 00000 11111 000000 10 00000 1010101010 111111 11111 000 111 111111 000000 110010 111111010 000000 000 00000 111 1111111010 000000 000000 111111 000000 10101010 111111 111111 00000 10 111111 10 11111 101000000 1111110 101000000 00000 00000 11111 11111 000000 010 00000 11111 0 1 1 0 0 1 0 1 111111 11111 00000 111111 000 000000 00000 101100 111111 11111 000000 11001100 1001100 111 111111 000000 00000 11111 0 000000 1 111111 1 1 0 00000 101010 111111 11111 000000 101010 11111 00000 101100 111111 000000 1010 00000 11111 000000 101010 111111 1 11111 0 00000 0 000000 1 111111 000 111 000000 10101010 10 000000 10 11111 00000 000 111 1011010 111111 11111 00000 10101010 111111 000000 1100 111111 000 111 010 10010 0 000000 1 111111 00000 11111 000 111 1 0 1 10 111111 00000 000000 000 111 11111101010 00000 111111 00000 000000 1010 11111 110010 11001 11111 111111 0000000 0000000 11 0 1111111010 10 11111 000000 000000 111111 101010 111111 110 000000 00 11 00000010 111111 111111 000000 000 111 000000 1100 111111 0010 11 10101100 111111 000000 00 11 11111 00000 11001 000000 1010 111111 001010 111111 11 11111 00000 0 000000 111111 000000 101010 111111 000000 00 11 10 111111 11111 00000 11 000000 1010 111111 000000 1010 00 11111 00000 000000 111111 000000 10 0010 111111 11 111111 000000 1010 111111 000000 10 00 11 1010 1010 111 10 000 000 10 111 000 111 000 111 000 111 000 111 000 111

11111 00000 11111 00000 11111 00000 11111 00000

Fig. 16.2

000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 10 101010101 1001010 101010 101 1001 0

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10101010 1010 11010 01010101010 101010 101 0

0011 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 00 1011 00 1011 00 1011 00 1011 11 00 1011 00 1010 11 00 11 00 11 00 11 00

000000 00 111111 11 11 00 00 11 00 11 000000 111111 0010 11 000000 111111 00 11 000000 111111 001100 11 000000 111111 00 11 10 000000 111111 00 11 00 11 00 11 10101 01010 1111 0000 1111 0000 1011010 1111 0000 111111 000000 1111 0000 111111 000000 10010

11 00 00 11 00 11 00 101011 00 11 1011 00 1011 0010 00 11 0010 11 00 11 00 11

0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 011010 101 1001 0 1010 1100

00 11 00 11 00 11 00 11 00 11 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 11 00 011 00 11011 00 10011 00 111 00 1011 00 11 00 11 00 11 00 11 00

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0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11

11001 1001 100 10

000000 111111 111111 000000 000000 111111 00 10011 11 00 11011 00 11 00 101011 00 11 00 101 11 00 0 11 00 11 00 11 00

0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0110

1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 00 11 00 11 000 11011 00 11 1011 00 101011 00 00 11 011 00 11011 00 00 11 00 11

11111 00000 11111 00000 11111 00000 11111 00000 00 11 11 00 00 11 1011 00 100 10011 00 11 1011 00 00 11 00 11 00 11 00 11

0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 11 00 11 00 11 0010 11 00 10 11 00 11 000110 11 00 11 0001 11 00 11 00

00 11 11 00 00 11 00 11 001010 11 00 11 110010 10

111111 000000 111111 000000 111111 000000 111111 000000

Fig. 16.3

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2_16

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Figure 16.4 shows a portion of an imaginary town. The black circles represent cylindrical houses and the white ones landing platforms for helicopters. To every black circle a white one is assigned, tangent to it. Naturally, no black circle is allowed to overlap with another black circle or with a white one. But the white circles can overlap with each other in part or completely, so that several houses can share a common landing platform. All white circles are mutually congruent and so are the black ones, and their radii are prescribed. Under these conditions, the densest packing of the black circles is to be determined. Molnár [196, 198] and Jucovič [172] obtained a series of very nice arrangements of circles as the solution of this problem and of its variations. The same problem on the sphere was studied by Molnár [200]

Fig. 16.4

16.2 Packing Barrels

Fig. 16.5 Let us imagine that in a large area we want to place as many equal-sized barrels as possible, and so that each barrel can be moved away from the area without disturbing the other barrels. Thus the densest blocking-free packing of the plane with congruent circles is desired, where a packing is blocking-free if within the part of the plane not covered by the circles every circle can be moved arbitrarily far from its original position. G. Fejes Tóth, who raised this problem, stated the conjecture that in the best packing the circles are arranged in double rows, separated by aisles √ √𝜋 . Heppes [145] further (Figure 16.5). The density of this packing equals 5−1 2 12 generalized the problem, asking only that the barrels be 𝑟-accessible, meaning that a cellarer whose vertical shadow on the floor is a circle of radius 𝑟 can freely come

16.3 Covering with a Margin

315

into contact with every barrel. He conjectured that the densest 𝑟-accessible packing of unit circles consists of appropriately placed double-rows. He gave a density bound which brought him very close to the solution of the problem. The correctness of the conjecture was proved in a series of papers by G. Blind [31, 32, 33]. A simpler proof was given by G. Blind and R. Blind [36]. G. Blind [34] extended the result to the sphere. The analogous problem in space was considered by G. Blind and R. Blind [35].

16.3 Covering with a Margin A covering of the plane with unit circles has margin 𝜇 ∈ [0, 1] if the uncovered space created by the removal of any one of the circles can be covered by a circle of radius 1 − 𝜇. Obviously, a covering with a large margin cannot be too thin. Hence, it is natural to ask: Determine the minimum density of a covering with unit circles with margin 𝜇 and the covering that attains this density. A covering with margin 0 is a double covering, while margin 1 does not mean any additional restriction for the covering. Thus, this problem connects the problems of thinnest covering and thinnest double covering by unit circles. A. Bezdek and W. Kuperberg [18], who raised the problem about thinnest covering with a margin, solved it restricted to the special case when the arrangement is lattice-like. For 0 ≤ 𝜇 ≤ 𝜇1 = 0.56408 . . . the optimal arrangement is a triangular lattice, for 𝜇1 ≤ 𝜇 ≤ 𝜇2 = 0.78608 . . . the solution is a square lattice, and for 𝜇2 ≤ 𝜇 ≤ 1 the optimal lattice remains unchanged; it coincides with the thinnest lattice-like double covering by unit circles. Since the thinnest double covering by unit circles is not lattice-like, the solution of the problem is not lattice-like in general. However, it is conjectured that the solution is lattice-like for sufficiently small values of 𝜇. We note that for lattice arrangements the hole resulting by removing a circle is symmetric about the center of the removed circle, thus the smallest circle that can cover the hole will be centered here as well. In this case the problem can be interpreted as searching for the most economical distribution of transmitting towers over a large area, all towers having the same circular range, under the requirement that the region should be covered even if due to a partial power loss the range of radius of one of the towers is reduced by a factor of 1 − 𝜇. Heppes [150] considered a dual problem. A packing of unit circles has expendability 𝜀 > 0 if for every circle 𝐶 of the packing there is a circle of radius 1 + 𝜀 intersecting 𝐶 but not overlapping with any of the other circles. Thus, removing 𝐶 and replacing it with the larger circle still creates a packing. Heppes determined the densest lattice packing of unit circles with expendability 𝜀 for all 𝜀 > 0.

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16.4 Finite Packing and Covering in 2 Dimensions Nice, particular problems arise when trying to pack a given number of congruent circles of maximum radius in a specific region or to cover the region with congruent circles of minimum radius. Most often, the chosen container is the square, the circle, or the equilateral triangle. The extensive literature on this subject consists mainly of articles treating a single, specific case of the general problem, too numerous to be listed here. Among them are several algorithmic results that present some good arrangements, however not confirmed to be optimal. A major goal in this field is to find algorithms that give or approximate the optimal solutions. Thus far, the only algorithm of this type was constructed by Peikert, Würtz, Monagan and de Groot [213] (see also Peikert [212]) for finding the densest packing of congruent circles in a square. Figure 16.6 illustrates the optimal packing of up to 20 congruent circles in a square container.

Fig. 16.6 Consider a densest packing with a large number of translates of a convex disk in a large container, or under some other constraints that force finiteness. While it is expected that such a packing should be close to a cluster taken from the densest packing of the whole plane or space, it seldom is identical to such a cluster. Schürmann [225] proved that the solutions to several finite packing problems are

16.5 Finite Arrangements in Higher Dimensions

317

non-lattice if the number of the translates is sufficiently large. In particular, he proved this for the packings of circles that minimizes the diameter of their union, thereby confirming a conjecture of Erdős. For further literature on packings in bounded containers, see Szabó et al. [232] and Melissen [193].

16.5 Finite Arrangements in Higher Dimensions There are several results about packing congruent balls in various containers. Schaer [221, 222, 223, 224] considered the problem of densest packing of 𝑘 congruent balls in a cube, and solved it for 𝑘 ≤ 10. The notoriously difficult case of packing 14 that if 14 points congruent balls in a cube was settled by Joós in [164] by proving √ are placed in the unit cube, then two of the points are no more than 2/2 away from each other. The results of Schaer and Joós confirm some of the conjectures stated by Goldberg [127]. Golser [128] studied the problem of packing 𝑘 congruent balls in the regular octahedron and solved it for 𝑘 ≤ 7. Böröczky Jr. and Wintsche [49] generalized Golser’s result to higher dimensions. They proved that the maximum radius of 𝑘 ≤ 2𝑛 + 1 congruent balls packed in the regular 𝑛-dimensional cross-polytope does not depend on 𝑛, and for 4 ≤ 𝑘 ≤ 2𝑛 the radius is constant. K. Bezdek [20] solved the problem of packing 𝑘 congruent balls in a regular tetrahedron, for 𝑘 = 5, 8, 9 and 10. W. Kuperberg [184] considered the problem of finding the maximum radius of 𝑘 ≤ 2𝑛 + 2 𝑛-dimensional congruent balls packed in a spherical container. While for 𝑘 ≤ 𝑛 + 1 and 𝑘 = 2𝑛 + 2, the optimal configurations of balls are unique (see Davenport and Hajós [62] and Rankin [217]), W. Kuperberg described the structure of the non-unique configurations for 𝑛 + 2 ≤ 𝑘 ≤ 2𝑛 + 1, in which the radius remains constant. An alternative characterization of these configurations was given by Musin [203]. The problem of thinnest covering of the cube with 𝑘 congruent balls was solved by Joós [166, 167] for 𝑘 = 5 and 6. Joós [163, 165, 168] √︁proved that the minimum 5/12 in 4 dimensions and radius of 8 congruent balls that can cover the unit cube is √︁ 2/3 in 5 dimensions. The problem of covering the 𝑛-dimensional cross-polytope with 𝑘 congruent balls of minimum radius was studied by Böröczky, Jr., Fábián and Wintsche [47] who found the solution for 𝑘 = 2, 𝑛, and 2𝑛. Remarkably, the solution of the cases 𝑘 = 2 and 𝑘 = 𝑛 is substantially different for 𝑛 = 3 and 𝑛 ≠ 3. Another finite packing problem asks to arrange 𝑘 non-overlapping unit balls so that the convex hull of their union is of minimum volume. L. Fejes Tóth [94] conjectured that in dimension 𝑛 ≥ 5 the balls’ centers should be collinear, so that the convex hull of the union of the balls forms a “sausage-like” solid of length 2𝑘 (see Figure 16.7). The conjecture, known as the sausage conjecture, attracted great interest and generated intensive research on finite packing and covering (see Gritzmann and Wills [132] and Böröczky Jr. [42]). The conjecture was verified for 𝑛 ≥ 13, 387

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16 Miscellaneous Problems About Packing and Covering

Fig. 16.7 by Betke, Henk and Wills [10]. Betke and Henk [9] lowered the bound for the dimension to 𝑛 ≥ 42.

16.6 Slab, Cylinder, Torus What is the densest packing√ of unit circles in a strip of width 𝑤 ≥ 2? √ The answer is trivial for 2 ≤ 𝑤 ≤ 2 + 3 but it becomes difficult for 𝑤 √≥ 2 + 3. Extending √ a result by Kertész [179], who gave the √ solution for 2√+ 3 ≤ 𝑤 ≤ 2 + 2 2, Füredi [125] solved the problem for 2 + 3 ≤ 𝑤 ≤ 2 + 2 3. Molnár conjectured that the maximum density is   (𝑛 + 1) (𝑛 + 2)𝜋 𝑤−2 , » √ , 𝑛= √ 3 2𝑤(𝑛 + 1) + 2 1 − (𝑤 − 3) 2 √ and observed that for 𝑤 = 2 + 𝑛 3, 𝑛 = 1, 2, . . . , his conjecture follows from the theorem of Groemer [133] mentioned on page 200. Horváth and Molnár [158] studied the problem of densest packing of unit balls in a slab of space bounded by a pair of parallel planes. They showed, among other things, that such packings consisting of two hexagonal or square layers of balls are extremal in a slab of the corresponding width. Molnár [201] extended this result by finding the densest packing of unit balls in a slab of every width between 2 and √ 2 + 2. Horváth [155] investigated the problem of densest packing of unit balls in √ a 4-dimensional slab of width 2 < 𝑤 ≤ 2 + 2. Packing and covering with congruent circles on the surface of the infinite circular cylinder was considered by L. Fejes Tóth [85], Bleicher and L. Fejes Tóth [29] and Mughal and Weaire [202]. L. Fejes Tóth [93] gave an upper bound for the number of points with given minimal distance on the surface of polyhedra. The papers by Dickinson, Guillot, Keaton and Xhumari [67, 68], Connelly and Dickinson [55], Connelly, Shen, and Smith [58], Connelly, Funkhouser, V. Kuperberg and Solomonides [56], Heppes [147], Musin, Nikitenko [204], Przeworski [216] and Brandt, Dickinson, Ellsworth, Kenkel and Smith [50] investigate packings of circles on the torus. Joós and Nagy [170] determined the smallest upper bound for the radius of 𝑘 ≤ 4 congruent balls packed in the 3dimensional cubical flat torus. The dual problem of the thinnest covering of the torus by congruent circles was treated by Joós [169].

16.7 Close Packings and Loose Coverings

319

16.7 Close Packings and Loose Coverings L. Fejes Tóth [96] defined another measure of efficiency, alternate to density. He considered the supremum of the radii of the circles that can be placed in the complement of the packing. The smaller that number is, the more close, or efficient, is the packing. The closeness of the packing is measured by the inverse of this supremum, and a close packing is one with largest possible closeness. Looseness of a covering is similarly determined by the inverse of the supremum of the radii of circles that can be placed in the intersection of two members of the covering, and a loose covering is one with largest possible looseness. L. Fejes Tóth [97] proved that the closeness of a packing by translates of a convex disk 𝐾 cannot exceed the closeness of the closest lattice packing of 𝐾. He also remarked that this remains true if positively-homothetic copies of another convex disk instead of a circle are used to measure closeness. This is a result analogous to the corresponding theorem about density. On the other hand, he produced a centrally symmetric convex disk and a packing consisting of translates of the disk and a rotated copy of it, with closeness greater than that of the closest lattice packing. An alternative example was constructed by A. Bezdek [11]. Linhart [189] observed that the natural way of measuring closeness of a packing and looseness of a covering with translates of a convex disk 𝐾 is by using the largest negatively-homothetic copy of 𝐾 instead of a circle. Then the problems of close packing and loose covering become equivalent. He proved that, measuring closeness and looseness in this way, the triangle is the worst for both close packing and loose covering, with closeness 2 and looseness 3. This result corresponds to the theorems of Fáry concerning the “worst case” for densest packing and thinnest covering with translates of a convex disk, where again the triangle is the worst one in both cases. Linhart also conjectured that the worst case among centrally symmetric convex disks is the affine-regular octagon. Specifically, he conjectured that every centrally symmetric convex disk can pack the plane by translates with closeness√ at √ least 3 + 2 2, and can cover the plane by translates with looseness at least 4 + 2 2. Zong [244] considered the problem of closest packing, with closeness measured the Linhart way, though phrased in a slightly different manner, and he confirmed Linhart’s conjecture about the affine regular octagon. The relation between Linhart’s approach to closeness and Zong’s so-called simultaneous packing and covering constant is as follows: For a given convex disk 𝐾, let 𝑐(𝐾) denote the minimum closeness (in Linhart’s sense) of a packing with translates of 𝐾, and let 𝛾(𝐾) denote Zong’s minimum homothety coefficient for a transition from a packing to a covering with translates of 𝐾. Then 𝛾(𝐾) = 1 + 𝑐(𝐾) −1 . Similarly to his simultaneous packing and covering constant Zong [243] introduced the simultaneous lattice packing and covering constant 𝛾 ∗ (𝐾) and he proved that 𝛾 ∗ (𝐾) ≤ 7/4 for every three-dimensional convex body 𝐾. Confirming a conjecture of L. Fejes Tóth [96], Böröczky [40] proved that the closest packing with unit balls in space is unique and is obtained by placing the centers of the balls √ at the vertices and at the centers of all cubes of a cubic lattice of edge-length 4/ 3. Since for balls the problems of closest packing and loosest

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16 Miscellaneous Problems About Packing and Covering

covering are equivalent, Böröczky’s solution settles both. Böröczky [41] defined edge-closeness of a packing of congruent balls as the supremum of the distance between a line of an edge of some Dirichlet cell and the center of the corresponding √︁ ball. He showed that the minimum edge-closeness of a packing of unit balls is 3/2. Again, the unique optimal arrangement is the body-centered cubic lattice. Using techniques of Delone and Ryškov [66] and Ryškov and Baranovski˘ı [219, 220], Horváth [156, 157] solved the problem of the closest lattice packing with unit balls in dimension 4 and 5. Schürmann and Vallentin [226] designed an algorithm for approximating the loosest lattice sphere covering with arbitrary accuracy. In 6-dimensional space, the algorithm produced the best known lattice for loose sphere covering.

16.8 Arranging Regular Tetrahedra Since no integer multiple of the dihedral angle 𝜑 = arccos(1/3) = 1.23 . . . at the edges of the regular tetrahedron 𝑇 equals 2𝜋 (5𝜑 = 6.15 . . . is just slightly smaller than 2𝜋), we know that space cannot be tiled with congruent copies of 𝑇, hence 𝛿(𝑇) < 1. Then, as Hilbert [152] asked, how densely can space be packed with congruent regular tetrahedra? The question is also of interest in areas other than mathematics, e.g., physics (compacting loose particles), chemistry (material design), etc. The past few years brought an exciting development: A series of articles appeared, each providing a surprisingly dense—denser than any previously known—packing. Conway and Torquato [59] presented a packing with density 0.717455 . . . , which is almost twice the lattice packing density of the tetrahedron. After improvements by Chen [53], Haji-Akbari, Engel, Keys, Zheng, Petschek, Palffy-Muhoray and Glotzer [136], Kallus, Elser and Gravel [175], and Torquato and Jiao [235], a packing of the currently highest known density, namely 4000/4671 = 0.856347 . . . , was obtained by Chen, Engel and Glotzer [54]. While it was known for a long time that the value of 𝛿(𝑇) must be smaller than 1, no explicit non-trivial (i.e., strictly below 1) upper bound for the packing density of the regular tetrahedron was presented until Gravel, Elser and Kallus [131] gave such a bound, about 1 − 10−24 . They also gave a similar upper bound for the packing density of the regular octahedron. The gap between the density of the best known packing and the upper bound remains quite wide, and it may be very hard to narrow it down substantially. On the other hand, there is hope for the determination of the translational packing density of tetrahedra, for which Zong [245] suggested a computer approach. The papers by Lagarias and Zong [185] and Ziegler [240] survey the history of packing regular tetrahedra. The thinnest known covering by regular tetrahedra constructed by Conway and Torquato [59] has density 9/8. Fiduccia, Forcade and Zito [111] and independently Dougherty and Faber [70] found a body 𝑇84 that admits a lattice tiling of 𝐸 3 and is inscribed in a tetrahedron 𝑇 of volume vol(𝑇) = 125 63 vol(𝑇84 ). It follows that

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321

125 𝜗𝐿 (𝑇) ≤ 125 63 . Forcade and Lamoreaux [122] conjectured that 𝜗𝐿 (𝑇) = 63 and support the conjecture by proving that the lattice corresponding to the tiling by copies of 𝑇84 represents a local minimum of the density. Fu, Xue and Zong [124] proved 25 and also showed that the lattice covering density of the four-dimensional 𝜗𝐿 (𝑇) ≥ 18 343 . There is no non-trivial lower bound for the covering density simplex is at least 264 of the regular tetrahedron, but Xue and Zong [239] established such a bound for lattice arrangements: The lattice covering density of a simplex 𝑆 𝑛 in 𝐸 𝑛 satisfies

𝜗𝐿 (𝑆 𝑛 ) ≥ 1 +

1 . 23𝑛+7

16.9 Packing Cylinders The first known non-tiling 3-dimensional convex solid to have its packing density determined was the infinite circular cylinder 𝐶, that is, the Minkowski sum 𝐵 + 𝐿 where 𝐵 is a circle and √ 𝐿 is a line, see A. Bezdek and W. Kuperberg [16]. As expected, 𝛿(𝐶) = 𝜋/ 12, the maximum density being attained when all cylinders are parallel. Concerning packings of unit cylinders in which no two are parallel, A. Bezdek and W. Kuperberg [17] showed that for every such packing, the complement of the packing contains a ball of radius 𝑟 > 𝜌 = √2 − 1. In other words, the closeness (see 3 Section 16.7) of such a packing is greater than 1/𝜌. They also proved that√︁ every point 𝑝 lying in the complement of the packing is within a distance of at most 2/3 from the center of such a ball. Moreover, every ball of radius 𝜌 not intersecting any of the cylinders can be moved continuously from its original position to the position of any other such ball, while avoiding every cylinder during the motion. It appeared that if no two of the cylinders in a packing are parallel, then the density of the packing should be rather low, perhaps even zero. However, K. Kuperberg [183] constructed such a packing with positive density. Graf and Paukowitsch [130] improved the construction, reaching density 5/12. By a further improvement Ismailescu and Laskawiec [162] reached density 1/2. Moreover, they constructed packings of congruent cylinders with no two cylinders parallel to each other √ whose local density in a ball of sufficiently large radius is arbitrarily close to 𝜋/ 12.

16.10 Obstructing Light H. Hornich posed the question of how many material unit balls (meaning closed balls with mutually disjoint interiors) are needed to radially shield one such ball, in the sense that every ray emanating from the center of the shielded ball must meet a shielding one. The set of shielding balls is called a cloud. Let 𝐻 (𝑟) denote the smallest number of unit balls in a cloud for a ball of radius 𝑟. As a simple consequence

16 Miscellaneous Problems About Packing and Covering

322

of (5.1.2) (page 116), L. Fejes Tóth [83] showed the inequality 𝐻 (𝑟) >

12𝛼 , 6𝛼 − 𝜋

𝜋 1+𝑟 𝜋 < 𝛼 = arctan √ < , 2 6 2 6𝑟 + 3𝑟

which yields the lower bound 𝐻 (1) ≥ 19. The bound was subsequently improved to 𝐻 (1) ≥ 24 by Heppes [146] and then raised again by Csóka [60] to 𝐻 (1) ≥ 30. By a suitable construction, Danzer [61] showed that 𝐻 (1) ≤ 42. L. Fejes Tóth [83] showed that a point, and therefore also a sufficiently small ball, can be shielded by six unit balls. Grünbaum [135] proved that five balls do not suffice. This can be interpreted as the equality 𝐻 (0) = 6. Besides 𝑟 = 0, the value √ of 𝐻 (𝑟) is not known for any 𝑟. However, the above inequality yields 𝐻 (𝑟) > 8𝜋𝑟 2 / 27. For large values of 𝑟 the estimate is asymptotically exact. The notion of a cloud for a ball can be extended in various ways. We can consider clouds for balls of infinite radius, that is half-spaces. A cloud for a half-space is a packing in the complement of the half-space such that every line perpendicular to the bounding plane intersects a member of the packing. Of course, a cloud for a half-space contains infinitely many balls, so in this case, we are looking for the width of the cloud, that is for the minimal width of a slab containing the cloud. A packing disjoint from a set 𝑆 that intersects all rays issuing from the boundary of 𝑆 in the complement of 𝑆 is a dark cloud for 𝑆. If each such ray intersects the interior of a member of the cloud then the cloud is called deep. Finally, if the corresponding rays intersect at least 𝑘 members of the cloud, then we have a 𝑘-fold cloud.

Fig. 16.8 Consider 𝑘 consecutive rows of the densest√lattice packing of unit circles. These circles are contained in a strip of width (𝑘 − 1) 3 − 2 and form a 𝑘-fold cloud for the half-plane parallel to the rows. Heppes [144] proved that this is the narrowest 𝑘-fold cloud of unit circles shielding a half-plane. An alternative proof of this statement is due to Hajós [137]. Figure 16.8 shows the narrowest 4-fold cloud of circles. It is not difficult to show (L. Fejes Tóth [83]) that the√width of any cloud of unit balls shielding a half-space in 𝐸 3 must be at least 2 + 2. Equality holds only when the cloud consists of two square-pattern layers in contact with each other, so that each ball in one layer touches exactly four balls in the other layer. If we √ stack 𝑘 horizontal clouds of this kind, we obtain a 𝑘-fold cloud of width (2𝑘 − 1) 2 + 2. Heppes [144] improved this for every 𝑘 > 1 by constructing a 𝑘-fold cloud of unit   𝑘−1   √ balls of width 𝑘 + 3 3 + 2. Jucovič [171] gave bounds for the number of circles in a 𝑘-fold cloud for a point.

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323

For a convex body 𝐾, let 𝐶 (𝐾) and 𝐶𝑇 (𝐾) denote the minimum cardinality of dark clouds for 𝐾 consisting of congruent copies of 𝐾 and of translates of 𝐾, respectively. The papers by Böröczky and Soltan [44], Zong [241, 242], Talata [234] contain bounds for 𝐶𝑇 (𝐾) for convex bodies 𝐾 in 𝐸 𝑛 . Zong [241] proved that 𝐶𝑇 (𝐾) ≥ 2𝑛 with equality only for parallelotopes. Finding upper bounds for 𝐶𝑇 (𝐾) is related to the following problem: Given 𝜀 > 0, find the minimum number 𝑠(𝑛, 𝜀) such that for some packing of unit balls in 𝑛 dimensions, every line segment of length greater than 𝑠(𝑛, 𝜀) must be closer than 𝜀 to one of the centers of the balls. The bounds for 𝑠(𝑛, 𝜀) given by Henk and Zong [141] were improved by Böröczky, Jr. and Tardos [48], whose result implies that 2 2 𝐶𝑇 (𝐾) ≤ 3𝑛 +𝑜(𝑛 ) for every convex body 𝐾 ∈ 𝐸 𝑛 and 𝐶𝑇 (𝐾) ≤ 2𝑛

2 +𝑜(𝑛2 )

for every centrally symmetric convex body 𝐾 ∈ 𝐸 𝑛 . Szabó and Ujváry-Menyhárt [233] proved that the minimum cardinality of a deep cloud for a convex disk is at most 9 with equality only for the circle. Heppes [143] observed that a lattice packing of balls in 𝐸 + 3 is always penetrable by lines in three linearly independent directions, implying that no lattice packing of balls can form a dark cloud for a half-space. This shows that the existence of a dark cloud of congruent balls is not so trivial. Nevertheless, Böröczky [39] succeeded in constructing such a cloud of a relatively small width. Take four consecutive hexagonal-pattern layers of balls, 𝑠0 , 𝑠1 , 𝑠2 , 𝑠3 from the densest lattice packing of balls. Böröczky proved that these layers, adjoined by the mirror images 𝑠−1 , 𝑠−2 , 𝑠−3 of 𝑠1 , 𝑠2 , 𝑠3 reflected in the middle plane of 𝑠0 , form a dark cloud, and he asserted without proof that the same is true for 𝑠−2 , 𝑠−1 , 𝑠0 , 𝑠1 , 𝑠2 already. The aforementioned theorem of Heppes was strengthened by Hortobágyi [153], who proved that every lattice packing of unit balls can be penetrated in√three lin3 2 early independent directions by a cylindrical beam of light of radius −1 = 4 0.06065 . . . . As the example of the densest lattice packing of balls shows, this constant cannot be replaced with a larger one. This last statement is quite non-trivial, but it follows from Theorem II of Bambah and Woods [5], see also Makai and Martini [192], where a gap in the proof of this theorem was filled. The observation of Heppes inspired further research. Hausel [138], Henk and Zong [141], Henk, Ziegler and Zong [140] and Henk [139] estimated the greatest number 𝑘 (𝑛) with the property that for every lattice packing of the 𝑛-dimensional ball there exists a “free 𝑘-dimensional plane”, that is, a 𝑘-dimensional plane contained √ in the complement of the packing. Hausel [138] proved that 𝑘 (𝑛) ≤ 𝑛 − 𝑐 𝑛, for some constant 𝑐 > 0. The best lower bound up to date, given by Henk [139], is 𝑛/log2 𝑛 ≤ 𝑘 (𝑛), for 𝑛 sufficiently large. Horváth and Ryškov [159, 160] estimated the maximum radius of a cylinder around a line that can be inserted in the void of

324

16 Miscellaneous Problems About Packing and Covering

every lattice packing of the 𝑛-dimensional unit ball, and conjectured that their result is sharp for 𝑛 = 4. Their conjecture was refuted by Makai and Martini [192]. A related problem, posed by G. Fejes Tóth [74], concerns the thinnest lattice arrangement of balls that intersects every 𝑘-dimensional plane. For 𝑘 = 0 the problem is about the thinnest lattice covering, hence it is solved for 𝑛 ≤ 5. For 𝑛 = 2, 𝑘 = 1, the problem was solved by L. Fejes Tóth and Makai [105] by proving √ that the density of a lattice packing of circles intersecting every line is at least 3𝜋/8. For 𝑘 = 𝑛 − 1 the problem turned out to be equivalent to finding the densest lattice packing of balls. This is a consequence of the following result of Makai [191], also found independently by Kannan and Lovász [176]: Let 𝜌(𝐾) denote the infimum of the density of a lattice arrangement of a convex body 𝐾, such that every hyperplane b denote the polar body intersects one of the members of the arrangement, and let 𝐾 1 of 2 (𝐾 − 𝐾). Then   b . b = vol(𝐾)vol 1 𝐾 𝜌(𝐾)𝛿 𝐿 ( 𝐾) 4 This solves the problem for balls in dimensions 𝑛 ≤ 8 and 𝑛 = 24. For 0 < 𝑘 < 𝑛 − 1 only the case 𝑘 = 1, 𝑛 = 3 has been solved: Bambah and Woods [5] showed that the thinnest lattice arrangement of balls intersecting every line arises √ from the densest lattice packing by enlarging the balls’ radius by a factor of 3 2/𝜋. Kannan and Lovász [176] and González Merino and Schymura [129] investigated the case 0 < 𝑘 < 𝑛 − 1 further.

16.11 Avoiding Obstacles If convex disks are packed in a parallel strip of the plane, we call it a layer of disks. Let 𝑤 be the width of the strip and 𝑙 be the length of a path that connects the two edges of the strip without penetrating any of the disks. (L. Fejes Tóth [87]) defined the permeability 𝑝 of the layer as 𝑝 = 𝑤/inf 𝑙. √ He showed that for the permeability 𝑝 of a layer of congruent circles 𝑝 > 27/2𝜋 = 0.82699 . . . holds. For layers consisting of a large number of rows from the densest lattice packing of circles, 𝑝 comes arbitrarily close to this lower bound. For incongruent circles the value of inf 𝑝 is not known. However, by a rather complicated construc√ tion in the same article a layer of incongruent circles with 𝑝 = 0.82322 . . . < 2𝜋/ 27 was found. The situation is quite different if one considers layers of squares instead of circles. For layers consisting of congruent squares, inf 𝑝 = 2/3 holds. But the same holds even for layers of squares of arbitrary sizes and orientations. Moreover, it was shown by L. Fejes Tóth [91] that to any layer of similar copies of a parallelogram 𝑃 there is a layer consisting of translates of a replica of 𝑃 with the same permeability. It is an interesting question to consider which convex disks share this property with

16.11 Avoiding Obstacles

325

the parallelogram and which behave like the circle. Florian and Groemer [119] showed that for every 𝑚 ≥ 39, regular 𝑚-gons belong to the latter group, that is, there exists a layer of homothetic copies of them whose permeability is smaller than the infimum of the permeability of all layers of congruent regular 𝑚-gons. The above mentioned results of L. Fejes Tóth were sharpened by Bollobás [37] and Florian [114], Bollobás determining the infimum of the permeability of a layer of given width of similar copies of a given parallelogram and Florian determining the infimum of the permeability of a layer of given width of unit circles. Hortobágyi [154] proved√that the permeability of a layer of translates of a disk of constant width is at least 27/2𝜋. Florian [115] proved that the permeability of a layer of translates of a regular hexagon, and also of a layer of translates of a regular triangle, is at least 3/4. Subsequently, Florian [116] observed that these results are special cases of a general theorem. Namely, the infimum of the permeability of layers by translates of a convex disk 𝐾 equals the infimum of the permeability of layers by translates of the difference body 𝐾 − 𝐾. A survey on permeability can be found in Florian [117]. L. Fejes Tóth [98] also raised the following, natural problem. Given a packing of the plane with convex “obstacles” and two points not in the interior of any of the obstacles, find or estimate the greatest possible size of a necessary detour caused by the obstacles in traveling from one point to the other. The first result in this direction was given by Pach [209], who proved that for any packing of square obstacles of sides at most 1, any pair of points at distance 𝑑 outside √ the squares can be connected by an obstacle-avoiding path of length at most 23 𝑑 + 𝑑 + 1. G. Fejes Tóth [75] √ improved this bound to (3𝑑 + 1)/2 and proved the bound (2𝜋/ 27)(𝑑 − 2) + 𝜋 for unit circular obstacles. Both of these bounds are sharp for infinitely many values of 𝑑. The shortest path problem for balls was treated in G. Fejes Tóth [77]. It turned out that in 𝐸 𝑛 , even for a packing of balls with arbitrary but bounded radii, where the obstacles’ density might be 1, we need not make a detour greater than 𝑂 (𝑑/𝑛) (𝑑 → ∞) in order to connect two points lying at distance 𝑑 outside the balls by a path avoiding the balls. For a packing of congruent balls in 𝐸 𝑛 the detour we have to make approaches zero exponentially with the dimension. Algorithmic aspects of problems of this type have been studied by Papadimitriou and Yannakakis [210, 211], Chan and Lam [52], and A. Bezdek [14]. A. Bezdek also obtained a bound for the length of the shortest path in space with cubical obstacles. An interesting variation of the obstacle-avoiding path problem was considered by L. Fejes Tóth [101]: From a point outside the obstacles, one tries to escape in any direction to a distance 𝑑 away from the point. He proved that for any set of convex, open obstacles an escape path exists of length at most 𝑑 2 + 21 ln 𝑑 + 𝑐, and that some   sets of obstacles require the length of 𝑑 2 − 𝜋3 + 𝜋3 𝑑 + 𝑐. A dual problem arises when we consider a family of convex disks covering the plane and we wish to travel only within the part of the plane covered at least twice. G. Fejes Tóth [77] stated the following conjecture. If the plane is covered by a

326

16 Miscellaneous Problems About Packing and Covering

family of unit circles, then for any two points, each covered at least twice, there is a path contained in the multiply covered region, connecting one point with the other, √ and of length at most 𝑑 2 + 𝑐, where 𝑑 is the distance between the points and 𝑐 is a constant. Baggett and A. Bezdek [2] confirmed the conjecture in the case when the circles form a lattice covering. Roldán-Pensado [218] showed that two points at distance 𝑑 apart lying in the multiply covered part of the plane can be connected by a path that remains√in the part of the plane covered at least twice and whose length is at most (𝜋/3 + 3)𝑑 + 𝑐 for some constant 𝑐 < 17. A. Bezdek and Yuan [19] investigated a related problem, where they measured the length of the path by the number of passings from one circle into another.

16.12 Stability We say that a packing of (not necessarily congruent) circles is stable if every circle is immobilized by the others, that is, if in every circle the central angle 𝜆 based on the largest “free” arc, that is, containing no contact point, is smaller than 𝜋. The instability of the packing is defined as Λ = sup 𝜆, and its stability as 𝜋 − Λ. We wish to find a thinnest packing among those with prescribed stability. The following theorem of L. Fejes Tóth [84] is related to this problem:

Fig. 16.9

16.12 Stability

327

Given a stable packing of the plane with circles whose radii have a positive lower bound and a finite upper bound. If 𝑑 is the density and Λ is the instability of the packing, then 𝜋 𝑑≥ , 𝑛 = ⌊2𝜋/Λ⌋. Λ 𝑛 tan 2 + tan 2 𝜋−𝑛Λ 2 This bound is reached in the case of congruent circles whose centers form the vertices of one of the tilings {3, 6}, {4, 4}, {6, 3}, (4, 8, 8), or (3, 12, 12). The corresponding instabilities are Λ = 𝜋/3, 2𝜋/3, 3𝜋/4 and 5𝜋/6. Surprisingly, Böröczky [38] succeeded in constructing stable packings of density 𝑑 = 0 consisting of congruent circles (Figure 16.9). His example shows that the above inequality is sharp even in the limiting case Λ = 𝜋. Kahle [174] modified Böröczky’s construction to obtain thin stable packings of congruent circles in rectangles and regular hexagons. Dominyák [69] gave bounds for the instability of stable circle packings in the spherical and hyperbolic plane. 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11111111111111111111 00000000000000000000 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 0000000000000000000 1111111111111111111 0000000000000000000 1111111111111111111 0000000000000000000 1111111111111111111

Fig. 16.10

Fig. 16.11

De Bruijn [64,65] proved that the members of a packing of starlike sets in 𝐸 𝑛 can be moved apart arbitrarily far from each other, each translated continuously without overlapping with any of the others (Figure 16.10), unless each of the sets’ star-center is unique, and they all coincide in the packing (Figure 16.11). He also proved that in a finite packing of convex disks, for every direction, there is a disk that can be translated in that direction arbitrarily far from its original position without intersecting other members of the packing. L. Fejes Tóth and Heppes [103] independently discovered the same results, and the result about starlike sets was also noticed by Dawson [63]. In contrast to the case of the plane, in space there are finite packings of convex bodies such that no single member can move rigidly without disturbing the others. The example given by L. Fejes Tóth and Heppes consists of 12 tetrahedra packed around a rhombic dodecahedron. They conjectured that the tetrahedra alone, without the central dodecahedron, have the stated property. This was confirmed by Snoeyink and Stolfi [228]. Shephard [227] constructed a packing of twelve centrally symmetric polyhedra with the same property.

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16 Miscellaneous Problems About Packing and Covering

By the theorem of De Bruijn, the members of every finite packing of convex bodies can be moved arbitrarily far without disturbing the others by simultaneous translations. Natarajan [207] conjectured that the members of such a packing can even be separated by translations with two hands that is, a proper subset of the packing exists that can be translated to infinity by applying a common translation to them without disturbing the members in the complement. This conjecture turned out to be false: Snoeyink and Stolfi [228] gave a counterexample of 6 bodies and showed that a packing of at most 5 convex bodies can indeed be separated with two hands.

16.13 Minkowskian Arrangements A Minkowskian arrangement of similar copies of a centrally symmetric convex disk was defined by L. Fejes Tóth [86] as an arrangement in which no member contains the center of another√one. He proved that the density of such an arrangement of circles cannot exceed 2𝜋/ 3. A densest Minkowskian arrangement of circles consists of congruent circles, and is obtained by replacing every circle in a densest packing of congruent circles by a concentric one, with twice as large radius. This result is an extensive generalization of inequality (3.2.1). In fact, L. Fejes Tóth [86] proved that if finitely many circles form a Minkowskian arrangement then the density of the √ circles in their union cannot exceed 2𝜋/ 3. In a subsequent paper [89] he gave an upper bound for the total area of such an arrangement, which is sharp in many cases. Molnár [197] extended that result to the sphere and to the hyperbolic plane.

Fig. 16.12 The problem of the densest Minkowskian arrangement of circles was generalized by L. Fejes Tóth [88] as follows. Let 𝜇 be a positive number smaller than 1. We consider a set of circles 𝑐 1 , 𝑐 2 . . . of radii 𝑟 1 , 𝑟 2 . . .. With each circle 𝑐 𝑖 we associate a concentric circle with radius 𝜇𝑟 𝑖 , and we call it the kernel of 𝑐 𝑖 . In a generalized Minkowskian arrangement of circles of order 𝜇 none of the circles is

16.14 Saturated Arrangements

329

allowed to √ overlap the kernel of another. L. Fejes Tóth [88] conjectured that for 𝜇 ≤ 𝜇¯ = 3 − 1 the densest arrangement consists of congruent circles, and each of them touches six kernels (Figure 16.12). In this conjecture 𝜇¯ denotes the greatest value of 𝜇 under which this particular arrangement is a covering. Molnár [199] and Florian [112] gave density bounds under some condition for the homogeneity of the arrangement. Böröczky and Szabó [45] proved the conjecture in full generality. Kadlicskó and Lángi [173] gave a sharp bound for the total area of the √ circles in a finite generalized Minkowskian arrangement of circles of order 𝜇 ≤ 3 − 1. For the density of Minkowskian arrangement of balls of order 𝜇 ≤ 𝑛1 in 𝐸 𝑛 Böröczky 2

2 𝑑𝑛 and Szabó [46] proved the upper bound (1+𝜇) 𝑛 . Here 𝑑 𝑛 denotes the simplex bound defined in 10.2. Minkowskian arrangements of circles on the sphere were treated by L. Fejes Tóth [102] who gave an upper bound for the total area of 𝑛 spherical caps in a Minkowskian arrangement, sharp for 𝑛 = 3, 4, 6 and 12. The centers of the circles in the optimal arrangement form an equilateral triangle inscribed in a great circle, a regular tetrahedron, a regular octahedron and a regular icosahedron inscribed in the sphere, respectively. Also, the bound is asymptotically sharp for large 𝑛. The density of a Minkowskian arrangement of homothetic copies of a centrally symmetric convex body in 𝐸 𝑛 is at most 2𝑛 . On the other hand, allowing similar copies, there is no universal upper bound for the density. This was noticed already by L. Fejes Tóth [86], who also observed that in order to achieve high density, the members of the arrangement must occur in many different orientations: A Minkowskian arrangement of similar bodies in 𝐸 𝑛 with at most 𝑚 distinct orientations can have density at most 𝑚2𝑛 . Bleicher and Osborn [30] showed that there are Minkowskian arrangements in 𝐸 𝑛 even of congruent copies in at most 𝑚 distinct orientations with densities arbitrarily close to 𝑚2𝑛 . What is the maximum number 𝑀 (𝑛) of pairwise intersecting homothetic copies of a centrally symmetric convex body forming a Minkowskian arrangement in 𝐸 𝑛 ? The example of the cube shows that 3𝑛 ≤ 𝑀 (𝑛). Füredi and Loeb [126], who first considered this problem, proved that 𝑀 (𝑛) ≤ 5𝑛 . This upper bound was subsequently improved by Naszódi, Pach and Swanepoel [205] to 𝑂 (3𝑛 ln 𝑛), by Polyanskii [215] to 3𝑛+1 , by Naszódi and Swanepoel [206] to 2 · 3𝑛 . For translated copies of a centrally symmetric convex bodyFöldvári [121] proved the sharp bound 3𝑛 .

16.14 Saturated Arrangements Minkowskian arrangements of circles are in a certain sense dual to saturated collections of circles. Let S be a collection of closed circular disks and let 𝑟 > 0 be the infimum of their radii. We say that S is saturated if the part of the plane not covered by the circles contains no circle of radius 𝑟. It was conjectured by L. Fejes Tóth [88] that√the density of a saturated collection of circles is always greater than or equal to 𝜋/ 108. The conjecture was confirmed by Eggleston [73] for families of mutually non-overlapping circles, and proved in general by Bambah and Woods [4]. Thus, a

330

16 Miscellaneous Problems About Packing and Covering

thinnest saturated arrangement of circles arises by replacing the circles in a thinnest covering of the plane by concentric circles of half size. This is a generalization of the inequalities (3.2.2) and (3.2.5). A corresponding theorem for a packing of homothetic copies of a centrally symmetric convex disk in place of circles was proved by Bambah and Woods [3]. Dumir and Khassa [71] strengthened the above result for circles, and in [72] for arbitrary centrally symmetric convex disks as follows: No saturated arrangement of homothetic copies of a centrally symmetric convex disk 𝐾 can cover a smaller portion of the plane than 𝜗(𝐾)/4. For the density of a saturated arrangement of homothetic copies of a (not necessary symmetric) convex disk 𝐾, Khassa [181] proved the upper bound area(𝐾)/𝑡 (𝐾), where 𝑡 (𝐾) denotes the area of the largest triangle contained in 𝐾. For the density of saturated packings of balls in 3 dimensions Khassa [182] established the upper bound 3/32. Recall from Chapter 10 that a packing with congruent copies of a set 𝐾 is 𝑘saturated, if deleting 𝑘 − 1 members of the packing never creates a void large enough to pack in it 𝑘 copies of 𝐾. It is natural to ask for the infimum Δ 𝑘 (𝐾) of the densities of all 𝑘-saturated packings with replicas of 𝐾. G. Fejes Tóth, G. Kuperberg and W. Kuperberg [79] proved the asymptotic bound Δ 𝑘 (𝐾) ≥ 𝛿(𝐾) − 𝑂 (𝑘 −1/𝑛 ) for every body 𝐾 in 𝐸 𝑛 . The determination of these quantities is difficult even for 𝐾 = 𝐵2 . The only result in this direction is due to Heppes [149], who determined the infimum of the densities of 2-saturated √ lattice √ packings of circular disks, supporting the conjecture that Δ2 (𝐵2 ) = 𝜋(3 − 5)/ 27 = 0.461873 . . . . Another notion of higher-order saturation was studied by L. Fejes Tóth and Heppes [104] and A. Bezdek [13]. Here, we formulate the concept only for the special case of packings of congruent circular disks. A packing of disks of radius 𝑟 is saturated of order 𝑘 if every disk of radius 𝑟 intersects at least 𝑘 members of it. Saturation of order 1 means just saturation, and it is easily seen that the order of saturation of a packing of congruent circles is at most 3. L. Fejes Tóth and Heppes proved √ that the density of an order 3 saturated packing of congruent circles is at least 𝜋/(2 − 3), and A. Bezdek proved √ that the density of an order 2 saturated packing of congruent circles is at least 𝜋/( 27). The thinnest order 3 saturated packing arises by placing the centers in the vertices of a tiling (3, 3, 4, 3, 4), and a thinnest order 2 saturated packing consists of the face-incircles of the tiling {3, 6}.

16.15 Compact Packings A packing of the plane is said to be compact if each member 𝐾 of the packing satisfies the following three conditions: (i) 𝐾 has a finite number of neighbors; (ii) all neighbors of 𝐾 can be ordered cyclically so that each of them touches its successor; (iii) the union of the neighbors of 𝐾 contains a polygon enclosing 𝐾.

16.15 Compact Packings

331

L. Fejes Tóth [100] originated the study of compact packings by proving that if a compact packing of the√plane with circular disks has positive homogeneity then its density is at least 𝜋/ 12. Further, if a compact packing of the plane with homothetic centrally symmetric convex disks has positive homogeneity, then its density is at least 3/4, where equality occurs only for packings with affine regular hexagons. A. Bezdek, K. Bezdek and Böröczky [15] proved that if a compact packing of the plane with positively homothetic copies of a convex disk has positive homogeneity, then its density is at least 1/2, and equality occurs for various packings with homothetic triangles. The only compact packing of congruent circular disks is the hexagonal lattice. Kennedy [178] considered compact packings of circular disks of two different radii, 1 and 𝑟 < 1, and proved that there are only nine values of 𝑟 for which such compact packings exist. He also described all packing configurations in the nine cases. Messerschmidt [194] proved the upper bound 13617 for the number of pairs (𝑟, 𝑠) that allow a compact packing by disks of radii 1, 𝑟 and 𝑠 (𝑟 < 𝑠 < 1). In fact, there are much fewer such pairs: Fernique, Hashemi and Sizova [109] enumerated all 164 compact packings consisting of three different sizes of circular disks. Messerschmidt [195] proved that for every 𝑛 there exist only finitely many tuples (𝑟 1 , . . . , 𝑟 𝑛 ) with 0 < 𝑟 1 < . . . < 𝑟 𝑛 = 1 that can occur as the radii of the disks in any compact packing of the plane with 𝑛 distinct sizes of disk. It can be expected that compact packings of circles are the densest among all packings with the given radii. It was conjectured by Connelly, Gortler, Solomonides, and Yampolskaya [57] that if disks with 𝑛 different radii allow a saturated compact packing in which at least one disk of each size appears, then the maximal density over all the packings by disks with these 𝑛 radii is reached for a compact packing. The hypothesis of saturation is necessary for 𝑛 ≥ 3. The conjecture was proved for some cases, including all of the nine pairs of radii allowing compact packings of disks with two different sizes by Heppes [148, 151], Kennedy [177], Bédaride and Fernique [7] and Fernique [106], however Fernique and Pchelina [110] showed that the density of one of the compact circle packings with three different radii is not maximal. Florian [118] considered compact packings with circular disks on the sphere and in the hyperbolic plane. Compact packings in higher dimensions were investigated by K. Bezdek [21] and K. Bezdek and Connelly [22]. A packing in 𝐸 𝑛 is compact if each member 𝐴 of the packing is enclosed by its neighbors in the sense that any curve connecting a point of 𝐴 with a point sufficiently far from 𝐴 intersects the closure of a neighbor of 𝐴. K. Bezdek and Connelly [22] proved that the density of a compact packing in 𝐸 𝑛 consisting of homothetic centrally symmetric convex bodies with bounded homogeneity is at least (𝑛 + 1)/2𝑛, and there is a compact lattice packing of centrally symmetric convex bodies where equality holds. Fernique [108] proposed a different generalization of the concept of compact packings. The contact graph of a compact packing of circles, i.e., the graph that connects the centers of adjacent circles, is a triangulation. By analogy, Fernique calls a packing of balls in 𝐸 𝑛 compact if its contact graph is the 1-skeleton of a

332

16 Miscellaneous Problems About Packing and Covering

face-to-face tiling by simplices. Since regular tetrahedra do not tile the space, there is no compact packing of congruent balls. Fernique [108] determined the unique compact packing with two different sizes of balls and in [107] he described all the four compact packings with balls of three different sizes.

16.16 Totally Separable Packings A packing of convex bodies is totally separable if each pair of the bodies can be separated by a hyperplane not intersecting the interior of any of the bodies. Given a convex disk, what is the maximum density of a totally separable packing with its congruent copies? G. Fejes Tóth and L. Fejes Tóth [78] proved that the density of an arbitrary totally separable packing with congruent copies of a convex disk cannot exceed the ratio between the area of the disk and the minimum area of a quadrilateral containing the disk. If the disk is centrally symmetric, then that ratio is actually the maximum density of such a packing. Namely, the minimum area of the quadrilateral containing a centrally symmetric disk can always be attained by a parallelogram, and congruent parallelograms admit a totally separable tiling. In particular, this yields that the density of a completely separable packing with congruent circles cannot exceed 𝜋/4. A. Bezdek [12] proved this density bound under an assumption weaker than total separability, requiring only the packing to satisfy the following local separability condition: For every triple of circles there is a line separating one of them from the other two. He also showed that, in general, for non-circular disks local separability does not imply the similar parallelogram bound. K. Bezdek and Lángi [23] proved an analogue of Oler’s inequality for totally separable packings of translates of a convex disk. Vermes [237] investigated totally separable tilings and totally separable packings of circles in the hyperbolic plane. The interesting problem of maximum density of a totally separable packing of (not necessarily congruent) circular disks remains open. One can rephrase this problem in the following way: Start with a configuration of lines partitioning the plane into bounded regions and place a circle in each of the regions. What is the maximum density of a circle packing so obtained? In this phrasing, the analogous question about the minimum density of the covering by the circumcircles of the resulting cells can be asked. It is conjectured that the configuration of lines providing each of these extreme densities partitions the plane into the Archimedean tiling (3, 6, 3, 6) (see Figure 16.13). G. Fejes Tóth [76] proved that neither of the extreme densities in question are equal to 1. K. Bezdek and Lángi [24] and Vásárhelyi [236] investigated totally separable packings and coverings of circles on the sphere. The problem of the densest totally separable packing of 𝐸 3 with congruent balls was solved by Kertész [180], who proved that if a cube of volume 𝑉 contains 𝑁 unit balls forming a totally separable packing, then 𝑉 ≥ 8𝑁. Consequently, the cubic lattice packing is the densest one among all totally separable packings of 𝐸 3 with congruent balls, and the maximum density is 𝜋/6.

16.18 Connected Arrangements

333

Fig. 16.13

16.17 Point-Trapping Lattices

An arrangement of sets is point-trapping if every component of the complement of the union of the sets is bounded. It is natural to ask: What is the minimum density of a point-trapping lattice arrangement of any 𝑛-dimensional convex body? Confirming the chessboard conjecture of L. Fejes Tóth [95], Böröczky, Bárány, Makai and Pach [43] proved that the minimum is 1/2, attained in the “chessboard” lattice arrangement of cubes. The problem about the minimum density of a point-trapping lattice of 𝐾 can be posed for any specific convex body 𝐾. Bleicher [28] proved that the minimum density of a point-trapping lattice arrangement of three-dimensional unit balls is equal » √ 1.1104 . . . , and the extreme lattice is generated by to 128𝜋/3 7142 + 1802 17 =» √ 1 three vectors, each of length 2 7 + 17 = 1.6676 . . . and each two forming an angle of arccos

√ 17−1 8

= 67.021◦ . . . .

16.18 Connected Arrangements An arrangement of sets is said to be connected if the union of the sets is connected. The problem of the minimum density 𝑐(𝐾) of a connected lattice arrangement of an 𝑛-dimensional convex body 𝐾 has been explored by Groemer [134], who proved the inequalities 1 𝜋 𝑛/2 ≤ 𝑐(𝐾) ≤ 𝑛 𝑛 . 𝑛! 2 Γ( 2 + 1) Both inequalities are sharp. The value 𝑐(𝐾) = 1/𝑛! is attained when 𝐾 is a simplex or a cross-polytope, and the other extreme value of 𝑐(𝐾) is attained when 𝐾 is a ball. Groemer characterized those centrally symmetric bodies 𝐾 for which 𝑐(𝐾) = 1/𝑛!. Extending Groemer’s investigation, L. Fejes Tóth [92] characterized all 𝑛-dimensional convex bodies 𝐾 for which the inequality 𝑐(𝐾) ≥ 1/𝑛! turns into

334

16 Miscellaneous Problems About Packing and Covering

equality: They are the topological isomorphs of the regular cross-polytope and their limiting polytopes.

16.19 Points on the Sphere In Chapter Í10 we mentioned results about problems of finding the optima of sums of the form 𝑖≠ 𝑗 𝑓 (|𝑥𝑖 − 𝑥 𝑗 |) for a given number 𝑁 of points {𝑥1 , . . . , 𝑥 𝑁 } on the unit sphere. We continue to survey problems of this type. We start with two problems of this kind, of interest from the geometric point of view, and remarkably easy to solve for every value of 𝑁 ≥ 2. Distribute points 𝑥1 , . . . , 𝑥 𝑁 on 𝑆 2 so that: Í 1. the sum (|𝑥𝑖 − 𝑥 𝑗 | 2 is as large as possible;   Í 1 2. the sum 𝑥d 𝑖 𝑥 𝑗 = 2 arcsin 2 |𝑥 𝑖 − 𝑥 𝑗 | is the 𝑖 𝑥 𝑗 is as large as possible, where 𝑥d spherical distance between 𝑥𝑖 and 𝑥 𝑗 . Í Concerning the first problem, we have (|𝑥 𝑖 − 𝑥 𝑗 | 2 ≤ 𝑁 2 and equality occurs only if the vectors from the sphere’s center to the points 𝑃𝑖 are in equilibrium (see L. Fejes Tóth [80]). For the solution of the second problem, the parity of 𝑁 plays a role. If 𝑁 = 2𝑘, Í 2 then 𝑥d of points 𝑖 𝑥 𝑗 ≤ 𝜋𝑘 , and equality occurs precisely when the configuration Í is symmetric about the sphere’s center. For 𝑁 = 2𝑘 + 1 we have 𝑥d 𝑖 𝑥 𝑗 ≤ 𝜋𝑘 (𝑘 + 1). Equality occurs here precisely when all points without an antipodal partner are distributed on a great circle in such a way that two open semicircles determined by any point contain the same number of points. The case 𝑁 = 4 was solved by Frostman [123], the cases 𝑁 = 5 and 6 by L. Fejes Tóth [82], the cases 𝑁 = 2𝑘 (𝑘 = 1, 2, . . .) by Sperling [229], and the general case by Nielsen [208] (see also Larcher [186]). L. Fejes Tóth [82] also considered the corresponding problem in the elliptic plane, that is, maximizing the sum of the non-obtuse angles formed by 𝑁 lines. He conjectured that the maximum of the angle sum is attained when the lines are evenly distributed between the three coordinate axis in which case the sum of the angles is 2 asymptotically 𝑁6 𝜋 = 𝑁 2 · 0.523 . . . . He solved the problem for 𝑁 ≤ 6 and proved the upper bound 𝑘 ( 𝑁5−1) 𝜋 on the sum of the angles of 𝑁 lines. Fodor, Vígh and Zarnócz [120] gave an improvement of this bound that is asymptotically equal to 3𝑁 2 𝜋 = 𝑁 2 · 0.589 . . . as 𝑁 → ∞. Bilyk and Matzke [27] further improved the 16 69 )𝑁 2 = 𝑁 2 · 0.555 . . . and also gave a bound for the corresponding bound to ( 𝜋4 − 300 sum in higher dimensions. Lim and McCann [187, 188] asked for the maximum of the sum of the 𝛼-th power of the angles between the lines and proved the existence of constants 𝛼𝑑 > 1 such that in 𝐸 𝑑 for 𝛼 > 𝛼𝑑 the maximum of the sum of the 𝛼-th power of the angles between the line is attained when the lines are evenly distributed between the coordinate axis.

16.20 Arrangements of Great Circles

335

Alexander [1] considered another problem of this kind: Find the configurations of 𝑁 points on the 𝑛-dimensional unit sphere for which the sum of all distances between the points attains its maximum 𝑆(𝑁, 𝑛). Through an elegant integral-averaging technique he obtained the bounds √ 2 2 1 2 𝑁 − 10 𝑁 ≤ 𝑆(𝑁, 2) ≤ 𝑁 2 − . 3 3 2 Stolarsky [230] extended Alexander’s result for powers of the distances between the points and generalized it to all dimensions. In [231] Stolarsky proved a remarkable invariance theorem stating that the sum of all the distances between the points of a given set of 𝑁 points on the 𝑛-dimensional sphere plus the discrepancy of the set is a constant independent from the distribution of the points. Using this result, Stolarsky gave a sharper bound for 𝑆(𝑁, 𝑛). If 𝑐 0 (𝑛) denotes the average distance from a variable point to a fixed one on the surface of the sphere, then there is a constant 𝑐 1 (𝑛) such that 𝑆(𝑁, 𝑛) < 𝑐 0 (𝑛)𝑁 2 − 𝑐 1 (𝑛)𝑁 1−1/𝑛 . Beck [6] showed exactness of the constant 𝑐 0 by proving that 𝑐 0 (𝑛)𝑁 2 − 𝑐 2 (𝑛)𝑁 1−1/𝑛 < 𝑆(𝑁, 𝑛) with a suitable constant 𝑐 2 (𝑛). Alternative proofs, generalizations and applications to problems of energy optimization of Stolarsky’s invariance theorem were given by Brauchart and Dick [51], Bilyk, Dai and Matzke [26] and Bilyk and Dai [25]. In a similar vein, Witsenhausen [238] stated the following problem: Under the constraint that the diameter of the set of points 𝑥1 , 𝑥2 , . . . , 𝑥 𝑘 in 𝑛-dimensional space Í must be smaller than 1, find their configuration that maximizes the sum |𝑥𝑖 − 𝑥 𝑗 | 2 . 𝑖, 𝑗

Witsenhausen conjectured that the maximum, denoted by 𝑀 (𝑛, 𝑁), is attained when the points are distributed among the 𝑛 + 1 vertices of a regular simplex of edge-length 1 and supported the conjecture by proving the inequality 𝑀 (𝑛, 𝑁) ≤ 𝑁 2 𝑛/(𝑛 + 1). The conjecture was confirmed for 𝑛 = 2 by Pillichshammer [214] and in arbitrary dimensions by Benassi and Malagoli [8].

16.20 Arrangements of Great Circles Maehara [190] considered the following problem: Arrange 𝑘 great circles so that the maximum spherical distance between a point of the sphere and the nearest crossing point of the great circles is as small as possible. He solved the problem for 𝑘 = 3 and 4. The optimal arrangement in these cases occur when each circle is divided into 2(𝑘 − 1) equal arcs by the other 𝑘 − 1 great circles.

336

16 Miscellaneous Problems About Packing and Covering

L. Fejes Tóth [81] showed that the longest edge of a spherical tiling formed by four or five great circles takes its minimum for the Archimedean tiling (3, 4, 3, 4) and (3, 5, 3, 5), respectively. Heppes [142] studied a related problem: He showed that the area of the face with the smallest area of a tiling formed by four great circles takes its maximum for the cuboctahedron (3, 4, 3, 4). A conjecture of L. Fejes Tóth [99] about arrangements of great circles on a sphere states: If 𝑘 great circles are in general position (no three of them have a common point), then the ratio of the greatest among the areas of the regions into which the circles partition the sphere to the smallest one tends to infinity as 𝑘 → ∞. In the same article a lower bound is given: For sufficiently large 𝑘 the ratio is greater than 7.43. Motivated by this conjecture, Ismailescu [161] considered an analogous problem on the plane and proved the following: Consider an arrangement of 𝑘 lines such that no three are concurrent and all intersection points lie inside a unit circular disk. Then among the 1 + 𝑘 (𝑘 − 1) bounded cells of the subdivision of the plane by the lines, 𝜋 there is one whose area is at least 4𝑘 . As a corollary, it follows that the ratio 𝑞 of the greatest area to the smallest area of cells is at least (𝑘 + 1)/8. The question of whether the ratio 𝑞 tends to infinity as 𝑘 → ∞ if we do not restrict the points of intersections to lie in a circle, remains open.

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Name Index

Aberth, O., 227, 229 Abrosimov, N.V., 229, 230 Abu-Affash, A. K., 215 Ádám, I., 30 Agarwal, P.K., 218 Akimova, I.Ya., 205 Akiyama, S., 226 Akopyan, A., 192, 214, 306, 307 Alexander, R., 223, 306, 309, 335 Alexandrov, A.D., 29, 30, 193 Alon, N., 216, 285, 288, 289 Alsina, C., 191 Ambartzumian, R.V., 197 Ambrus, G., 239, 290, 296 Amenta, N., 216 Andreanov, A., 259 Andreev, N.N., 256–258 Anstreicher, K.M., 232, 238, 283 Archimedes, xv, 159 Arias-de-Reyna, J., 287 Artstein-Avidan, S., 312 Aste, T., 242 Atariah, D., 228 Atiyah, M., 232 Atzema, E.J., 193 Audet, Ch., 189, 201, 244–247 Averkov, G., 289 Avez, A., 191 Babenko, V.F., 205

Bacher, R., 252 Bachoc, Ch., 232, 233, 256, 284 Baggett, D.R., 326 Balázs, J., 217 Bálint, V., 295 Bálintová, A., 295 Balitski, A., 214 Ball K., 287 Ball, K.M., 249, 306 Ball, W.W. Rouse, 110 Balla, I., 235 Ballinger, B., 258 Bambah, R.P., 96, 185, 207, 210–212, 240, 323, 324, 329, 330 Bandelt, H.J., 288 Bandle, C., 191 Bang, Th., 98, 155, 306 Bankoff, L., 191 Bannai, E., 283 Baranovski˘ı, E.P., 218, 237, 240, 253, 259, 320 Bárány, I., 201, 216, 261, 267, 281, 333 Barlow, W., xv Barnes, E.S., 240 Baron, H.J., 30 Barrow, D.F., 191 Bateman, P., 96, 201, 214, 215 Beardwood, J., 217

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2

421

422

Beck, J., 335 Bédaride, N., 331 Beltrami, E., 155 Benassi, C., 335 Bencze, M., 222 Berger, S.B., 229 Berkes, J., 222 Berman, J.D., 224 Bern, M., 308 Berstein, F., 143 Besicovitch, A.S., 97, 113, 196, 218, 227, 228 Best, M.R., 261 Betke, U., 223, 241, 318 Bezdek, A., 191, 200, 201, 204, 211, 233, 237, 240, 242, 244, 246, 254, 260, 261, 267, 279, 284, 290, 294, 295, 306–308, 315, 319, 321, 325, 326, 330–332 Bezdek, K., 190, 191, 214, 230, 233, 239, 240, 242, 267, 279, 280, 284, 286, 288, 294, 295, 306, 308–312, 317, 331, 332 Bianchi, G., 197 Bieberbach, L., 243 Bieri, H., 245 Bilyk, D., 334, 335 Bingane, Ch., 245, 247 Biniaz, A., 215 Bisztriczky, T., 311 Blachman, N.M., 274 Blaschke, W., 29–31, 42, 47, 54, 55, 98, 109, 113, 151, 243 Blåsjö, V., 191 Bleicher, M.N., 207, 318, 329, 333 Blekherman, G., 258 Blichfeldt, H.F., 95, 174, 185, 236, 252–254, 256, 259, 260, 273, 274 Blind, G., 203, 204, 207, 217, 315 Blind, R., 203, 207, 217, 315 Blinovsky, V., 274 Bloh, È.L., 252

Name Index

Blundon, W.J., 272 Boerdijk, A.H., 169, 177, 185 Bogataya, S.I., 307 Bogaty˘ı, 307 Bognár, M., 306 Böhm, J., 193, 226 Boju, V., 279 Bol, G., 29, 136 Bolle, U., 271, 272, 274, 275 Bollobás, B., 205, 236, 325 Boltjanski˘ı, W.G., 193, 218, 306, 310–312 Bolton, R., 205 Bolyai, J., 155 Bondarenko, A., 305, 311 Bongiovanni, E., 242 Bonnesen, T., 29 Borchardt, C.W., 229 Böröczky, K., 203, 209, 225, 226, 230, 232–234, 237, 240, 251, 253, 263, 264, 269, 277, 282, 283, 285, 319, 320, 323, 327, 329, 331, 333 Böröczky, K. Jr., 189, 201, 205, 209, 218, 222, 223, 225, 226, 230, 242, 253, 317, 323 Borsuk, K., 303–305 Bose, P., 215 Bottema, O., 192 Bourgain, J., 287, 305 Bourne, D.P., 205 Bowen, L., 268–270, 279 Boyvalenkov, P., 256 Bozóki, S., 290 Brabant, H., 191 Brandt, M., 318 Branická, M., 295 Brass, P., 189, 218, 279, 286, 289, 291, 295, 296 Brauchart, J.S., 335 Bravais, A., xv Brøndsted, A., 193 Bronshte˘ın, E.M., 228 Brouwer, A., 305

Name Index

Brubach, B., 298 Brückner, M., 21, 30 Brunn, H., 29 Buchta, Ch., 207 Burago, Yu.D., 191, 193 Butler, G.J., 251 Cantor, G., 40 Capoyleas, V., 309, 310 Carmi, P., 215 Casselman, B., 239 Cassels, J.W.S., 242 Cauchy, A.-L., 32, 35, 36 Chabauty, C., 251 Chaikin, P.M., 262 Chakerian, G.D., 195, 295, 300, 301, 303 Chalk, J.H.H., 241 Chan, K., 325 Chan, S.H., 275 Chen, E.R., 320 Cheong, O., 278, 279 Chepoi, V., 288 Choi, M., 290 Chung, F.R.K., 217 Chvátal, V., 251, 280 Clinton, J.D., 223 Cohn, H., 236, 242, 256–259 Cohn, M.J., 274 Cohn-Vossen, S., 185 Connelly, R., 203, 267, 286, 309, 310, 318, 331 Conway, J.H., 235, 236, 242, 262, 306, 320 Courant, R., 113, 218 Coxeter, H.S.M., 30, 110, 174, 185, 192, 227, 240, 252, 253, 262, 264 Croft, H.T., 218, 228, 306 Crofton, M.W., 53 Csikós, B., 190, 205, 308–310 Csirik, J., 302 Csóka, G., 322 Culpin, D., 245 Dai, F., 335

423

Danzer, L., 215, 216, 232, 271, 287, 322 Daróczy-Kiss, E., 239 Datta, B., 244 Dauenhauer, M.H., 237 Davenport, H., 185, 186, 306, 317 Dawson, R., 327 de Bruijn, N.G., 327, 328 Debrunner, H., 193, 215, 216 Degen, W., 192 de Groot, C., 316 Dekster, B.V., 201, 289, 304, 305, 311 de Laat, D., 257, 284 De Loera, J.A., 216 Delone, B.N., 199, 259, 320 Delsarte, P., 255, 284 Demar, R.F., 191 de Oliveira Filho, F.M., 213, 260 Dergiades, N., 191 Diaz, A., 242 Dick, J., 335 Dickinson, W., 318 Ding, R., 298, 307 Dinghas, A., 193 Dirichlet, G.L., 29, 198 Djordjević, R.Ž., 192 Dodunekov, S., 256 Doheny, K.R., 219 Dolbilin, N.P., 267 Dolmatov, S.L., 277 Dol’nikov, D.L., 304 Dominyák, I., 327 Donahue, J., 229 Donev, A., 262 Dorninger, D., 204 Dostert, M., 241, 260, 283, 284, 286 Dougherty, R., 320 Dowker, C.H., 37, 54, 88, 194, 195 Dragan, S., 215 Dragnev, P.D., 258 Dräxler, F., 235 Dumer, I., 251 Dumir, V.C., 211, 272, 330 Dumitrescu, A., 190, 215, 296, 300

424

Dupin, C., 150, 152 Eckhoff, J., 216 Edelsbrunner, H., 217 Eggleston, H.G., 193–196, 227, 229, 303, 329 Elkies, N.D., 250, 256 Ellsworth, A.V., 318 Elser, V., 320 Engel, M., 320 Engelstein, M., 226 Ennola, V., 218 Eppstein, D., 286 Erdős, P., 15, 30, 96, 122, 191, 201, 214–216, 242, 246, 250, 279, 287, 288, 304, 305, 317 Erickson, J., 288 Euler, L., 17, 18, 162 Faber, V., 320 Fábián, I., 317 Falconer, K.J., 218, 304, 306 Fáry, I., 102, 113, 218, 319 Fedorov, E.S., xv Fejér, L., 30 Fejes Tóth, G., 194, 195, 197, 201–211, 214, 218, 226, 232, 234, 242, 250, 254, 260, 262, 266, 268, 271, 272, 274, 280, 284, 285, 314, 324, 325, 330, 332 Fejes Tóth, L., 30, 189–192, 194, 195, 197, 199, 200, 203, 204, 206–211, 213, 216, 218, 223–227, 230–235, 242, 252, 253, 260, 266–268, 274, 277, 278, 280–286, 295, 300, 307, 313, 317–319, 322, 324–334, 336 Fekete, S.P., 298–300 Fenchel, W., 29, 98, 193, 306 Ferguson, S.P., 238, 239 Fernique, T., 331, 332

Name Index

Few, L., 216, 217, 240, 253, 262, 273, 274 Fiduccia, C.M., 320 Firsching, M., 291 Flatley, L., 283 Florian, A., xiii, 194–197, 203–206, 222–224, 228, 253, 254, 264, 266, 274, 282, 313, 325, 329, 331 Florian, H., 224, 282 Fodor, F., 194, 201, 215, 239, 244, 246, 260, 307, 308, 334 Folkman, J.H., 210 Földvári, V., 288, 329 Föppl, L., 153 Forcade, R.W., 320, 321 Foster, J., 245 Fourier, J.-B.J., 40, 54 Fournier, X., 201 Frankl, N., 274, 289 Frettlöh, D., 192 Freudenthal, H., 193 Frostman, O., 334 Fruchard, A., 227 Fu, M., 299, 321 Funar, L., 279 Funkhouser, M., 318 Füredi, Z., 233, 250, 281, 287, 298, 299, 318, 329 Gács, P., 282 Gale, D., 303 Galiev, Sh.I., 274 Gallagher, W.G.., 226, 242 Gallai, T., 97, 215, 216 Gardner, M., 219 Gashkov, S., 244–246 Gáspár, Zs., 233 Gaudio, J., 217 Gauss, C.F., 149, 236, 237, 240, 241 Gelfand, I.M., 197 Gericke, H., 191, 223 Gersho, A., 205 Ghang, W., 226, 242 Ghosh, M.N., 217

Name Index

Giansiracusa, N., 258 Gindikin, S.G., 197 Glazyrin, A., 192, 207, 232, 253, 283, 286, 296 Glotzer, S.C., 320 Goddyn, L.A., 217 Goethals, J.M., 255 Gohberg, I.Z., 193, 310–312 Goldberg, M., 152, 232, 317 Goldstein, A.S., 217 Golovanov, A., 282 Golser, G., 317 González Merino, B., 289, 324 Goodman, A.W., 96, 214 Goodman, R.E., 96, 214 Gorbovickis, I., 309, 310 Gori, M., 191 Gortler, S., 331 Graev, M.I., 197 Graf, C., 321 Graham, R.L., 210, 217, 245, 246 Gravel, S., 320 Gravin, N., 275 Gregory, D., 169, 277 Gerencsér, B., 288 Grešák, P., 295 Grey, J., 305 Griffiths, D., 245 Grigorev, M., 214 Gritzmann, P., 201, 251, 317 Groemer, H., 193, 198, 200, 209, 211–213, 236, 241, 254, 260, 261, 274, 275, 278, 287, 292, 294, 295, 301, 302, 318, 325, 333 Gross, W., 40 Gruber, P.M., 193, 205, 218, 228, 242, 291 Grünbaum, B., 192, 193, 215, 216, 229, 278, 286, 287, 289, 303, 304, 306, 322 Guillot, D., 318 Gupta, U., 300 Gurunathan, V., 300 Guy, R.K., 218, 223, 288, 306

425

Guzmán, C., 260 Haantjes, J., 235 Haas, A., 272 Habicht, W., 153, 169, 236, 308 Hadwiger, H., xvi, 30, 54, 91, 93, 96, 153, 185, 193, 215, 216, 226, 278, 295, 304, 305, 310, 312 Haimovich, M., 205 Haji-Akbari, A., 320 Hajós, G., xvi, 30, 186, 200, 322 Hales, T.C., 206, 213, 218, 226, 238–240, 285 Halton, J.H., 217 Hammer, J., 242 Hammersley, J.M., 154, 217 Han, M., 275 Han, X., 298 Hanes, K., 224 Hansen, P., 189, 201, 244–247 Hans-Gill, R.J., 211, 272 Harangi, V., 288 Harborth, H., 278, 279, 285 Hardin, R.H., 235, 236 Hare, K.G., 244 Har-Peled, S., 215 Hárs, L., 232, 267 Hashemi, A., 331 Hausel, T., 304, 323 Heilbronn, H., 98, 99, 217 Helly, E., 97, 216 Henk, M., 223, 239, 241, 318, 323 Henley, C.L., 213 Henrion, D., 246 Heppes, A., 191, 193, 202, 204, 206–208, 211, 213, 216, 227, 229, 236, 240, 242, 261, 266, 267, 271, 272, 282, 283, 291, 303, 304, 314, 315, 318, 322, 323, 327, 330, 331, 336 Hertel, E., 193 Hilbert, D., 185, 236, 320 Hinrichs, A., 268, 305

Name Index

426

Hirsch, J., 226 Hlawka, E., 185, 249, 250, 261 Hoehner, S., 229 Hoffmann, H.-F., 298 Hölder, O., 54 Holmsen, A., 216 Hornich, H., 321 Hortobágyi, I., 278, 323, 325 Horváth, Á.G., 224 Horváth, J., 272, 309, 318, 320 Horváth, M., 309, 310 Hou, X., 258 Hoylman, D.J., 241 Hrinko, I., 295 Hsiang, W.-Y., 238, 239 Hu, D., 226, 242 Hua, L.K., 260 Huang, H., 311 Hujter, M., 305, 306 Hunter, H.F., 306 Iglesias-Ham, M., 217 Imre, M., 266 Ismailescu, D., 195, 196, 219, 220, 321, 336 Ivanov, I., 312 Iwama, K., 298 Jaillet, P., 217 Janić, R.R., 192 Januszewski, J., 212, 295–300 Jenrich, T., 305 Jensen, J.L.W.V., 33, 35, 36, 54, 62, 65, 84, 86, 91, 117, 124 Jenssen, M., 251, 283 Jessen, J., 217 Jiang, M., 215, 296, 300, 307 Jiang, Z., 230, 235, 307 Jiao, Y., 259, 262, 320 Jin, W., 262 Johnson, N.W., 192, 193 Joós, A., 200, 287, 317, 318 Joos, F., 251, 283 Joswig, M., 239 Jucovič, E., 204, 234, 314, 322 Juhnke, F., 223

Juneja, K., 300 Kabatjanski˘ı, G.A., 254–256, 260, 283 Kadets, M., 307 Kadlicskó, M., 329 Kahle, M., 327 Kahn, J., 305 Kakkar, A., 242 Kalai, G., 305, 306 Kallus, Y., 213, 218, 219, 259, 320 Kanagasabapathy, P., 273 Kang, J.-H., 250 Kannan, R., 324 Kaplan, H., 215 Karasev, R., 306, 307 Karloff, H.J., 216 Katz, M.J., 215 Katzarowa-Karanowa, P., 304 Kazarinoff, D.K., 122, 191, 222 Kazarinoff, N.D., 192, 222 Keaton, A., 318 Keevash, P., 235 Keldenich, Ph., 299, 300 Kellerhals, R., 265 Kelly, E., 258 Kelvin, Lord (Sir William Thomson), 185, 241, 242 Kemnitz, A., 277, 278 Kenkel, J., 318 Kenn, D., 201 Kennedy, T., 331 Kepler, J., xv, 236, 239 Kershner, R., 95, 96, 198, 205, 221, 271 Kertész, G., 190, 207, 211, 230, 266, 284, 286, 318, 332 Keys, A., 320 Khan, M.A., 280, 312 Khassa, D.S., 330 Kim, B., 219 Kim, S., 288 Kim, S.-J., 215 Kind, B., 246 Kiss, Gy., 311

Name Index

Kitrick, C.J., 223 Klamkin, M.S., 277 Klee, V., 216, 218, 229, 287, 288, 309 Klein, A., 246 Kleinschmidt, P., 246 Kleist, L., 300 Kleitman, D.J., 216 Kneser, M., xvi, 308–310 Knop, O., 232, 258 Kobos, T., 289 Koch, M., 278 Kołodziejczyk, D., 305 Kolountzakis, M., 275 Kolpakov, A., 283, 286 Kolushov, A.V., 257, 258 Kömhoff, M., 229 Komlós, J., 217, 218 Komornik, V., 191 König, D., 169 Koolen, J., 288, 289 Korkine, A., 259, 284 Kosiński, A., 293, 295 Kovács, Á., 226, 253 Kozma, R.T., 265 Krammer, G., 229, 231 Kumar, A., 256–259 Kunszenti-Kovács, D., 310 Kupavskii, A., 301 Kuperberg, G., 212, 213, 219, 268, 285, 286, 304, 330 Kuperberg, K., 279, 321 Kuperberg, V., 318 Kuperberg, W., 195–197, 208, 212, 213, 218–220, 237, 242, 260–262, 268, 279, 290, 291, 296, 297, 315, 317, 321, 330 Kupitz, Y.S., 279 Kürschák, J., 29 Kusner, R., 242, 283, 288 Kusner, W., 213, 260, 283 Lagarias, J.C., 238, 239, 283, 320 Lagrange, J.L., 229

427

Lam, T.W., 325 Lamoreaux, J., 321 Landau, H.J., 232 Lange, L.H., 195 Lángi, Zs., 189, 190, 192, 214, 224, 279, 288, 305, 306, 312, 329, 332 Larcher, H., 334 Larman, D.G., 244, 286, 287, 305 Laskawiec, P., 321 Lassak, M., 295–298, 302, 305, 311, 312 László, Z., 221 Laurent, M., 288 Lawlor, G., 289 Lázár, D., 55 Ledermann, W., 113 Lee, H., 191, 279 Lee, M., 278 Lee, T.-L., 290 Leech, J., 232, 249–251, 258, 283 Legg, D.A., 258 Leichtweiß, K., 193, 197 Lekkerkerker, C.G., 185, 242 Lemmens, B., 289 Lenhard, H.Ch., 194 Lenz, H., 245, 303, 304 Lešo, J., 204 Leuenberger, F., 194 Lev, N., 275 Leven˘ste˘ın, V.I., 254–256, 260, 283 Levi, F.W., 310–312 Lewis, T., 277 Lexell, A.J., 25, 26, 118, 128 L’Huilier, S., 12, 13, 71, 134, 135 Li, B., 229 Li, K., 226 Li, Sh., 262 Lian, Y., 299 Liebisch, Th., 185 Lillington, J.N., 223, 225 Lim, T., 334 Lindelöf, E.L., 135–137, 173 Lindenstrauss, J., 305 Lindsey, J.H., 237

428

Linhart, J., 222, 225, 227, 229, 230, 272, 277, 281, 282, 307, 319 Litsyn, S., 274 Littlewood, J.E., 289, 290 Litvak, A.E., 222 Liu, A., 277 Liu, B., 275 Liu, L., 250, 262 Liu, X., 299 Livshyts, G., 295, 311 Lobachevski˘ı, 155 Loeb, P.A., 329 Loeb, P.H., 287 Longuet-Higgins, M.S., 192 Loomis, P., 211 Lovász, L., 250, 324 Lu, M., 299 Lyusternik, L.A., 193 Macbeath, A.M., 55, 220, 295 Maehara, H., 193, 232, 283, 284, 335 Magnanti, T.L., 205 Mahalanobis, P.C., 217 Mahler, K., 96, 106, 113, 199, 218 Makai, E. Jr., 189, 223, 229, 230, 237, 240, 280, 301, 304, 323, 324, 333 Makeev, V.V., 241, 289, 304 Malagoli, F., 335 Maligranda, L., 194 Maman, M., 215 Mao, Q.J., 223 Marchal, C., 238, 239 Marciniewicz, J., 96 Marcuccio, A., 226 Markus, A.S., 310, 311 Marshall, T.H., 265, 266 Martin, G.J., 265, 266 Martin, Z., 226, 242 Martini, H., 191, 193, 306, 311, 312, 323, 324 Matoušek, J., 261 Matzke, R.W., 334, 335 Mauldin, R.D., 293

Name Index

Maurmann, Q., 226 McCann, R.J., 334 McKean, H.P. Jr., 207 McLaughlin, S., 238 McMullen, P., 193, 261, 275 Mednyik, A.D., 229, 230 Meir, A., 294, 297 Melissen, H., 215, 317 Melnyk, T.W., 232, 258 Melzak, Z.A., 217, 222, 229, 303, 304 Meschkowski, H., 232 Messer, P.W., 192 Messerschmidt, M., 331 Messine, F., 189, 201, 244–247 Meyer, M., 189 Miller, J.C.P., 192 Miller, M., 226, 242 Miller, S.D., 257, 259 Milman, V.D., 289 Minkowski, H., xv, 29, 44, 113, 135–137, 173, 185, 210, 241, 249, 250, 260, 275 Mitrinović, D.S., 192 Mittelmann, H.D., 256 Mixon, D.G., 236 Miyamoto, Y., 265 Moese, H., 306 Möller, M., 277, 278 Molnár, J., xvi, 154, 169, 194, 199, 200, 202, 204, 230, 231, 236, 254, 314, 318, 328, 329 Monagan, M., 316 Montejano, L., 306 Moon, J.W., 293, 294 Moran, S., 217 Mordell, L.J., 15, 30, 122, 191 Morgan, F., 205, 206, 242, 289 Morić, F., 207 Morin, P., 215 Morr, S., 300 Moser, L., 293, 294, 297 Moser, W.O.J., 189, 218, 294–296, 306

Name Index

Mossinghoff, M.J., 244–247 Mount, D.M., 210, 213 Moussong, 308 Moustrou, P., 284 Muder, D.J., 237 Mügge, O., 185 Mughal, A., 318 Mulzer, W., 215 Muny, H., 192 Musin, O.R., 232, 233, 256, 258, 283–285, 317, 318 Mustafaev, Z., 191 Mutoh, N., 224 Nagy, B.v. Szőkefalvi, 29, 30 Nagy, J., 274, 318 Nakprasit, K., 215 Napier, J., 28 Naszódi, M., 190, 250, 252, 274, 280, 295, 296, 306, 310, 312, 329 Natarajan, B.K., 328 Nazarov, F.L., 218 Nelsen, R.B., 191 Newton, I., 169, 277 Nielsen, F., 334 Nikitenko, A.V., 318 Nikonorov, Yu.G., 229, 230 Nikonorova, Yu.V., 230 Nilli, A., 305 Ninin, J., 246, 247 Novotný, P., 295 Odlyzko, A.M., 250, 256, 283 Ogievetsky, O., 291 Ohmann, D., 307 Oler, N., 210, 332 Oliveros, D., 306 Oliwa, M., 280 Oppenheim, A., 191 Ordentlich, O., 251 Ortega-Moreno, O., 308 Osborn, J.M., 329 Osserman, R., 191

429

Pach, J., 189, 190, 218, 273, 281, 288, 295, 296, 301, 310, 325, 329, 333 Pál, J., 243, 303 Palffy-Muhoray, P., 320 Pálvölgyi, D., 273 Pankov, P.S., 277 Papadimitriou, Ch.H., 205, 325 Papadoperakis, I., 312 Papadopoulos, A., 193 Parshall, H., 236 Parsons, Ch., 289 Paukowitsch, P., 321 Payne, L.E., 191 Pchelina, D., 331 Pečarić, J.E., 192 Peikert, R., 316 Peletier, M.A., 205 Pelsmajer, M.J., 215 Perkins, W., 251, 283 Perlstein, A., 304 Perpetua, B., 226, 242 Perron, S., 246 Petrov, F., 306 Petschek, R.G., 320 Petty J., 226 Petty, C.M., 195, 222, 288, 289 Peyerimhoff, N., 225 Pfender, F., 239, 256 Phelan, R., 242 Pick, G., 42 Pierre, M., 203 Pikhitsa, P.V., 290, 291 Pikhitsa, S., 291 Pikhurko, O., 305 Pillichshammer, F., 335 Pinchasi, R., 207, 304 Pintz, J., 217, 218 Pirl, U., 215 Plaisted, D.A., 216 Plato, xv Pleijel, A., 191 Poincaré, H., 53, 54 Pólya, G., 191

430

Polyanskii, A., 214, 230, 274, 307, 329 Poulsen, E.T., 308–310 Prachar, K., 196 Pritchard, T., 226 Prosanov, R., 198 Prymak, A., 311, 312 Przeworski, A., 265, 266, 318 Pudlák, P., 288 Purdy, G.B., 273 Radchenko, D., 257, 259, 311 Rademacher, H., 30 Radin, Ch., 268, 269 Rado, R., 96 Radó, T., 30, 96 Radziszewski, K., 295 Ra˘ıgorodski˘ı, A.M., 305, 306 Rankin, R.A., 174, 185, 236, 252, 260, 317 Rax, O., 312 Regev, O., 251 Reid, S., 279 Reifenberg, E.R., 96, 214 Reingold, E.M., 216, 217 Reinhardt, K., 106, 113, 218, 244–247 Reisner, S., 189 Ren, D.L., 197 Reuter, G., 279 Revenko, S., 215 Révész, P., 304 Richardson, T.J., 298 Richter, Ch., 268, 305 Riesz, F., 29 Rissling, A.S., 304, 305 Robins, G., 286, 287 Robins, S., 275 Robinson, R.M., 233 Roditty, L., 215 Rodríguez-Arias Fernández, M., 195 Rogers, C.A., xiii, 96, 186, 199, 207, 210, 211, 236, 237, 240, 249–251, 253, 261, 262, 305, 311

Name Index

Roldán-Pensado, E., 326 Rolfes, J.H., 250 Rónyai, L., 290 Ros, A., 191 Rosenfeld, M., 235 Rosenthal, A., 243 Rosta, V., 307 Rote, G., 228–230, 297 Roth, K.F., 98, 217, 218 Rush, J.A., 250, 251 Rusza, I.Z., 201 Rutishauser, H., 169 Ryškov, S.S., 259, 320, 323 Sachs, H., 284 Sah, A., 260 Sahai, A., 308 Sallee, G.T., 223, 303 Salowe, J.S., 286, 287 Sansone, G., 154, 222 Santaló, L.A., 30, 53, 54, 197 Sardari, N.T., 256 Sas, E., 39, 54, 105, 113, 196 Sauer, N., 286 Sawhney, M., 260 Schaake, G., 260 Schaer, J., 317 Schäffer, J.J., 245 Scheffer, Ch., 299, 300 Scherer, K., 290 Schläfli, L., xv, 19, 186 Schmidt, W.M., 218, 249, 261, 274 Schmitt, P., 214 Schneider, R., 193, 196, 197, 223, 308 Schoenberg, I.J., 255 Schoenflies, A.M., xv, 29, 185 Schopp, J., 215, 222, 277, 303 Schramm, O., 285, 286, 305, 311 Schreiber, M., 30, 207 Schrijver, A., 288, 289 Schulte, E., 193, 304 Schürmann, A., 239, 258, 259, 261, 289, 316, 320

Name Index

Schütte, K., 157, 164, 165, 169, 176, 201, 232–234, 277, 283 Schwartz, R.E., 258 Schwarz, H., 36, 149 Schymura, M., 324 Scott, R.C., 229 Segre, B., 96, 199 Seidel, J.J., 201, 235, 255 Seiferth, P., 215 Şerban, M., 232 Shah, S., 300 Shannon, C.E., 251 Shao, J., 258 Sharir, M., 215 Shen, J.D., 318 Shepelska, V., 312 Shephard, G.C., 193, 228, 249, 311, 327 Shermer, T.C., 215 Shiryaev, D., 275 Shlosman, S., 283, 291 Sidelńikov, V.M., 254 Siegel, A., 190, 191 Silverman, R., 210 Simonič, A., 193 Sizova, O., 331 Skilling, J., 192 Skokan, J., 215 Slepian, D., 232 Sloane, N.J.A., 233, 235, 236, 242, 250, 251, 256, 262, 283 Slomka, B.A., 311, 312 Smale, S., 290 Smith, A.D., 318 Smith, E.H., 196, 241 Smith, H., 318 Smith, M.J., 291 Smith, W.R., 232, 258 Smurov, M.V., 307 Smyth, C., 288 Snoeyink, J., 327, 328 Snyder, T.L., 217 Soberón, P., 216 Soifer, A., 295, 298 Solomonides, E., 318, 331

431

Soltan, P.S., 288, 306, 311, 312 Soltan, V., 215, 296, 312, 323 Sopov, S.P., 192 Sothanaphan, N., 242 Sperling, G., 334 Sriamorn K., 275 Sriamorn, K., 212, 272 Stachó, L., 215 Stacho, M., 295 Starostin, E.L., 291 Steele, J.M., 217 Stehling, T., 192 Stein, S.K., 211, 228 Steiner, J., 7, 29, 54, 121, 134–136, 152, 240 Steinerberger, S., 217, 308 Steinhagen, P., 223 Steinhaus, H., 206 Steinig, J., 191 Steinitz, E., 29, 30, 136, 226 Stern, N., 205 Stillinger, F.H., 262 Stolarsky, K.B., 335 Stolfi, J., 327, 328 Stoner, D., 260 Strachan, C., 312 Straszewicz, S., 304, 306 Stroilˇa, C., 232 Su, W., 193 Su, Z., 299 Subak, H., 272 Sudakov, B., 235 Sudakov, H., 309 Sullivan, J.M., 242 Supnick, F., 185 Supowit, K.J., 216 Süss, W., 295 Sutcliffe, P., 232 Swanepoel, K.J., 280, 287–289, 304, 329 Swinnerton-Dyer, H.P.F., 291 Szabó, L., 232, 233, 240, 278, 284, 285, 291, 312, 323, 329 Szabó, P.G., 317 Szabo, T., 246

432

Szalkai, B., 279 Szalkai, I., 280 Szász, O., 243 Szegő, G., 191 Szekeres, G., 97 Szemerédi, E., 217, 218 Szirmai, J., 264, 265 Szpiro, G.G., 239 Szűcs, A., 304 Szüsz, P., 206 Talata, I., 281, 286, 287, 292, 311, 323 Talenti, G., 191 Tammela, P., 218 Tammes, R.M.L., 221, 232 Tamvakis, N.K., 244, 305 Tanner, R.M., 222 Tarasov, A.S., 233, 283, 285 Tardos, G., 273, 323 Tarnai, T., 233 Tarski, A., 98, 306, 308 Taylor, M., 283 Temesvári, Á.H., 272 Theil, F., 205, 283 Thomson, T.M., 250 Thue, A., xvi, 95, 96, 198, 199, 221, 271 Tidor, J., 235 Tikhomirov, K., 295, 311 Tkocz, T., 311 Tomor, B., 224 Torquato, S., 259, 262, 320 Tóth, G., 273 Townsend, D.W., 258 Tumanov, A., 259 Uvjáry-Menyhárt, Z., 278, 285, 323 Ulam, S., 219 Ungár, P., 97 Vásárhelyi, É„ 332 Väisälä, J., 289 Vallentin, F., 213, 232, 233, 241, 250, 256, 257, 259, 260, 284, 320

Name Index

van der Waerden, B.L., xvi, 153, 157, 164, 165, 169, 176, 193, 232, 233, 236, 277, 283 van Lint, J.H., 235 Vance, S., 249 van der Corput, J., 260 Vásárhelyi, É., 207, 296, 298, 299 Vasić, P.M., 192 Végh, A., 272 Vegter, G., 228 Veldkamp, G.R., 191 Venkatesh, A., 249, 254 Venkov, B.A., 261, 275 Verblunsky, S., 96, 97, 216 Verger-Gaugry, J.-L., 251 Vermes, I., 265, 266, 332 Viazovska, M., 256, 257, 259 Vígh, V., 260, 307, 308, 334 Vilenkin, N.Ya., 197 Villa R., 287 Villa, R., 289 Vincze, S., 246 Vlăduţ, S., 291 Volenec, V., 192, 223 von Wolf, M.R., 211 von Höveling, S., 300 Voronoi, G., 185, 199 Vritsiou, B.H., 311 Wagon, S., 218, 309 Wang, Y., 215 Waruhiu, S., 226, 242 Waterman, D., 222 Weaire, D., 242, 318 Wegner, G., 200, 201, 208, 278, 281, 282, 285 Weierstrass, K., 9, 11, 12, 29, 49, 52 Weil, W., 197 Weiss, B., 251 Weiss, G.H., 207, 222 Weißbach, B., 223 Wenger, R., 216 Wenk, C., 291 Wessler, M., 246 Wetayawanich, A., 272

Name Index

White, S., 307 Whitworth, J.V., 241 Whyte, L.L., 169 Wieacker, J.A., 197 Wilker, J.B., 201 Willert, M., 215 Wills, J.M., 193, 201, 261, 317, 318 Wimmer, L., 234 Wintraecken, M.H.M.J., 228 Wintsche, G., 251, 317 Wise, M.E., 185 Wisewell, L., 307 Witsenhausen, H.S., 210, 335 Woeginger, G.J., 297, 302 Wojzischke, D., 278 Woo, J., 236 Woods, A.C., 210, 212, 323, 324, 329, 330 Wu, Y.-D., 222 Würtz, D., 316 Wyner, A.D., 251 Xhumari, S., 318 Xing, C.P., 250 Xiong, J., 245 Xu, C., 298 Xu, J., 277 Xu, L., 287 Xue, Ch., 226 Xue, F., 212, 321 Yaglom, I.M., 193 Yakovlev, N.N., 272 Yampolskaya, M., 331 Yang, L., 223

433

Yang, Q., 275 Yannakakis, M., 325 Yao, Y., 235 Youngs, J.W.T., 277 Yuan, L., 298, 326 Yuan, Y., 262 Yudin, V.A., 257, 258 Zador, P.L., 205 Zalgaller, V.A., 192, 193 Zamfirescu, T., 214, 227 Zargar, M., 256 Zarnócz, T., 307, 308, 334 Zassenhaus, H.J., 210, 211, 237 Zeitler, H., 264 Zhang, G., 298 Zhang, J., 223, 296 Zhang, S., 235 Zhang, Y., 299, 307 Zhao, L., 277 Zhao, Y., 235, 256, 260, 308 Zheng, X., 320 Ziegler, G.M., 193, 239, 286, 320, 323 Zielonka, Ł., 299 Zielonka, L., 296–298 Zito, J.S., 320 Zolotareff, G., 259, 284 Zong C., 275 Zong, C., 218, 241, 242, 260, 261, 275, 286, 287, 306, 311, 319–321, 323 Zuo, Q.R., 223 Zygmund, A., 96

Subject Index

600-cell, 256, 257 absolute value, 26 affine distance, 98 affine isoperimetric inequality, 52 affine length, 42, 47, 89 affine perimeter, 42, 197 affine regular polygon, 7 affine surface area, 151, 197 affinity, 7 allied dictators problem, 232, 234, 236 𝛼-theory, 290 antipodal points of a set, 287 antipodal set, 287 antiprismatic bipyramid, 234 antiprisms, 22 approximability, 149 approximating a ball by polyhedra, 120 approximating a convex surface by polyhedra, 149 approximation theory, 268 Archimedean antiprism, 158 Archimedean polyhedra, 19 nondegenerate, 22 Archimedean solids, 19, 192 area density, 59 area-deviation, 33 arrangements of great circles, 335

arrangements of points with minimum potential energy, 257 arranging houses, 313 arranging regular tetrahedra, 320 average number of pieces, 92 average value of a functional, 58 avoiding obstacles, 324 ball packings in hyperbolic space, 263 ball polytope, 304 Banach–Mazur distance, 289 Bang’s plank theorem, 155 basic parallelogram, 102 belt bodies, 311 bendings, 149 Blaschke’s kinematic formula for simply connected domains, 54 Blaschke’s selection theorem, 31, 54 Blichfeldt’s bound, 252, 254 blocking-free packing, 314 body-centered cubic lattice, 173 Borsuk number, 305 Borsuk’s conjecture, 303, 305 for smooth convex bodies, 304 bounded sequence of convex bodies, 301

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Fejes Tóth et al., Lagerungen, Grundlehren der mathematischen Wissenschaften 360, https://doi.org/10.1007/978-3-031-21800-2

435

436

bounds for the covering density of convex bodies, 259 bounds for the packing density of convex bodies, 259 box, 293 cage, 227 cap body, 312 Cauchy sequence of convex disks, 32 Cauchy’s inequality, 36 cell, 62, 171 center set, 309 centrally symmetric domains, 105 chessboard conjecture, 333 circle-convex region, 202 circumcircle, 5 circumradius, 5 circumscribed ellipsoid, 133 circumscribed polygon, 5 circumsphere, 6 clipped Reuleaux polygon, 244 close neighbors, 179 close packing, 319 closeness of a packing, 319 cloud, 321 dark, 322 deep, 322 for a half-space, 322 𝑘-fold, 322 width, 322 compact packing, 330, 331 completely monotonic function, 257 completely reduced covering, 268 completely saturated packing, 268 concave function, 33 concentric ellipsoid, 133 connected arrangement of sets, 333 contact graph of a packing of circles, 331 converging sequence of disks, 31 convex bodies (theory of), 193 convex body, 6 convex curve, 5 convex disk, 5 convex disk-polygon, 194

Subject Index

convex disks of constant width, 243 convex function, 33 of two variables, 35 convex hull, 5 convex polygon, 5 spherical, 17 convex polyhedra with regular faces, 192 convex polyhedron, 6 uniform, 192 convex set, 5 convex spherical polygon, 17 convex spherical region, 155 convex surface, 6 convex uniform polyhedron, 192 Conway’s fried potato problem, 306 covering a convex body by smaller homothetic copies, 310 covering a convex hexagon with congruent convex disks, 86 covering codes, 251 covering convex disks with congruent circles, 66 covering cubes and boxes, 294 covering density, 198 of convex bodies, 240 covering efficiency, 108 of sequences of disks, 108 covering measure, 141 covering properties of sequences of convex bodies, 293 covering space, 300 covering the sphere with congruent spherical caps, 116 covering with a margin, 315 covering with circles, 81 covering with fragmented disks, 111 covering with slabs, 301 crate, 227 crossing pairs of disks, 207 crystals (230 classes), xv cubo-octahedron, 185 cuboctahedron, 24, 185 cylinders touching a ball, 291

Subject Index

dark cloud, 322 decomposition of multiple arrangements, 273 deep cloud, 322 degenerate polyhedra, 19 Delone cell, 265 Delone triangulation, 199 densest ball packing, 174 densest circle packing, 60 densest packing with circles, 59 density, 57, 141 density of a family of domains, 59 density of an invariant measure, 269 density of arrangements of domains, 57 Dido arrangement, 190 difference body, 102 difference of vectors, 26 dihedra, 19 Dirichlet cell, 199 Dirichlet–Voronoi region, 199 dissecting a convex domain into convex parts, 71 dissecting a convex hexagon into convex polygons, 85 dissecting the sphere into convex parts of equal area, 142 distributive property, 26 dodecahedral conjecture, 238 dodecahedron, 22 double bee-cell, 172, 185 double-lattice arrangement, 212 Dowker’s theorem, 37, 194 dual polyhedra, 19 Dupin curve, 150 Dupin indicatrix, 152 𝐸8 root lattice, 259 economic coverings, 249 economic packings, 249 edge curvature, 7 edge-closeness of a packing of congruent balls, 320 𝜀-quasi-twelve-neighbor packing, 240

437

equifacial (Archimedean) polyhedra, 21 equilateral set, 289 equilateral subset of a metric space, 288 equilateral triangular lattice, 57 equivertical (Archimedean) polyhedra, 21 Erdős–Mordell inequality, 15, 122, 191 ergodic measure, 269 error correcting code, 250 error-correcting code, 255 estimates for incongruent circles, 74 Euler’s theorem on convex polyhedra, 17 Euler’s theorem on polyhedra, 17, 18, 162 expendability of a packing of unit circles, 315 external angle, 6 extremum problems, xv extremum properties of regular polygons, 9 extremum properties of triangles, 101 extremum property of the ellipse, 39 face-centered cubic lattice, 172 Fáry’s inequality, 218 fatness of convex disks, 208 finite arrangements in higher dimensions, 317 finite covering in two dimensions, 316 finite packing in two dimensions, 316 flyspeck, 238 football, 266 fractional illumination number, 312 fundamental characteristics of convex bodies, 7 fundamental domain, 101 Gaussian curvature, 149 Gegenbauer polynomials, 255

438

generalized Minkowskian arrangement of circles, 328 generalized superball, 250 geometry of numbers, xv grade of saturation of a packing of congruent circles, 267 graph, 158 associated with a family of points, 157 Groemer packing, 200 Hadwiger number, 278, 286, 287 higher-order, 283 Hadwiger–Levi covering problem, 312 Hadwiger’s inequality, 226 Hajós lemma, 200 Hausdorff distance, 31, 194 Heilbronn’s conjecture, 217 hexagonal family of circles, 63 hexahedron, 22 higher-order Hadwiger numbers, 283 higher-order neighbors, 282 Hölder–Jensen inequality, 54 homogeneity of an arrangement, 203 homothetic, 7 ellipsoid, 133 homothety, 7 honeycomb cells, 242 honeycomb conjecture, 206 honeycomb-like ball packing, 174 honeycomb-like covering, 63 honeycomb-like family of circles, 63 honeycomb-like packing, 63 honeycomb-like space partition, 180 horoball, 263 horocycles, 231 hostile dictators problem, 232–234 hyperball, 265 hyperbolic geometry, 155 hyperconvex disk, 194, 199 hypersphere, 265 icosahedron, 22 identical vectors, 26

Subject Index

illuminated boundary point of a convex body, 310 illumination number, 312 incircle, 5 inequalities linking the inradius and circumradius of polyhedra, 131 inequalities on triangles, 14 inner parallel domain, 13 inner product, 26 inradius, 5 inscribed ellipsoid, 133 inscribed polygon, 5 inseparable family of domains, 96 insphere, 6 instability of a circle packing, 326 integral geometry, 53 integral of the mean curvature, 6, 7 intrinsic isometries, 149 𝑖-region, 202 irreducible graph, 158, 170 isolated point, 158 isometric covering, 293 isometric packing, 293 isomorphic polyhedra, 121 isoperimetric equation, 54 isoperimetric inequality, 190, 191 isoperimetric problem, 11, 243 for 𝑛-gons, 12 for polygons, 243 for polyhedra, 134 for space partitions into cells of equal volume, 241 isoperimetric problem for polygons, 247 Jensen’s inequality, 33 Kelvin conjecture, 242 Kepler conjecture, 236, 239 Kepler–Poinsot star-polyhedra, 224 kernel, 13 of a set of circles, 328 𝑘-fold Borsuk number, 306 𝑘-fold cloud, 322 𝑘-fold covering of sets, 271

Subject Index

𝑘-fold packing of sets, 271 𝑘-fold tiling of bodies, 275 kinematic density, 53 in space, 183 kissing number, 277 Kneser–Poulsen conjecture, 308–310 𝑘-reduced covering, 268 𝑘-saturated packing, 268, 330 𝑘-th neighbor, 282 𝑘-th Newton number, 282 𝑘-truncation of a packing, 267 lattice arrangements of balls, 259 lattice ball packings, 249 lattice covering, 101 density, 198 lattice of sets, 101 lattice packing, 101 density, 198 lattice packing density, 241 lattice-like arrangement, 57 layer of disks, 324 Leech lattice, 249, 256–259 Lexell’s circle, 25 Lexell’s theorem, 25, 193 for the hyperbolic plane, 193 Lexell’s theorem, 25, 26 L’Huilier’s inequality, 12, 13 limited semi-convex domain, 211 Lindelöf–Minkowski theorem, 136, 137 line of dislocation, 206 line-element, 98 linear programming bound, 254–257 local separability, 332 loose covering, 319 looseness of a covering, 319 Lovász local lemma, 250 lower density, 58 magnitude, 26 margin of a covering with unit circles, 315 maximal configuration, 159, 161, 164 maximal graph, 158

439

maximal packing of congruent convex disks, 282 mean value formula, 91, 92 in space, 183 𝑚-gon of a graph, 158 minimal angle at the boundary of a convex disk, 281 minimal spherical shell, 121 minimum closeness (Linehart), 319 minimum homothety coefficient (Zong), 319 Minkowski arrangement generalized, 328 Minkowskian arrangement, 328, 329 Minkowski–Hlawka theorem, 249 mixed product, 26 moment lemma, 205, 223 moment of a region with respect to a point, 205 moment-sum theorem, 205 𝑚-periodic arrangement, 259 multiple arrangements, 271 in space, 273 on the plane, 271 multiple tiling, 275 mutually touching cylinders, 289 Napier’s rule, 28 natural equation, 43 𝑛-dimensional isoperimetric inequality, 226 neighbors, 234, 277 higher-order, 282 in lattice packings, 291 Newton number, 277 of balls, 283 Newton versus Gregory, 169, 232, 277 𝑛-neighbor packing, 280 𝑛+ -neighbor packing, 280 non-regular isogonal spherical tilings, 193 non-regular polyhedra with congruent faces, 193

440

nondegenerate Archimedean polyhedra, 22 normal sequence of convex disks, 108 normal tiling, 239 of the plane, 192 𝑛-th Borsuk number, 305 number density, 58 number of elements, 53 obstructing light, 321 octahedron, 22, 24 one-sided kissing number, 284 on-line covering problem, 296 on-line packing problem, 296 optimally dense packing, 269 𝑜-region, 202 orthocenter, 229 orthocentric simplex, 229 outer product, 26 packing a convex domain with circles of different sizes, 72 packing a convex hexagon with congruent convex disks, 86 packing barrels, 314 packing convex disks with congruent circles, 66 packing cubes and boxes, 294 packing cylinders, 321 packing density, 198, 269 of convex bodies, 240 of the ball, 249 of the regular cross-polytope, 260 of the regular octahedron, 241, 320 of the regular pentagon, 213 of the regular tetrahedron, 320 packing efficiency, 108 of sequences of disks, 108 packing in space, 171 packing properties of sequences of convex bodies, 293 packing space, 300

Subject Index

packing the sphere with congruent spherical caps, 116 packings of superballs, 250 parallel domain, 6 parallelohedron, 261, 275 pentagram, 192 perfect configuration, 235 perimeter deviation, 33, 195 periodic arrangement of sets, 214 periodic Reuleaux polygon, 244 permeability of a layer, 324 𝑝-hexagon, 196, 219 piercing number, 215 Platonic solids, 192, 224 pocket, 279 point of support, 5 point of tangency, 5 points on the sphere, 334 point-trapping arrangement of sets, 333 point-trapping lattice, 333 polar triangle, 25 polarity, 7 positive-definite kernels, 255 potato-sack problems, 293 potential energy, 257 power mean, 76 prisms, 22 proper triangular polyhedron, 129 push-away, 158 𝑞-adic online algorithm, 297 quarter-convex disk, 212 quasi-periodic set, 275 𝑟-accessible, 314 Ramsey’s theorem, 290 region enclosed by disks, 190 regular convex polygon, 5, 192 regular convex polyhedron, 19 regular family of balls, 172 regular pentagon, 213 regular polyhedra, 19 regular set of points, xv regular star-polygon, 192 regular vertex, 19

Subject Index

regular zonotope, 230 regularity of optimal arrangements, 214 Reinhardt polygon, 244 Reinhardt–Mahler conjecture, 218 Reuleaux polygon, 244 clipped, 244 periodic, 244 sporadic, 244 𝑟-fat convex disk, 208 𝜚-convex region, 202 rhombic dodecahedral arrangement, 176 rhombic dodecahedral crystal structure, 284 rhombic dodecahedron, 24, 172 Rogers’ theorem, 210 saturated arrangement of circles, 329 saturated arrangements, 329 saturated family of congruent circles, 61 saturated family of domains, 99 saturated packing of order 𝑘, 330 sausage conjecture, 317 scalar product, 26 Schwarz’s inequality, 36 Scottish Book, 293 second law of cosines, 27 segment-area, 48 Segre–Mahler polytope, 254 semi-convex domain, 211 semidisk, 213 semiregular antiprisms, 192 semiregular polyhedra, 19, 21 semiregular prisms, 192 semiregularity, 192 separable family of domains, 96 separation by translation with two hands, 328 separation of a family of sets, 87 set of closed circular disks, 329 similarity, 7 simplex bound, 252, 254, 263

441

simultaneous lattice packing and covering constant, 319 simultaneous packing and covering constant, 319 slab of cube, 241 smooth convex surface, 129 smoothed octagon, 106 solid circle covering, 266 solid circle packing, 266 of order 𝑘, 267 special Jacobi polynomials, 255 sphere packing problem, 236, 238 spherical code, 254, 255 spherical convex disk, 55 spherical law of cosines, 27 spherical law of sines, 28 spherical triangle, 17 spherical trigonometry, 27 sporadic Reuleaux polygon, 244 stability, 326 of a circle packing, 326 stable packing of circles, 326 Stacey, B.C., 235 Steiner minimal tree, 216, 217 Steiner symmetrization, 152 Steiner–Lindelöf condition, 135, 136 Steiner’s conjecture, 136 stereochemistry, 258 strictly antipodal, 287 strictly antipodal set, 287, 288 strictly concave function, 33 strictly convex function, 33 string of balls, 237 strong dodecahedral conjecture, 239 strongly solid circle packing, 267 strongly translationally solid, 267 sum of vectors, 26 superball, 250 generalized, 250 supporting line, 5 supporting plane, 6 surface area, 6, 7 symmetry operation of a family of balls, 172

442

Tammes problem, 221, 283 tangent line, 5 Tarski’s plank problem, 306, 308 tetrahedron, 22 theorema egregium, 149 thinnest circle covering, 59, 60 thinnest saturated packing of spherical caps, 147 tiling domain, 66 total length of the edges of a polyhedron, 143 totally separable packing of convex bodies, 332 touching number, 287 touching pairs in finite packings, 279 translates of a Jordan disk with a common point, 279 translational covering, 210 translational packing, 210 translative covering density, 198 translative kissing number, 278 translative packing, 293 density, 198 trapezo-rhombic dodecahedral crystal structure, 284 traveling salesman tour, 217 truncated octahedron, 173, 185 tube, 266 tubes touching a ball, 291 tubes touching another tube, 291 twelve spheres problem, 283 twisted cuboctahedron, 283 two-flat zonotope, 275

Subject Index

ultraspherical polynomials, 255 uniform contraction, 310 uniform polyhedron, 192 uniform star-polyhedra, 192 uniformly stable covering, 267 uniformly stable packing, 267 unitary ellipse, 42 universal covering set, 252 universally optimal arrangement of points, 257 upper density, 58 valence, 158 vector, 26 vector algebra identities, 26 vector product, 26 vertex-and-edge-truncation, 22 vertex-truncation, 22 visible vertex, 137 volume, 7 of a circumscribed polyhedron, 125 of an inscribed polyhedron, 128 product, 26 Voronoi region, 199 weighted area deviation, 194 weighted covering, 312 number, 312 weighted illumination, 312 number, 312 zone, 230, 307 zonotope, 230