# Bodies of constant width. An introduction to convex geometry with applications 9783030038663, 9783030038687

269 4 4MB

English Pages 486 Year 2019

Preface......Page 5
Contents......Page 7
1 Introduction......Page 12
1.1 First Properties of Bodies of Constant Width......Page 13
1.2 Mathematical Content of the Book......Page 17
1.3 Convexity and Constant Width in University Teaching......Page 19
2.1 Introduction......Page 24
2.2 Basic Concepts......Page 25
2.3 The Convex Hull......Page 29
2.4 Support Function and Minkowski Addition......Page 30
2.5 Blaschke's Selection Theorem......Page 33
2.6 Steiner Symmetrization......Page 35
2.7 The Brunn–Minkowski Theory......Page 38
2.8 Radon, Carathéodory, and Helly......Page 40
2.9 Classification of Boundary Points......Page 45
2.11 The Problems of Busemann–Petty and Shephard......Page 48
2.12.1 Brunn's Theorem......Page 52
2.12.2 Blaschke's Theorem......Page 55
2.12.3 The False Center of Symmetry......Page 61
3.1 Diameters, Binormals, and Diametral Chords......Page 70
3.2 The Minimum Width Condition......Page 76
3.3 Projections and Sections......Page 78
3.4 Circumsphere and Insphere......Page 79
3.5 Minkowski Sum and Central Symmetry......Page 81
4.1 Characterizations......Page 86
4.2 The Mizel Conjecture......Page 88
4.3 Alexander's Conjecture......Page 89
4.4 The Makai–Martini Characterization......Page 91
4.5 Intersection Properties of the Boundaries......Page 99
5.1.1 The Parametrization in Terms of the Pedal Function......Page 105
5.1.2 Areas, Perimeters, and the Cauchy Formula......Page 108
5.2.1 The Envelope......Page 111
5.2.2 Leaving on the Same Side Regions of the Same Area......Page 112
5.3.1 Main Properties of Systems of Externally Simple Lines......Page 115
5.3.2 A Parametrization of Figures of Constant Width......Page 119
5.3.3 The Analytic Curve of Constant Width due to Rabinowitz......Page 121
5.3.4 Evolutes and Euler's Constant Width Curve......Page 122
5.4.1 Introduction......Page 124
5.4.2 A Figure Floating......Page 126
5.4.3 Floating with Density One Half and Constant Width......Page 130
6.1 Spindle h-Convexity......Page 137
6.2 Ball Polytopes......Page 142
6.3 The Vázsonyi Problem......Page 144
6.4 Reuleaux Polytopes......Page 146
7.1 Introduction......Page 153
7.2 The Theorems of Meissner and Pál......Page 154
7.3.1 Minkowski Difference and Strong Convexity......Page 157
7.3.2 Pairs of Constant Width......Page 158
7.4 Reduced Convex Bodies......Page 159
7.5 Regular Constant Width Hulls......Page 164
8.1 Reuleaux Polygons......Page 176
8.2 The Reuleaux Tetrahedron......Page 178
8.3.1 Original Meissner Bodies......Page 180
8.3.2 Performing Surgery to the Reuleaux Tetrahedron......Page 182
8.3.3 A Description of Meissner Bodies......Page 183
8.3.4 Meissner Polyhedra......Page 185
8.4.1 The Bull; A Concrete Example......Page 187
8.4.2 Constructing Bodies of Constant Width from Reuleaux Polygons......Page 191
8.4.3 The General Case......Page 193
8.5 Algebraic Constant Width Bodies......Page 195
9.1 Concurrent Constant Width Sections......Page 205
9.2 Thickness of a Body......Page 210
9.3 Which Bodies Are Sections?......Page 212
10.1 Introduction......Page 216
10.2.1 mathcalG-Diameters......Page 220
10.2.2 The Homothety Theorem......Page 222
10.3 Complete Bodies in Minkowski Spaces......Page 225
10.4 Strong mathcalG-Convexity and Perfect Norms......Page 229
10.5 Reduced Bodies in Minkowski Spaces......Page 231
10.6 The Borsuk Conjecture in the Minkowski Plane......Page 234
11.1 The Support Parametrization in Terms of the Gauss Map......Page 253
11.2.1 Introduction and Preliminaries......Page 255
11.2.2 Positive Constant Gaussian Curvature......Page 259
11.3 The Curvature of a Body of Constant Width......Page 262
11.4 Baire Category and Curvature......Page 265
11.5.1 A Characterization of Constant Width in Terms of the Hessian......Page 268
11.5.2 The Local Geometry of the Boundary of a 3-Dimensional Convex Body......Page 269
11.6 Constant Width in Affine Geometry......Page 273
12.1 Mixed Volumes......Page 284
12.2 Surface Areas and Mixed Surface Areas......Page 293
12.3 The Projection Formulas of Cauchy......Page 295
12.4 Mixed Volume Inequalities......Page 296
13.1.1 Two Theorems on Spherical Integration......Page 303
13.1.3 Firey's Variant of Minkowski's Theorem......Page 305
13.1.4 A Characterization of Central Symmetry......Page 306
13.1.5 Aleksandrov Type Theorems for Sections......Page 307
13.2 Fourier Series and Spherical Harmonics......Page 308
13.3.1 Centrally Symmetric Bodies That Float in Equilibrium......Page 313
13.3.2 Convex Bodies of Constant Brightness......Page 314
13.3.3 Constant Outer k-Measure Bodies......Page 317
14.1 Isoperimetric Inequalities......Page 325
14.2 The Blaschke–Lebesgue Problem......Page 327
14.3 Measures of Asymmetry......Page 331
14.4 Inequalities Involving the Circumradius......Page 332
14.5 A Bonnesen-Type Isoperimetric Inequality......Page 334
15.1 Helly's Theorem and Constant Width......Page 347
15.1.1 The (p,q)-Property and the Piercing Number......Page 348
15.1.2 Hyperplane Systems and the Kneser Conjecture......Page 350
15.2.1 Introduction......Page 352
15.2.2 Universal Covers in the Plane......Page 353
15.2.3 Universal Covers in n-Space......Page 355
15.2.4 Minimal Universal Covers......Page 358
15.3.1 Packing and Covering......Page 359
15.3.2 The Borsuk Conjecture......Page 360
16.1 The Topology of Certain Hyperspaces......Page 372
16.1.1 The Hilbert Cube......Page 374
16.1.2 Hilbert Cube Manifolds......Page 375
16.1.3 Topology of Some Hyperspaces of Convex Sets......Page 379
16.2 Transnormal Manifolds......Page 382
16.3 Fiber Bundles and Universal Covers......Page 386
16.4.2 Schubert Cycles and Cocycles......Page 389
16.4.3 The Geometry of Grassmannians......Page 391
16.4.4 Recognizing Bodies of Constant Width......Page 392
16.4.5 Recognizing Balls......Page 395
17.1.2 Rotors in Polygons......Page 402
17.1.4 The Relationship to Immobilization Problems......Page 413
17.2 Billiards......Page 416
17.2.1 The Bezdek–Bezdek Approach to Billiards......Page 417
17.2.2 Symplectic Geometry and Mahler's Conjecture......Page 421
18.2 Bodies of Constant Width in Art and Design......Page 428
18.3 Bodies of Constant Width in Engineering......Page 431
18.4 Cams and the Old Film Projector......Page 433
18.5 The Square-Hole Drill......Page 436
18.6 The Wankel or Rotary Engine......Page 438
18.7 Kenichi Miura's Water Wheel......Page 441
18.8 Noncircular Wheels......Page 442
Figure Credits......Page 447
Bibliography......Page 448
Index......Page 481
##### Citation preview

Horst Martini Luis Montejano Déborah Oliveros

Bodies of Constant Width An Introduction to Convex Geometry with Applications

Horst Martini Luis Montejano Déborah Oliveros •

Bodies of Constant Width An Introduction to Convex Geometry with Applications

Horst Martini Faculty of Mathematics Chemnitz University of Technology Chemnitz, Sachsen, Germany

Déborah Oliveros Instituto de Matemáticas Universidad Nacional Autónoma de México, Campus Juriquilla Querétaro, México

Luis Montejano Instituto de Matemáticas Universidad Nacional Autónoma de México, Campus Juriquilla Querétaro, México

ISBN 978-3-030-03866-3 ISBN 978-3-030-03868-7 https://doi.org/10.1007/978-3-030-03868-7

(eBook)

Library of Congress Control Number: 2018962136 Mathematics Subject Classification (2010): 52-01, 52A15, 52A21, 52A39, 52A40, 14M15, 70B10, 33C55, 53C45, 53C65 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book might well have been titled Some Geometry and Convexity instead of Bodies of Constant Width. Convexity is a small branch of classical mathematics located at the confluence of geometry, analysis, and combinatorics. Its origins can be traced back to Archimedes. In his works On the Sphere and the Cylinder and On Plane Equilibriums he defined the concepts, which we call today curves and convex surfaces. In these writings he postulated that every convex body contains its center of mass, a claim which was proved much later by H. Minkowski as a generalization of Cauchy’s well-known formulas for calculating perimeters and areas of curves and convex surfaces using the lengths and areas of their projections. The first mathematician who studied sets, curves, and surfaces characterized solely by their convexity properties was H. Brunn, in two articles published in 1887 and 1889: On ovals and convex surfaces and On curves without inflection points. These articles contained many results, some of them intuitively obvious, presented without proofs and with less rigor than would be demanded today, but in many cases quite deep. It was H. Minkowski who, appreciating the originality and profound nature of Brunn’s results, fleshed them out and shaped them into what is known today as the Brunn–Minkowski Theory. Along with results by A. D. Aleksandrov in the 1930s, this work ushered in the modern theory of convexity. Convexity has broadened considerably since then, new areas have opened up, and others “once forgotten” have been revitalized; examples are combinatorial aspects of convexity, the theory of convex polyhedra, and the local theory of Banach spaces. In addition, convexity has had a great influence on applications by way of its relationship to optimization and linear programming. The majority of books and expository articles on convexity mention bodies of constant width in one chapter or section. The authors found themselves falling under the spell of the mysterious beauty of bodies of constant width, and planned this book thinking in the transversality of the topic. Yet, in the writing, the subject, having started as bodies of constant width, became convexity. This book here had an antecedent, namely, a textbook published in Spanish “25 years ago” by one of the authors. But this textbook covered only some aspects of bodies of constant width. The present book deals with the most classical and representative results and techniques of standard convexity, including mixed volumes, spherical integration, Cauchy formulas, and others. Hence it might be considered as a textbook, but it also covers, mainly due to the notes at the end of each chapter, the existing material about bodies of constant width. Furthermore, the set of exercises at the end of every chapter reinforces both vocations because some of the exercises are simple and others can demand some research effort. We subdivided them into three classes regarding the degree of difficulty; they are correspondingly marked by stars (the most difficult exercises are marked with two stars). v

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Curves of constant width and their properties have been known for centuries. Leonhard Euler, in fact, studied them under the name “orbiforms” from the Latin word for circle-shaped curves. Euler was interested in figures of constant width, whose boundaries could be represented as the evolute of a hypocycloid. Nearly, a hundred years later, in 1875, Franz Reuleaux published a book on kinematics in which he mentioned curves of constant width and gave some examples. He later gave the construction of what might be considered as the simplest constant width curve which is not a circle, and today bears his name. Interest in bodies of constant width grew significantly near the beginning of the twentieth century. H. Minkowski, A. Hurwitz, and “shortly thereafter” E. Meissner were among the first who contributed to the area. In 1911, the Schilling Verlag [1030] published a collection of mathematical models, which included some constant width curves and models of constant width bodies, molded in plaster and inspired by Meissner’s examples. The list of further renowned mathematicians who have helped to extend the theory of constant width shapes contains the names of W. Blaschke, H. Lebesgue, K. Reidemeister, and, more recently, V. G. Boltyanski, A. S. Besicovitch, G. D. Chakerian, H. Groemer, and R. Schneider. It is evident from the number of recent research articles on bodies of constant width and closely related notions that the field is increasing. There exists a broad, diverse body of knowledge on bodies of constant width supported by an extensive, sophisticated theoretical framework. Many famous mathematicians have worked in the area, and the success of their popularization is due to the fascinating geometric nature of the topic. It is surprising, therefore, that this may be the first book ever dedicated exclusively to bodies of constant width. It is the hope of the authors that this book fulfills its goal as a textbook on geometry and convexity, but furthermore, that it is successful in conveying the attraction that constant width bodies have for those who fall under their spell. For the great help in the translation of parts of this book and the care of language style we are grateful to Margaret Schroeder. We thank Isaac Arelio and Juan Carlos Díaz Patiño who produced the figures. We also give special thanks to Natalia Jonard-Pérez for writing the section about hyperspaces. In addition, we wish to thank our colleagues and friends Vitor Balestro, Endre Makai, Jr., Zokhrab Mustafaev, Edgardo Roldan-Pensado, Valeriu Soltan, Konrad J. Swanepoel, Deyan Zhang, and Senlin Wu for reading parts of the book critically, to correct and enrich them! Chemnitz, Germany Querétaro, México Querétaro, México June 2018

Horst Martini Luis Montejano Déborah Oliveros

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 First Properties of Bodies of Constant Width . . . . . . . . 1.2 Mathematical Content of the Book . . . . . . . . . . . . . . . 1.3 Convexity and Constant Width in University Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Convex Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Convex Hull . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Support Function and Minkowski Addition . . . . 2.5 Blaschke’s Selection Theorem . . . . . . . . . . . . . . 2.6 Steiner Symmetrization . . . . . . . . . . . . . . . . . . . 2.7 The Brunn–Minkowski Theory . . . . . . . . . . . . . 2.8 Radon, Carathéodory, and Helly . . . . . . . . . . . . 2.9 Classification of Boundary Points . . . . . . . . . . . 2.10 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 The Problems of Busemann–Petty and Shephard 2.12 Ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12.1 Brunn’s Theorem . . . . . . . . . . . . . . . . . . 2.12.2 Blaschke’s Theorem . . . . . . . . . . . . . . . . 2.12.3 The False Center of Symmetry . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Basic Properties of Bodies of Constant Width . . . . . . 3.1 Diameters, Binormals, and Diametral Chords 3.2 The Minimum Width Condition . . . . . . . . . . 3.3 Projections and Sections . . . . . . . . . . . . . . . . 3.4 Circumsphere and Insphere . . . . . . . . . . . . . . 3.5 Minkowski Sum and Central Symmetry . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Figures of Constant Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Mizel Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.3 Alexander’s Conjecture . . . . . . . . . . . . . . 4.4 The Makai–Martini Characterization . . . . 4.5 Intersection Properties of the Boundaries . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Systems of Lines in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Pedal Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The Parametrization in Terms of the Pedal Function . . . . . 5.1.2 Areas, Perimeters, and the Cauchy Formula . . . . . . . . . . . 5.2 Systems of Lines and Their Envelopes . . . . . . . . . . . . . . . . . . . . . . 5.2.1 The Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Leaving on the Same Side Regions of the Same Area . . . . . 5.3 Systems of Externally Simple Lines . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Main Properties of Systems of Externally Simple Lines . . . 5.3.2 A Parametrization of Figures of Constant Width . . . . . . . . 5.3.3 The Analytic Curve of Constant Width due to Rabinowitz 5.3.4 Evolutes and Euler’s Constant Width Curve . . . . . . . . . . . 5.4 Figures Which Float in Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 A Figure Floating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Floating with Density One Half and Constant Width . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Spindle Convexity . . . . . . . . . 6.1 Spindle h-Convexity . . . 6.2 Ball Polytopes . . . . . . . 6.3 The Vázsonyi Problem . 6.4 Reuleaux Polytopes . . . Notes . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . .

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Complete and Reduced Convex Bodies . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Theorems of Meissner and Pál . . . . . . . . . . . . . 7.3 The Adjoint Transform . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Minkowski Difference and Strong Convexity 7.3.2 Pairs of Constant Width . . . . . . . . . . . . . . . 7.4 Reduced Convex Bodies . . . . . . . . . . . . . . . . . . . . . . 7.5 Regular Constant Width Hulls . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Examples and Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Reuleaux Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Reuleaux Tetrahedron . . . . . . . . . . . . . . . . . . . . . . . 8.3 Meissner Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Original Meissner Bodies . . . . . . . . . . . . . . . . . . . 8.3.2 Performing Surgery to the Reuleaux Tetrahedron 8.3.3 A Description of Meissner Bodies . . . . . . . . . . . .

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. . . 8.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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182 184 186 189 194

Sections of Bodies of Constant Width . . . . . 9.1 Concurrent Constant Width Sections 9.2 Thickness of a Body . . . . . . . . . . . . 9.3 Which Bodies Are Sections? . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .

8.4

8.3.4 Meissner Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.5 Meissner Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 The Bull; A Concrete Example . . . . . . . . . . . . . . . . . . 8.4.2 Constructing Bodies of Constant Width from Reuleaux Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic Constant Width Bodies . . . . . . . . . . . . . . . . . . . . . .

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197 197 202 204 205 207

10 Bodies of Constant Width in Minkowski Spaces . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Characterizations of Constant G-Width . . . . . . . 10.2.1 G-Diameters . . . . . . . . . . . . . . . . . . . . . . 10.2.2 The Homothety Theorem . . . . . . . . . . . . 10.3 Complete Bodies in Minkowski Spaces . . . . . . . 10.4 Strong G-Convexity and Perfect Norms . . . . . . . 10.5 Reduced Bodies in Minkowski Spaces . . . . . . . . 10.6 The Borsuk Conjecture in the Minkowski Plane . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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209 209 213 213 215 218 222 224 227 228 244

11 Bodies of Constant Width in Differential Geometry . . . . . . . . . . . . . . . . . . . . . . 11.1 The Support Parametrization in Terms of the Gauss Map . . . . . . . . . . . 11.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Positive Constant Gaussian Curvature . . . . . . . . . . . . . . . . . . . . 11.3 The Curvature of a Body of Constant Width . . . . . . . . . . . . . . . . . . . . . 11.4 Baire Category and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 The Geometry of a Body of Constant Width . . . . . . . . . . . . . . . . . . . . . 11.5.1 A Characterization of Constant Width in Terms of the Hessian . 11.5.2 The Local Geometry of the Boundary of a 3-Dimensional Convex Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Constant Width in Affine Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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247 247 249 249 253 256 259 262 262

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263 267 270 276

12 Mixed Volumes . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Mixed Volumes . . . . . . . . . . . . . . . . . . . 12.2 Surface Areas and Mixed Surface Areas . 12.3 The Projection Formulas of Cauchy . . . . 12.4 Mixed Volume Inequalities . . . . . . . . . .

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279 279 288 290 291

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x

Contents

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 13 Bodies of Constant Width in Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Spherical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Two Theorems on Spherical Integration . . . . . . . . . . . . 13.1.2 The Theorem of Minkowski . . . . . . . . . . . . . . . . . . . . 13.1.3 Firey’s Variant of Minkowski’s Theorem . . . . . . . . . . . 13.1.4 A Characterization of Central Symmetry . . . . . . . . . . . 13.1.5 Aleksandrov Type Theorems for Sections . . . . . . . . . . 13.2 Fourier Series and Spherical Harmonics . . . . . . . . . . . . . . . . . . 13.3 A Third Theorem of Spherical Integration . . . . . . . . . . . . . . . . 13.3.1 Centrally Symmetric Bodies That Float in Equilibrium 13.3.2 Convex Bodies of Constant Brightness . . . . . . . . . . . . . 13.3.3 Constant Outer k-Measure Bodies . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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299 299 299 301 301 302 303 304 309 309 310 313 314 320

14 Geometric Inequalities . . . . . . . . . . . . . . . . . . . . . 14.1 Isoperimetric Inequalities . . . . . . . . . . . . . . 14.2 The Blaschke–Lebesgue Problem . . . . . . . . 14.3 Measures of Asymmetry . . . . . . . . . . . . . . 14.4 Inequalities Involving the Circumradius . . . 14.5 A Bonnesen-Type Isoperimetric Inequality . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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321 321 323 327 328 330 332 341

15 Bodies of Constant Width in Discrete Geometry . . . . . . . . . . . . 15.1 Helly’s Theorem and Constant Width . . . . . . . . . . . . . . 15.1.1 The ðp; qÞ-Property and the Piercing Number . . . 15.1.2 Hyperplane Systems and the Kneser Conjecture . 15.2 Universal Covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Universal Covers in the Plane . . . . . . . . . . . . . . 15.2.3 Universal Covers in n-Space . . . . . . . . . . . . . . . 15.2.4 Minimal Universal Covers . . . . . . . . . . . . . . . . . 15.2.5 Strong Universal Covers . . . . . . . . . . . . . . . . . . 15.3 Packing, Covering, Lattice Points . . . . . . . . . . . . . . . . . . 15.3.1 Packing and Covering . . . . . . . . . . . . . . . . . . . . 15.3.2 The Borsuk Conjecture . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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343 343 344 346 348 348 349 351 354 355 355 355 356 359 366

16 Bodies of Constant Width in Topology . . . . . . . . . . . . . . . . . . . 16.1 The Topology of Certain Hyperspaces . . . . . . . . . . . . . . 16.1.1 The Hilbert Cube . . . . . . . . . . . . . . . . . . . . . . . . 16.1.2 Hilbert Cube Manifolds . . . . . . . . . . . . . . . . . . . 16.1.3 Topology of Some Hyperspaces of Convex Sets . 16.2 Transnormal Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Fiber Bundles and Universal Covers . . . . . . . . . . . . . . . 16.4 Recognizing Constant Width . . . . . . . . . . . . . . . . . . . . .

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369 369 371 372 376 379 383 386

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Contents

xi

16.4.1 16.4.2 16.4.3 16.4.4 16.4.5

A Little Bit of Homology and Cohomology . Schubert Cycles and Cocycles . . . . . . . . . . . The Geometry of Grassmannians . . . . . . . . . Recognizing Bodies of Constant Width . . . . Recognizing Balls . . . . . . . . . . . . . . . . . . . .

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386 386 388 389 392 394 397

17 Concepts Related to Constant Width . . . . . . . . . . . . . . . . . . . . 17.1 Rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2 Rotors in Polygons . . . . . . . . . . . . . . . . . . . . . 17.1.3 Rotors in Regular Polyhedra . . . . . . . . . . . . . . 17.1.4 The Relationship to Immobilization Problems . 17.2 Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 The Bezdek–Bezdek Approach to Billiards . . . . 17.2.2 Symplectic Geometry and Mahler’s Conjecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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399 399 399 399 410 410 413 414 418 420 423

18 Bodies of Constant Width in Art, Design, and Engineering . 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Bodies of Constant Width in Art and Design . . . . . 18.3 Bodies of Constant Width in Engineering . . . . . . . . 18.4 Cams and the Old Film Projector . . . . . . . . . . . . . 18.5 The Square-Hole Drill . . . . . . . . . . . . . . . . . . . . . . 18.6 The Wankel or Rotary Engine . . . . . . . . . . . . . . . . 18.7 Kenichi Miura’s Water Wheel . . . . . . . . . . . . . . . . 18.8 Noncircular Wheels . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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425 425 425 428 430 433 435 438 439 441 442

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Figure Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

Chapter 1

Introduction

In the great garden of Geometry, everyone can pick up a bunch of flowers, simply following his taste. David Hilbert

The circle is the geometric locus of all points equidistant from a fixed point called its center. It is precisely due to this property that wheels are round or circular in shape. An axle placed at the center of the wheel does not move up and down when the circle turns. It only moves laterally at a constant height from the ground, and this is because every ray of the circle from the axle to the edge of the wheel has the same length. Before using it for the wheel, the circle was applied to transportation in a more primitive way, the roller. A heavy object may be moved by placing it on top of a number of round rollers (Figure 1.1). When these rotate, the object moves without bobbing up and down. Surprisingly, these two applications of the circle—wheel and roller—are based on radically different principles. The characteristic of the circle which makes it suitable as a wheel is quite different from the property which makes it work as the cross section of a roller. In fact, wheels can only be round, but there are non-round rollers which work just as well as round ones. Let us examine what property of the circle causes round rollers to fulfill their function; it has nothing to do with the center of the circle. What matters is that as the roller rotates, it holds the load at a constant height from the ground because its width is the same in every direction. It is clear that, if the rollers had elliptical cross sections, the load would bob up and down as they turned, and soon it would slide to one side or the other and fall off the rollers. This is because the ellipse has different widths in different directions (see Figure 1.2).

Figure 1.1 © Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_1

1

2

1 Introduction

Figure 1.2

There are, however, other shapes which are not circles but still have the property that their width is the same in every direction; rollers with cross sections of such shapes work every bit as well as round rollers.

1.1 First Properties of Bodies of Constant Width Let us begin by defining what is meant by the width of a figure φ in a given direction. Choose a direction and a pair of support lines perpendicular to this direction. These are lines which grip φ and hold it fast. The distance between these two parallel support lines is the width of φ in the chosen direction (Figure 1.3). Figures that have the same width in every direction are called figures of constant width, and they are known for a long time along with some of their properties. In fact, Leonhard Euler [323] studied them under the name orbiforms, from the Latin word for circle-shaped curves. Euler was interested in constant width curves whose boundaries could be represented as the evolute of a hypocycloid. It was nearly a hundred years later, in 1875, that Franz Reuleaux, an engineer and mathematician who taught in the Royal Technical School in Berlin, published the book [968] on kinematics in engineering, in which he treated also figures of constant width. In this book, he described what is perhaps the simplest constant width shape after the circle: the circular-arc triangle shown in Figure 1.5. He described the first mechanical usage of this geometric figure to Hornblower, the inventor of the compound steam engine.

Figure 1.3 The width of a figure ψ in a direction d

1.1 First Properties of Bodies of Constant Width

3

Figure 1.4 Windows in Gothic Cathedrals

This figure, which today bears his name (namely, Reuleaux triangle), had already been known for a long time (Figure 1.4), but Reuleaux was the first to focus attention on its constant width properties. This figure, the well-known Reuleaux triangle, is constructed as follows: Let abc be an equilateral triangle with sides of unit length. Draw a circular arc with unit radius from b to c with center at a (Figure 1.5). Now draw another circular arc from c to a with center at b, and finally, a circular arc with unit radius from a to b with center at c. The resulting figure, whose convex hull can also be described as the intersection of three disks of radius 1 centered at a, b, and c, is called a Reuleaux triangle. The series of diagrams in Figure 1.6 may convince the reader that a set of rollers with cross sections shaped like Reuleaux triangles will work just as well as round rollers. As the Reuleaux triangle rotates, it always touches the ground and the block. The triangle is first supported at the vertex c while the block glides over the arc ab. In the second position, the arc bc rolls on the ground while a holds up the block. There are related applications of Reuleaux triangles in our everyday life. For example, in the 1950’s the Philadelphia Fire Department started to use the shape of Reuleaux triangles for the fire hydrant shaft, to keep it safe from pranksters who want to chill out in the summer heat. Namely, the parallel jaws of an ordinary wrench would slip around these “curves of constant width”, in the same way as they would do it also around a shaft of circular shape (and so nothing could be turned this way). Clearly, firefighters have suitably shaped wrenches (see p. 84 of [988]). There are, in fact, many figures of constant width. For example, it is possible to construct a figure of constant width based on regular polygons with an odd number of sides (Figure 1.7); there are infinitely

Figure 1.5 The Reuleaux triangle

4

1 Introduction

Figure 1.6 A Reuleaux triangle rolling between the ground and a block

many non-regular polygons suitable for such constructions (Figure 1.9). And there are even many more possibilities to construct figures of constant width. One of these possibilities is to start with an irregular star-shaped polygon, whose vertex number is odd and whose sides are all of the same length like the seven-pointed star-shaped polygon in Figure 1.8. Draw circular arcs whose centers are the vertices that connect every pair of adjacent opposite points.

Figure 1.7 A figure of constant width based on an equilateral pentagon

The corners of the figure may also be rounded by extending all the sides of the star-shaped polygon to the same length and joining their ends with circular arcs whose centers are the opposite points of the star. Another example of a figure of constant width may be constructed from the starting point of a square abcd whose diagonal ac is of length one, say (see Figure 1.9), and adding points e and f such that the distances ed, ec, a f , and d f are all of unit length. Then, by drawing the circular arc e f with center at d which passes through b, the arc ae with center at c and radius one, the arc f c with center at a and, finally, the arcs dc and ad with centers at e and f , respectively, we obtain our desired figure. The figures of constant width described up to this point are all constructed by circular arcs. As we shall see later in Chapter 5, there are figures of constant width that may be constructed in such a way that they do not contain circular arcs in any portion of their boundary. In fact, it is possible to construct

Figure 1.8 A curve of constant width based on a star-shaped polygon and a smoothened version of it

1.1 First Properties of Bodies of Constant Width

5

Figure 1.9

a curve of constant width, different from the circle, which is analytic. That is, there exists a polynomial equation whose zeros are the points of a noncircular curve of constant width. As in the planar case, solid sets of constant width also exist in higher dimensions. In these cases one defines the width of a body φ in a given direction by choosing a pair of parallel support hyperplanes, which hold φ tight and are orthogonal to that direction. Then the distance between these two hyperplanes is the width of φ in this direction. Hence a body of constant width is a convex body with the property that its width is the same in every direction. A nice example of a 3-dimensional body of constant width is the solid of revolution obtained from the Reuleaux triangle rotating it around one of its axes of symmetry, see Figure 1.10. As we shall see later in Chapter 8, the construction analogous to that of the Reuleaux triangle in dimensions higher than two does not result in a body of constant width. For instance, if we consider the regular tetrahedron and take the intersection of the four solid unit spheres, whose centers are the vertices of this simplex, it turns out that the result is not a body of constant width. It is, however, possible to round three of its curved edges (see [1204] and [818]) to obtain two 3-dimensional analogues of the Reuleaux triangle called the Meissner solids (see Figure 1.11). Later in the book we will be able to construct 3-dimensional constant width bodies with the help of special embeddings of self-dual graphs, and also to construct constant width bodies with analytic boundaries. Bodies of constant width have diameters (chords of maximal length) in every direction. However, unlike the diameters of a ball, those of a general body of constant width do not always meet at a single point, and when they do it, it is because the body is indeed a solid ball, see Chapter 3. The failure to recognize that non-round shapes may have constant width can, and in fact has had, disastrous results in practice. One example arises in testing the roundness of a submarine hull during construction of a vessel. For instance, it might be thought that it is sufficient to measure its width in all directions. But, in fact, the cross section of the submarine may be grossly distorted, and yet pass such a test. It is for this reason that the roundness of a submarine hull is always tested using circular templates. In fact, one of

Figure 1.10

6

1 Introduction

Figure 1.11 A Meissner solid (University of Toronto Libraries)

the contributing causes of the 1986’s Challenger Space Shuttle accident was the failure of the O-rings to seal the joint between cylindrical-shaped rocket motor segments. The report [984] on the accident mentioned that the segments were significantly non-round at the failed joint. The fit of the segments had been checked by measuring their diameters at six positions, 30 degrees apart, but equal diameters are not guaranteeing a circular shape. The report recognized that it would have been necessary for the midpoints of all the diameters to coincide at a single point for the rocket motor segments to have had circular cross sections. In fact, we will see in Chapter 3 that for bodies of constant width the notions of normal chords, affine diameters, and diameters coincide.

1.2 Mathematical Content of the Book The aim of this book is to study bodies of constant width from different points of view. Being the central theme of this book, bodies of constant width are related to many areas of mathematics, for example: analysis, convex geometry, topology, differential geometry, combinatorics, and others. This textbook shows in practice the amazing fact that all these areas of mathematics are related to each other. For example, from the differential geometry point of view, the notion of system of lines is developed to parametrize convex curves, and this will be used to construct the constant width curve of Euler (Figure 1.12), to study constant width figures as duals of figures that float in equilibrium in every position with density one half, and to prove interesting further results, for example, that any figure of constant width contains at least three semi-disks of the same diameter (Chapter 5). Furthermore, the notion of curvature is studied extensively in Chapter 11. It is proved, for example, that the sum of the radii of curvature at the endpoints of a diameter of a body of constant width equals the width, and that a typical convex body of constant width has the property that at almost all points in its boundary all radii of curvature are equal to the width. We will submerge into the fascinating world of the study of local geometry of the boundary of a convex body.

1.2 Mathematical Content of the Book

7

Figure 1.12 Euler’s constant width curve

From the discrete geometry point of view, the connection between bodies of constant width and Helly’s theorem is established (Chapter 15). Interesting and relevant topics from discrete geometry are treated, for example: the theory of universal covers in polyhedral theory, the Kneser Conjecture, ( p, q)-problems in combinatorics, etc. From the topological point of view, in Chapter 16 we study the hyperspace of constant width bodies and use the theory of fiber bundles to prove that every set of diameter 1 in 3-space is contained in the rhombic dodecahedron that circumscribes the sphere of diameter 1. Furthermore, the topology of Grassmannians is used to recognize constant width by means of some sections. Two powerful techniques, mixed volumes (Chapter 12) and spherical harmonics (Chapter 13), are developed in the book to prove properties of bodies of constant width. For instance, the perimeters of all orthogonal projections of a 3-dimensional body of constant width are equal. The converse is also true. This deep result, originally proved by Minkowski, stands at the origin of the theory of spherical harmonics. Another interesting result is that for bodies of constant width 1 the volume depends linearly on the surface area. It is highlighted in this book, like also the fact that these techniques allow us to relate the study of bodies of constant width with well-known conjectures, such as the Borsuk conjecture, the Mahler conjecture and the Blaschke–Lebesgue conjeture. The behavior of bodies of constant width in finite-dimensional real Banach spaces (Minkowski spaces) is studied in Chapter 10; it is of special interest for us not only by its conceptual wealth, but also for the important relations to other areas of geometry. Staying with this motivation, we give at the end of this chapter also a brief overview to what has been done about constant width sets in other non-Euclidean geometries. We also show the close relation of bodies of constant width to other charming and important mathematical objects, such as bodies of constant brightness (Chapter 13), rotors, billiards (Chapter 17), and transnormal manifolds (Chapter 16). Curves of constant width, in particular the Reuleaux triangle, have been exploited by engineers to design a number of ingenious devices and mechanisms. In Chapter 18 we will describe several different engineering applications of these curves; these are cams, drills which produce square holes and pistons in rotary internal combustion engines, as well as the charming water wheel of Kenichi Miura and the amazing devices that allow noncircular figures of constant width to be used as wheels (not rollers), yet allowing the vehicle to run smoothly on a plane surface. Of course, the basic tool for the study of bodies of constant width is convex geometry. Although we do not present a monograph on convexity, in Chapter 2 we introduce the classical material, in particular the Theory of Brunn–Minkowski, the notions of Steiner symmetrization, support functions, and Hausdorff metric, also Helly’s Theorem and its relatives, a classification of boundary points, the problem of Busemann–Petty, ellipsoids, etc.

8

1 Introduction

The interest in bodies of constant width grew significantly around the beginning of the twentieth century. Minkowski, Hurwitz and, shortly thereafter, Meissner were among those who contributed first to the area. Two books and a survey should be mentioned for their marked influence on the theory of bodies of constant width. The first is the delightful exposition by Rademacher and Töplitz [958], which includes a section devoted to constant width shapes and their basic properties. The second is the classical work on convexity by Bonnesen and Fenchel [160], which contains nearly everything that was known about constant width bodies at the time of writing it. The survey of Chakerian and Groemer [238] (updated by [527] and [793]) can be considered as an excellent guide. Other renowned mathematicians who have helped to extend the theory of constant width shapes are Blaschke [130], Lebesgue [707], Reidemeister [964] and, more recently, Boltyanski [1204], Besicovitch [106], Chakerian [230], Schneider [1039], and many others. The 18 chapters in this book pretend to cover most of the materials regarding bodies of constant width, traveling from the eighteenth century until today through many important areas of mathematics, such as: discrete and differential geometry, topology, Fourier analysis, spherical harmonics, variational methods, fiber bundles, Banach spaces, and convex geometry, showing throughout the book important geometric techniques, analytical tools, and relevant problems that remain open. We added many exercises, which could be also inspiring for further research. It is our goal to make clear that the interest in bodies of constant width created and creates an important and expanding area of mathematics, offering a convenient approach to larger fields (like, e.g., convexity or Banach space theory).

1.3 Convexity and Constant Width in University Teaching Our main purpose is to give a comprehensive representation of all the knowledge about sets of constant width and closely related topics, showing also (as good as it is possible for us) the state of the art regarding this classical notion and its modern generalizations and modifications. Thus, from this point of view we try to be close also to recent research topics, e.g., by giving interesting open research problems. But on the other hand, namely seeing how broad the subject is, we also try to stay close to all kinds of readers interested in this wonderful partial field of convexity, i.e., particularly also to nonspecialists. For that reason we are naturally motivated (and also obligued) to give at least suggestions on how to use this book in the framework of university teaching. Having this in mind, we propose now two structurized ways “through the book” which can be used for two conceivable university courses. Namely, we guide the reader through respectively selected chapters and sections containing basic topics which would be important for designing courses in convexity. The first conceivable course would be suitable for undergraduates, and the second one basically for graduates. Both these courses might be planned for a frame time of around 46 hours, in each case Course 1: Undergraduate Convex Geometry Course The following guide through sections of the book suggested contents and their natural sequence suitable for an undergratuate course on Basic Convex Geometry. Chapter 2 Sections 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10

1.3 Convexity and Constant Width in University Teaching

9

Chapter 3 Sections 3.1, 3.2, 3.5

Chapter 5 Sections 5.1, 5.2

Chapter 7 Sections 7.1, 7.2, 7.3

Chapter 18 All sections

Course 2: Two Graduate Courses in Convex Geometry The following two lists of chapters present a conceivable sequence of topics suitable for a two-parts course in convex geometry for graduates. The green selection is suggested for people with more discrete geometry flavor, whereas the red one is intended for people with more analysis type of inclinations. Advanced convex geometry course Chapter 2 Chapter 3 Sections 3.2, 3.5 Chapter 10

Advanced convex geometry course with discrete geometry flavor Chapter 15 Chapter 6 Chapter 8 Chapter 17

Advanced convex geometry course with analysis flavor Chapter 11 Chapter 12 Chapter 13 Chapter 14

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1 Introduction

Further possible courses, in which this book could be used as (basic or supplementary) literature, are Convex Geometry and Geometric Tomography (mainly based on the monographs [401] and [1039], and certainly benefiting from Chapters 2, 7, 9, and 10) and Harmonic Analysis and Geometry (e.g., based on [649], perhaps using our Chapters 12, 13, and 14). And also undergraduate courses like “Euclidean Geometry” or “Fundamental Concepts in Geometry” would certainly benefit from the first four chapters of our book. The reader is additionally referred to the following Notes, in which also references are given which are close to teaching aspects of the field.

Notes One can see in the text above that curves and bodies of constant width have, as famous geometric figures, many inspiring properties, aspects, and applications. Even more: behind a seemingly simple and “narrow” definition of a class of geometric figures an unexpectedly large variety of properties and applications is “hidden”, not visible at first glance and discussed here in view of more than thousand (directly or indirectly related) mathematical references! This is sufficiently potential to give the nonexpert also some feeling how intuitive and refreshing notions and methods from convexity are, and how beautiful this field of geometry is. Thus, constant width sets might play a very important role in the following sense. They present an excellent starting field to become familiar with convex geometry! Even the authors were surprised when they observed how many research papers related to this topic exist! We will try to reflect the widespread occurrence of constant width sets in the literature, partially also in a compressed way, i.e., the vast amount of references on certain aspects of constant width sets is mainly concentrated in concise notes at the end of each chapter of the book. Clearly, historically we have to start already with medieval times, namely with Gothic church windows. But coming to mathematics, we first have to mention three famous names. Namely, Euler (cf. [323]), Minkowski (see [837] and the interesting second appendix of [835]), and Lebesgue (see [707]) looked already explicitly at the phenomenon of noncircular constant width sets. More precisely, Euler was most likely the first who explicitly looked for mathematical properties of curves of constant width and called them orbiforms (see also the historical study [1199]). He was interested in the construction of closed convex curves (called catoptrices) having the property that a ray, starting from an interior point of the curve, reaches this point again after two reflections at the curve (the ellipse is a well-known example). He started with so-called triangular curves having three concave sides and three cusps, and he showed that the evolvent of them is a curve of constant width, which then was used to construct a catoptrix. In large parts, Minkowski created the fundaments of convexity and, via the geometry of numbers, also of parts of discrete geometry. Many basic notions (like support function, see, e.g., [836]) are due to him, and in his early works one finds also investigations of bodies of constant width (see [837] and the second appendix of [835]). Lebesgue [707] proved that the Reuleaux triangle has the smallest area among all curves of the same constant width (the circle clearly yields the other extremum), and one year later this was independently reproved by Blaschke [131]. Here we also mention that the astronomer and mathematician Barbier proved already in 1860 that all planar sets of fixed constant width h have the same perimeter, namely π h, see [75]. Another historical milestone is the contribution of Reuleaux, a German professor in mechanical engineering. In section 22 of his classical book [969] (see also the German original [968]) on the theory of machines he described the curves called after him, and their properties were developed in later sections of that interesting monograph. More historical background regarding Reuleaux and other contributors to the field of kinematics in engineering is given in the nice book [854]. We mention here also the interesting paper [648] about Burmester, distinguished professor of descriptive geometry and kinematics at the University of Munich. He was responsible for much of the development of scientific mechanical engineering in the nineteenth century, and the article [648] interestingly illuminates also the rivalry between Burmester and Reuleaux.

1.3 Convexity and Constant Width in University Teaching

11

We continue now with mentioning books and expository articles in which constant width sets are discussed. The scheme of our discussion is (simply following the occurrence in the literature) structured by the sequence classical geometry, recreational mathematics, differential geometry, and convexity. Thus, we note first that, with their beauty and impressive variety of shapes, curves and bodies of constant width are well established in classical geometry, occurring in correspondingly famous monographs like in § 32 of [540] and in subsection 12.10.5 of [100]. They are also discussed in history of mathematics (see, e.g., § 7.1 of [1053]) and occur as interesting mathematical models in many collections and publications, cf. [1029], [818], [274], [362], and chapter 2 of [143] (the latter containing also a broad mathematical discussion). Books close to recreational mathematics or education in mathematics, suitable for students and also for interested non-mathematicians, are [550] and [26], not to forget the treasure of related publications of Martin Gardner appearing often as sections in his popular, nice books (see, e.g., [398]). Typical goals in such books are the repeatedly occurring descriptions of nice properties of Reuleaux polygons, constructions also of general constant width sets and their parallel curves, and applications (like rotors). Such a detailed discussion of the construction of plane curves of constant width is, for example, given in the nice elementary book [950], pp. 107–123; discovering mathematical facts is presented there in the form of discussions between Salviati, Sagredo, and Simplicio in the style of Galileo’s famous book. Here we mention also the paper [537], in which the authors found a way to define and investigate sets of constant width on a chessboard with n rows and n columns. From the viewpoint of differential geometry and classical curve theory, plane and spatial curves or surfaces of constant width are interesting geometrical objects, see Chapter 11 and also [731], to give also an early comprehensive reference. Unfortunately, we are not aware of a monograph discussing in a systematic way properties of surfaces of constant width in the spirit of differential geometry (but we refer, however, to Chapters 11 and 17 here). Only the planar case is handled in several classical books. With their relation also to Zindler curves, plane curves of constant width are presented in the first chapter of [1103], and in connection with isoperimetric properties and vertex theorems they occur in Chapter 3 of [830]. Extremal properties of plane curves of constant width are discussed in Chapter 2 of [561], and two further books to be mentioned here are [1147], written as a classical course in differential geometry and studying also curves of constant width in the first chapter, and [18], where the concept of constant width can be found in the fifth chapter. We come now to books discussing constant width sets in the spirit of convexity. Early ones are [132] (see appendices III and IV there, where also a body of constant brightness is presented), [958, § 20b], and [160, § 15], followed by the monograph [1204]. Three problem books, in which constant width sets and closely related topics occur directly or indirectly several times, are [311], [272], and [635]. Textbooks in convexity, suitable for teaching undergraduates or graduates and containing sections about constant width sets are [312], [1140], [98] (containing also a chapter on Minkowski geometry), [737], [706], [618], [1134], [152], [654], [1163], [850] and [719] (the latter together with [720]). Concrete subjects discussed therein are, e.g., constructions, the Blaschke–Lebesgue theorem, Reuleaux polygons, completeness, related notions (e.g., equichordality), Barbier’s theorem, the incircle–circumcircle relation, double normals, and sometimes also 3-dimensional analogues. There are also books from neighboring fields, where constant width sets are shortly presented. As example we mention the field of geometric probability, with the famous monograph [1021] (see Section 4 in Chapter 1 there). Monographs being closer to recent research on bodies of constant width and related notions are [1039, Chapter 3], [464, Chapter 5], [401, Chapters 3, 4, 6, 7, and 9], and [1124, Chapter 4]. Subjects and applications discussed there in a deeper way are projection and section functions, rotors, completeness, geometric inequalities, related notions like constant brightness or equichordality, constant width in Minkowski spaces, and others. One should also underline once more the historical importance of the book [160, § 15] regarding research on constant width sets in the first half of the twentieth century.

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1 Introduction

Finally we mention surveys and expository articles on sets of constant width and their main aspects. The most comprehensive representation of the field with a large variety of references (after [160] and until 1983) is the basic survey [238], covering practically all aspects of constant width sets. Ordered with the same arrangement of aspects, this survey was updated by [1192](1991) and [527] (1993), and in 2004 (again with the same arrangement of aspects) an analogous survey on constant width sets in Minkowski spaces followed, see Section 2 of [793]. A wonderful expository paper on curves and surfaces of constant width is [970] which, unfortunately, was not published until now. Many aspects are discussed there, and various new results are also derived. Also in the exposition [85] curves and surfaces of constant width in two and three dimensions are discussed. Two older surveys on constant width sets are [598] and [99]. In the first one, basic properties (mainly in the spirit of our Chapter 3) are reproved, and the second one gives an update of § 15 of [160]; basic properties of planar constant width sets and inequalities for them are derived, and also curves of constant width in space and in Minkowski geometry are presented. Two further papers, presenting known results in a really nice way, are [137] and [363]. For example, the elementary paper [363] is based on the characterization of constant width sets in terms of Fourier series, setting the used context at a level suitable for undergraduates; it is also presented how to construct sets of constant width from midpoint curves. Two older papers, in which known results on curves and surfaces of constant width are presented in a new analytic way (e.g., via support functions) and their mechanical applicability regarding the transformation of alternating movements is shown, are [1127] and [1128]. The authors of [161] and [1161] had certainly similar aims; parts of their contributions are also close to mathematical education. The survey [612] describes (with some history and nice figures) the state of the art regarding the conjecture that Meissner’s bodies are of minimal volume among all 3-dimensional bodies of the same constant width. The paper gives a good overview to the problem, is at the same time readable for experts and non-experts, and collects known results supporting the conjecture (here we mention also the problem paper [609] and [272, A 22]). Two further elementary surveys are [1130] and [85]. The first one refers only to basic properties in the planar case, and in the second also properties of 3-dimensional constant width bodies are derived, where the support function is the main tool used there. The problem collection [1106] and the surveys [302] and [722] refer to problems from convexity and contain parts discussing nicely constant width sets, see also [823] and [825].

Chapter 2

Convex Geometry

Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things. Isaac Newton

2.1 Introduction Convexity is a classical branch of mathematics tucked in between geometry, analysis, and combinatorics. It is also closely related to many other areas of mathematics such as number theory and graph theory, and of course differential geometry and classical analysis, the calculus of variations, functional and convex analysis, algebraic geometry, topology, and mathematical physics, among others. Its origins can be traced back to Archimedes, who, in his works On the Sphere and the Cylinder and On Plane Equilibriums, defined the concepts which we today call curves and convex surfaces. He postulated that every convex body contains its center of mass, claim that was not proved until later by Minkowski, when generalizing Cauchy’s well-known formulas for calculating perimeters and areas of curves and convex surfaces using the length and area of their projections. One of the first mathematicians who studied sets, curves, and surfaces characterized solely by their convexity properties was Brunn. In two articles, On ovals and convex surfaces and On curves without inflection points published in 1887 and 1889, he listed results without proofs and with less rigor than would be required today. But these papers were nevertheless quite deep. It was Minkowski who, appreciating the originality and profound nature of Brunn’s results, put them into a context and shaped them into what is today known as the Brunn–Minkowski Theory. This work, in combination with that of Aleksandrov in the 1930s, ushered in the modern theory of convexity. Convexity has broadened considerably since then; new areas have opened up, and others, “once forgotten”, have been revitalized, such as combinatorial aspects of convexity, the theory of convex polytopes, and the local theory of Banach spaces. In addition, convexity has had a great influence on applications through its relationship with fields like optimization and linear programming. For a profound treatment of convexity, we have [98], [1204], [312], [706], [1140], and [477] and the survey articles [278], [347], [633], and [471], see also our notes at the end of this chapter. A more complete study of convexity and the Brunn–Minkowski Theorem may be found in [1039]. We can particularly recommend the excellent classic work on convexity by Hadwiger [502]. For applications, [973] and [978] may be consulted. Finally, a two-volume handbook on convexity edited by Gruber and Wills presents important results on the topic [483]. Although we will introduce notation and terminology as necessary, in this section we start by saying that En denotes the Euclidean n-space with origin o, scalar product ·, · and the induced norm | · |. We will denote by B n ( p, r ) the n-dimensional ball with center at p and radius r . When the dimension © Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_2

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is not essential, we simply denote it by B( p, r ), and when the center or the radius are not important we will denote it by B(r ) or B( p), or simply by B. Similarly, Sn−1 ( p, r ) will denote the boundary of B n ( p, r ), and S( p, r ) the boundary of B( p, r ), and S(r ) or S( p) the boundary of B(r ) or B( p), respectively. When there is no danger of confusion we will also denote by Sn−1 the (n − 1)-dimensional unit sphere Sn−1 (o, 1). Usually (but not always) elements in En or points in general are denoted by lower case letters, subsets are denoted by capital letters, and the greek letters φ or ψ denote convex sets.

2.2 Basic Concepts We say that a set in Euclidean n-space En is convex if for every two points of the set, the segment joining them is completely contained within the set (Figure 2.1). Specifically, a set φ ⊂ En is convex if given two points p, q ∈ φ, the line segment between p and q, pq = {(1 − t) p + tq | 0 ≤ t ≤ 1}, belongs to the set φ. Examples of convex sets are line segments, planes, half-spaces and (now always meant with their interior points) ellipses, parallelograms, triangles, tetrahedra, cubes, and balls. More examples can be constructed by intersecting several convex sets and taking the parts in common to all of them. Namely, if we take any two points in the common region, the segment joining them is in all of them (since they are convex) and is therefore in their intersection (Figure 2.2). In other words the following theorem is true. Theorem 2.2.1 The intersection of a family of convex sets is a convex set. Intuitively, we think of a convex body as one which is not “dented”. Picture a “dented” plane figure (Figure 2.3), note that it is precisely at the dent where a line segment could lie with its ends in the figure yet without being itself contained entirely within the figure. In contrast, where the boundary of the plane figure is not dented, it is possible to draw a line touching the boundary such that all of the figure is entirely on one side of the line. We will give a name to these lines; a line that touches the figure such that the figure is entirely contained in one of the two half-planes defined by the line will be called a support line of the figure, see Figure 2.4. Faithful to the intuitive idea that such a figure has no dents, we are tempted to define a figure as convex if through every point of the boundary passes a support line. In fact, we shall see below that this is the case.

Figure 2.1 Convex sets

2.2 Basic Concepts

15

Figure 2.2 The intersection of convex sets is convex

We will assume that the reader is familiar with the basic concepts and terminology of analysis such as the concepts of interior, exterior, and boundary points of a set, the terms open and close, and the basic definitions of limits and continuity, as well as the concepts of bounded and unbounded sets.

Figure 2.3

In particular, the proofs of the following lemmas, linking these concepts to the concept of convex sets, are left as exercises to the reader. Lemma 2.2.1 Between a point p of a set P and a point q not in P there is always a boundary point. That is, there is always a boundary point of P in the segment pq. Lemma 2.2.2 All points between two interior points p and q of a convex set φ are interior points.

Figure 2.4

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Figure 2.5 Dented sets are not separated by hyperplanes

Lemma 2.2.3 All points strictly between an interior point p and a boundary point q of a closed convex set φ are interior points. Lemma 2.2.4 Strictly between two boundary points p and q of a closed convex set φ, either all the points are boundary points or all the points are interior points. We shall denote by bd φ and intφ the boundary and interior, respectively, of a closed, convex set φ. Recall that a closed, bounded set is called a compact set, and a compact, convex set is called a convex body. Finally, a convex body φ ⊂ E2 is also called a convex figure. Lemma 2.2.5 If a line L passes through an interior point p of a convex body φ, then L intersects the boundary of φ at exactly two points. Returning to the intuitive idea that convex bodies do not have dents, we note another characterizing property of convex sets. If a set is not dented, then any point not in the figure can be separated from it by a line (in the plane) or a hyperplane. Observe that this does not happen with a dented set, because the dent forms a hollow whose points cannot be separated from the set by a hyperplane (see Figure 2.5). Given two closed sets in Euclidean n-space, we say that they are separated by the hyperplane H if they lie in different open half-spaces determined by H . The following characteristic property of convex bodies can be stated in a theorem. Theorem 2.2.2 A closed set φ is convex if and only if for any point p not in φ there exists a hyperplane that separates p from φ. Proof Let us take a closed, convex set φ and any point not in it; call it p. Since φ is closed, of all the points of φ, let us consider the one nearest to p, which we will call q (see Figure 2.6). If we consider the ball B( p, r ) centered at p and having radius r , equal to the length of the segment pq, then the interior of B( p, r ) does not contain points of φ; otherwise there would be a point of φ whose distance from p would be less than the distance from p to q. Therefore, by convexity of φ, the point q is unique. We will show that the hyperplane H  tangent to B( p, r ) at q has the property that there is no point of φ on the same side as p. If there were, say, a point s of φ on the same side as p, then, since φ is convex, the segment qs would intersect the interior of B( p, r ), and there would then be a point of φ whose distance from p would be less than the distance from q to p. But this would be impossible because q was chosen precisely to be the point of φ closest to p. Now it is clear that the hyperplane H parallel to H  through the midpoint of pq separates φ from p. Assume now that we have a closed set φ with the property that, for any point p not in φ, there is a hyperplane that separates φ from point p. We wish to show that φ is a convex body. To do so, it will be necessary to take two points a and b of φ and to show that the line segment joining them is

2.2 Basic Concepts

17

Figure 2.6

also in φ. We claim that the segment ab is in φ. Suppose there is a point c between a and b that is not in φ. If this is so, then there is a hyperplane that separates {a, b} from c, which is impossible.  The following theorem is a corollary of Theorem 2.2.2. Theorem 2.2.3 Let φ be a proper closed convex subset of En . The intersection of all closed half-spaces containing φ is precisely φ. Proof First note that if φ is not in En , there is at least one closed half-space that contain φ. Clearly, φ is contained in the intersection of all the half-spaces that contain it, but this intersection could be bigger than φ. Every point not in φ can be separated from φ by a hyperplane. Therefore, there exists a closed half-space that contains φ but not this point. Thus the intersection of all closed half-spaces that contain φ is contained in φ.  A hyperplane H is called a support hyperplane of a closed set in En if H contains points of the set, but the set entirely lies in one of the closed half-spaces determined by H . If a closed set is bounded, then for each direction, there are two (not necessarily distinct) support hyperplanes. These two hyperplanes bound the thinnest slab, in this direction, that contains the set. Such a slab is called the support slab of the set. This does not mean that a support hyperplane passes through every boundary point of a set. As mentioned above, it is only true for convex bodies with non-empty interior. Theorem 2.2.4 A closed set φ ⊂ En with non-empty interior is convex if and only if a support hyperplane passes through each point of its boundary. Proof Let us start by proving that for each boundary point p of a closed convex set φ passes a support hyperplane H . Let pi be a sequence of points outside φ, with the property that they converge to p, in notation ({ pi }∞ 1 → p). By Theorem 2.2.2, through every i ≥ 1, there is a hyperplane Hi that separates pi from φ. We may assume, without loss of generality, that {Hi }∞ 1 → H. By construction, the hyperplane H contains the point p but also leaves φ entirely in one the closed half-spaces determined by it. That is, H is a support hyperplane of φ through p. Suppose now that a support hyperplane passes through every boundary point of a closed set φ with non-empty interior. To verify that φ is convex, it is necessary to prove that the segment joining any pair of points of φ is also contained in φ. We may begin by noting that all the points between an interior point a and any other point b of φ are interior points. If it were not so, then there would be a boundary point x of φ between a and b. We know, there is a small ball centered at a, completely contained in φ, and we also know that there is a support hyperplane of φ through x that will leave φ entirely on one side of the hyperplane. This is impossible.

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2 Convex Geometry

Figure 2.7

Next, we show that all points between two boundary points p and q of φ also belong to φ. Consider the line L that passes through p and q. As φ has interior points, there is an interior point c that is not on the line L. All points between c and p are interior and therefore all points between any point of the segment cp and the point q are interior points. That is, all interior points of the triangle pcq belong to the figure φ, in particular the segment pq must belong to the figure φ (see Figure 2.7). 

2.3 The Convex Hull The smallest convex set that contains a set A is called the convex hull of A. Of course, in the case of a convex set, the set and its convex hull coincide, see Figure 2.8. The convex hull of A, denoted by cc(A) is obtained by taking the intersection of all possible convex sets that contain A. Being the intersection of convex sets, cc(A) is also convex. Moreover, it naturally contains A. In fact, cc(A) is the smallest convex set that contains A, if B is a convex set containing A, it is obvious that cc(A) is contained in B. Note that cc(A) is equal to A if and only if A is a convex set and that if A1 is contained in A2 then cc(A1 ) is contained in cc(A2 ). It follows directly from the definition that a hyperplane is a support hyperplane of a compact set if and only if it is a support hyperplane of its convex hull. A first corollary of this fact is that the convex hull of a compact set is the intersection of all closed half-spaces that contain it. A second corollary is the following theorem that will be used later to prove that a closed set of constant width, with connected complement is convex. Theorem 2.3.1 If the complement of a compact set A ⊂ En is connected and every support hyperplane touches the boundary of A at a single point, then A is a convex body. Proof Since En \ A is connected, it will be enough to prove that every point a in the boundary of the convex hull of cc(A) belongs to A. By Theorem 2.2.4, there is a support hyperplane H of cc(A) at a, see Figure 2.8. Since A is closed, this hyperplane must touch A; otherwise there should be a hyperplane separating a from cc(A), contradicting the fact that a ∈ cc(A). By hypothesis, H ∩ A consists of a single point that must be a. 

2.3 The Convex Hull

19

Figure 2.8

2.4 Support Function and Minkowski Addition Given a convex body φ in Euclidean n-space En . The support function of φ is defined as the continuous real function P φ : Sn−1 → E such that for every unit vector u ∈ Sn−1 Pφ (u) = max{a, u | a ∈ φ},

(2.1)

where max is the maximum value of a, u for all a in φ, see Figure 2.9. For every unit vector u ∈ Sn−1 the support hyperplane Hu in the direction u is defined as Hu = {x ∈ En | x, u = Pφ (u)}. Note that Hu is a support hyperplane orthogonal to u. That is, Hu intersects φ but leaves it entirely contained in one of the two half-spaces determined by it. Of course, every support hyperplane of φ is of the form Hu for some unit vector u ∈ Sn−1 . Furthermore, we have exactly two parallel support hyperplanes orthogonal to u, Hu , and H−u . They define a strip denoted by S(φ, u) whose width is precisely P(u) + P(−u). Let us define, for every u ∈ Sn−1 and every convex body φ, the width function as w(φ, u) = P(u) + P(−u).

Figure 2.9 The support function

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2 Convex Geometry

A closed, convex set φ is strictly convex if φ does not contain line segments in its boundary or, equivalently, if every support hyperplane intersects φ at a single point. If this is the case for every direction u ∈ Sn−1 , the support hyperplanes Hu and H−u intersect φ, respectively, at u + and u − . The line segment u + u − is called the affine diameter or diametral chord of φ in the direction u. Note that between all chords of φ parallel to u + u − the affine diameter u + u − is the largest chord of φ. For integers 0 < m < n, we may consider Em as those points of En with the last n − m coordinates equal to zero. Let  : En → Em be the orthogonal projection over this first n coordinates. The following lemma will be useful later in the book. Lemma 2.4.1 Let φ ⊂ En be a convex body. Then Pφ |Sm−1 = P(φ) . Proof For every u ∈ Sm−1 and a ∈ En , we have that a, u = (a), u. Therefore max{a, u | a ∈ φ} = max{a, u | a ∈ (φ)}.  Given two sets S, T ⊂ En in Euclidean n-space and a real number λ ∈ E, the dilatation of S by a factor of λ is the set λS = {λa | a ∈ S}, (2.2) while the Minkowski sum of S and T is the set S + T = {a + b | a ∈ S, b ∈ T }.

(2.3)

Sometimes it is useful to think on the Minkowski sum S + T as the union of all translated copies of T by vectors of S. That is:  (a + T ). a∈S

This gives us a kinematic interpretation that may help to see S + T intuitively as the set that is covered if T undergoes all translations by vectors of S, see Figure 2.10. Directly from the definition it follows that the dilatation (2.2) and Minkowski sum (2.3) satisfy the following properties.

Figure 2.10 The Minkowski sum of ψ and φ

2.4 Support Function and Minkowski Addition

21

Lemma 2.4.2 For given sets R, S, T ⊂ En and real number λ, ν ∈ E we have a) b) c) d) e)

S + T = T + S. (S + T ) + R = S + (T + R). λ(S + T ) = λS + λT. λS + ν S = (λ + ν)S, provided S is convex. If λ < ν and R ⊂ T , then λS ⊂ ν S, and R + S ⊂ T + S. Furthermore, the dilatation and the Minkowski sum are operations closed under convex sets.

Lemma 2.4.3 If ψ and φ are convex sets, so are ψ + φ and λψ. Proof Let us prove first that λψ is convex whenever ψ is. Let λa, λb ∈ λψ. Note that a point c in the closed segment with extreme points λa  and λb has the representation c = (1 − t)λa + tλb, where t ∈ [0, 1]. Since (1 − t)λa + tλb = λ (1 − t)a + tb , by convexity we have that c ∈ λψ. Similarly, given two points a + b, a  + b ∈ ψ + φ, a point c in the closed segment with extreme points  a + b and a  + b has the representation c = (1 − t)(a + b) + t (a  + b ) = (1 − t)a + ta  ) +    (1 − t)b + tb . Therefore, by convexity of ψ and φ, the point c ∈ ψ + φ. The next theorem gives us the relation between the support function and the operations of dilatation and Minkowski sum. Theorem 2.4.1 Given two convex bodies ψ, φ ⊂ En and a real number λ ∈ E. Then we have a) Pφ+ψ = Pφ + Pψ and b) Pλφ = λPφ . Proof For every unit vector u ∈ Sn−1 , we have that Pφ+ψ (u) = max{a + b, u | a ∈ φ, b ∈ ψ} = max{a, u | a ∈ φ} + max{b, u | b ∈ ψ} = Pφ (u) + Pψ (u). Similarly, Pλ (u) = max{λa, u |  a ∈ φ} = λmax{a, u | a ∈ φ} = λPφ (u). The following corollary follows directly from the definitions. Corollary 2.4.1 The Minkowski sum, dilatation, and support function associated to the ball satisfy a) λB( p, r ) = B(λ p, λr ), whenever λ ≥ 0, b) B( p, r ) + B(q, s) = B( p + q, r + s), and c) P B(o,r ) = r , where r is the constant function. For a convex set φ and a real nonnegative number r ≥ 0 define the exterior parallel body φr as the union of all balls of radius r and with center at every point p ∈ φ. By definition, φr = φ + B(o, r ). The inner parallel body φ−r is defined as the set of all points p of φ with the property that the ball of radius r and with center at p is contained in φ. The maximum of the radius among all balls contained in φ is called the inradius of φ. Of course, φ−r is non-empty provided r is smaller than or equal to the inradius of φ. If this is the case, by definition, φ−r + B(o, r ) ⊂ φ. Furthermore, φr and φ−r are convex whenever φ is convex. The proofs of the following three lemmas we leave as exercises for the reader. Lemma 2.4.4 Given a convex set φ ⊂ En and nonnegative numbers r, s ≥ 0. Then we have: a) b) c) d)

(φr )s = φr +s , (φ−r )−s = φ−r −s , (φr )−s = φr −s , and (φ−r )s = φs−r , s > r.

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The next lemma gives us information about the support function of a parallel body. Lemma 2.4.5 Given a convex body φ ⊂ En and the constant function r with r ≥ 0. Then we have: a) Pφr = Pφ + r and b) Pφ−r ≤ Pφ − r , whenever r is smaller than or equal to the inradius of φ. As a consequence of the above considerations we have the following technical lemma that will be useful later. Lemma 2.4.6 Let φ ⊂ En be a convex body and suppose B( p, r ) ⊂ φ ⊂ B( p, s) for some p ∈ En . Then, for every  > 0 and λ > 1, we have that   φ; φ ⊂ 1 + r

λφ ⊂ φ(λ−1)s .

Usually, it is important to extend the support function of φ to the whole space En , by using exactly the same formula. That is, Pφ : En → E is extended naturally as follows: for any vector u ∈ En Pφ (u) = max{a, u | a ∈ φ}. This will allow to characterize those functions h : En → E which are support functions of a unique convex body. For a proof see Theorem 4.3 of [477]. Theorem 2.4.2 A continuous function h : En → E is the support function of a unique convex body if and only if 1. h(λu) = λh(u) for u ∈ En , λ ≥ 0, and 2. h(u + v) ≤ h(u) + h(v) for u, v ∈ En .

2.5 Blaschke’s Selection Theorem Now we introduce a measure of proximity between the collection of convex bodies of Euclidean n-space En called the Hausdorff metric. During this section Kn we will denote the family of all convex bodies in En . Given two convex bodies ψ, φ ∈ Kn , we define the Hausdorff distance d(ψ, φ) as the infimum of all  > 0 such that ψ ⊂ φ and φ ⊂ ψ . As expected, the notion of distance in Kn satisfies the axioms of a metric distance. Lemma 2.5.1 Given ψ, φ, χ ∈ Kn . Then we have a) d(ψ, φ) = 0 if and only if ψ = φ, b) d(ψ, φ) = d(φ, ψ), and c) d(ψ, φ) + d(φ, χ) ≤ d(ψ, χ) (triangle inequality).

2.5 Blaschke’s Selection Theorem

23

Note that the Hausdorff metric is invariant under parallelism, that is: d(φδ , ψδ ) = d(φ, ψ).

(2.4)

∞ converges to a convex body ψ if and only if We say that a sequence of convex bodies {ψi }i=1 limi→∞ {d(ψi , ψ)} = 0, and we denote this by

{ψi }∞ 1 → ψ. Furthermore, a real function F : Kn → E is continuous if ∞ {F(ψi )}∞ 1 → F(ψ) whenever {ψi }1 → ψ.

Theorem 2.5.1 The Minkowski sum (2.3) and the dilatation (2.2) are continuous operations, that is: ∞ ∞ If {ψi }∞ 1 → ψ, {φi }1 → φ, and {λi }1 → λ, then ∞ {ψi + φi }∞ 1 → ψ + φ and {λψi }1 → λψ.

Proof The proof of the theorem follows from the following two observations: 1) If d(ψ  , ψ) <  and d(φ , φ) < , then d(ψ  + φ , ψ + φ) < 2. 2) If d(ψ  , ψ) < , then d(λψ  , λψ) < λ. Furthermore, 1) and 2) follow immediately from the definitions and Lemma 2.4.2.



The following theorem, known in the literature as Blaschke’s Selection Theorem, is essential for many extremal geometrical problems. Theorem 2.5.2 (Blaschke’s Selection Theorem) Given a bounded and infinite collection F of convex bodies in Kn , there exists a countable subsequence of elements of F, {ψm }∞ m=1 , that converges to a convex set ψ ∈ Kn . Proof Let S = {φ0 , φ1 , φ2 , . . . } be an infinite sequence of bounded elements in F. Then we may assume, without loss of generality, that all our convex bodies are contained in the n-cube Q of sides of length, say, one. We say that a sequence of convex bodies {φm }∞ m=1 is a Cauchy sequence if, given  > 0, there is an integer N sufficiently large such that for every i, j > N the Hausdorff distance d(φi , φ j ) < . The proof of the theorem consists of two steps: The first step is to show that any Cauchy sequence is convergent, and the second step consists of exhibiting a Cauchy subsequence of S. We begin by proving that every Cauchy sequence of convex bodies {φm }∞ m=1 converges to some ∞ φ ). Note that {ϕ } is a nested sequence of convex bodies, that convex body φ. Let ϕi = cc( ∞ m i m=i i=1 is, ϕi+1 ⊂ ϕ i. ∞ ϕi . Clearly, φ is a non-empty convex body. We shall prove next that {φm }∞ Let φ = i=1 m=1 → φ. ∞ Since {φm }m=1 is a Cauchy sequence, then, given  > 0, there is an integer N sufficiently large such that for every i, j > N , d(φi , φ j ) <  and hence that φ j ⊂ (φi ) , for j > i > N . By construction of ϕi , we also have that ϕ j ⊂ (φi ) , for j > i > N , and therefore φ ⊂ (φi )ε , for every i > N . Then it is only left to show that φi ⊂ (φ) for i > N . But we know that φi ⊂ ϕi . Thus there is a number M sufficiently large such that if i > M, then ϕi ⊂ (φ) and, moreover, φi ⊂ (φ) . Next we will show that, given S = {φ0 , φ1 , φ2 , . . .√}, there exists a Cauchy subsequence of S. We start by considering the n-cube Q 1 := Q of diameter n, then we subdivide Q 1 into smaller cubes in a way that every edge is partitioned into two√ equal parts. Thus, we obtain a second generation of 2n n-cubes called Q 2 , all of them of diameter 2n . If we repeat the process, we subdivide every n-cube

24

2 Convex Geometry √

of Q 2 to obtain the third generation Q 3 , with altogether 22n n-cubes, each of them of diameter 22n . Inductively, we obtain the m-th √ generation of n-cubes Q m by subdividing every edge of all n-cubes of n Q m−1 , all of them of diameter 2m−1 . Note that in every generation there are | Q m |= 2(m−1)n cubes. (m−1)d Qm of such elements. Given Let 2 be the family of all subsets of Q m , and observe that we have 22 any convex body φ ∈ F, it is possible to assign to it its m-cumulus, (φ)m , consisting of all elements φ. It is left to the reader to show (see Exercise 2.32) that if (φ)m = (ψ)m , then of Q m that intersect √ n d(φ, ψ) ≤ 2m−1 . Start from S = {φ0 , φ1 . . . }. By assumption all its elements are contained in Q 1 . Next, for k = 2 consider the 2-cumulus associated to each of the elements in S. Being the number of 2-cumulus finite, and the sequence S infinite, there should be one of the 2-cumulus, say α2 , that has an infinite family of convex bodies from S associated to it. Take the subsequence S2 = {φ2,1 , φ2,2 , φ2,3 . . . } of all convex sets of S associated to α2 . Inductively, suppose Sm = {φm,1 , φm,2 , φm,3 . . . } is the subsequence of all convex sets of Sm−1 associated to the m-cumulus αm . Letting the number of (m + 1)-cumulus finite, and the sequence Sm infinite, there should be one of the (m + 1)-cumulus, say αm+1 , that has an infinite family of convex bodies from Sm associated to it. Define Sm+1 = {φm+1,1 , φm+1,2 , φm+1,3 . . . }, the subsequence of all convex sets of Sm associated to α√m+1 . Thus, every sequence Sm is a subsequence n of Sm−1 . Consequently, for j, i > N , d(φi,k , φ j,l ) < 2 N −1 . ∞ Select the sequence {φm,m }1 and note that by construction it is easily checked that it is a Cauchy sequence and hence converges to a convex body. With this the proof is finished. 

2.6 Steiner Symmetrization In this section we study the so-called Steiner symmetrization that will be used later to prove the isoperimetric and the Brunn–Minkowski inequalities. Let H be a hyperplane through the origin in En . Given a convex body φ ∈ Kn , the Steiner symmetral of φ with respect to H is a compact set S(φ) with the following properties (see Figure 2.11): i) S(φ) is symmetric with respect to H , ii) for every line L orthogonal to H , L ∩ φ = ∅ if and only if L ∩ S(φ) = ∅, and iii) for every line L orthogonal to H that intersects φ, the length of L ∩ φ is exactly the length of L ∩ S(φ). The procedure to generate S(φ) from φ is called Steiner symmetrization of φ at H . The following theorems summarize the main properties of the Steiner symmetrization.

Figure 2.11 Steiner symmetrization of a non-convex figure

2.6 Steiner Symmetrization

25

Theorem 2.6.1 1. 2. 3. 4. 5. 6.

S(φ) ⊂ S(ψ) whenever φ ⊂ ψ. S(B(o, r )) = B(o, r ). S(λφ) = λS(φ). S(φ) + S(ψ) ⊂ S(φ + ψ). For every convex set φ ∈ Kn , its Steiner symmetral S(φ) is also convex. The Steiner symmetrization S : Kn → Kn is a continuous, volume preserving function.

Proof The proof of 1, 2 and 3 are left as exercises to the reader. Proof of 5. By definition, it is clear that S(φ) is closed and bounded. Given two points p and q in S(φ), they lie in a trapezium T  determined by the convex hull of the intervals S(φ) ∩ L p and S(φ) ∩ L q , where L p and L q are the lines orthogonal to H through p and q, respectively. Consider the trapezium T determined by the convex hull of φ ∩ L p and φ ∩ L q . Since φ is convex, it follows that T is contained in φ. On the other hand, it is easy to see that S(T ) = T  , and therefore, by 1, T  ⊂ S(φ). Consequently, since pq ⊂ T  ⊂ S(φ), we have that S(φ) is convex. Proof of 6. For the proof of the continuity of the Steiner symmetrization, suppose {φi } → φ. Without loss of generality, we may assume that the origin is an interior point of φ. Therefore, there are α > 0, β > 0 such that B(o, α) ⊂ φ ⊂ B(o, β). By 1 and 2, and since {φi } → φ, we have that there is a number M sufficiently large such that for every i > M, B(o, α) ⊂ S(φ) ⊂ B(o, β) and B(o, α) ⊂ S(φi ) ⊂ B(o, β). Given  > 0, we need to show that for N sufficiently large, the following two claims are true: a) S(φi ) ⊂ (S(φ)) β , for every i > N , and α b) S(φ) ⊂ (S(φi )) β , for every i > N . α

a) By hypothesis there is N > M such that φi ⊂ (φ) . Thus by Lemma 2.4.6 and by 1 and 3, φi ⊂ (1 + α )φ. Hence, we have that S(φi ) ⊂ (1 + α )S(φ), and again by Lemma 2.4.6 that S(φi ) ⊂ [S(φ)] β , for every i > N . α b) By hypothesis, φ ⊂ (φi ) , for every i > N . This implies by Lemma 2.4.6 that φ ⊂ (1 + α )φi , and hence by 1 and 3, that S(φ) ⊂ (1 + α )S(φi ). Finally, by Lemma 2.4.6 we obtain that S(φ) ⊂ (S(φi )) β , α for every i > N . From the above and Cavalieri’s Principle we have that the Steiner symmetrization is a continuous volume preserving function. We consider now the proof of 4. Let r be a point of S(φ) + S(ψ), where r = p + q with p ∈ S(φ) and q ∈ S(ψ). Let L r , L p and L q be the lines orthogonal to H through r, p, and q, respectively. Evidently, r ∈ (S(φ) ∩ L p ) + (S(ψ) ∩ L q ). Note that (S(φ) ∩ L p ) + (S(ψ) + L q ) is an interval whose length is the sum of the lengths of the intervals (S(φ) ∩ L p ) and (S(ψ) ∩ L q ). Moreover, the middle point of (S(φ) ∩ L p ) + (S(ψ) ∩ L q ) ⊂ L r lies in H . On the other hand, (φ ∩ L p ) + (ψ ∩ L q ) ⊂ L r is an interval whose length is the sum of the lengths of the intervals φ ∩ L p and ψ ∩ L q . It is therefore equal to the length of the interval (S(φ) ∩ L p ) + (S(ψ) ∩ L q ). This implies that S[(φ ∩ L p ) + (ψ ∩ L q )] = (S(φ) ∩ L p ) + (S(ψ) ∩ L q ) , but now, by 1, r ∈ S[(φ ∩ L p ) + (ψ ∩ L q )] = (S(φ) ∩ L p ) + (S(ψ) ∩ L q ) ⊂ S(φ + ψ).



Equality in 4, holds if and only if one of the convex bodies is a dilatation of the other, see Exercise 2.33. Let u ∈ Sn−1 be a unit vector, and Hu be the hyperplane through the origin orthogonal to u. We will denote by Su (φ) the Steiner symmetral of the convex body φ with respect to Hu .

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Lemma 2.6.1 Let φ be a convex body contained in the ball B(0, r ). If φ is not B(0, r ), then there is a finite sequence of successive symmetrizations of φ on different directions, that is, there are unit vectors u 1 , . . . , u m such that Su m (. . . (Su 2 (Su 1 (φ)))) ⊂ int B(0, r ) . Proof Let S(o, r ) be the (n − 1)-sphere of radius r centered at the origin, i.e., the boundary of B(o, r ). A cap of S(o, r ), C = H + ∩ Srn−1 , is the intersection of an open half-space H + with S(o, r ). Furthermore, an -cap C is a cap such that the (n − 1)-dimensional surface area of C is . Note that, given two -caps C, C  of S(o, r ), there is a hyperplane  through the origin such that C is symmetric to C  with respect . Since φ is contained but not equal to B(0, r ), there is one -cap C0 such that C0 ∩ φ = ∅, for  sufficiently small. Next choose an open cover of S(0, r ) = C0 ∪ C1 ∪ · · · ∪ Cm of -caps and let u i ∈ Sn−1 be such that C0 is symmetric to Ci with respect Hu i , for i = 1, . . . , m. Note a simple but important fact: if p ∈ S(o, r ) \ φ and p  ∈ S(o, r ) is symmetric to p with respect Hu i , then neither p nor p  belong to Su i (φ). With this in mind, we see that Su 1 (φ) ∩ (C0 ∪ C1 ) = ∅, but  similarly Su 2 (Su 1 (φ)) ∩ (C0 ∪ C1 ∪ C2 ) = ∅, and finally Su m (. . . (Su 2 (Su 1 (φ)))) ∩ S(o, r ) = ∅. Let φ ⊂ Kn be the collection of all convex bodies obtained from a finite sequence of successive symmetrizations of φ in different directions. Note that by Theorem 2.6.1 all convex bodies of φ have the same volume as φ. The main symmetrization theorem is the following. Theorem 2.6.2 Let φ ⊂ En be a convex body that contains the origin in its interior. Then we may select a sequence {φi } ⊂ φ such that {φi }∞ 1 → B(o, r ), where B(o, r ) has the same volume as φ. Proof For every ψ ∈ φ , let ρ(ψ) be the infimum of all radii t such that ψ ⊂ B(o, t). Furthermore, let ρ0 = Inf {ρ(ψ) | ψ ∈ φ }. If ρ0 = ρ(ψ), for some ψ ∈ φ , then, by Lemma 2.6.1, ψ = B(o, ρ0 ). Therefore in this case the theorem trivially holds, because ρ0 = r . If the infimum ρ0 is never reached for an element of φ , then there is a sequence {φi } ⊂ φ such that {ρ(φi )}∞ 1 → ρ0 . By Blaschke’s Selection Theorem 2.5.2, we may assume that, without loss of generality, {φi }∞ 1 → ψ, where ψ is a convex body not necessarily in φ . We claim that ψ is the ball B(o, ρ0 ). Note that at the limit ψ ⊂ B(o, ρ0 ), so if ψ is not B(o, ρ0 ), by Lemma 2.6.1, there are unit vectors u 1 , . . . , u m such that Su m (. . . (Su 2 (Su 1 (ψ)))) ⊂ int B(o, ρ0 ). But now, since the Steiner symmetrization is a continuous function, there is an integer i sufficiently large such that Su m (. . . (Su 2 (Su 1 (φi )))) ⊂ int B(o, ρ0 ) and consequently ρ(Su m (. . . (Su 2 (Su 1 (φi ))))) < ρ0 . Since Su m (. . . (Su 2 (Su 1 (φi )))) ∈ φ , this is a contradiction to the definition of ρ0 . Thus {φi }∞ 1 → B  (o, ρ0 ). Corollary 2.6.1 Let {φi }k1 be a finite family of convex bodies in En that contain the origin in its interior. For i = 1, . . . , k, let B(o, ri ) be the ball whose volume is the volume of φi . Then, given  > 0, there is a finite sequence of successive Steiner symmetrizations such that B(o, ri − ) ⊂ Su m (. . . (Su 2 (Su 1 (φi )))) ⊂ B(o, ri + ), for every i = 1, . . . , k. Proof The proof is by induction on k. If k = 1, the corollary follows directly from Theorem 2.6.2. Supposing the theorem is true for k − 1, we shall prove it for k. Given  > 0, by induction there is a finite sequence of successive symmetrizations such that B(o, ri −) ⊂ Su m (. . . (Su 2 (Su 1 (φi ))))⊂B(o, ri + ), for every i = 1, . . . , k − 1. Apply now Theorem 2.6.2 to the convex body Su m (. . . (Su 2 (Su 1 (φk )))). Then there is a finite sequence of successive symmetrizations such that

2.6 Steiner Symmetrization

27

B(o, rk − ) ⊂ Su m+λ (. . . (Su m+1 (Su m (. . . (Su 2 (Su 1 (φk )))))) . . . ) ⊂ B(o, rk + ) and, by Theorem 2.6.1, 1) and 2), B(o, ri − ) ⊂ Su m+λ (. . . (Su m+1 (Su m (. . . (Su 2 (Su 1 (φi ))))))) ⊂ B(o, ri + ), for every i = 1, . . . , k − 1.



2.7 The Brunn–Minkowski Theory One of the most interesting developments in the area of convexity is, together with the theory of mixed volumes, the Brunn–Minkowski theory. From the latter, perhaps one of the deepest results is the claim that the volume of a convex combination of convex sets is a concave function. Recall that a function f : [0, 1] →  E is concave if for every 0 ≤ a < b ≤ 1 we have that (1 − t) f (a) + t f (b) ≤ f (1 − t)a + tb . Remember that if, in addition, f (t) is twice differentiable, we have that f  (t) ≤ 0, for every t ∈ [0, 1]. Theorem 2.7.1 Let φ and ψ be convex bodies in En , and denote by V (φ) the n-dimensional volume of a convex body φ. Then the function    n1 F(t) = V (1 − t)φ + tψ is a concave function, where t ∈ [0, 1]. Observe that, replacing ψ by the convex set φ1 = φ + B(o, 1), we obtain that the function F(t) = 1 V (φt ) n is a concave function, where φt is the outer parallel body and t ∈ [0, 1]. Let us start by examining some consequences of this fact in the case n = 2. Clearly, in this case the 2-dimensional volume V (φ) must be interpreted as the area A(φ) of a convex figure φ, thus 1 F(t) = A(φt ) 2 . Next, the reader is easily convinced (by Figure 2.12) that the area of the outer parallel body φt with t ∈ [0, 1] is given by A(φt ) = A(φ) + t P(φ) + πt 2 , where P(φ) is the perimeter of the convex figure φ.

Figure 2.12

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This implies that the function 1

1

F(t) = (A(φ) + t P(φ) + πt 2 ) 2 = (A + t P + πt 2 ) 2 , t ∈ [0, 1], is concave, and therefore its second derivative is always negative. Indeed, F  (t) = −

P 2 − 4 Aπ ≤ 0, t ∈ [0, 1]. 4F(t)3

As an immediate consequence of this fact, we obtain the following inequality. Theorem 2.7.2 (Isoperimetric inequality in the plane) Let φ ⊂ E2 be a convex figure of area A and perimeter P. Then P 2 − 4 Aπ ≥ 0 , and equality holds only for a 2-dimensional circular disk. In Euclidean 3-space E3 , the volume of the outer parallel body is in fact a polynomial function of degree three with the following coefficients: the volume, the area, the integral of the mean curvature (see [502]), and the volume of the unit sphere. Indeed, for a convex body φ ⊂ E3 V (φt ) = V + St + Mt 2 + Ct 3

(2.5)

holds, where V is the volume of φ, S is the surface area of φ, M is the integral of the mean curvature denotes the volume of the unit ball in E3 . of φ, and C = 4π 3 1 As before, this implies that F(t) = (V + St + Mt 2 + Ct 3 ) 3 is a concave function, and by taking the second derivative, the following two inequalities could again be obtained, see Exercise 2.35: S 2 − 3M V ≥ 0,

(2.6)

M 2 − 3C S ≥ 0.

(2.7)

A direct consequence of these two inequalities is the following important theorem. Theorem 2.7.3 (Isoperimetric inequality in 3-space) Let φ ⊂ E3 be a convex body of volume V and surface area S. Then 4π S 3 − 27 V 2 ≥ 0 , 3 where equality holds only for a 3-dimensional ball. In general the n-dimensional volume of the outer parallel body in En is also a polynomial of degree n, called the Steiner polynomial, where the coefficients are by definition the intrinsic volumes or quermassintegrals, see Section 12.1.    n1 In order to prove that F(t) = V (1 − t)φ + tψ is a concave function, we need to consider first the Brunn–Minkowski inequality.

2.7 The Brunn–Minkowski Theory

29

Theorem 2.7.4 (Brunn–Minkowski inequality) Let φ and ψ be two convex bodies in En . Then 1

1

1

V (φ) n + V (ψ) n ≤ V (φ + ψ) n , and equality holds if and only φ is a dilatation of ψ. Proof Let r1 , r2 , and r3 be the radii of the balls with the same volume as φ, ψ, and φ + ψ, respectively. It will be enough to prove that r1 + r2 ≤ r3 . Let  > 0 be given. By Corollary 2.6.1, there is a finite sequence of successive symmetrizations such that B(o, r1 − ) ⊂ Su m (. . . (Su 2 (Su 1 (φ)))) ⊂ B(o, r1 + ), B(o, r2 − ) ⊂ Su m (. . . (Su 2 (Su 1 (ψ)))) ⊂ B(o, r2 + ), B(o, r3 − ) ⊂ Su m (. . . (Su 2 (Su 1 (φ + ψ)))) ⊂ B(o, r3 + ). Moreover, by Theorem 2.6.1 4), Su m (. . . (Su 2 (Su 1 (φ)))) + Su m (. . . (Su 2 (Su 1 (ψ)))) ⊂ Su m (. . . (Su 2 (Su 1 (φ + ψ)))). Then B(o, r1 − ) + B(o, r2 − ) ⊂ B(o, r3 + ), B(o, r1 + r2 − 2) ⊂ B(o, r3 + ), and therefore r1 + r2 ≤ r3 + 3. Since this holds for every  > 0, we obtain our desired inequality.  1    n is a concave function Proof of Theorem 2.7.1 We want to prove that F(t) = V (1 − t)φ + tψ By the Brunn–Minkowski inequality 2.7.4, we have that (1 − t)F(0) + t F(1)) ≤ F(t). Given 0 ≤ a < b ≤ 1, define  = (1 − a) + a and   = (1 − b) + b. By using again the Brunn–Minkowski inequality for  and   and a simple calculation, we obtain   that (1 − t)F(a) + t F(b) ≤ F (1 − t)a + tb , thus proving that F(t) is a concave function. 

2.8 Radon, Carathéodory, and Helly In this section, the classical theorems of Helly, Carathéodory, and Radon will be examined. As a comprehensive treatment of this subject, the article by Danzer, Grünbaum and Klee [278] is recommended, and an excellent continuation is the survey [306]. We begin by stating Helly’s Theorem [528].

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2 Convex Geometry

Figure 2.13

Theorem 2.8.1 (Helly’s Theorem) Assume that F is a family of at least n + 1 convex sets in En . Suppose, moreover, that either F is finite or each member of F is compact. If every n + 1 members of F have a point in common, then there exists a point common to all the members of F. This theorem is best possible in all possible ways. First observe that if the sets are not convex, then the intersection of the family may be empty; see, for instance, Figure 2.13 were every three sets have a point in common, but there is not a point in the intersection of the four of them. Next, if the number n + 1 is reduced to a smaller number, then also the implication does not hold. For example, if a family of three convex sets in the plane E2 with the property that every pair of them intersects, then there is not necessarily a point in the intersection of the three of them, see Figure 2.14. Moreover, if the family is not finite and consists of non-compact convex sets, then Helly’s Theorem is not true either. This can by observed be taking the family of half-planes in E2 given by F := {Ft }∞ t=0 , where t is a natural number and Ft = {(a, b) ∈ E2 |b ≥ t}. Although the proof of Helly’s theorem will be given in this section, we will start by analyzing the case n = 1. If one considers a finite family of closed intervals, say, [α1 , β1 ], . . . , [ατ , βτ ], with the property that every pair of them has a point in common, then it is clear that the minimum of all the βi s is a point which is located in all these intervals. Helly’s Theorem is closely related to the following two important theorems. Theorem 2.8.2 (Carathéodory’s Theorem) Let T be a subset of En . Given a point x ∈ cc(T ), there are points x1 , . . . , xn+1 (or less) in T such that x ∈ cc({x1 , . . . , xn+1 }). Theorem 2.8.3 (Radon’s Theorem) Every set with n + 2 or more points of En can be expressed as the union of two disjoint subsets whose convex hulls have a point in common.

Figure 2.14

31

For example, given four points in the plane, either one of them is in the triangle defined by the other three, or the interval determined by a suitable pair of them intersects the interval whose endpoints are the other two. These three theorems, Helly, Radon, and Carathéodory, are equivalent. However, in this section Radon’s Theorem will be proved first and then be used to prove Helly and Carathéodory’s. The following lemma will provide the necessary linear algebra tools. Lemma 2.8.1 Let {x1 , . . . , xr } be a finite family of points in En . Then cc({x1 , . . . , xr }) = {x ∈ En | x = λ1 x1 + · · · + λr xr , λ1 + · · · + λr = 1 and λi ≥ 0}. Proof Suppose x = λ1 x1 + · · · + λr xr , λ1 + · · · + λr = 1 and λi ≥ 0. We claim that x ∈ cc({x1 , . . . , xr }). The proof is by induction on r . The claim is trivially true for r = 1. Suppose it is λr −1 λ1 x1 + · · · + 1−λ xr −1 . Then x  ∈ cc({x1 , . . . , xr −1 }) ⊂ cc({x1 , . . . , xr }). true for r − 1 and let x  = 1−λ r r  On the other hand, x = (1 − λr )x + λr xr ∈ cc({x1 , . . . , xr }). This implies that {x ∈ En | x = λ1 x1 + · · · + λr xr , λ1 + · · · + λr = 1 and λi ≥ 0} ⊂ cc({x1 , . . . , xr }). For the converse, it will be enough to prove that {x ∈ En | x=λ1 x1 + · · · + λr xr , λ1 + . . . +λr = 1 and λi ≥ 0} is a convex set which contains {x1 , . . . , xr }. Let x = λ1 x1 + · · · + λr xr , λ1 + · · · + λr = 1, λi ≥ 0, and y = τ1 x1 + · · · + τr xr , τ1 + · · · + τr = 1, τi ≥ 0. Then the point t x + (1 − t)y, t ≥ 0, is the interval whose endpoints are x and y and can be expressed as 

   tλ1 + (1 − t)τ1 x1 + · · · + tλr + (1 − t)τr xr ,

      where tλi + (1 − t)τi ≥ 0 and tλ1 + (1 − t)τ1 + · · · + tλr + (1 − t)τr = 1. This completes the proof.



Now we come back to the proof of Radon’s Theorem. Proof of Radon’s Theorem. Let T = {x1 , . . . , xr } be a set of points of En with r ≥ n + 2. Consider the following system of n + 1 homogeneous linear equations. τ1 + · · · + τr = 0 , τ1 x j,1 + · · · + τr x j,r = 0

(1 ≤ j ≤ n),

where the coordinates of a point xi are given by (x1,i , . . . , xn,i ). Since r > n + 1, the system has a be the set of all indices nontrivial solution (τ1 , . . . , τr ). Let U i for which τi ≥ 0, V the set of all indices i for which τi < 0, and let c = (i∈U ) τi > 0. Then (i∈V ) τi = −c; moreover,

 τi 

 τi  xi = xi . − c c (i∈U ) (i∈V ) This means that U and V generate a partition of T into two subsets TU = {xi |i ∈ U }, TV = {xi |i ∈ V }  for which cc(TU ) ∩ cc(TV ) is different from the empty set. Now we will use Radon’s Theorem to prove the Theorems of Helly and Carathéodory. Proof of Helly’s Theorem 2.8.1. Helly’s Theorem concerns two types of families; those which are finite and those whose members are compact. We begin by proving the theorem for the case of a finite family F with m elements, using induction on m, the number of members of F. Observe first that the theorem is obvious for a family of n + 1 sets. Suppose then, that the theorem is true for a family of j − 1 convex sets, j ≥ n + 2, and consider a family F of j convex

32

2 Convex Geometry

sets with the property that any n + 1 of them have a point in common. By the induction hypothesis, for every φi ∈ F, i ∈ I = {1, . . . , m}, there exists a point pi common to all members of F \ {φi }. By Radon’s Theorem, there exists a partition of I into two subsets U and V such that there exists a point x ∈ cc({ pi | i ∈ U }) ∩ cc({ p j | j ∈ V }). It is now easy to see that, since all the members of F are convex sets, the point x is common to all members of F (see Figure 2.15). This proves the finite version of Helly’s Theorem. Suppose now that we have an infinite family F of compact convex sets in En with the property that any n + 1 of them have a point in common. By the finite version of Helly’s Theorem, any finite subfamily of F is intersecting, and since all these sets are compact, the finite intersection property implies that there exists a point in common to all the members of F.  Proof of Carathéodory’s Theorem. Let T be a subset of En . Consider the union of the convex hull of all finite subsets of T . It is clear that this set must be the convex hull of T since it contains T , is convex and is, by definition, contained in any convex subset which contains T . With this in mind, let x be a point of cc(T ). Then, by the above, there are points x1 , . . . xm ∈ T such that x ∈ cc({x1 , . . . , xm }). By Lemma 2.8.1 we have that x = α1 x1 + · · · + αm xm , where αi ≥ 0 and α1 + · · · + αm = 1. Let m be the minimum integer for which this is true and suppose, by contradiction, that m ≥ n + 2. By Radon’s Theorem, there exists a partition TU , TV of the set X = {x1 , ...xm } such that x ∈ cc(TU ) ∩ cc(TV ). By Lemma 2.8.1, there are numbers λ1 , . . . , λm , not all zero, such that λ1 x1 + · · · + λm xm = 0 and λ1 + · · · + λm = 0. Let I be the set of i’s such that λi < 0, and let i 0 ∈ I be such that α j /λ j is a maximum. Then we have that   αj  αj  λ1 x1 + · · · + αm − λm x m , x = α1 − λj λj where all the coefficients are positive and add up to one; furthermore, the coefficient of x j = 0. This implies that x is in the convex hull of m − 1 points of T , which contradicts our assumption.  The following result is an interesting consequence of Carathéodory’s Theorem. Corollary 2.8.1 The convex hull of a compact subset φ of En is also compact.

Figure 2.15 U = {1, 3} and V = {2, 4}

33

Proof Let n be the standard n-simplex in En+1 , that is, the set of all points α with coordinates α = (α0 , . . . , αn ) such that αi ≥ 0 and α0 + · · · + αn = 1. Clearly, n is compact and therefore, since a product of compact sets is a compact set, n × φn+1 is also compact, where φn+1 is the Cartesian product of n + 1 copies of φ, contained in En(n+1) . Let f : n × φn+1 → En be the function defined as follows: f (α, x0 , . . . , xn ) = α0 x0 + · · · + αn xn . By Carathéodory’s Theorem cc(φ) = f (n × φn+1 ), and since the continuous image of a compact set is compact, we have that cc(φ) is also compact.  There are many interesting applications of Helly’s Theorem. Here we present two of them; Jung’s Theorem [599] relating the circumradius and the diameter of a set, and Rado’s Central Point Theorem [959]. Theorem 2.8.4 (Jung’s Theorem) Let T be a subset of En with diameter 1. Then, there exists a ball n of radius 2n+2 that contains T . Proof Let us start proving the case in which T consists of n + 1 or fewer points. Assume that, without loss of generality, B(o, σ) is the ball of minimal radius containing T , and that φ ∩ bdB(o, σ) = {x1 , . . . , xm }, where m ≤ n + 1. Note that the origin is in the convex hull of {x1 , . . . , xm }; otherwise we could decrease σ. Hence λ1 x1 + · · · + λm xm = 0, where λ1 + · · · + λm = 1 and λ1 , . . . , λm ≥ 0. Then 1 − λk =

i=k

λi ≥

λi | xi − xk | = 2

i=k

m

  λi | xi |2 + | xi |2 −2 < xi , xk >

1

= 2σ 2

m

λi −

2
= 2σ 2 .

1

m−1 2m

n . 2n+2

n + 1 points of T For the general case, by the above, T is a subset of E with the property that any n n n are contained in the ball of radius 2n+2 . Consider the family of all balls with radius 2n+2 centered at some point of T . We shall prove now that this family is intersecting. Indeed, if we take n + 1 of n these balls, say with centers {x0 , x1 , . . . , xn } ⊂ T , then there exists a ball of radius 2n+2 with center at x that contains these centers. It is obvious that x is located in the intersection of these n + 1 balls. n By Helly’s Theorem, there exists a point y in the intersection of all balls of radius 2n+2 and with n centers in T . Now it is obvious that the ball of radius 2n+2 and with center at y contains T . 

Theorem 2.8.5 (Rado’s Central Point Theorem) Let T be a collection of m points in En . Then there exists a point p (not necessarily in T ) with the property that every closed half-space H containing p m points of T . contains also at least n+1

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2 Convex Geometry

n Proof Let F = {cc(G) | G ⊂ T and the size of G is larger than n+1 m}. We start by proving that every , subfamily of n + 1 elements of F has a point in common. Let G 0 G1 , . . . , Gn be such that Gi ⊂ T .  Then we shall prove n0 Gi = ∅. Observe that

|T\

n 0

Gi |=|

n 

(T \ Gi ) |≤

0

n

| T \ Gi |< m

0

  and | T |= m. Then n0 Gi = ∅ and also n0 cc(Gi ) = ∅. By Helly’s Theorem 2.8.1, there is a point p ∈ G∈F G. Let us now prove that p is a central point. m points of T . Then Suppose not; hence there is a closed half-space H + containing p and less than n+1 n +  m points. Clearly, the open half-space complementary to H contains a subset G of T of more than n+1  p is not in the convex hull G  , which is a contradiction.

2.9 Classification of Boundary Points In this section we will classify all the different types of points in the boundary of a convex body with non-empty interior. If more than one support hyperplane goes through a boundary point p of a convex body φ, we say that p is a singular point of the boundary of φ. If on the contrary only one support hyperplane passes through it, the boundary point is called regular. First of all, note that the set of regular points is dense in bdφ. This is indeed the case because given any open set U ⊂ En intersecting bdφ, choose B of sufficiently small radius contained in the interior of φ ∩ U . Then there is a translate of B contained in φ that touches bdφ ∩ U at some point p. Clearly, p is regular in B and is therefore regular in φ and has a unique support hyperplane at p. The following theorem is a natural consequence of the previous discussion. Recall that a nowhere dense set is a set whose closure has empty interior. Theorem 2.9.1 The set of singular points of the boundary of a convex body φ is a countable union of nowhere dense sets. Proof Since the complement of the singular points is dense in the boundary of φ, it will be enough to prove that the set of singular points is a countable union of closed sets. For that purpose, let i be the set of all points p of the boundary of φ with the  property that p admits two support hyperplanes making an angle of at least 1i between them. Clearly, ∞ 1 i is the set of singular points, and a simple  compactness argument shows that every i is closed. From the viewpoint of Baire category (see Section 11.4), the set of singular points φ is meager or of the first category [808]. We say that a unit vector u ∈ Sn−1 is normal at p if the hyperplane through p, orthogonal to u, is a support hyperplane of φ; that is, the vector u points outside of φ. By Theorem 2.2.4, p has at least one normal unit vector, but if it is singular it has more than one. Let ( p) ⊂ Sn−1 be the set of normal vectors of φ at p. Observe that by definition the closed set ( p) has no antipodal pair of points. In fact, the set ( p) is spherically convex. This means that, given two vectors u, v ∈ ( p), the shortest sub-arc of the circle between u and v is contained in ( p). ˜ p) be the convex cone with apex p generated by the vectors of ( p). The dimension of Let ( this cone enables to classify the boundary points. In fact, we say that p is an r -singular point if ˜ p)) = n − r . We will denote by Sr the set of all r -singular points of the boundary of φ. In the dim(( case of r = 0, the 0-singular points are also called vertex points.

2.9 Classification of Boundary Points

35

Let V be the collection of vertex points of a convex body φ. By Exercise 2.41, {int( p)} p∈V is a collection of pairwise disjoint open spherically convex sets in Sn−1 . By taking a point with rational coordinates in every member of {int( p)} p∈V , it is possible to verify that V must be a countable set. Then the following theorem is true. Theorem 2.9.2 A convex body has at most a countable number of 0-singular or vertex boundary points. A generalization of this theorem for general values of r states that the set Sr of all r -singular points has σ-finite r -dimensional Hausdorff measure. In order to prove this result we need to recall the definition of the nearest point mapping. Given a convex body φ, define the function ℘φ : En → En , where ℘φ ( p) is the unique point in φ nearest to p. Clearly, ℘φ ( p) = p if and only if p ∈ φ and, ˜ p). From this fact it is easy moreover, for a point p in the boundary of φ, ℘φ−1 ( p) is the convex cone ( n n to see that the nearest point mapping ℘φ : E → E is non-expansive, that is, | ℘φ ( p) − ℘φ (q) |≤| p − q | . Having this in mind, we are ready for the proof of the following theorem that measures the set of r -singular points of the boundary of a convex set. For the definition of Hausdorff measure see [867]. Theorem 2.9.3 The set Sr of r -singular points of the boundary of a convex body φ is a countable union of finite r -dimensional Hausdorff measurable sets. Proof Let Er be the countable union of all r -dimensional planes of En , each of which is determined by r + 1 affinely independent points with rational coordinates. Take also m sufficiently large such that φ ⊂ int B(a, m) for some a. Obviously, Er ∩ B(a, m) has σ-finite r -dimensional Hausdorff measure. Since ℘φ : En → En is a non-expansive map, for the proof of the theorem it is sufficient to prove that Sr ⊂ ℘φ (Er ∩ B(a, m)). Let now p ∈ Sr and take q ∈ int B( p, m) in the relative interior of the (n − r )-dimensional convex cone ℘φ−1 ( p). Consider the r -plane H through q orthogonal to the (n − r )-dimensional cone ℘φ−1 ( p). Choose r + 1 affinely independent points in H and choose other r + 1 points with rational coordinates, affinely independent and close enough to the previous ones. Then these points determine an r -plane H  that intersects the relative interior or the (n − r )-dimensional convex cone ℘φ−1 ( p) in some point q  . Obviously, q  ∈ H  ∩ B( p, n), and furthermore ℘φ (q  ) = p. The proof is now complete because  this shows that Sr ⊂ ℘φ (Er ∩ B(a, m)). Assume the boundary of a convex body φ ⊂ En has only regular points. In this case, φ will be called a smooth body. If this is so, we can define the Gauss map as follows:  : bdφ → Sn−1 ,

(2.8)

where for every p ∈ bdφ, ( p) is the unique normal vector of φ at p. The continuity of this map follows immediately from the fact that the support hyperplanes through the points of a convergent sequence of boundary points must converge to the unique support hyperplane at the limit point. In fact, the Gauss map is a diffeomorphism (invertible, and its inverse is differentiable as well). In fact, a much deeper result is true. In [14], Aleksandrov proved that the boundary of any convex body is almost everywhere twice differentiable. A convex set without line segments on its boundary is called strictly convex. In other words, a closed convex set φ ⊂ En is strictly convex if every support hyperplane intersects φ at a single point. In this case, we may define the continuous inverse Gauss map

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2 Convex Geometry

γ : Sn−1 → bdφ ,

(2.9)

where γ(u) is the unique point p ∈ bd φ for which u is the outward normal at p. Let A be a subset of En . A point p ∈ A is called an extreme point of A if p is never in the relative interior of an interval with endpoints in a body φ. The set of extreme points of A will be denoted by Ext(A). For convex sets, every extreme point is also a boundary point, but not conversely, although every vertex point is of course an extreme point. We shall prove at the end of this section that a convex body is the convex hull of its extreme points. The proof of the following characterization of extreme points is left to the reader, see Exercise 2.42. Lemma 2.9.1 Let φ be a convex set of En . The point p is an extreme point of φ if and only if the set φ \ { p} is also convex. The following lemma ensures the existence of extreme points. Lemma 2.9.2 Let φ be a closed non-empty subset of En that does not contain any line. Then φ has at least one extreme point. Proof The proof is by induction on the dimension n. If n = 0, the result is trivial. Assume that every non-empty subset of En−1 that does not contain any line contains at least one extreme point, and let φ be a non-empty subset of En . Let H be a support hyperplane of φ. By the induction hypothesis there is an extreme point p of H ∩ φ. It is easy to see that p is also an extreme point of φ. With this observation, the proof of the lemma is complete.  Lemma 2.9.3 Let T be closed non-empty subset of En . Then Ext(cc(T )) ⊂ Ext(T ). Proof Let x ∈ Ext(cc(T )). We will first prove that x ∈ T . Since x ∈ cc(T ), by Carathéodory’s Theorem 2.8.2 there are points x0 , x1 , . . . , xn ∈ T such that x ∈ cc({x0 , x1 , . . . , xn }). Since, moreover, x is an extreme point, we have that x = x j for some j = 0, . . . , n, which implies that x ∈ T . Once we know that x is a point of T , it is quite easy to see that x is an extreme point of T . Otherwise, it would not be an extreme point of cc(T ) either.  Theorem 2.9.4 (Krein–Milman Theorem) Let T be a compact non-empty subset of En . Then the convex hull of the set of extreme points of T is also the convex hull of T ; that is, cc(Ext(T )) = cc(T ). Proof We first prove, by induction, that if T is a convex body, then cc(Ext(T )) = T . The claim is obvious for sets of dimension zero. Assume that it is true for convex bodies of dimension n − 1. Clearly, cc(Ext(T )) ⊂ cc(T ) = T . To prove that T ⊂ cc(Ext(T )) it suffices to prove that bdT ⊂ cc(Ext(T )). Let a ∈ bdT and let H be the support hyperplane of T at a. The set T ∩ H is closed, bounded, and convex, and therefore, by the induction hypothesis, a ∈ T ∩ H = cc(Ext(T ∩ H )). It is now not difficult to see that every extreme point of T ∩ H is necessarily an extreme point of T , which implies that a ∈ cc(Ext(T )). This proves the theorem for the convex case. Now, in general, if T is any compact set of En , then by Corollary 2.8.1 we have that cc(T ) is a convex body and therefore, by the convex case of this theorem, we have that cc(T ) = cc(Ext(cc(T ))). By Lemma 2.9.3, we know that cc(Ext(cc(T ))) ⊂ cc(Ext(T )), which implies that cc(T ) ⊂ cc(Ext(T )). Since the proof of the opposite inclusion is obvious, this concludes the proof of the theorem. 

2.10 Duality

37

2.10 Duality For every nonzero vector x ∈ En define the dual half-space of x as x ∗ = {y ∈ En | x, y ≤ 1}.

(2.10)

This gives a one to one correspondence between closed half-spaces containing the origin and nonzero vectors of En . We note that o∗ = En . Given a set A ∈ En , we define the polar dual of A as A∗ = ∩{x ∗ | x ∈ A} = {y ∈ En | x, y ≤ 1 for all x ∈ A}.

(2.11)

For each set A its polar dual A∗ is a closed convex set containing the origin. The polar duals ∅∗ and o are both En , while the polar dual of En is {o}. The proof of the following lemma is left as an exercise. ∗

Lemma 2.10.1 Let A and B be subsets of En and λ > 0 a real number. Then the following statements hold: a) If A ⊂ B, then B ∗ ⊂ A∗ , b) (λA)∗ = ( λ1 )A∗ , and c) B(o, 1)∗ = B(o, 1). Theorem 2.10.1 Let A be a closed subset of En . Then A∗∗ = cl(cc(A ∪ {o})). In particular, if A is a closed convex set containing the origin, then A∗∗ = A. Proof For all a ∈ A and x ∈ A∗ , we have a, x ≤ 1, whence a ∈ A∗∗ and A ⊂ A∗∗ . Since A∗∗ is a closed convex set containing the origin and A, we have cc(A ∪ {o}) ⊂ A∗∗ . We now establish the opposite inclusion. Suppose z ∈ / cc(A ∪ {o}). By Theorem 2.2.2, there is a hyperplane H that separates z from cc(A ∪ {o}). Thus, since H cannot pass through the origin, there exists u ∈ En such that u, z > 1 / a∗∗. Hence A∗∗ ⊂ cc(A ∪ {o}), and so and u, a < 1 for all a ∈ A. This shows that u ∈ A∗ and z ∈ ∗∗  A = cc(A ∪ {o}). We conclude this section by observing that if we consider the set of all convex sets containing the origin as an interior point, then the assignment φ → φ∗ is a continuous bijection.

2.11 The Problems of Busemann–Petty and Shephard In this small section we will discuss the so-called Busemann–Petty problem, referring to volumes of hyperplane sections of centrally symmetric convex bodies, and Shephard’s problem which is somehow “dual” since it refers to projections of centrally symmetric convex bodies onto hyperplanes. Although these problems are not directly related to bodies of constant width, there are several reasons to discuss also them in our book. First of all, both are typical problems from geometric tomography, a field which is closely related to parts of our Chapters 3, 9, and 13, and for which constant width is an

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important notion (see Gardner’s book [401]; already the definition of bodies of constant width has an inherent connection to geometric tomography). The second reason refers to the first occurrence of the Busemann–Petty problem in the literature; it was posed as one of ten problems in the paper [206] (the other nine problems are still unsolved). All of these ten problems are formulated in terms of classical convexity, but are also interpreted in the language of Minkowski geometry (i.e., the geometry of finitedimensional real Banach spaces), therefore useful for a better understanding of the local structure of Finsler spaces (which is Minkowskian). This viewpoint is closely related to our Chapter 10. And third we are convinced that this chapter also exemplifies a topic from convexity where recent research intensively goes on. As already mentioned, the purpose of this section is to compare the volumes of two convex bodies. More precisely, we want to study conditions on their sections or projections which guarantee that the volume of one is larger than the volume of the other. Given a convex body φ ⊂ En and u ∈ Sn−1 , we denote by φ|u the (n − 1)-dimensional body obtained by orthogonally projecting φ into Hu = {x ∈ En | x, u = 0}. Furthermore, we denote by υ(. ) the (n − 1)-dimensional volume. The Busemann–Petty Problem [206] Given two convex bodies ,  ⊂ En , both symmetric with respect to the origin such that υ( ∩ Hu ) ≤ υ( ∩ Hu ), for each u ∈ Sn−1 . Does it follow that

V () ≤ V ()?

This problem (like also the other nine problems from [206], which are completely unsolved for more than 60 years) attracted the interest of many specialists. But especially over the last 25 years (roughly) it was in the focus of many researchers, also after its first complete solution. A short introduction is given in [272, A 9]. The answer is yes if n ≤ 4 (see [399] and [1221]) and no if n ≥ 5 (see [687], [69], [400], [416], [915], and [1220]). A unified solution for all dimensions using Fourier analysis was given by Gardner, Koldobsky and Schlumprecht [402]; the solution uses the following interesting result by Lutwak [743] based on the notion of intersection body. The solution of the Busemann–Petty problem in En is positive if and only if every convex body in E with non-empty interior and symmetric with respect to the origin is an intersection body. Let φ be a convex body containing the origin as an interior point. Then the radial function ρ(u) of the intersection body of φ is given by n

ρ(u) = υ(φ ∩ Hu ), for each u ∈ Sn−1 . It is interesting to know that if, instead of requiring an inequality, the equality υ( ∩ Hu ) = υ( ∩ Hu ) for each u ∈ Sn−1 yields  = . See Theorem 16.4.1. Equally intuitive is Shephard’s Problem [1059] Given two convex bodies ,  ⊂ En , both symmetric with respect the origin, such that υ(|u) ≤ υ(|u)

2.11 The Problems of Busemann–Petty and Shephard

for each u ∈ Sn−1 . Does it follow that

39

V () ≤ V ()?

We start with the Aleksandrov Projection Theorem [12]. Theorem 2.11.1 (Aleksandrov) Given two convex bodies ,  ⊂ En , centrally symmetric with respect to the origin and such that υ(|u) = υ(|u) for each u ∈ Sn−1 . Then  is a translate of . A positive solution of Shephard’s problem is far from being true. There are convex bodies ,  ⊂ En , symmetric with respect to the origin such that υ(|u) < υ(|u) for each u ∈ Sn−1 , but satisfying V () > V (). Examples were provided by Petty [930] and Schneider [1031]. They also proved that the answer is positive if the body  is a zonoid. A convex polytope P is a zonotope if it is the Minkowski sum of line segments, that is, if it can be represented in the form P = J1 + · · · + Jm , where each Ji is a line segment. A convex body which is the limit of a convergent sequence of zonotopes is a zonoid. For further information regarding this incredibly interesting family of convex bodies see the two excellent surveys [1043] and [444]. Examples for zonotopes are the translative space fillers in 3-space (e.g., the affine cube and the rhombic dodecahedron), and all ellipsoids are special zonoids, as well as the 3-dimensional spindleshaped body of revolution obtained by rotating (around the x-axis) the part of the sine curve between 0 and π. A characterization of zonotopes is the following. Theorem 2.11.2 A convex polytope P is a zonotope if and only if P is centrally symmetric and every 2-dimensional face of P is a centrally symmetric polygon. From this one gets that the class of zonotopes coincides with the class of totally centrally symmetric polytopes (i.e., of those convex polytopes all whose faces of all dimensions have a center of symmetry). To get this class of n-dimensional polytopes, it is sufficient to demand central symmetry for all r -faces, for only one r with 2 ≤ r ≤ n − 2, see [809]. The projection body of a convex body is again a convex body (this way uniquely determined) whose support function value in direction u gives the (n − 1)-dimensional volume of the orthogonal projection of the original body in a hyperplane with normal u. It turns out that any projection body is (clearly symmetric about the origin, and) a zonoid, and for a given convex polytope particularly a zonotope. On the other hand, any n-dimensional zonoid centered at the origin is the projection body of a certain class of convex bodies, hence motivating interesting tomographic questions (see Chapter 4 of [401] and [783]). That is, given a convex body φ, the projection body φ is the unique zonoid (for polytopes the unique zonotope) whose support function is given by P φ(u) = υ(φ|u) for each u ∈ Sn−1 . In Shephard’s problem projection bodies play a similar role as intersection bodies play in the Busemann–Petty problem. Due to the following fact, discovered by Lindquist, the study of zonoids plays an important role in the study of Shephard’s problem. Theorem 2.11.3 A convex body is a projection body if and only if it is a zonoid centered at the origin.

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Figure 2.16 Projection body

The projection bodies of affinely equivalent convex bodies are also affinely equivalent. More precisely, we have Theorem 2.11.4 Let G : En → En be a linear isomorphism. Then (Gφ) = |det G|G −1 (φ). If φ is a convex body in En , there will be many other convex bodies whose projection bodies are also φ; only the parallelotopes are uniquely determined in this sense. For example, any body of constant brightness equal to one has the unit ball as its projection body (see Section 13.3.2). Among these, by Aleksandrov’s Theorem 16.4.1 there is a unique (up to translation) centrally symmetric body  whose projection body is φ. Under the hypothesis of the Busemann–Petty problem or the Shephard problem it is possible to prove that V () ≤

√ nV ().

While in the case of projections no essential improvements are found, in the case of sections be replaced by smaller quantities. It is even possible that the following is true.

n can

Slicing Problem Does there exist a universal constant c > 0 such that the following is true? Given two convex bodies ,  ⊂ En , symmetric with respect to the origin, such that υ( ∩ Hu ) ≤ υ( ∩ Hu ) for each u ∈ Sn−1 . Does this imply that V () ≤ cV () ? For more information about related results see Gardner [401] and Martini [783].

2.12 Ellipsoids

41

2.12 Ellipsoids In this section ellipsoids are defined as images of balls under an affine transformation. Therefore, the boundary of an ellipsoid is a closed (n − 1)-dimensional quadric surface. The purpose of this section is to develop the convex geometric tools to achieve deep characterizations of ellipsoids. For surveys in this partial field we refer to [932], [478], and [527]. A very recent, excellently written expository paper is [1081]. And also we wish to thank Efren Morales-Amaya for letting us use his notes about ellipsoids.

2.12.1 Brunn’s Theorem The purpose of this section is to prove a classical characterization of the ellipsoid given by Brunn. We will see that this theorem somehow presents the origin of almost every other characterization of the ellipsoid in convexity. Theorem 2.12.1 Let φ be a convex body in Euclidean space En . Then φ is an ellipsoid if and only if in any direction the midpoints of all chords of φ parallel to this direction lie in a hyperplane. An ellipsoid satisfies our hypothesis due to the following three observations: i) the balls satisfy our hypothesis, ii) if φ satisfies our hypothesis and is an affine transformation, then (φ) also satisfies our hypothesis, and iii) every ellipsoid is the image of a ball under an affine transformation. The proof of the converse requires some preliminary results on affine geometry. Corresponding original references can be found in the surveys mentioned above. Minimal Ellipsoids The following Theorem is also known in a more general form; but in the present one it is sufficient for our purposes here. Theorem 2.12.2 (Löwner’s Ellipsoid Theorem) Let φ be a compact set in Euclidean space En . Among all ellipsoids containing φ and centered at the origin, there is a unique one of minimum volume.

Figure 2.17

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Proof The proof is by induction on n. The theorem is clearly true for n = 1. Suppose it is true for dimension n − 1, to prove it for dimension n. Recall that for an ellipsoid E ⊂ En with axis lengths {2γ1 , ..., 2γn } the volume can be calculated by the following formula:

V (E) = κn γi , where κn is the volume of the n-dimensional ball of radius one. Since φ is a compact set, there exists σ > 0 such that σ = min{t > 0 | t = V (E)}, where E is an ellipsoid with center at the origin that contains φ. Suppose now that φ ⊂ E 1 ∩ E 2 , where E 1 and E 2 are ellipsoids centered at the origin o with volume σ. We shall show that E 1 = E 2 . If σ = 0, then φ is contained in a hyperplane. In this case the theorem follows by induction. Consequently, let be a linear isomorphism of determinant one that preserves areas and is such that it sends E 1 to a ball E 1 . Now let φ and E 2 be the images of φ and E 2 , respectively, under . Since E 1 is a ball centered at the origin o, we can take coordinates in such a way that E 2 :

x2 i ≤1 bi2

and

E 1 :

x2 i ≤ 1. a2

xi2 Since preserves areas, V (E 2 ) = V (E 1 ) = σ. Note that for all x ∈ φ we have ≤ 1 and bi2  1 xi2 ≤ 1, and then xi b2 + a12 ≤ 2. a2 i √ xi2  Therefore, φ ⊂ E 3 : ≤ 1, where ci = 2 √ ab2 i 2 . Calculating the volume of E 3 , we have that c2 a +bi

i

V (E 3 )2 = κn2



(abi )



(

 2abi  2abi 2abi   2 ) = V (E )V (E ) ( ) = σ ). ( 2 2 1 2 2 a 2 + bi a 2 + bi a + bi2

i Note that a2ab 2 +b2 ≤ 1 with equality only if a = bi . Thus, V (E 3 ) ≤ σ, which implies that V (E 3 ) = σ. i But then a = bi , and therefore E 1 = E 2 . Consequently we have that E 1 = E 2 , which is what we wished to prove. 

Remarks 2.12.1 A result closely related to Löwner’s Theorem is John’s Ellipsoid Theorem, which claims the following: Let φ ⊂ En be a centrally symmetric convex body. Then there exists a unique ellipsoid of maximal volume contained in φ. The proof of this theorem requires more work and will not be treated in this book. For more related results about ellipsoids consult [472]. As a corollary of Löwner’s Ellipsoid Theorem we have our first characterization. Theorem 2.12.3 Let φ be a convex body in En . Then φ is an ellipsoid if and only if, given any two points x and y in the boundary of φ, there is an affine isomorphism such that (φ) = φ and (x) = y. Proof Let E be the ellipsoid of minimal volume centered at the origin and containing φ, and let x ∈ bdφ∩ bd E. Let us take y ∈ bdφ. We shall show that y ∈ bd E. By hypothesis, there is a linear isomorphism such that (φ) = φ and (x) = y. Since (φ) = φ, the determinant of is ±1. Therefore preserves volumes. Thus, (E) is an ellipsoid that contains (φ) = φ, and whose volume is the volume of E. By Löwner’s Ellipsoid Theorem (E) = E, and therefore y ∈ bdE. We have shown that bdφ ⊂ bdE, and therefore, by Exercise 2.52, we have that E = φ. 

2.12 Ellipsoids

43

Remarks 2.12.2 Theorem 2.12.3 is the affine version of the following characterization of the ball: Let φ be a convex body in En . Then φ is a ball if and only if, given any two points x and y in the boundary of φ, there is an isometry such that (φ) = φ and (x) = y. The proof follows essentially the proof of Theorem 2.12.3, if we substitute the notion of minimal ellipsoid by the notion of circumsphere. This result allows us, for example, to give a very elegant proof of the fact that a convex body with two axes of revolution is a ball. We shall first prove Brunn’s Theorem when our convex body is centrally symmetric. Lemma 2.12.1 Let φ ⊂ En be a centrally symmetric convex body with the property that for any direction the midpoints of all chords of φ parallel to this direction lie in a hyperplane. Then φ is an ellipsoid. Proof Without loss of generality we may assume that the center of φ is the origin. Our purpose is to use Theorem 2.12.3 to conclude that φ is an ellipsoid. Let x, y∈ bdφ be any two points. By hypothesis there is a hyperplane containing the midpoints of all chords of φ parallel to the interval with endpoints x and y. Clearly, the origin lies in H . The reflexion along H in the direction x y is a linear isomorphism that sends φ into itself; furthermore it sends x into y. Theorem 2.12.3 implies that φ is an ellipsoid.  The next step is to prove our theorem in its full generality. Let us do it first for n = 2. Lemma 2.12.2 Let φ ⊂ E2 be a convex body with the property that in any direction the midpoints of all chords of φ parallel to this direction lie in a line. Then φ is centrally symmetric and, therefore, is an ellipse. Proof Let us choose a direction u 1 ∈ S1 , and let L 1 be the line containing the midpoints of all chords of φ parallel to u 1 . Let 1 be the reflexion along L 1 in the direction u 1 . Clearly, 1 (φ) = φ. Now let L 2 be the line containing the midpoints of all chords of φ parallel to L 1 noting that 1 sends any chord parallel to L 1 to some other chord parallel to L 1 . Furthermore, since linear isomorphism preserves midpoints 1 sends the midpoint of a chord I parallel to L 1 to the midpoint of the parallel chord (I). This implies that 1 (L 2 ) = L 2 . Let 2 be the reflection along L 2 in the direction L 1 . Similarly, we obtain 2 (L 1 ) = L 1 . Let us assume, without loss of generality, that L 1 and L 2 are the two orthogonal principal coordinate axes of E2 . Then it is easy to see that 2 1 (x) = −x for every  x ∈ E2 , and thus φ = −φ. Moreover, by Lemma 2.12.1 φ is an ellipse. Our previous lemma proves that, if φ ⊂ En is a convex body such that in any direction the midpoint of all chords of φ parallel to this direction lie in a hyperplane, then every 2-dimensional section of φ is an ellipse. To conclude the proof of Brunn’s Theorem it will be enough to prove the following theorem. Theorem 2.12.4 Let φ ⊂ En be a convex body, and let p0 be a point in its interior. Suppose that for every 2-dimensional plane H through p0 the section H ∩ φ is an ellipse. Then φ is an ellipsoid. Proof We shall prove the theorem for n = 3. The proof for n > 3 follows with the same ideas and will be left as an exercise to the reader. Choose q1 q2 , a chord of φ through p0 , with the property that p0 is the midpoint of q1 q2 , and let H1 and H2 be support planes of φ through q1 and q2 . Finally, let L = H1 ∩ H2 . If is the line through the origin parallel to q1 q2 , then the plane  = L + through L does not intersect φ. If we add to E3 the points at infinity and consider the projective isomorphism which sends the plane  to the plane at infinity, then the image of our convex body φ also satisfies the hypothesis of our theorem. But now p0 is the midpoint of q1 q2 , and the support planes H1 and H2 are parallel. Furthermore, by using an affine isomorphism we may assume, without loss of generality, that the chord q1 q2 is orthogonal to the support planes H1 and H2 . Let H0 be the plane through p0 parallel to H1 and H2 , and let E be the ellipse H0 ∩ φ. For every plane H  through the chord q1 q2 the ellipse E  = H  ∩ φ has H  ∩ H1 and H  ∩ H2 as support lines

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both, orthogonal to q1 q2 . This implies that the chord q1 q2 is an axis of the ellipse E  and p0 is the center of E  . Moreover, the chord H  ∩ H0 of E  is its second axis and has also p0 as its midpoint. Hence, p0 is the center of the ellipse E. By using an affine isomorphism we may assume that, without loss of generality, E is a circle with center at p0 and with diameter the length of the chord q1 q2 . If this is so, the ellipse E  is also a circle with center at p0 , and since this also holds for any plane H  containing q1 q2 , we have that φ is a ball. This concludes the proof of the theorem.  Theorem 2.12.4 allows us to characterize ellipsoids via properties of their sections. The next theorem characterizes ellipsoids by their orthogonal projections. We present hereby the proof of Chakerian in [229], which follows the proof of Süss [1110] for the case n = 3. Theorem 2.12.5 Let φ ⊂ En be a convex body. Suppose that for every hyperplane H through the origin the orthogonal projection of φ onto H is an ellipsoid. Then φ is an ellipsoid. Proof The proof of the theorem consists of three steps: In the first step we prove that the property that every orthogonal projection of φ is an ellipsoid is an affine property. In the second step we prove that, orthogonal to pq, then H ∩ φ and if pq is a binormal chord of φ and H is the hyperplane through p+q 2 the orthogonal projection of φ onto H coincide. In the third step we conclude the proof of the theorem. For  the first step let be an affine isomorphism and let L be any line. We have to prove that (φ) + L ∩ L ⊥ is an ellipsoid, where L ⊥ is the hyperplane orthogonal to the line L. Con   ⊥ sider φ + −1 (L). By hypothesis φ + −1 (L) ∩ −1 (L) is an ellipsoid. By Exercise 2.57,   φ + −1 (L) ∩ −1 (L ⊥ ) is an ellipsoid contained in the hyperplane −1 (L ⊥ ). Then its image under   an affine isomorphism (φ) + L ∩ L ⊥ is an ellipsoid contained in L ⊥ . For the second step, first note that by hypothesis φ is strictly convex. Let H be the hyperplane orthogonal to pq and consider a hyperplane  parallel to pq. Let φ∗ be the orthogonal through p+q 2 projection of φ onto . Then p  q  , the orthogonal projection of pq onto , is a binormal of φ∗ . Since by hypothesis φ∗ is an ellipse, we have that p  q  is one of its axes. The other axis of φ∗ lies in H . Hence, for any support hyperplane  of φ∗ ⊂  parallel to pq the support point  ∩ φ∗ lies in H . If  is the support hyperplane parallel to pq and orthogonal to , then  projects orthogonally onto  . Consequently, it intersects φ in a single point belonging to H . Since the same holds true for every support hyperplane of φ parallel to pq, we have that H ∩ φ and the orthogonal projection of φ onto H coincide. We are ready to prove our theorem. Let pq be a diameter of φ. Consider H as the hyperplane orthogonal to pq. Then H ∩ φ and the orthogonal projection of φ onto H coincide, and through p+q 2 by hypothesis, S = H ∩ φ is an ellipsoid, which, from the first part of the proof, we may assume to be, without loss of generality, a ball. Note that every diameter r t of the ball S is a binormal of φ, and therefore, by the second step of the proof, E =  ∩ φ is an ellipsoid where  is the hyperplane through pq orthogonal to r t. Note that E =  ∩ φ is an ellipsoid of revolution with axis on the chord pq. Since the same holds true for every hyperplane through pq, we conclude that φ is an ellipsoid of revolution. 

2.12.2 Blaschke’s Theorem The Shadow Boundary of a Convex Body We start with the following definitions. A line L is tangent to a convex body φ ⊂ En if the points of L ∩ φ lie in the boundary of φ. Let us consider a direction u ∈ Sn−1 . The union of all points of the boundary of φ, which lie in all tangent

2.12 Ellipsoids

45

lines of φ parallel to u, will be called the shadow boundary of φ in the direction u, and it will be denoted by S∂(φ, u). If is a line through the origin in the direction u, we denote by T ∂(φ, ), or T ∂(φ, u), the union of all tangent lines of φ parallel to . Therefore, we have that T ∂(φ, u) ∩ φ = S∂(φ, u) and S∂(φ, u) + = T (φ, u). Moreover, we will denote the shadow boundary of φ in the direction u by S∂(φ, ), where u is parallel to . If φ is strictly convex, then for every tangent line of φ parallel to u we have a unique point in the shadow boundary of φ and also a unique point in the boundary of the orthogonal projection of φ in the direction u. This shows that for a strictly convex body the shadow boundary of φ is a topological (n − 2)-dimensional sphere embedded in bdφ. For a ball the shadow boundaries are precisely the equators, i.e., in all dimensions the analogues of all great circles. For an ellipsoid the shadow boundaries are the affine equators. Thus they lie in some hyperplane. Blaschke’s Theorem is precisely the converse of this statement. Theorem 2.12.6 (First version of Blaschke’s Theorem) Let φ ⊂ En be a convex body with the property that every shadow boundary lies in a hyperplane. Then φ is an ellipsoid. For the proof of Blaschke’s Theorem we need the following characterization of central symmetry. Theorem 2.12.7 Let φ be a strictly convex body. Then φ is symmetric with respect to the point x0 as its center if and only if the diametral chords of φ are concurrent in x0 . Proof If φ is centrally symmetric, then obviously all diametral chords are concurrent in the center. First suppose φ is a strictly convex figure in E2 . Let r (ϑ) and s(ϑ) be the pedal functions of φ, and −φ, respectively. See Section 5.1. Then both r (ϑ) and s(ϑ) are Lipschitz functions, and therefore they are differentiable almost everywhere. The point x(ϑ) = r (ϑ)u(ϑ) + r  (ϑ)u (ϑ) in the boundary of φ has a support line parallel to u (ϑ). Analogously, with −φ the point y(ϑ) = s(ϑ)u(ϑ) + s  (ϑ)u (ϑ) in the boundary of −φ has a support line parallel to u (ϑ). Therefore, by hypothesis there exists t1 > 0 is a Lipschitz function, such that t1 x(ϑ) = y(ϑ) for almost every ϑ. This implies that f (ϑ) = x(ϑ) y(ϑ) whose derivate is zero almost everywhere. Therefore there exists t2 > 0 such that t2 r (ϑ) = s(ϑ). This proves that φ is a homothetic copy of −φ. By Exercise 2.58 this is enough to conclude that φ is centrally symmetric. We are ready to prove our theorem for higher dimensions. Let H be a 2-dimensional plane through x0 . Note that, if every chord of φ through x0 is a diametral chord of φ, then every chord of H ∩ φ through x0 is a diametral chord of H ∩ φ. This implies that H ∩ φ is centrally symmetric with its center at x0 , and from Exercise 2.59 it follows that φ is centrally symmetric with its center at x0 .  Remarks 2.12.3 Another variant of Theorem 2.12.7, namely when φ is not strictly convex, is the following. If every chord through x0 ∈ intφ is a diametral chord, then φ is centrally symmetric with its center at x0 . The proof of this new, slightly different version of Theorem 2.12.7 is considerably more difficult and is based on the following statement. Let φ1 and φ2 be two convex bodies containing the origin on their interiors. Suppose that for every ray through the origin there are parallel support lines to φ1 and φ2 at ∩ bd φ1 and ∩ bd φ2 , respectively. Then φ1 is directly homothetic to φ2 through the origin, see Theorem 16.7 in Burton [200]. The Proof of the First Version of Blaschke’s Theorem The proof of the theorem is by induction on the dimension n. We shall first prove the theorem for n = 3. The plan of the proof is the following: In the first step we prove that φ is centrally symmetric with its center at p0 . In the second step we prove that every section of φ is centrally symmetric, and we specify

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2 Convex Geometry

where the center of the section lies. In the third step we show that every plane through p0 generates a shadow boundary. Finally, the purpose of the fourth step is to prove, using Brunn’s theorem, that every section through p0 is an ellipse, thus proving by Theorem 2.12.4 that φ is an ellipsoid. First note that φ is strictly convex. Using Theorem 2.12.7, we shall prove that φ is centrally symmetric. For this purpose we start proving that any two diametral chords have no empty intersection. Let q1 q2 and q1 q2 be two diametral chords, and let H and H  be planes through the origin with the property that q1 + H and q2 + H are support planes of φ. Similarly, q1 + H  and q2 + H  are support planes of φ. If = H ∩ H  , it is clear that the four lines q1 + , q2 + , q1 + and q2 + are support lines of φ. Therefore, by definition {q1 , q2 , q1 , q2 } ⊂ S∂(φ, ). The fact that S∂(φ, ) lies in a plane implies that the diametral chords q1 q2 and q1 q2 have a point p0 in common. Let q1 q2 be another diametral chord, which is not in the plane determined by the first two diametral chords. Then q1 q2 intersects both q1 q2 and q1 q2 , and hence p0 lies in q1 q2 . Consequently, every diametral chord, which is not in the plane generated by q1 q2 and q1 q2 , passes through p0 . This is enough to conclude that all diametral chords are concurrent in p0 , and thus φ is centrally symmetric. From now we assume, without loss of generality, that p0 is the origin. For the next step, let H be any plane which intersects the interior of φ, and let H1 and H2 be the support planes of φ parallel to H . Furthermore, let {q1 } = H1 ∩ φ and {q2 } = H2 ∩ φ. We shall prove that H ∩ φ is centrally symmetric with its center at H ∩ q1 q2 . By Theorem 2.12.7 it will be enough to prove that every diametral chord of H ∩ φ, intersects q1 q2 . Indeed, let p1 p2 be a diametral chord of H ∩ φ, and let be a line through the origin parallel to H , such that p1 + and p2 + are support lines of H ∩ φ. Then the four lines p1 + , p2 + , q1 + , and q2 + are support lines of φ, and consequently, by definition, { p1 , p2 , q1 , q2 } ⊂ S∂(φ, ). Our hypothesis implies that S∂(φ, ) lies in a plane, and therefore p1 p2 intersects q1 q2 . The proof of the third step follows the same ideas. Let H be a plane through the origin and remember that φ is centrally symmetric with respect to the origin. We want to find a line through the origin such that H = S∂(φ, ). For this purpose let q1 q2 be a chord of φ through the origin contained in H , and let H1 be a plane through the origin, such that q1 + H1 and q2 + H1 are support planes of φ. Let p1 p2 = H ∩ H1 ∩ φ. Since the origin lies in the chord p1 p2 and since we know that the origin is the center of symmetry of the figure H1 ∩ φ, there is a line trough the origin such that p1 + and p2 + are support lines of H1 ∩ φ. This implies that the four lines p1 + , p2 + , q1 + , and q2 + are support lines of φ. Consequently, by definition, { p1 , p2 , q1 , q2 } ⊂ S∂(φ, ). Since by hypothesis S∂(φ, ) lies in a plane and { p1 , p2 , q1 , q2 } ⊂ H , then S∂(φ, ) = H. It is time to prove that every section through the center of symmetry of φ is an ellipse because it satisfies the hypothesis of Brunn’s Theorem. Let  be a plane through the origin, which is the center of φ, and let u be the unit vector contained in . We shall prove that the midpoints of every chord of  ∩ φ parallel to u lie in a line. We proved in the third step that  ∩ φ = S∂(φ, ) for some line through the origin. Let L 1 , L 2 ⊂  be the support lines of  ∩ φ parallel to u, and let Hi, = L i + , i = 1, 2. Obviously, H1 and H2 are parallel support planes of φ. Let q1 q2 be the diametral chords of φ, such that qi = Hi ∩ φ, i = 1, 2. We proved in the second step that for every plane parallel to Hi the figure H ∩ φ is centrally symmetric with its center at H ∩ q1 q2 . Then the midpoints of the chords of  ∩ φ in the direction u lie in q1 q2 . This proves that  ∩ φ is an ellipse for every plane  through the origin, and hence by Theorem 2.12.4 φ is an ellipsoid. This concludes the proof of the theorem when n = 3. If n > 3 and H is a hyperplane through the interior of φ, then in every direction u parallel to H we have that S∂(φ, u) ∩ H = S∂(H ∩ φ, u). Consequently, if every shadow boundary of φ lies in a hyperplane, then every shadow boundary of the (n − 1)-dimensional convex body H ∩ φ lies in a hyperplane, and by induction H ∩ φ is an ellipsoid. Finally, by Exercise 2.56 we have that φ is an ellipsoid. 

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Second Version of Blaschke’s Theorem The following version of Blaschke’s theorem is especially useful for dealing with convex bodies that a priori are not necessarily strictly convex. Theorem 2.12.8 (Second Version of Blaschke’s theorem) Let φ ⊂ En be a convex body, n ≥ 3, and let x0 be an interior point of φ. If for every hyperplane H through x0 the boundary of H ∩ φ is contained in a shadow boundary of φ, then φ is an ellipsoid. The case n = 3. The proof consists of several steps. In the first step we prove that φ is centrally symmetric with its center at x0 , and in the second step we prove that φ is strictly convex. The third step is devoted to proving that every shadow boundary is a section. Therefore, by the first version of Blaschke’s theorem, φ is an ellipsoid. For the first step, by Remark 2.12.3, it will be enough to prove that every chord through x0 is a diametral chord. To see this, let L be a line through x0 and let H and  be two different planes through L. We know that there are vectors v H and v such that bd(H ∩ φ) ⊂ S∂(φ, v H ) and bd( ∩ φ) ⊂ S∂(φ, v ). Since L ∩ bd φ consists of two points p and q, and { p, q} ⊂ S∂(φ, v H ) ∩ S∂(φ, v ), there is a support plane of φ at p generated by v H and v , the same holds true for q. This proves that L ∩ φ is a diametral chord and, therefore, that φ is centrally symmetric. For the second step we first prove that there are no plane disks in the boundary of φ. For this purpose let  be a support plane with the property that the figure  ∩ φ has relative interior points, and let ab be an interior chord of  ∩ φ, which is not a diametral chord. Let H be the plane through x0 and the chord ab, and by hypothesis suppose that there is a line L H through the origin, such that H ∩ bd φ ⊂ S∂(φ, L H ). Since ab = φ ∩  ∩ H , that is, ab is the intersection of a support line with bd(H ∩ φ), we have that ab + L H = T ∂(φ, L H ) ∩ . Then  ∩ φ ⊂ ab + L H is a contradiction to the fact that ab is not a diametral chord of  ∩ φ. Now we prove that there are no close segments I = ab, a = b, in the boundary of φ. Let H be a plane through x0 that intersects φ transversally. Then H ∩ bd φ ⊂ S∂(φ, v0 ), where v0 is the unit vector parallel to I . The reason is that by hypothesis there is a unit vector v such that H ∩ bd φ ⊂ S∂(φ, v). Obviously v = v0 , otherwise there would be a plane disk in the boundary of φ. This is a contradiction because, if H ∩ bd φ ⊂ S∂(φ, v0 ) for every plane H through x0 that intersects transversally I , then bd φ ⊂ S∂(φ, v0 ), which is impossible. This proves that φ is strictly convex. Once we know that φ is strictly convex we have that every section generates a shadow boundary. We need to prove that every shadow boundary is generated by a section. For this purpose let p ∈ S∂(φ, u), and L be the line through x0 and p. Suppose L ∩ bd φ = { p, q}. Obviously { p, q} ⊂ S∂(φ, u). Let m ∈ S∂(φ, u) \ { p, q}, and H be the plane generated by L and m. By hypothesis H ∩ bd φ = S∂(φ, v) for some v. If u = v, there is nothing to prove. But if u = v, then there is a support plane of φ at m generated by the vectors u and v. This is a contradiction to the fact that exactly the same holds true for p and q.  k-Shadow Boundaries and the Case n > 3 We are going to develop the notion of k-shadow boundaries. This notion will allow us to generalize several results for dimensions higher than 3. In particular, it is essential for the proof of Theorems 2.12.8 and 2.12.12 for dimensions greater than 3. As always, we say that a k-plane, or k-dimensional plane, H is tangent to or supporting a convex body φ ⊂ En if H intersects φ, but not its interior. Observe that if φ is strictly convex, a support k-plane intersects φ in a single point. Let  be a k-plane through the origin. Let us consider all the points in the boundary of φ for which there is a support k-plane of φ parallel to . This set will be called the k-shadow boundary of φ in the direction , and will be denoted by S∂(φ, ).

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When φ is strictly convex, the orthogonal projection of En onto the (n − k)-dimensional plane orthogonal to  sends the k-shadow boundary of φ onto the boundary of the projection of φ. Thus, S∂(φ, ) is a topological (n − k − 1)-sphere embedded in the boundary of φ. Finally, we say that the k-shadow boundary is flat if there is a (n − k)-plane that contains it. The analogue of the Blaschke theorem is the following. Theorem 2.12.9 Let φ ⊂ En be a convex body. If every k-shadow boundary of φ is flat, then φ is an ellipsoid. The proof of this theorem is completely analogous to the proof of Blaschke’s theorem, but using the notion of k-shadow boundary instead of the notion of shadow boundary. Finally, the complete proof of Theorem 2.12.8 follows the ideas of the case n = 3, but using the notion of k-shadow boundary. Applications of Blaschke’s Theorem Our first application of Blaschke’s Theorem deals with the following property of ellipsoids: every two parallel sections are homothetic. The fact that ellipsoids have this property is not immediate because the notion of homothety is not an affine notion. Therefore it is not preserved under linear isomorphisms. Thus, in order to prove that ellipsoids satisfy this property we need the following definition. Let φ1 and φ2 ⊂ En be two strictly convex bodies. For a unit vector u ∈ Sn−1 the u-diametral chord of φ1 is the chord pq of φ1 with the property that p + u ⊥ and q + u ⊥ are the support hyperplanes of φ1 orthogonal to u. We say that φ1 is projectively homothetic to φ2 , if for every direction u the u-diametral chords of φ1 and φ2 are parallel. Clearly, the notion of projective homothety is an affine notion, and homothetic bodies are projectively homothetic. The opposite is not true. For example, a ball and a body of constant width are projectively homothetic, but not necessarily homothetic. In fact, it is not difficult to prove the following theorem. Theorem 2.12.10 Let φ1 and φ2 ⊂ En be two smooth, strictly convex bodies. Then φ1 is projectively w(φ1 ) of its width functions is constant. homothetic to φ2 if and only if the ratio w(φ 2) The proof of this theorem depends basically on the fact that for a smooth and strictly convex body the angle between a vector u and its u-diametral chord is the derivate of the width function. For two convex bodies that are additionally symmetric with respect to the origin, their support functions are half the width functions. Consequently, the notion of homothety and projective homothety coincide. From the fact that the notion of projective homothety is an affine notion and coincides with the notion of homothety for ellipsoids, we have that any two parallel sections of an ellipsoid are homothetic ellipsoids. On the other hand, this property characterizes ellipsoids. We give the proof for dimension three. The proof for higher dimensions is straightforward and requires the concept of k-shadow boundaries. Theorem 2.12.11 Let φ ⊂ E3 be a convex body. Every two parallel sections of φ are homothetic if and only if φ is an ellipsoid. Proof We leave it to the reader to verify that under this hypothesis φ is strictly convex. Let u be a unit vector. We shall prove that S∂(φ, u) lies in a plane. Let S be the set of all unit vectors orthogonal to u, and let v ∈ S. Suppose H is a plane orthogonal to v, which cuts the interior of φ. As we assume that φ is strictly convex, then H ∩ S∂(φ, u) has exactly two points, say a and b. If this is the case, we say that the chord ab of S∂(φ, u) is a v-chord of S∂(φ, u). We know that the support lines of H ∩ φ parallel to u pass precisely through a and b. Therefore the chord ab of H ∩ φ is the corresponding diametral chord in the direction u. If H1 and H2 are two

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planes orthogonal to v, the v-chords H1 ∩ S∂(φ, u) and H2 ∩ S∂(φ, u) of S∂(φ, u) are parallel, since Hi ∩ S∂(φ, u) is a chord of Hi ∩ φ in the direction u, i = 1, 2, and H1 ∩ φ is homothetic to H2 ∩ φ. This proves that for any v ∈ S there is a line L v through the origin, such that all the v-chords of S∂(φ, u) are parallel to L v . Our intuition tells us that this would be enough to claim that S∂(φ, u) lies on a plane. Indeed, this is the case. To prove this, our next step is to show that there is a plane T through the origin, such that for every v ∈ S L v = T ∩ v⊥. Let v1 , v2 in S and T be the plane generated by L v1 and L v2 . Let us consider now the inverse Gauss map γ : S2 → bd φ, which assigns to every unit vector w the unique point γ(w) of bdφ with the property that w is a normal vector to bd φ at γ(w). Note that S∂(φ, u) = {γ(v) | v ∈ S}. Take c ∈ S∂(φ, u) between γ(v1 ) and γ(v2 ) and let ca be the v1 -chord of S∂(φ, u) and cb be the v2 -chord of S∂(φ, u). Let H be the plane containing the chord ab of S∂(φ, u) and parallel to u, and suppose that w is orthogonal to H . By definition w ∈ S, ab is a w-chord of S∂(φ, u), and at least for this unit vector w ∈ S our premise L w = T ∩ w ⊥ = T ∩ H is satisfied. Clearly, w depends continuously on the point c while moving along S∂(φ, u). Note that when c approximates γ(v2 ), a also approximates γ(v2 ). Hence w approximates v2 . Similarly, when c approximates γ(v1 ) also b does, therefore w approximates v1 . Consequently, while c travels between γ(v1 ) and γ(v2 ), w travels continuously between v1 and v2 . It is easy to verify that while c travels through S∂(φ, u), the corresponding w takes all the directions of S. This proves that for every unit vector w ∈ S, L w = T ∩ w ⊥ . Finally, this gives us enough information to prove that S∂(φ, u) lies on a plane because, if p ∈ S∂(φ, u) is a fixed point, then S∂(φ, u) \ { p} = {q ∈ bd φ | pq is a v-chord, v ∈ S \ {γ −1 ( p)}}. Given that pq ⊂ p + L v ⊂ p + T , we have that S∂(φ, u) = ( p + T ) ∩ bd φ. Since the same happens for every shadow boundary S∂(φ, u) by Blaschke’s theorem, we have that φ is an ellipsoid.  The last theorem can be generalized as follows: Its poof is essentially the proof of the 3-dimensional case, but now using the notion of k-shadow boundaries. Theorem 2.12.12 Let φ ⊂ En be a convex body and 2 ≤ k < n. If for every two parallel k-planes H and  that intersect the interior of φ we have that H ∩ φ and H ∩  are homothetic, then φ is an ellipsoid. The next characterization of the ellipsoid is essential, too. Theorem 2.12.13 Let φ ⊂ En be a convex body. If for every hyperplane H the section H ∩ φ is either empty or centrally symmetric, then φ is an ellipsoid. Proof We start proving that φ is centrally symmetric. Let q1 q2 be a diameter of φ. Any hyperplane  through q1 q2 intersects φ in a centrally symmetric body, and since q1 q2 keeps being a diameter of  ∩ φ then, by Exercise 2.65 the midpoint x0 of q1 q2 is the center of  ∩ φ. Since the same holds true for every section through q1 q2 , we have that φ is centrally symmetric with its center at x0 , see Exercise 2.66. The next step is to use the second version of Blaschke’s theorem. With this purpose we need to prove that for any hyperplane H through x0 the boundary of H ∩ φ is contained in a shadow boundary of φ. Without loss of generality, let x0 be the origin, and u be a normal unit vector to H. The convex bodies φi = (H + 1i u) ∩ φ and φ−i = (H − 1i u) ∩ φ are centrally symmetric. By the symmetry of φ

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with respect to the origin we have that φ−i = −φi , and by Exercise 2.58 and the central symmetry of φi we have that φ−i is a translated copy of φi . Assume that φi = φ−i + vi . Without loss of generality, we may assume that the vectors vvii  converge to a vector v. We shall prove bd(H ∩ φ) ⊂ S∂(φ, v). With this purpose let x ∈ bd(H ∩ φ) and consider the line {x + tv | t ∈ R}. Suppose this line is not a support line of φ. If this is the case, then we may assume, without loss of generality, that for i ≥ η0 , {x + tv | t ∈ E+ }∩intφi = ∅. But then there is η1 > η0 such that for i ≥ η1 , {x + tvi | t ∈ E+ }∩intφη0 = ∅. This implies that {x + tvη1 | t ∈ E+ }∩intφη1 = ∅, which is a contradiction. Consequently, for every x ∈ bd(H ∩ φ) the line {x + tv | t ∈ R} is a support line of φ. Therefore bd(H ∩ φ) ⊂ S∂(φ, v). By the second version of Blaschke’s theorem φ is an ellipsoid. 

2.12.3 The False Center of Symmetry We just proved that if all sections of a convex body are centrally symmetric, then this body must be an ellipsoid. We would like to improve this result considerably restricting the number of sections involved. For example: What can we say about a convex body φ, if all its sections through a fixed point p0 are centrally symmetric? Not too much; perhaps φ is centrally symmetric and p0 is its center. Indeed, this is the most uninteresting case. For example, if p0 is any point, not necessarily the center of an ellipsoid E, then all sections of E through p0 are ellipsoids, and consequently all of them are centrally symmetric. This phenomenon motivates the following definition. Let φ ⊂ En be a convex body, and p0 be a point of En , n ≥ 3. We say p0 is a false center of symmetry of φ, if p0 is not a center of symmetry of φ, yet any 2-dimensional section of φ through p0 is either empty or centrally symmetric. In the case of an ellipsoid any point is a false center of symmetry. Note that for a false center of symmetry p0 of φ the centers of symmetry of sections of φ through p0 do not coincide with p0 . What can we say about a convex body φ with a false center of symmetry? Is it at least centrally symmetric? The last part of this section will be devoted to answering these questions. Two Theorems of Rogers About Central Symmetry Following the ideas of Rogers [980], we start with the following naive statement. Let p1 ∈ intφ1 and p2 ∈ int φ2 be two interior points of the two respective convex bodies. If every section of φ1 through p1 is a translate of the corresponding parallel section of φ2 through p2 , then φ1 is a translate of φ2. This is intuitively clear, and the amazing issue is that the translation that sends φ1 to φ2 does not necessarily send p1 to p2 . This is exactly what happens when p1 ∈ intφ1 , − p1 ∈ int(−φ1 ), and p1 is a false center of symmetry of φ1 . Theorem 2.12.14 Roger’s theorem. Let φ1 , φ2 ⊂ En be two convex bodies, n ≥ 3, and p1 ∈ intφ1 and p2 ∈ intφ2 . Suppose that for every hyperplane H1 through p1 the section H1 ∩ φ1 is directly homothetic to H2 ∩ φ2 , where H2 is the parallel hyperplane of H1 through p2 . Then φ1 is directly homothetic to φ2 . Proof We start the proof of the theorem with the following claim, whose proof is left as an exercise to the reader.

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Claim. Let C1 ⊂ C2 be two directly homothetic, strictly convex bodies. Suppose they have a common support hyperplane H for which H ∩ C1 = H ∩ C2 = {x0 }. Then there is a positive homothety with center at the point x0 that sends C1 onto C2 . Let us first prove the theorem in a particular case. Assume that φ1 ⊂ φ2 are strictly convex and satisfy the hypothesis of the theorem. Furthermore, suppose the following two additional properties: the first is that p1 = p2 , and the second is that our convex bodies have a common support hyperplane H for which H ∩ φ1 = H ∩ φ2 = {x0 }. If this is so, we shall prove that there is a positive homothety that sends φ1 onto φ2 . Let L be the line through x0 and p1 = p2 . By this claim for every plane H through L there is a positive homothety with center at x0 that sends H ∩ φ1 onto H ∩ φ2 . If the ratio of the homothety for each of the sections is the same as for the homothety sending φ1 onto φ2 . This is so because each one of the homotheties has its center at x0 and sends L ∩ φ1 onto L ∩ φ2 . We are now ready for the proof of the theorem for the case in which φ1 and φ2 are strictly convex. Note first that if φ1 is a direct homothetic copy of φ1 then φ1 also satisfies the hypothesis of the theorem. By the previous arguments for the proof it is sufficient to verify the existence of a positive homothety that sends φ1 onto φ1 in such a way that φ1 and φ2 satisfy the conditions under which the theorem was previously proved. First of all, by using a positive homothety we may assume that p1 = p2 and also that φ1 ⊂ intφ2 . Now simply dilate φ1 through a positive homothety until the boundary of φ1 touches the boundary of φ2 in some point, which will be called x0 . This concludes the proof of the strictly convex case. It is an easy exercise to prove the theorem in full generality.  The following theorem tells us that a convex body with a false center has a center of symmetry. Theorem 2.12.15 Let φ ⊂ En be a convex body, n ≥ 3, and let p0 ∈ intφ be a false center. Then φ is centrally symmetric. Proof Simply note that p0 ∈ intφ and − p0 ∈ int(−φ) satisfy the hypothesis of 2.12.14. Therefore φ is directly homothetic to −φ, which implies that φ is centrally symmetric.  Remarks 2.12.4 By Exercise 2.69 the previous theorem holds true also when p0 is not necessarily an interior point of φ. The following notable theorem due to Rogers [983] is crucial in the proof of the false center theorem. Theorem 2.12.16 (The Equichordal Theorem of Rogers) Let φ be a plane convex figure and let p1 and p2 be two points with the property that for any line L 1 through p1 the length of the interval L 1 ∩ φ is equal to the length of the interval L 2 ∩ φ, where L 2 is the line parallel to L 1 passing through p2 . Then φ is a centrally symmetric figure. Furthermore, its center is the midpoint between p1 and p2 . Our next purpose is to prove the false center theorem. Theorem 2.12.17 (The False Center Theorem) Only ellipsoids admit false centers. That is, let φ ⊂ En be a convex body, n ≥ 3, and let p0 ∈ En be a false center. Then φ is an ellipsoid. The proof of the theorem requires a technical lemma. Lemma 2.12.3 Let φ ⊂ E3 be a centrally symmetric convex body with center at the origin, and let p ∈ E3 be a false center of φ. Then for every plane H through the origin, such that neither p nor the origin is a center of the non-empty section ( p + H ) ∩ φ, there is a line L H with the property that bd(H ∩ φ) ⊂ S∂(φ, L H ).

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Proof Let H be a plane through the origin such that neither p nor the origin is a center of the non-empty section ( p + H ) ∩ φ, and let v H be its center of symmetry. For the proof of this lemma it will be sufficient to prove that, for t > 0 sufficiently small, the section (H + tv H ) ∩ φ is centrally symmetric with its center at tv H . Indeed, if this is the case, by Exercise 2.71, (H + tv H ) ∩ φ − 2tv H = (H − tv H ) ∩ φ, which implies that bd(H ∩ φ) + {λv H | λ ∈ E} = T ∂(φ, v H ), and therefore bd(H ∩ φ) ⊂ S∂(φ, L H ). See the last part of the proof of Theorem 2.12.13. In order to prove that (H + tv H ) ∩ φ is centrally symmetric with its center at tv H we shall use Rogers Equichordal Theorem 2.12.16. Let us fix the points ( p − v H + tv H ) and (v H − p + tv H ) in the plane H + tv H . We shall prove that parallel chords of (H + tv H ) ∩ φ through each one of these points have the same length. The following notation will be convenient during the last part of this proof. Let L ⊂ H be a line through the origin. For each x ∈ E3 , such that (x + L) ∩ φ = ∅, let us denote by [x] the chord (x + L) ∩ φ, and by |[x]| its length. Let  be the plane through L and v H . Note that [ p] is a chord of ( p + H ) ∩ φ and also of ( p + ) ∩ φ. Since ( p + H ) ∩ φ is centrally symmetric with respect to v H we have that |[ p]| = |[2v H − p]|, and by symmetry of φ we have that |[ p]| = |[ p − 2vh |. Note that [ p] and [2v H − p] are parallel chords of ( p + ) ∩ φ with the same length. Consequently, one of the two situations holds true: • either the center of ( p + ) ∩ φ lies in [ p − v H ] ⊂ H, or • for −1 ≤ λ ≤ 1, |[ p − v H + λv H ]| = |[ p]|. In the first case, since ( p + ) ∩ φ is centrally symmetric and the center lies in [ p − v H ] we have that |[ p − v H + tv H ]| = |[ p − v H − tv H ]|, and by symmetry of φ, we have that |[ p − v H + tv H ]| = |[v H − p + tv H ]| for 0 ≤ t ≤ 1. In the second case, by symmetry of φ, |[v H − p − λv H ]| = |[ p]|, for −1 ≤ λ ≤ 1. In any of the two cases, we have that |[ p − v H + tv H ]| = |[v H − p + tv H ]|, and since this holds true for any line L ⊂ H through the origin that cuts φ, the Equichordal Theorem 2.12.16 of Rogers implies that (H + tv H ) ∩ φ is symmetric with respect to tv H for 0 ≤ t ≤ 1. Consequently,  bd(H ∩ φ) ⊂ S∂(φ, L H ). This completes the proof of the lemma. Proof of the False Center Theorem Our final purpose is to give the proof of the False Center Theorem. We focus on the case n = 3 and p0 ∈ intφ. By Theorem 2.12.15 we may assume that φ is symmetric with respect to the origin. Our purpose is to prove that for a dense set of planes  through the origin the boundary of the section  ∩ φ is contained in a shadow boundary. If this is so, by continuity and the second version of Blaschke’s theorem φ is an ellipsoid. Let  be the collection of all planes H through the origin, such that the center of ( p + H ) ∩ φ is p. By the above and Lemma 2.12.3 it will be enough to prove that for  ∈ int the boundary of the section   ∩ φ is contained  in a shadow boundary. Let H1 , H2 ∈ . By Exercise 2.71, ( p + Hi ) ∩ φ − 2 p = (− p + Hi ) ∩ φ i = 1, 2. It follows that if T is a support plane of φ at x ∈ ( p + H1 ) ∩ ( p + H2 ) ∩ bdφ, then T − 2 p is a support plane of x − 2 p ∈ bdφ. This is so because T is tangent to both sections ( p + H1 ) ∩ bd φ and ( p + H2 ) ∩ bd φ. Hence, since ( p + Hi ) ∩ bd φ is a translate of (− p + Hi ) ∩ bd φ, then T − 2 p is tangent to both curves (− p + H1 ) ∩ bd φ and (− p + H2 ) ∩ bd φ. This implies that T is parallel to the vector p, and therefore x − 2t p ∈ bdφ for 0 ≤ t ≤ 1. Suppose now H1 ∈ int. Then, given any line L through the origin contained in H , there is H2 ∈  such that H1 ∩ H2 = L. With this in mind, we conclude from the same argument of the last paragraph that (1 − 2t) p + bd(H ∩ φ) ⊂ bdφ for 0 ≤ t ≤ 1, which implies that bd(H1 ∩ φ) ⊂ S∂(φ, p/ p). This completes the proof of the theorem when n = 3. Now let n > 3 and p0 ∈intφ. For every 3-dimensional plane  through p0 the section  ∩ φ satisfies the required hypothesis to conclude that  ∩ φ is an ellipsoid. Therefore for every 2-dimensional plane H through p0 the section H ∩ φ is an ellipse. By Theorem 2.12.4 we have that φ is an ellipsoid. 

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Notes Convexity in the widest sense (e.g., also with its combinatorial, discrete, and stochastic aspects) is an interesting research field in the heart of mathematics, since many mathematical disciplines (like optimization, convex analysis, functional analysis, computer science (via computational geometry), discrete mathematics, combinatorics,...) need basic notions, tools, and methods from there. The field is mainly geometric in nature; therefore also some basic books on general classical geometry contain chapters/parts referring to convexity (see, e.g., [100, Chapter 12] and [156]). The refreshing character of the field mainly comes from the fact that one has immediate approach to the studied objects (e.g., strictly convex bodies, smooth convex bodies, convex polytopes, etc.), which are at first glance not too complicated and geometrically defined. (Strongly enriching the geometric part, in Minkowski geometry a second convex body, the unit ball, comes additionally into the game.) But regarding the used methods the field is very wide; these methods can come from analysis, stochastic, discrete geometry, combinatorics, integral and differential geometry, classical geometry, etc.! In other words, we have a clearly defined, geometrically inspiring family of studied objects, and these bring together mathematicians from very different fields to look at “the same geometric objects” with very different viewpoints (a property that, although being even “more narrow”, also constant width sets have, see Chapter 1 above). An additional, but related reason for the refreshing beauty of convexity is given by the richness of fascinating geometric properties that already certain special types of convex bodies have. We mention some of them, in all cases also citing basic surveys or books presenting these families in a broad manner. Namely, we have, for example, simplices (see [1040] and [527, Section 2], classes of polytopes with different degrees of symmetry, like uniform polytopes, semiregular polytopes, or isogonal ones (see, e.g., [540, § 23], [268], [986, pp. 378–382], [100, § 12.5], [595], [784], [812], and [527, § 1.1]), balls (cf. [540, § 32], [160, § 16], [125], [100, Chapters 10, 18, and 20], [419], and Subsection 1.2 in [527]), ellipsoids (cf. [160, § 70], [932], [478], [100, Chapters 14 and 15], [527, § 1.3]), and [82], zonoids and zonotopes (here we refer to the excellent surveys [1043] and [444]), and the three closely related classes sets of constant width (see [238], [527, Section 5] and, for Minkowski spaces, [793, Section 2]), complete sets (seemingly no comprehensive survey is existent, but the topic is also discussed in [238], [527], and [793]), and reduced sets (cf. [701] and, for normed spaces, [702]). It would be nice to have a monograph on all these (and more) classes of convex sets listing their interesting properties and applications, because they are useful and essential in many directions. For example, they can be important as special unit balls of Banach spaces, as inspiring objects to understand phenomena in high dimensions, as projection bodies (zonoids and, in the polytopal case, zonotopes centered at the origin), and so on. The history of convexity started in ancient times, with problems in the spirit of the isoperimetric question, and connected with famous geometers like Euclid, Archimedes, and others. Later, the growing theoretical building of this field was enriched bycontributions of Kepler, Euler, Cauchy, Steiner, Brunn, Minkowski, Blaschke, Busemann, Eggleston, Hadwiger, A. D. Aleksandrov, Coxeter, L. Fejes Tóth, Klee, Grünbaum, Schneider as well as many other excellent mathematicians (it is impossible to write a complete list of really important contributors). Regarding detailed representations of the history of convexity we refer to the articles [347], [475], [627], and [473], recommendable refreshing expository articles on convex sets in general are [471], [101], [633], and [70]. A very important, influential book that gave directions for further research, fundamental and also responsible for the way that convexity has chosen over the 20th century, was [160]. In this monograph the field itself was represented and summarized for the first time. Later everything was growing faster, and leading experts tried to structurize and organize the field by books like the Proceedings to the 1965 Colloquium on Convexity in Copenhagen (Kobenhavns Univ. Mat. Inst., Copenhagen, 1967), [630], [1129], and [482], which are collections of important surveys and articles on different aspects and subfields of convexity. Then, in 1993, even a Handbook of Convex Geometry (see [483]) appeared, filled again with many excellent surveys which cover the most important partial fields of convexity. We mention that there are also other Handbooks

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(on Banach spaces, or on discrete and computational geometry, etc.) containing a lot of material closely related to convexity. And also the books of the Hungarian series usually named “Intuitive Geometry” contain very many contributions to convexity. More modern milestones are the already mentioned monographs [1039], [401], [461], [1124], and [477]. Further books which illuminate special aspects of convexity or have textbook character were already mentioned in the Notes of Chapter 1, and we shortly repeat them here: early examples are [132] and [160], later continued by [1204], [312], [737], [1140], [98] (containing also a chapter on Minkowski geometry), [737], [706], [618], [1134], [654], [1163], and [719]. And we also have to mention the problem books [629], [311], [272], [635], and [182]. In particular, the problem book [272] contains (after its preface) a useful list of further related problem collections and standard references from convexity. This list should be extended by books on general convexity which are not mentioned in Chapter 1 (since bodies of constant width do not occur in these books). These are [15], [205], [502], [503], [493], [711] (also partially considering Minkowski spaces), [82], [714], and [1080]. Clearly, in all these books various aspects and properties of convex sets are discussed with different focal points. Since our book contains also aspects of discrete and combinatorial geometry of convex bodies and polytopes (see, e.g., Chapters 6 and 15), we mention here also books going into these more special directions. Regarding polytope theory, there are the basic monographs [16], [489], [813], [188], and [1223], whereas the books [505], [151], and [152] refer to the combinatorial geometry of convex bodies. Furthermore, [345], [981], [479], [163], [801], and [113] are important in discrete geometry and related fields. Analytic aspects of convex sets, widely presented in [461] and [464], are also discussed in the more recent important publications [649], [650], and [992]. Referring to finite-dimensional real Banach spaces in the spirit of our Chapter 10 and the monograph [1124], we also mention the books [938], [186], and [41]; several survey articles in the Handbook [596] are also closely related. And for all these aspects, we again refer to further excellent surveys in the Handbook [483]. The most comprehensive and also deepest monograph referring to the central topics of the field is [1039]. There are also geometrical fields that, at least some decades ago, had almost no direct connections to convex geometry. As non-specialist, one could guess that algebraic geometry is one of them. However, the monographs [897] and [325] clearly show that there are many deep cross-connections between both fields, e.g., referring to polytope theory, mixed volumes, Ehrhart polynomials, and further topics. See, for example, Section 12.4, and also the excellently grown field of tropical geometry should be mentioned here. Finally we mention here at least one well-known example for applications of convexity. Convex analysis (see the classical book [978] and also [546]) originally belonged to pure mathematics. But roughly since 1960, applied aspects started to play an increasingly important role, also in connection with non-smooth analysis (since convexity allows the usage of a differential calculus going further than the classical one). Due to this, convex analysis became more important for optimization, approximation theory, inverse problems, etc.; a suitable related reference still close to classical convexity in our sense here is [1100]. Ellipsoids Besides bodies of constant width, ellipsoids represent a very important class of special convex bodies. Therefore also here (like in Section 2.12) they deserve to be discussed separately. Ellipsoids have many captivating properties and have been studied for their own intrinsic interest. Many extremal problems of affine nature have ellipsoids as extremal bodies, and perhaps this is the reason for the extensive research. In Minkowski geometry, Hilbert geometry, and affine differential geometry, ellipsoids play an important role, see also Subsection 11.6. A good example is the already mentioned article [206]. All ten problems posed there (only one of them being solved) refer to possible ellipsoid characterizations in Euclidean space, but the motivations come from Minkowski geometry. Surveys on ellipsoids in convex geometry are the papers of [932], [478], and Section 3 in [527].

2.12 Ellipsoids

55

The first proof of the False Center Theorem can be found in [685], although the proof presented here is in [852]. Concerning characterizations of ellipsoids by means of their sections or projections there has been a considerable amount of research. Before Larman’s paper, among others, there is work by Rogers [980] and Aitchison [5]. After Larman’s paper, there is a series of publications by Burton [200], Burton and Larman [201], Burton and Mani [202] and Larman, Montejano and Morales [686], where an interesting theory has been developed. See also the related work of Gruber and Ódor [480]. Here we also mention a nice result of Burton [199] generalizing a theorem of Aitchison: a convex body of dimension n > 2 all whose hyperplane sections cutting from it a piece of sufficiently small volume are centrally symmetric, is the Minkowski sum of an ellipsoid and a zonotope. Further important characterizations of ellipsoids were obtained in [469] and in [1079]. The uniqueness of the largest ellipsoid in a convex body and the smallest ellipsoid containing it was first proved in the 2-dimensional case by Behrend [95]. The case of uniqueness of the smallest ellipsoid containing a given convex set was deduced by John [594] from general results on extremum problems with inequalities as constraints. The name Löwner ellipsoid was first used by Busemann [203], but refers to the ellipsoid with given center and smallest volume containing a convex body, see Theorem 2.12.2. Busemann mentioned that Löwner did not publish his result. Proofs for the Löwner and John ellipsoid were given by Danzer, Laugwitz and Lenz [279]. Gruber [472] proved that most convex bodies in En , in the sense of the Baire category, touch the boundaries of their John and the Löwner ellipsoid in precisely n(n + 3)/2 points.

Exercises 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9.

Show that between a point p of a set P and a point s not in P there is always a boundary point. That is, there is always a boundary point of P in the segment pq (Lemma 2.2.1). Show that all points between two interior points p and q of a convex set φ are interior points (Lemma 2.2.2). Show that all points between an interior point p and a boundary point q of a closed convex set φ are interior points (Lemma 2.2.3). Show that between two boundary points p and q of a closed convex set φ either all the points are boundary points or all the points are interior points (Lemma 2.2.4). Show that if a line L passes through an interior point p of a convex body φ, then L intersects the boundary of φ at exactly two points (Lemma 2.2.5). For a closed, convex set φ with non-empty interior, show that the topological closure of intφ coincides with φ. Show that cc(S) is equal to S if and only if S is a convex set. Prove that if S1 is contained in S2 then cc(S1 ) is contained in cc(S2 ). Let A and B be closed sets. Prove the following properties: a) b) c) d) e) f)

2.10. 2.11. 2.12.

If A ⊂ B, then A ⊂ cc B. If A ⊂ B and B is convex, then cc(A) ⊂ B. cc(A) ∪ cc(B) ⊂ cc(A ∪ B). cc(A ∩ B) ⊂ cc(A) ∩ cc(B). Find examples to show that equality need not hold in cases c) and d). cc(A + B) = cc(A) + cc(B).

Suppose S ⊂ En is a compact set whose complement is connected. Prove that if the boundary of the convex hull of S is contained in S, then S is convex. Prove that the convex hull of a bounded set is bounded. Prove that the union of an increasing sequence of convex sets is convex.

56

2.13. 2.14.

2.15*. 2.16.

2.17. 2.18. 2.19. 2.20. 2.21. 2.22. 2.23.

2 Convex Geometry

Prove that a convex body has no line segments inside its boundary if and only if every support hyperplane intersects it at a single point. Let φ, ψ two convex bodies and let p ∈ φ and q ∈ ψ be the pair of points that minimizes the distance between two points of φ and ψ. Prove that the hyperplanes orthogonal to the line segment pq through p and q are parallel support hyperplanes of φ and ψ, respectively. Prove that the intersection of two strictly convex bodies is strictly convex, but the intersection of infinitely many strictly convex bodies may not be strictly convex. Let H1 and H2 be two parallel support hyperplanes of a strictly convex body φ and assume {ai } = φ ∩ Hi , i = 1, 2. Prove that the chord a1 a2 is the longest chord of φ among all chords parallel to a1 a2 . Prove the basic properties of dilatation and Minkowski sum given in Lemma 2.4.2. Give an example of a non-convex set φ for which λφ + νφ = (λ + ν)φ. Prove the properties of balls stated in Corollary 2.4.1. Show the basic properties of exterior and interior parallel bodies given in Lemma 2.4.4. Show the properties involving the support function and the exterior and interior parallel bodies given in Lemma 2.4.5. Prove Lemma 2.4.6. Prove that the support function of a ball is given by P B(x,r ) = r + x, u .

2.24.

Let E be the ellipsoid E = {(x1 , . . . , xn ) ∈ En | of E is PE (x1 , . . . , xn ) =

n

xi 1 ai

n 

≤ 1}. Prove that the support function ai2 xi2

 21

.

1

2.25. 2.26. 2.27*. 2.28. 2.29. 2.30. 2.31.

2.32.

2.33*. 2.34.

Prove that if pq is a diameter of the closed set S, then the hyperplanes orthogonal to the line segment pq through p and q are support hyperplanes of S. Find a line L and a closed convex set φ in E3 such that L ∩ φ = ∅, but each plane through L intersects φ. Let B(o, r ) be a ball and φ a convex body. Suppose that φ + B(o, r ) is a dilatation of φ. Prove that φ is also a ball. Prove the triangle inequality for the Hausdorff metric. Prove that for convex bodies the Hausdorff metric is invariant under parallelism, that is, d(φδ , ψδ ) = d(φ, ψ). ∞ Suppose {φi }∞ 1 is an infinite collection of convex bodies. Prove that if {φi }1 → ψ, then ψ is convex. Suppose {φi }∞ 1 is a collection of convex bodies that converges to the ball B( p, r ). Prove that given  > 0, there is a number N sufficiently large such that B( p, r − ) ⊂ φi ⊂ B( p, r + ), for every i > N . m Let 2 Q be the family of all subsets of Q m , the mth generation of subdivision of the cube (see the proof of Theorem 2.5.2). Given any convex body φ ∈ F, assign to it its m-cumulus m of all elements of 2 Q that intersect φ. Show that if (φ)m = (ψ)m , then (φ)m , consisting √ n d(φ, ψ) ≤ 2m−1 . Given two convex sets φ and ψ. Prove that their Steiner symmetrals satisfy S(φ) + S(ψ) = S(φ + ψ) if and only if one of the convex bodies is a dilatation of the other. Let φ ⊂ E2 be a plane convex body of area A(φ) and perimeter P(φ). Suppose P(φ)2 − 4 A(φ) = 0. Prove that φ is a 2-dimensional disk.

Exercises

2.35. 2.36. 2.37. 2.38.

2.39.

2.40. 2.41. 2.42. 2.43. 2.44. 2.45. 2.46.

57 1

By taking the second derivative of F(t) = (V + St + Mt 2 + Ct 3 ) 3 and using the fact that this function is concave, show that the equations 2.6 and 2.7 hold. Show that in Theorem 2.7.3 equality holds only if φ is a 3-dimensional ball.      n1 , then (1 − t)F(a) + t F(b) ≤ F (1 − t)a + tb , Show that if F(t)= V (1 − t)φ + tψ for every 0 ≤ a < b ≤ 1. Hint: let  = (1 − a) + a and   = (1 − b) + b. Consider a finite family of closed intervals, [α1 , β1 ], . . . , [ατ , βτ ], in the line E1 with the property that every pair of them has a point in common. Prove that the minimum of all the values βi is a point in all these intervals. Let F be a finite family of parallel closed intervals in the plane E 2 . Prove that if for every three of these intervals there is a line that crosses them, then there is a line which crosses all the intervals of F. Let ψ ⊂ En be a convex set and S ⊂ En . Suppose that every n + 1 points of S can be covered by a translate of ψ. Prove that S can be covered by a translate of ψ. Prove that if p and p  are vertex points of the boundary of a convex body, then int( p) ∩ int( p  ) = ∅. Let φ be a convex set in En (see Lemma 2.9.1). Show that the point p is an extreme point of φ if and only if the set φ \ { p} is also convex. Suppose H is a support hyperplane of a closed convex set φ at p. Prove that p is an extreme point of H ∩ φ if and only if p is an extreme point of φ. Prove that (℘φ−1 ( p) − p) ∩ Sn−1 = ( p), where ℘φ is the nearest point mapping and ( p) is the Gauss map. Prove that the nearest point mapping ℘φ : En → En is non-expansive. We say that a boundary point p of a convex set φ is an exposed point of φ if there is a support hyperplane of φ at p such that { p} = H ∩ φ. Prove or disprove the following: a) each exposed point is an extreme point of φ, and b) each extreme point of φ is an exposed point of φ.

2.47. 2.48.

Prove the basic properties of polar duals stated in Lemma 2.10.1. Let φ be a closed convex set containing the origin. Prove that a) φ is bounded if and only if the origin is an interior point of φ∗ , and b) φ∗ is bounded if and only if the origin is an interior point of φ.

2.49. 2.50*. 2.51. 2.52. 2.53.

2.54*. 2.55.

2.56.

2.57.

Prove the basic properties of polar duals stated in Lemma 2.10.1. Prove that any ellipsoid centered at the origin is a projection body (i.e., a zonoid symmetric with respect to the origin). Prove that if G is an isometry of En , then (Gφ) = G −1 (φ). Let A ⊂ B be convex sets such that bdA ⊂ bdB. Prove that A = B. Without using Brunn’s Theorem, prove that if φ ⊂ En is a convex body with the property that in any direction u ∈ Sn−1 the midpoint of all chords of φ parallel to u lie in a hyperplane orthogonal to u, then φ is a ball. Prove that a convex body with two axes of revolution must be a ball. Let φ ⊂ En be a convex body. A line L is called an axis of φ if there is a hyperplane H with the property that any section of φ parallel to H is a ball with center in L . Prove that if φ has two different axes if and only if φ is an ellipsoid. Prove the following: Let φ ⊂ En be a convex body and let p0 ∈ intφ, Suppose 1 < r ≤ n. If for every r -dimensional plane H through p0 the section H ∩ φ is an ellipsoid, then φ is an ellipsoid. Let E be an ellipsoid in En−1 × {0}, and let L ⊂ En be any line not parallel to En−1 × {0}. Prove that if H is a hyperplane not parallel to L, then H ∩ (E + L) is an ellipsoid.

58

2.58.

2.59.

2.60. 2.61*. 2.62. 2.63. 2.64. 2.65. 2.66.

2.67.

2.68. 2.69. 2.70.

2.71.

2 Convex Geometry

Let φ ⊂ En be a convex body. Prove that φ is centrally symmetric if and only if φ is a translated copy of −φ. Moreover, prove that if φ is a homothetic copy of −φ, then φ is centrally symmetric. Let φ ⊂ En be a convex body and let x0 ∈ intφ. Suppose that for every 2-dimensional plane H through x0 , the section H ∩ φ is centrally symmetric with center at x0 . Prove that φ is centrally symmetric with center at x0 . Let φ be an n-dimensional convex body, and let H be a hyperplane through the origin. Prove that for a unit vector u ∈ H , S∂(φ, u) ∩ H = S∂(H ∩ φ, u). Let φ ⊂ E3 be a convex body. If every two parallel sections of φ are homothetic, prove that φ is strictly convex. Prove that every convex body φ has a chord that intersects the interior of φ and is not a diametral chord. In fact, there is always such a chord, sufficiently close to a given direction. Prove Theorems 2.12.9 and 2.12.10. Use the notion of k-shadow boundary to prove Theorem 2.12.8 for n > 3, and Theorem 2.12.12. Prove that the center of a centrally symmetric convex body is the midpoint of every diameter. Let φ ⊂ En be a convex body and let L be a line that intersects its interior. Suppose that there is a point x0 ∈ L with the property that for every hyperplane H through L, the section H ∩ φ is centrally symmetric with respect x0 . Prove that φ is symmetric with respect to x0 . Let C1 ⊂ C2 be two directly homothetic, strictly convex bodies. Suppose they have a common support hyperplane H for which H ∩ C1 = H ∩ C2 = {x0 }. Prove that there is a positive homothety with center at the point x0 that sends C1 onto C2 . Prove Theorem 2.12.14 for the case in which φ1 and φ2 are not necessarily strictly convex. Verify that the proof of Theorem 2.12.14 also holds when the points p1 and p2 are any two points of En and not necessarily interior points. Let φ ⊂ En be a convex body, n ≥ 3, and let p1 and p2 two points of En with the property that, for any hyperplane H through the origin, the section ( p1 + H ) ∩ φ is inversely homothetic to the corresponding section ( p2 + H ) ∩ φ. Prove that φ is centrally symmetric. Let φ ⊂ En be a convex body symmetric with respect to the origin, H a hyperplane through the origin, and q ∈ intφ − H . Suppose that the section (q + H ) ∩ φ is symmetric with respect to q. Prove that ((q + H ) ∩ φ) − 2q = (−q + H ) ∩ φ.

Chapter 3

Basic Properties of Bodies of Constant Width

The only royal road to elementary geometry is ingenuity. Eric Temple Bell

3.1 Diameters, Binormals, and Diametral Chords One of the essential characteristics of bodies of constant width is that, like balls, they have a diameter in every direction. Diameters are those chords of a convex body that have maximum length, and it is their behavior which gives constant width bodies their basic properties. Unlike the diameters of a ball, those of a body of constant width do not always meet at a single point, but when they do so, it is because the body is indeed a ball. In order to identify the maximum length chords of a body of constant width h, it is necessary to start by determining how far apart two of its points can be. Note that the distance between any two points in a body of constant width h must always be less than or equal to h. This is because if there were two points, p and q, farther apart than the width of the body, say h, then the body would have width greater than h in direction pq, contradicting the fact that the (constant) width of the body is precisely h. As we have done before, a Greek letter like φ will usually denote a convex body, and throughout the rest of the book capital Greek letters will usually denote bodies of constant width. Lemma 3.1.1 Every support hyperplane H of the body of constant width  touches  at exactly one point. Proof To see this, take H  as the support hyperplane of  which is parallel to H .  is held fast in the strip delimited by H and H  , whose width is h. The hyperplane H  may touch  in many points, but let us focus on only one such point; call it q. As the distance between H and H  is h, there exists one point p in H such that the distance between p and q is h. In fact, p is the point of H with the property that the segment pq is perpendicular to the hyperplanes H and H  . The distance between q and any point in H other than p must be strictly greater than h. Therefore, since q is in , no other point m of H distinct from p can be in , for if m were in , the width of  in the direction of segment qm would be greater than h. This proves that H can touch  at only one point, p.  The proof gives another result as a bonus. Exchanging the roles of H and H  , we have that H  touches  at only one point, q, which implies the following theorem. Theorem 3.1.1 The line segment joining the points of contact between a body of constant width and two of its parallel support hyperplanes is perpendicular to them. © Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_3

59

60

3 Basic Properties of Bodies of Constant Width

Figure 3.1 Normal and binormal chords

This theorem has many and varied consequences. The four most important ones are as follows: i) Every body of constant width is strictly convex since any of its support hyperplanes touches it at only a single point. ii) The diameter of a body of constant width h is h because the body cannot contain two points that are farther apart than h apart. However, it does contain two points–the points of contact with two parallel support hyperplanes–which are exactly h apart. iii) If H is a support hyperplane of a body  with constant width h, then the ball with radius h tangent to H at the point of contact between H and  contains , because  has diameter h and the center of this ball is a point of , as it is precisely the point of contact between  and the support hyperplane H  parallel to H . iv) A body has of constant width if and only if it has a diameter in every direction. A chord of a body φ is called a normal of φ, if the hyperplane perpendicular at one on of the ends of the chord is a support hyperplane of φ. If the hyperplanes at each end of a chord and perpendicular to it are support hyperplanes, then the chord is called a binormal (or a double normal) of φ, see Figure 3.1. For example, it is not difficult to see that the diameter of a body is always a binormal, since the two balls that have this diameter as their radius will always contain the body. Moreover, by Theorem 3.1.1 binormals and diameters are the same in a constant width body. The following theorem will prove to be extremely important, see [316]. Theorem 3.1.2 A body  has constant width if and only if it has a binormal in every direction. Proof Suppose that  ⊂ En is a body of constant width; let u ∈ Sn−1 be any direction. Take the two support hyperplanes of  perpendicular to u. By Theorem 3.1.1, the segment joining the points of contact between  and these two parallel hyperplanes is a binormal parallel to u. We will need the following construction in order to proceed with the proof of sufficiency. From any body φ, we construct another body φ∗ as follows. Take a fixed point o and draw a ray from o in direction u ∈ Sn−1 . Take a point pu on this ray such that the distance between o and pu is precisely the width of φ in direction u. As we vary u, the point pu describes the boundary of a body φ∗ with the property that if m is a point in the boundary of φ∗ , the length of the segment om is the width of φ in direction om (Figure 3.2). The body φ∗ is, of course, different from φ. In fact, if φ is a body of constant width, then φ∗ is a ball, since the width of φ is the same in every direction. Conversely, if φ∗ is a ball with center o, then the width of φ is the same in every direction, and so φ is a body of constant. Having introduced this construction, we return to the original problem. Suppose the body  has a binormal in every direction. We wish to show that  is a body of constant width. To do this, it is sufficient to prove that ∗ is a ball. We will use the following result which is part of the folklore in this area (see Exercise 3.7): if the convex body ∗ has an interior point o and for every boundary point m the hyperplane orthogonal to om through m is a support hyperplane, then ∗ is a ball.

3.1 Diameters, Binormals, and Diametral Chords

61

-

Figure 3.2

Take any point m in the boundary ∗ and let H be the hyperplane passing through m perpendicular to om, see Figure 3.3. We wish to prove that H is a support hyperplane of ∗ . To do so, it will be sufficient to prove that, for every direction u ∈ Sn−1 , the segment opu does not cross H . If this is so, then we can be sure that ∗ is completely situated on one side of H . Let H1 and H2 be the two support hyperplanes of  perpendicular to the direction om, and let x and y, respectively, be the points at which H1 and H2 touch . We know that the segment x y is parallel, and equal in length, to the segment om. By identifying H with H1 (indeed, the two hyperplanes are parallel), that is, by superimposing  on ∗ in such a way as to make H1 coincident with H and the segment x y with om, we can see that H2 passes through o and that  is between H and H2 (see Figure 3.4). Consider any direction u ∈ Sn−1 . There is a binormal pq of  in direction u whose length is, of course, the width of  in direction u. Moreover, the binormal pq is contained in  and is therefore between H and H2 . As the segments pq and opu are parallel (since both are parallel to u) and are of the same length, it is easy to see that this implies that opu does not cross the hyperplane H . But since this is true for every direction u ∈ Sn−1 , we have that ∗ does not cross the hyperplane H . Therefore H , which is perpendicular to om at m, is  a support hyperplane of ∗ ; ∗ is therefore a ball and  thus a constant width body. In the following we present a classical result. Theorem 3.1.3 A body has constant width if and only if each of its normals is a binormal.

Figure 3.3

62

3 Basic Properties of Bodies of Constant Width

Figure 3.4

Proof Suppose that  is a body of constant width and that the segment pq is a normal of , that is, the hyperplane H perpendicular to pq at p is a support hyperplane of  (see Figure 3.5). We wish to prove that the hyperplane H1 , perpendicular to pq at q, is also a support hyperplane of . Consider H  , the support hyperplane of  parallel to H . We know that H  touches  at just one point—let us call it x—and that the segment px is perpendicular to both H and H  . The segments pq and px must therefore coincide, x must be q and the support hyperplane H  must be H1 . That is, pq is a binormal since the hyperplane H1 , perpendicular to pq at q, is a support hyperplane. Suppose now that every normal of the body  is a binormal. In order to be convinced that  has constant width, we shall construct a binormal in every direction. Let u ∈ Sn−1 be any direction. Consider the support hyperplane H of  perpendicular to u and suppose that H touches  at least in the point x (and perhaps also in further points). The chord of  perpendicular to H with one end at x is a normal of  and therefore a binormal parallel to direction u. This proves that  has a binormal in every direction, and therefore has constant width.  If φ is a strictly convex body and H1 and H2 are two parallel support hyperplanes of φ, then the chord whose two ends are the points where φ touches these hyperplanes is called a diametral chord of φ. The following theorem characterizes constant width bodies in terms of their diametral chords.

Figure 3.5

3.1 Diameters, Binormals, and Diametral Chords

63

Theorem 3.1.4 Let  be a strictly convex body. Then the following are equivalent: a) b) c) d)

all diametral chords of  are of the same length, every diametral chord of  is a diameter of , every diametral chord of  is a binormal of , and the convex body  is of constant width.

Proof The first two implications, a) ⇒ b) and (b) ⇒ c), follow easily from the fact that every body has at least one diameter and that every diameter is a binormal. If every diametral chord is a binormal, then every diametral chord is perpendicular to the support hyperplanes that define it; there is therefore a binormal in every direction. The implication c) ⇒ d) now follows from Theorem 3.1.2. Lastly, d) ⇒ a) follows from the definitions and from Theorem 3.1.1.  It is well known that if a convex body has the property that all its normals meet at a point, then it must be a ball. The following characterization is, however, less well known. Theorem 3.1.5 Let φ be an n-dimensional convex body, 3 ≤ n, with the property that every two of its normals intersect. Then φ must be a ball. Proof We shall show that all the normals of φ are concurrent. Let I1 and I2 be two normals of φ, not parallel, and let o be the point where they intersect. Every normal I  of φ not in the plane  determined by I1 and I2 must intersect both I1 and I2 and must therefore pass through the point o. If I  is now a normal of φ in , it must pass through o, since otherwise it would not intersect any normal not in . That is, all normals of φ pass through o, and therefore, by the characterization of the ball stated in Exercise 3.7, φ is a ball.  It is interesting to contrast this lemma with the following characterization of constant width figures. Theorem 3.1.6 A convex figure in the plane has constant width if and only if every two of its normals intersect. Proof Suppose first that φ is a figure of constant width h and let pq and r s be two normals of φ. Then they are also diameters of φ. Suppose that pq does not intersect r s. Then, since φ is convex, and the points p, q, r , and s are in the boundary of φ, we may assume that the diagonals ps and qr of the convex quadrilateral pr sq intersect at the point o. By the triangle inequality we have | po| + |oq| > h = | pq|, |r o| + |os| > h = |r s|, which implies that | ps| + |rq| > 2h, and therefore one of the diagonals ps or rq is longer than h, which is a contradiction, since the diameter of φ is h. By Theorem 3.1.3, if φ is not of constant width, then there exist parallel support lines L 1 and L 2 of φ such that the normal I1 of φ with respect to L 1 is different from the normal I2 of φ with respect to  L 2 . Since I1 and I2 are distinct parallel chords, they do not intersect (Figure 3.6). Let φ ⊂ En be a convex body. Remember that for every u ∈ Sn−1 the width of φ in the direction u, denoted by w(φ, u), is the distance between the support hyperplanes of φ orthogonal to u. So, w(, u)

64

3 Basic Properties of Bodies of Constant Width

Figure 3.6

is a continuous function of u. Define the thickness (or minimum width) of a body φ, denoted by (φ), as the minimum of w(φ, u), over all u ∈ Sn−1 . For example, if φ has constant width h, then the thickness of φ is h. Let H and H  be two parallel hyperplanes in En . Then S = conv(H ∪ H  ) is called a strip with boundary hyperplanes H and H  . If u ∈ Sn−1 is orthogonal to H and H  , we say that S is a strip in the direction u. If, in addition, H and H  are support hyperplanes of a convex body φ, then we say that S is a φ-strip, and the φ-strip orthogonal to u will be denoted by S(φ, u). Therefore, w(φ, u) is just the width of the strip S(φ, u). Lemma 3.1.2 Let φ ⊂ En be a convex body and suppose the φ-strip S(φ, u) with boundary hyperplanes H and H  has width (φ), the thickness of φ. Then there are points p ∈ H ∩ φ and p  ∈ H  ∩ φ such that the segment pp  is orthogonal to the strip S(φ, u). Proof Let  : En → H be the orthogonal projection in the direction u. Suppose the lemma is not true. Then (H ∩ φ) ∩ (H  ∩ φ) = ∅. If this is so, then there is an (n − 2)-dimensional affine plane L ⊂ H strictly separating (H ∩ φ) from (H  ∩ φ) in H . Let L  ⊂ H  be such that (L  ) = L. By parallel rotation of H and H  through L and L  , respectively, we obtain a φ-strip of width smaller than (φ), which is a contradiction.  A subset A ⊂ Sn−1 is spherically convex if A does not contain a pair of antipodal points and, given p, q ∈ A, the shortest maximal arc between p and q is contained in A. Remember from Section 2.9.2 that for a strictly convex body φ, we may define the inverse Gauss map (2.9) γ : Sn−1 → bd φ as follows: if u ∈ Sn−1 , then γ(u) is the unique point p in the boundary of φ with the property that there is a support hyperplane of φ at p orthogonal to u, where the vector u points outward φ. It is well known that γ : Sn−1 → bdφ is a continuous map and that γ −1 ( p) ⊂ Sn−1 is a spherical convex set, for every p in the boundary of φ. Following the spirit of Theorem 3.1.3, which is a classical characterization of bodies of constant width, we have one of the most interesting problems regarding constant width bodies. Conjecture 3.1.1 Suppose that the normal through every regular boundary point of a strictly convex body  is a binormal of length h. Then  has constant width. The following theorem, which partially solves Conjecture 3.1.1, will be useful in Section 8.3. to prove that the Meissner Solids are bodies of constant width.

3.1 Diameters, Binormals, and Diametral Chords

65

Theorem 3.1.7 Suppose that every normal of a strictly convex body  ⊂ En through a non-vertex point is a binormal of length h. Then  is a body of constant width h. Proof Our first purpose is to prove that for every vertex point p of the boundary of  and any unit vector v ∈ bd γ −1 ( p), we have that w(, v) = h, where bd γ −1 ( p) denotes the boundary of the spherically convex set γ −1 ( p) ⊂ Sn−1 . Suppose that there is v ∈ bd γ −1 ( p) such that w(, v) = h. Therefore, by continuity, there is a open neighborhood U of v in Sn−1 with the property that w(, u) = h for every u ∈ U . Since there are countably vertex points in the boundary of , let us denote them by { pi }i∞ . By hypothesis, ∞ many −1 U ⊂ 1 γ ( pi ) which is impossible unless U is contained in only one of these sets, because n−1 . But this last option {γ −1 ( pi )}∞ 1 is a countable family of pairwise disjoint closed subsets of S −1 is a contradiction to the fact that v ∈ bd γ ( p). Next it will be shown that the thickness of , (), is equal to h. For that purpose, let H and H  be parallel support hyperplanes with the property that the width of the strip bounded by H and H  is (), the thickness of . By Lemma 3.1.2, the diametral chord defined by the parallel hyperplanes H and H  is orthogonal to them. Suppose () = h, then take p ∈ H ∩  and q ∈ H  ∩ , and note that the segment pq is orthogonal to H and H  . By hypothesis, p and q are vertex points of the boundary of , otherwise () = h. Let u ∈ Sn−1 be the outer unit normal vector to H . If u ∈ bd γ −1 ( p), then w(, u) = h = (), as wished. Similarly, if −u ∈ bd γ −1 (q), again w(, u) = h = (). Assume hence that u ∈ relint γ −1 ( p) and −u ∈ relint γ −1 (q). If this is the case, we find a contradiction because, given p ∈ L ⊂ H and given q ∈ L  ⊂ H  , where L and L  are parallel (n − 2)-dimensional affine planes, it is always possible to slightly rotate H and H  parallel through L and L  , respectively, reducing the thickness of . Having this in mind, we are ready to conclude the proof of the theorem by showing that the diameter of  is h. Let ab be a diameter of  and let H and H  be parallel support hyperplanes through a and b, respectively, orthogonal to the diameter ab. We may assume that both a and b are vertex points of the boundary of , since otherwise, by hypothesis, the length of the diameter ab must be h. Given L and L  as parallel (n − 2)-dimensional affine planes with a ∈ L ⊂ H and b ∈ L  ⊂ H  , it is always possible to rotate H and H  parallel through L and L  , respectively, until either the normal vector of H is in bd γ −1 (a) or the normal vector of H  is in bd γ −1 (b). In any of the cases, we have that the width of the final position of the rotated strip is h, and since every diametral chord of a strip whose width is () has the property to be orthogonal to the strip, we have that the diameter ab has length h, the thickness of . 

3.2 The Minimum Width Condition For a boundary point p of a convex body φ ⊂ En define the minimum width at p, denoted by ω( p), to be the smallest distance between parallel support hyperplanes of φ when one of them passes through p. Of course, ω( p) is not a continuous function of p ∈ bdφ. We say that φ has constant minimum width if the minimum width at p is the same for all p ∈ φ. Constant width bodies have constant minimum width, but there are examples of convex bodies which have constant minimum width, but are not bodies of constant width. Such examples are the n-dimensional cube and the common part of the unit n-ball with the first orthant.

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Figure 3.7

A binormal pp  of φ whose length is the thickness of φ is called a thickness chord or a thickness binormal, and the corresponding φ-strip is a thickness strip. With this language, an immediate consequence of the definition is the following characterization of constant minimum width. Theorem 3.2.1 A strictly convex body φ has constant minimum width if and only every normal through a regular point of the boundary of φ is a thickness binormal. Proof If the strictly convex body φ has constant minimum width h, then h = (φ). Let p be a regular point of the boundary of φ. Then the unique φ-strip through p has width h = (φ), and therefore, by Lemma 3.1.2 and the fact that φ is strictly convex, the unique normal of φ through p is a thickness binormal. Suppose now that every normal through a regular point of the boundary of φ is a thickness chord. If p is a regular point of φ, then ω( p) = (φ). If p is not regular, we can approximate p with regular points { pi } with unit normal vectors {u i }. Without loss of generality, {u i } → u. So u is a normal vector of φ at p and, clearly, by the continuity of the width function, (φ) = w(φ, u). Therefore ω( p) = (φ).  As a corollary we have that if Conjecture 3.1.1 is true, then φ is a body of constant width if and only if φ is strictly convex and has constant minimum width. The following theorem follows this spirit. Theorem 3.2.2 Let  ⊂ En be a strictly convex body with only regular or vertex points at its boundary. Then  has constant minimum width if and only if  has constant width. In particular, a strictly convex plane figure  has constant minimum width if and only if  has constant width. A Packing Problem In this section we will discuss an interesting packing problem related to the notion of minimal width. We say that two convex bodies ψ and φ are similar if one is obtained from the other by a dilatation followed by a rotation. Consider squares of perimeter a and b, respectively, that are enclosed in a square of perimeter c, so that their interiors do not overlap. Then a + b ≤ c. If the squares are replaced by circles of perimeter a, b, and c, the same is true. Let φ be a convex figure and let φ1 and φ2 two figures similar to φ, with φ1 and φ2 contained in φ, and φ1 and φ2 having nonoverlapping interiors, see Figure 3.7. We say that φ is tight if for every such configuration the perimeters of φ1 and φ2 add up to no more than the perimeter of φ. For example, both the square and the circle are tight. Theorem 3.2.3 Every figure φ of constant width is tight.

3.2 The Minimum Width Condition

67

Proof One of the classical results about constant width figures is Barbier’s Theorem 5.1.2 claiming that the perimeter of a figure of constant width h is πh. With this in mind, let φ1 and φ2 figures similar to φ, with φ1 and φ2 contained in φ, and φ1 and φ2 having nonoverlapping interiors. Suppose that for i = 1, 2 the figure λi φ is congruent to φi . Let h be the width of φ. Hence the width of φi is λi h and its perimeter is λi hπ. So, in order to prove our theorem it will be enough to prove that λ1 + λ2 ≤ 1. For that purpose, let L be the line that separates φ1 from φ2 , and let L 1 and L 2 be support lines of φ parallel to L. Suppose, without loss of generality, that φ1 lies between L 1 and L while φ2 lies between  L and L 2 . Clearly, λ1 h + λ2 h ≤ h. Theorem 3.2.4 If a strictly convex figure φ is tight, then φ satisfies the minimum width condition. Proof The formal proof of this theorem is very technical. The interested reader can find the proof at [94]. Here we will only sketch the idea of the proof. Assume that ω( p1 ) > ω( p), where p and p1 are points of the boundary of φ. We will show that there are figures φ1 and φ2 similar to φ, such that φ1 ∪ φ2 ⊂ φ, the interiors of φ1 and φ2 are nonoverlapping, but the sum of the perimeters of φ1 and φ2 is greater than the perimeter of φ. Let p2 and parallel support lines L 1 and L 2 be chosen so that p1 ∈ L 1 ∩ bdφ, p2 ∈ L 2 ∩ bd φ and ω( p1 ) is equal to the distance between L 1 and L 2 . Let φ1 and φ2 be arrayed along the chord p1 p2 with parameter λ, where λ is very small and φ1 , the smaller one of φ1 and φ2 , is near p1 . Then φ \ φ2 , the set-theoretical difference of φ and φ2 , contains a piece in the vicinity of p1 which looks like a long strip with parallel sides and whose width is λω( p1 ). Since λω( p) < λω( p1 ), we could remove φ1 and replace it with φ , a congruent figure reoriented so as to fit into the “almost strip” without touching either the boundary of φ or φ1 . Let φ be an expanded copy of φ , just a little larger than φ and still not touching φ2 or the boundary of φ, though lying within φ. Obviously, the sum of the perimeters of  φ and of φ2 is larger than the perimeter of φ. As a corollary of Theorem 3.2.3 and Theorem 3.2.4 we have Corollary 3.2.1 A strictly convex plane figure  is tight if and only if  has constant width. In [94] Beck and Bleicher introduced the notion of minimum width with the purpose to classify all tight convex figures. They were able to prove that a figure is tight exactly if it is either a regular polygon or a curve of constant width.

3.3 Projections and Sections In this section, some general theorems on constant width bodies will be proved. In the following two theorems, the dimension of the body  is greater than or equal to three. Theorem 3.3.1 A body  is a body of constant width if and only if all its orthogonal projections are of constant width. Proof Once we know, by Lemma 2.4.1, that the restriction of the support function of a convex body to a great circle is the support function of the corresponding projection body, we conclude that the same is true for the width function and therefore that the statement of our theorem is equivalent to the following statement: a continuous function f : Sn−1 → E is a constant function if and only if the restriction of f to every great circle of Sn−1 is a constant function. 

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In Theorem 3.3.1 it was proven that the property of constant width is inherited under orthogonal projection. We might wonder whether the constant width property is inherited under sections. By a section of  we mean the intersection between  and a hyperplane. Is it true that if all sections of a body are constant width bodies, then the original body is necessarily constant width? The answer to this question is “yes”, but as we shall see, it is more interesting than a straightforward “yes”, as it implies that the body must be a ball. A body of constant width h will also be shown in Chapter 9 which has the property that none of its sections of diameter h are of constant width [276]. As already said, also in the following theorem of Süss the dimension of  is greater than or equal to 3. Theorem 3.3.2 If every section of a body  of dimension at least 3 is a constant width body, then  is a ball. Proof We begin by proving that  is a body of constant width by showing that  has a diameter parallel to every direction. Suppose that h is the diameter of , and let x and y be two points of  that are at distance h apart. Let L be the line that passes through x and y, and let H be any hyperplane that contains L. By hypothesis, H ∩  is a body of constant width whose diameter is precisely h, the diameter of . This implies that any diameter of H ∩  is a diameter of . By Theorem 3.1.1(iv), there exists a diameter of H ∩ , and therefore a diameter of  parallel to every direction of H . As H is any hyperplane that contains L,  has a diameter parallel to every direction. Consequently, again by Theorem 3.1.1(iv),  has constant width h. We now show that  is a ball by proving that every pair of its binormals intersect. Let pq and r s be two binormals, and therefore diameters of . If pq and r s are in the same plane, they intersect, by the proof of Theorem 3.1.7. However, if there is no plane that simultaneously contains pq and r s, it is not difficult to see that there exists a hyperplane H1 that contains r s, but does not intersect the line that passes through pq. This hyperplane naturally contains a line parallel to pq. By hypothesis, H1 ∩  is a body of constant width h, since it contains r s. By Theorem 3.1.1(iv), H1 ∩  has a diameter (which is therefore a diameter of ) that is parallel to pq, which is a contradiction, since two different diameters of φ cannot be parallel.  Since not every body of constant width is a ball, it is clear now that not every section of a nonspherical body of constant width has constant width. This being so, what shapes may sections of constant width bodies have? Could a very long, narrow convex body be a section of a constant width body? The answer to this question will be given in Chapter 9. Theorem 3.3.2 can actually be improved substantially, for the aim to conclude that for  being a ball it is not necessary to require all its sections to be of constant width, and to use only those which pass through a fixed point [845]. The proof of this statement is quite complicated and will be given in Chapter 9. Another improvement to both Theorems 3.3.1 and 3.3.2 is that it is not necessary to consider sections and orthogonal projections onto hyperplanes only; the theorems are also true for sections of orthogonal projections onto k-planes for 1 < k < d. The proofs follow immediately from the above theorems and their proofs.

3.4 Circumsphere and Insphere The boundary of the smallest ball that contains a body ψ is called the circumsphere of ψ, and the boundary of a largest ball contained in ψ the insphere of ψ. Of course, they need not necessarily be concentric. A rectangular box (which is not a cube) has a single circumsphere, but uncountably many inspheres. In general, a body may have many inspheres, but the circumsphere is always unique. If

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69

a body ψ had two circumspheres, say C1 and C2 , then ψ would be inside both C1 and C2 , and there would then be a smaller sphere that contains the part common to C1 and C2 , which would therefore also contain ψ. We will now prove that the insphere of a body of constant width h is unique, is concentric with the circumsphere, and that the sum of their radii is h. To do so, we need the following two technical lemmas. Lemma 3.4.1 Let ψ be a convex body with diameter h, and let B(r ) be a ball contained in ψ having radius r . Then the ball B(h − r ) concentric to B(r ) with radius h − r contains ψ. Proof Suppose B(h − r ) does not contain ψ. Let y ∈ ψ \ B(h − r ) and let L be the line through x and y. Take a point p ∈ L between x and y such that |x − y| = h − r and take a point q ∈ L such that |x − q| = r but q in not between x and y. Then | p − q| = h and hence |y, q| > h, contradicting the fact that the diameter of φ is h.  Lemma 3.4.2 Let  be a body of constant width h and let B(r ) be a ball with radius r containing , r ≤ h. Then the ball B(h − r ) concentric with B(r ) and having radius h − r is contained in . In particular, if h = r , the center of B(r ) is in . Proof Suppose there is a point p of B(h − r ) that is not in . Let H be a support hyperplane that strictly separates p from , and let H1 and H2 be hyperplanes with the following properties: i) H1 and H2 are parallel to H ; ii) H1 is tangent to B(r ) and H2 is tangent to B(h − r ) in such a way that the distance between H1 and H2 is h, iii) the body  and the hyperplane H are in the strip delimited by H1 and H2 (Figure 3.8). Note that  does not touch H2 , which is a contradiction, since in that case the width of  in a direction perpendicular to H is less than h.  Theorem 3.4.1 The insphere and circumsphere of a body of constant width h are concentric, and the sum of their radii is h.

Figure 3.8

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Proof Let  be a body of constant width, and C be its circumsphere. Suppose that the radius of C is R. Since the diameter of  is h, R ≤ h. Let B(h − R) be the ball concentric with C having radius h − R. By Lemma 3.4.2, B(h − R) is contained in . We will prove that B(h − R) is an insphere of  and is unique. Suppose that it is not; then there is a ball B(r  ) different from B(h − R), with radius r  ≥ h − R. Then, by Lemma 3.4.1, there exists a sphere C concentric with B(r  ) and having radius  h − r  , which is a contradiction, since h − r  ≤ r and C is different from C . We now examine the range of possible variation for the radius of the circumsphere of a body of constant width h. Theorem 3.4.2 If R is the radius of the circumsphere of an n-dimensional body of constant width h, then   2n 1/2 h ≤ 2R ≤ h . n+1 Proof Let r be the radius of the insphere. Then  by Theorem 3.4.1, h = r + R ≤ 2R. Furthermore, by Jung’s Theorem 2.8.4, 2R ≤ 2h

n 2n+2

=h

2n . n+1



The radius of the circumsphere can reach the limits given in the theorem. The lower limit is reached when the body is a ball, and the upper limit when it is a body of constant width h that contains a regular d-simplex of diameter h, because the diameter of the circumsphere of this set is exactly h(2n/n + 1)1/2 . As we shall see later (Section 14.3), the latter family of constant width bodies essentially consists of a single figure, the Reuleaux triangle, when n = 2. However, when n > 2, the family has many members. The Reuleaux triangle is thus the figure of constant width h with the greatest circumcircle and the smallest incircle. This is in contrast to the circular disk and (in higher dimensions) ball, the constant width set with the smallest circumcircle and the greatest incircle.

3.5 Minkowski Sum and Central Symmetry To conclude this section, we present the following simple but interesting theorem, a corollary of which characterizes constant width bodies in terms of Minkowski addition. Theorem 3.5.1 The ball is the only centrally symmetric body of constant width. Proof If ψ is any convex body of diameter h, symmetric with respect to a point o, then every diameter pq of ψ passes through o. Otherwise, if o is not in pq, then by the triangle inequality we have the following contradiction: h h + ≥ | po| + |oq| > | pq| = h. 2 2 From this it can be deduced that if  is a centrally symmetric constant width body, then all its diameters, and therefore all its normals, concur. This implies, by Exercise 3.7, that  is a ball.  Theorem 3.5.2 A convex body  has constant width if and only if the body  + (−) is a ball. Proof Let  be a convex body and let − = {x | −x ∈ }. Note that  + (−) is therefore a centrally symmetric constant width body, since if x + y ∈  + (−), then −y + (−x) ∈  + (−).

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Recall that the width of the sum of two bodies is the width of the Minkowski sum of their widths; from this we have that  + (−) is of constant width if and only if  is also constant width. If  + (−) is a ball, then  + (−), and therefore also , are of constant width. Suppose now that  is of constant width. Then  + (−) is a centrally symmetric body of constant width. This implies, by Theorem 3.5.1, that  + (−) is a ball. 

Notes Based on the obvious relation between width function and support function of a convex body, it is true that (even in any Minkowski space) a set  is of constant width h > 0 if and only if its central symmetral 21 ( + (−)) coincides with h2 B n (o, 1); see, e.g., [1113]. In [310], Eggleston proved that  is of constant width if and only if for each pair H1 , H2 of parallel support hyperplanes there is a diametral chord of  generated by, and orthogonal to H1 , H2 , from which also other parts of Theorem 3.1.4 follow. Thus, it is necessary and sufficient that all diametral chords have length h, i.e., that  is of constant diameter. Regarding this formulation, the planar case was solved in [1204], p. 73, and Burke [197] used this for constructions of constant width curves. In [511], Hammer and Sobczyk mentioned this fact explicitly when studying outwardly simple line families built from prolonged diametrical chords (however, Hammer [502] observed that this characterization fails in 3-space). Theorem 3.1.3, that ensures that every constant width body has the binormal property, was mentioned in [308], p. 126; Besicovitch [104] confirmed the planar case of the converse, and one can easily see the higher dimensional analogue by the invariance of that property when projecting such sets orthogonally into 2-dimensional planes. Eggleston [310] proved that any two parallel normals coincide if and only if a convex body has constant width. For all these comments see also Sections 3.1 and 3.5 for Euclidean space and Sections 10.1 and 10.2 for Minkowski spaces. In [663] the following was derived: For each set Q of n lines through the origin o in En , there exists a convex body K such that o ∈ intK , and the double normals of K are exactly the intersections K ∩ Q. For n ≤ 3, the set of all lengths of double normals is of measure zero, but for n ≥ 4 there exists an n-dimensional convex body K , not of constant width, such that every width of K is attained as the length of some double normal, and such that the set of all directions of double normals of K contains an arc connecting a minimum width direction to a maximum width direction. A further nonintuitive property of normals was shown by Heil (see [525] and [526]): Every 3-dimensional constant width body contains either a point through which infinitely many normals pass, or an open set of points through each of which at least 10 normals pass. By the double-normal property of diametral chords a light ray reflecting internally at the boundary of a smooth constant width body will cross a diametral chord repeatedly if any segment of its way lies in that chord. Generalizing a result from [1067] obtained for the 3-dimensional situation, in [1068] it was shown that an n-dimensional smooth convex body is a ball iff each path of a light ray reflecting inside that body lies in a 2-flat. Of course, this is related to billiard problems (see also Notes in chapter 5 and 17.2). It is also proved in [1068] that the successive reflection points inside a smooth constant width curve in the plane and not on a double normal follow a clockwise or counterclockwise order, that this property even characterizes such curves, and that therefore the path of such light rays is never ergodic. Somehow related to double normals is the usage of the farthest point mapping. In [572] the mapping F, associating to each point x of a convex hypersurface the set of all points at maximal intrinsic distance from x, is used to provide two large classes of hypersurfaces with the mapping F single-valued and involutive, and to show that if the mapping F of the double of a convex body satisfies the above properties, then this body has to be a smooth constant width set; a partial converse of this result is verified, too. The following results refer to potential theory see Chapter 18, but we mention them also here since they also use the farthest point mapping in the above sense. Let E be a compact planar set containing more than one point.

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The function d E (z) = maxt∈E z − t is strictly positive, and logd E is clearly subharmonic and the logarithmic potential of a unique probability measure σ E with unbounded support. Various properties of d E and σ E are established in [705], [396], and [397], e.g., that σ E (E) = 21 holds for sets E of constant width and some converse. Alternative proofs and generalizations are presented in [611], see also [609]. At the end of [611] also 3-dimensional examples for E are given, among them constant width sets, such as rotated Reuleaux triangles and Meissner bodies, see also [609]. In [94] it was proved that a planar convex body is of constant width if and only if it is strictly convex and has the property of constant minimum width, see Section 3.2 and Theorem 7.4.3. Furthermore, in [866] it was proved that a convex set in the plane is of constant width iff its boundary is the image of a continuous map of a circle, such that diameters are mapped to double normals. Concerning the perimeter of the shadow boundary (see Section 2.12.2) of a 3-dimensional √ body of constant width 1, Makeev [761] was able to prove that it is always smaller or equal that 2π. The following paragraph refers to projections of constant width bodies and is therefore strongly related to Chapter 13 and its notes. Already in [160] p. 127, one can find the well-known theorem that if for some k with n > k > 1 and n ≥ 3 all orthogonal projections of an n-dimensional convex body K onto k-dimensional subspaces are of constant width, then K is of constant width, too. One can take also other properties of projections into consideration. For example, Minkowski proved that a 3-dimensional convex body K is of constant width if and only if for all directions u its orthogonal projections K u⊥ in planes have equal perimeter (see [888], [160], p. 136, and [357], p. 39), i.e., if and only if that body K is of constant girth (to take another name for this). For a presentation of this theorem and its proof in modern terms we refer to [464], p. 219–221. See also Section 13.1. Denoting by Mn−1 (K u⊥ ) the mean width of the orthogonal (n − 1)-projection K u⊥ of K , the following theorem generalizes Minkowski’s theorem (note that Mn−1 (K u⊥ ) playing the role that the perimeter has in Minkowski’s theorem for n = 3): If K and L are convex bodies in En , n > 2, such that for all directions u the equality Mn−1 (K u⊥ ) = Mn−1 (L u ) holds, then K and L have equal width functions; if, in particular, Mn−1 (K u⊥ ) is constant, then K is of constant width ([464], p. 221). A stability version of Minkowski’s generalized theorem is given in [464], p. 225. We come back now to three dimensions. If we multiply the perimeter of the projection with the width of K in the same direction u, we get the “lateral surface area” of a compact cylinder circumscribed about K and with generators which are parallel to u. Firey [354] showed that K has constant width iff all such cylinders circumscribed about K have equal lateral surface area. See Section 13.1.3. Weissbach [1189] proved that for n ≥ 4 there is a body K of constant width 1 in En such that the circumradius of the orthogonal projection K u of K in direction u is strictly greater than 21 for all directions u. In a separate way, he confirmed this also for n = 3, reproving therefore a crucial lemma used in [309] for constructing minimal universal covers with arbitrarily large diameter. Continuing [309] and [1189], Brandenberg and Larman [181] proved that for any n ≥ 3 there exists an n-dimensional body of constant width such that any of its 2-dimensional projections is not spherical. Calling such a constant width body totally nonspherical, they showed that the circumradius of each 2-dimensional projection of any totally nonspherical constant width body is larger than half the diameter of it. For generalizing the concept of constant width suitably, the following notions are used in [176]: the outer j-radius, j = 1, . . . , n − 1, of a convex body K ⊂ En is the minimum of the circumradii of the projections of K onto all j-dimensional subspaces, and the inner j-radius of K is the radius of the largest j-dimensional ball that fits into K . The authors of [176] nicely construct in any dimension n ≥ 2 nonspherical convex bodies of constant inner and outer j-radii. For a given convex body K in En , Kiderlen [624] considered the Minkowski average (i.e., the convex body Pk (K ) whose support function is defined by integration of support functions of the orthogonal projections of K onto k-dimensional subspaces with respect to the rotational invariant probability measure on the corresponding Grassmannian manifold) of all such k-projections of K . Among other results, he confirmed that P2k+1 (K ) determines K among all bodies of constant width. Let K and L be convex bodies in En , symmetric with respect to the origin, and suppose that M(K u⊥ ) − M(L ⊥ u )2 ≤ ε for some ε > 0. Here M again denotes the mean width,

3.5 Minkowski Sum and Central Symmetry

73

and  · 2 is the L 2 -norm for functions on the unit sphere. It is proved in [441] that the L 2 -distance 2 h 2 (K , L) of K and L can be estimated by h 2 (K , L) ≤ 21 ε n (λn (δ(K ) + δ(L)) + ε2 )(n−2)/2n , where 2 λn is an explicitly given constant and δ = κ−1 d Wn−1 − Wn−2 (κn denotes the volume of the unit ball, and Wk is written for the k-th quermassintegral). It follows that if the mean width M(K u⊥ ) of K u⊥ is nearly constant, then there is a constant width set near K in the Hausdorff metric. For results on recovering the shape of at least 3-dimensional convex bodies from their projections onto 2-planes, Golubyatnikov [435] had to exclude that all these projections are of constant width. Continuing these investigations in the spirit of geometric tomography, Groemer [462] proved that a convex body K is of constant width iff all its orthogonal projections onto hyperplanes of a hyperplane bundle (i.e., of a family of hyperplanes having a line in common such that each point of En belongs to some of them, where the position of this line is arbitrary) are of constant width. For n = 3 and strictly convex bodies this uniqueness is actually valid if the definition of a hyperplane bundle the condition of having a common line is suitably weakened (see [848], where also further related results are obtained). Besides such uniqueness results, Groemer [462] also derived analogous stability results. In connection with polyhedral universal covers, Makeev [757] investigated aspherical orthogonal projections of constant 3 width bodies. In [762], the same author studied shadow boundaries √ of constant width bodies in E . He showed that they form rectifiable spatial curves of length at most 2π which cannot be improved. And there is one direction such that the corresponding orthogonal projection of such a body has length at most sin(π/10) + sin(π/20). Weakly related is the paper [826], where support functions of constant width bodies and projections of support functions of convex sets play a role. We continue with in- and circumspheres of bodies of constant width. It is well known that the insphere and the circumsphere of a body of constant width h are concentric, and that their respective radii r and R satisfy r + R = h as 1 1 well as h(1 − ( n2 (n + 1)) 2 ) ≤ r and R ≤ h( n2 (n + 1)) 2 , see [840], [308], and [821]. The extension to Minkowski spaces was given in [231]. Scott [1052] proved that a convex set in En is contained in a set of constant width with the same diameter and circumradius, and he derived from this several related inequalities, see Section 14.4. And the following nice theorem was proved in [1159]: If C is a set of diameter h contained in a ball B in En , then there exists a set  of constant width h such that C ⊂  ⊂ B. More on inspheres and circumspheres and the respective radii we discuss in the Notes of Chapter 14.

Exercises 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7*.

3.8.

3.9.

Prove that a body has constant width if and only if it has a diameter in every direction. Prove that if  and  are bodies of the same constant width and  ⊂ , then  = . Prove that a body of constant width h is the intersection of a family of solid balls of radius h. Prove that in a convex body every diameter is a binormal. Prove that in a body of constant width every binormal is a diameter. Let  be a body of constant width h, and p be a vertex point in boundary of . Prove that there is an open subset U of the sphere with center p and radius h such that U ⊂ bd . Let φ ⊂ En be a convex body with the following property: there exists a point o ∈ intφ such that for every m ∈ bd φ the hyperplane through m, orthogonal to om, is a support hyperplane of φ. Prove that φ must be a ball. Note that the proof is easier if we assume differentiability. Suppose φ is a convex body with the property that given two parallel support hyperplanes H and H  , we can choose p ∈ H and q ∈ H  in such a way that pq is orthogonal to H and H  . Prove that φ has constant width. Prove that a convex body  has constant width if and only if any two parallel normals coincide.

74

3.10. 3.11.

3.12*. 3.13. 3.14.

3.15. 3.16. 3.17. 3.18*. 3.19. 3.20. 3.21. 3.22.

3.23.

3 Basic Properties of Bodies of Constant Width

Prove that a 3-dimensional convex body  is of constant width if and only if the line perpendicular to each pair of nonparallel normals intersects . Let φ ⊂ En be a convex body, and let u ∈ Sn−1 be a direction. Consider the chord pq of φ with the maximal length between all chords of φ in the direction u. Prove that pq is a diametral chord. Note that this implies that there is a diametral chord parallel to every direction. Prove that if φ is strictly convex, then the inverse Gauss map γ : Sn−1 → bdφ is a continuous map. Prove that if φ is strictly convex and γ : Sn−1 → bd φ is the inverse Gauss map, then γ −1 ( p) is spherically convex, for every boundary point p of . If φ is strictly convex, γ : Sn−1 → bdφ is the inverse Gauss map and p is a vertex point of , then γ −1 ( p) has interior points. Use this to prove that a strictly convex body has countably many vertex points. Prove that a continuous function f : Sn−1 → E is a constant function if and only if the restriction of f to every great circle of Sn−1 is a constant function. Give an example of a convex body in which the minimum width w( p) is not a continuous function of p. Prove that the n-dimensional cube and the common part of the unit solid n-sphere with the first closed octant have constant minimum width. Give an example of a regular point p in the boundary of a convex body with constant minimum width whose normal is not a thickness binormal. Prove that if every regular point p in the boundary of a convex body  is a thickness binormal, then  has constant minimum width. Prove that a square is tight. Prove that any regular polygon is tight. Give an example of a figure of constant minimum width which is not tight. Consider X ⊂ En to be a compact set, not necessarily convex, of constant width. Prove that every support hyperplane intersects X in a single point. Use this to prove that the boundary of the convex hull of X is contained in X . Let X ⊂ En be a set of constant width and, suppose that En \X has only one component. Prove that X is convex.

Chapter 4

Figures of Constant Width

Geometry is the science of correct reasoning on incorrect figures. George Polya

In this chapter, bodies of constant width in the plane are studied. We call them figures of constant width. In studying them, it is important to recall from Section 3.1 that the concepts “normal”, “binormal”, “diameter”, and “diametral chord” coincide.

4.1 Characterizations We begin by stating a characterization of constant width figures which was given in Section 3.1. Theorem 4.1.1 Any two normals of a figure of constant width intersect. Moreover, this property characterizes constant width figures. Assume that two such normals, say pq and pr , intersect at the point p on the boundary of a figure  of constant width h. Then, since every normal is a binormal, the lines L q and L r , perpendicular to pq and to pr at p, respectively, are support lines of  (see Figure 4.1). The point p is therefore a vertex point of , since every line between L p and L q that passes through p is a support line of . The normals of  that correspond to these support lines are binormals and therefore diameters of ; that is, they all have length h. This implies that the portion of the boundary of  between q and r that does not contain p is an arc of a circle of radius h. Now, if the boundary of  contains a circular arc γ with radius h  , then by Theorem 3.1.1 iii), h  is less than or equal to h. If h  = h, every radius of γ is a diameter of , and the center of γ is therefore a vertex point on the boundary of . If h  < h, the radii of γ are contained in the normals of  in γ and therefore, since every normal of  is a diameter of , the circular arc with radius h − h  concentric with γ but antipodal to it is contained in the boundary of . Now, fix a vertex point p in the boundary of  and let L q and L r be support lines of  that pass through p. We will prove that the angle between L q and L r that contains  is always greater than or equal to 120 degrees, and equals 120 degrees only when  is a Reuleaux triangle. Let pq and pr be normals of  perpendicular to L q and to L r through p. Then, the portion of the boundary of  between q and r that does not contain p is a circular arc γ with radius h. If the angle between L q and L r that contains  is greater than 120 degrees, then the angle qpr is greater than 60 degrees. The circular arc γ therefore contains two points m and n farther than h apart, which is a contradiction. If the angle between L q and L r that contains  has 120 degrees, then the angle qpr measures 60 degrees, and the © Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_4

75

76

4 Figures of Constant Width

Figure 4.1

equilateral triangle qpr with sides of length h is therefore contained in . It follows that qr and qp are two binormals of  that intersect at the boundary point q of , and therefore the portion of the boundary between p and r that does not contain q is a circular arc with radius h. In a similar way, the portion of the boundary of  between p and q that does not contain r is a circular arc with radius h, which implies that  is the Reuleaux triangle. Thus we have proved the following theorem. Theorem 4.1.2 The interior angle of a vertex point of a constant width figure is always greater than or equal to 120 degrees, and it equals 120 degrees only when the figure is a Reuleaux triangle. The following corollary is an interesting consequence of this theorem. Corollary 4.1.1 If a figure of constant width h contains an equilateral triangle with sides of length h, then it has to be a Reuleaux triangle. The following theorem of Heppes [531] characterizes constant width figures. Theorem 4.1.3 A convex figure  has constant width if and only if, given any line L that passes through the interior of , one of the sides of  determined by L has the property that the length of any of its chords is less than or equal to the length of the chord L ∩ . Proof We begin by proving sufficiency. Assume that  is not of constant width. Then by Theorem 3.1.1, there exist L p and L q , parallel support lines of , and points p in L p ∩  and q in L q ∩  such that the chord pq is not perpendicular to L p and L q . Let L be a line perpendicular to L p that strictly separates p from q, and let { p  } = L p ∩ L, {q  } = L q ∩ L and x y = L ∩  be such that the intervals x p  and yq  contain the chord x y (see Figure 4.2). It follows that   |x p| > x p   ≥ |x y| ,   |yq| >  yq   ≥ |yx| , which implies that there exist chords x p and xq of  on both sides of the line L that are longer than the chord x y of  determined by L. We now prove necessity. Assume that  is a figure of constant width, and L be any line that intersects the interior of . Let pq = L ∩ , and let α and γ be the curves into which L divides the boundary of  such that α and γ intersect only at p and q. If pq is a binormal of , and therefore a diameter, there is nothing left to prove. So we may assume, without loss of generality, that pq is not a binormal of . Let qα be a point of α with the property that the length of qqα maximizes the lengths of all the chords q x where x is any point in α. The circle with radius of length |qqα | and center q contains the

4.1 Characterizations

77

Figure 4.2

arc α. Thus, if qα is different from p, then qqα is a normal of  that passes through qα and is therefore a binormal of . We define qγ , pα , and pγ similarly. First assume that qα = p and pα = q. This implies that α is contained in the intersection of circles with radius of length | pq| and centers p and q. It is obvious that any chord of  whose ends are in α is shorter than or equal in length to the chord pq. Thus we may assume, without loss of generality, that qα is different from p. The point qγ is the same as p, otherwise, qα and qγ would be binormals of  that intersect at the boundary point q, which would imply, by the above argument, that qp is a binormal of , contradicting the assumption. Also pγ is equal to q, otherwise, ppγ and qqα would be two nonintersecting binormals of . By the argument of the preceding paragraph, this implies that every chord of  whose ends are in γ is shorter than or equal to the length of the chord pq. 

4.2 The Mizel Conjecture The Mizel conjecture states that a planar convex curve such that no rectangle has exactly three vertices on it is a circle. The conjecture was first proved by Besicovitch [104] using sophisticated methods, and later by Watson [1162] and Danzer [277]. We present here Danzer’s proof, which uses constant width figures. Theorem 4.2.1 Let φ be a convex plane figure. Suppose that every rectangle with three vertices in the boundary of φ also has its fourth vertex on bd φ. Then φ is a 2-dimensional disk. Proof The proof consists of two steps. In the first step, we prove that φ must be a figure of constant width. In the second, we conclude that φ is a 2-dimensional disk. For the proof of the first part, by Theorem 3.1.3, we have to show that every normal is a binormal. Let pq be a chord of φ which is normal at p. Let L be the support line of  at p orthogonal to pq, and let L  be the line orthogonal to pq at q, see Figure 4.3. If L  is a support line, we are finished with the proof; if not, then L  ∩ φ = rq, where r is a point of the boundary of φ. By hypothesis, since p, q and r belong to the boundary of φ, the fourth point s of the rectangle pqr s lies in the boundary of φ. Let now L  be the support line of φ parallel to L and let y ∈ L  ∩ φ. Consider yp  , the normal chord of φ, where yp  is orthogonal

78

4 Figures of Constant Width

Figure 4.3

to L  . It is obvious that p  lies in the interval sp. Finally, consider the three points y, p  , and p of the boundary of φ. By hypothesis, the point q must be the fourth vertex of a rectangle yp  pq, but hence q ∈ L  , and therefore pq is a binormal. Our next step is to prove that φ is a disk. Let a be a regular point of bd φ, and let  ⊂ bd φ be the set of points m with the property that am is the diagonal of a rectangle axmy inscribed in bd φ. Clearly,  is a closed set. We shall prove that  consists of a single point m. If this is so, then the boundary of φ is a circle because for every point x ∈ bd φ different from a and m we have that the angle axm is π/2. Indeed, if that were not the case, let n ∈ bd φ be a point with the property that the angle axn is π/2. By hypothesis, there should be a fourth point y  with the property that the rectangle axny  is inscribed in bd φ, and therefore, since  = {M}, we have that n = m. Let us now prove that  consists of a single point. Let m 1 be the first point of  in bd φ \ {a} and m 2 the last one. By hypothesis, there are points y, s ∈ bd φ such that the rectangle aym 1 s is inscribed in bd φ. Furthermore, for any point x ∈ bd φ, between a and m 1 , the angle axm 1 is larger than or equal to π2 ; otherwise, by hypothesis, m 1 would not be the first point of . If D is the disk with am 1 as diameter (see Figure 4.4), then it is clear that y ∈ bd D, and all the points of bd  between a and m 1 belong to D. Let  be the tangent line of D at y and note that, by the above,  is a support line of φ at y. This implies that the diagonal ys of the rectangle aym 1 s is a binormal of the constant width figure φ. But if this is so, then the diagonal am 1 is also a diameter and hence a binormal of φ. Similarly, arguing exactly in the same way but now with m 2 , we conclude that am 2 is a binormal of φ. The proof of the theorem is now complete by taking into account that a was a regular point of the boundary of φ admitting only one normal chord through it.  In [1213], Zamfirescu was able to show that any Jordan curve C with the property that every rectangle with three vertices in C also has its fourth vertex on C, is a convex curve and therefore a disk.

4.3 Alexander’s Conjecture The following characterization of the circle, proposed by Herda [533], was obtained independently in [234] and [841]. Let C be a simple, closed, and rectificable plane curve with perimeter P. For every point x of C, there exists a point x  of C with the property that the distance between x and x  over C is P/2. Let Sx be the length of the segment whose ends are x and x  . We call S, the minimum of Sx where x varies overall C, the pseudodiameter of C.

4.3 Alexander’s Conjecture

79

Figure 4.4

Theorem 4.3.1 Let C be a simple, closed, and rectificable plane curve with perimeter P and pseudodiameter S. Then P/S ≥ π, and P/S = π if and only if C is a circle. Proof It is clear that for every ϑ there exists a point x in C with the property that the segment with endpoints x and x  is parallel to u(ϑ) where u(θ) = (cos θ, sin θ). Therefore, S ≤ Sx ≤ w(θ), where w(θ) denotes the width function of the convex hull cc(C) of C. Clearly, the perimeter of cc(C) is less than P, and hence (see Section 5.1.2) 0≤

1 π



π 0

  P − S. w(ϑ) − S dϑ ≤ π

Moreover, since w(ϑ) ≥ S, the equality P/S = π is only true when w(ϑ) = S for every ϑ; that is, when ψ is a constant width figure and C its boundary. Note that, this means that ψ is a constant width figure with the property that its binormals divide the perimeter of ψ in half. This implies, by Corollary 5.1.4, that C, the boundary of ψ, is a circle.  It was Herda [533] who first asked for a proof of the above characterization. As we can see, the above proof can easily be adapted to prove the following generalization due to Chakerian [234]. Let C be a simple, closed, and rectificable plane curve with perimeter P, and let f : C → C be a continuous involution without fixed points. Let S = min{| f (x) − x| : x ∈ C}. Then P ≥ πS, and equality holds if and only if C is a curve of constant width and f is the involution such that each chord joining x to f (x) is a diametral chord. Following this spirit, we have the following theorem, first conjectured by Alexander [20] and proved by Falconer [329]. For a compact set K ⊂ En and x ∈ En , let d(x, K ) be the maximum of all distances from x to points of K . We know, for example, that by Reidemester’s Theorem 7.2.3, the set K is a convex body of constant width if and only if d(x, bd K ) is constant, for all x ∈ bd K .

80

4 Figures of Constant Width

Theorem 4.3.2 Let C be a simple, closed, rectificable plane curve with perimeter P. If d(x, C) ≥ λ for all x ∈ C, then P ≥ πλ, and equality holds if and only if C is a plane curve of constant width. We present here Falconer’s proof, which used the following interesting integral theorem of Santaló. See [1021, p. 112] for a proof. Theorem 4.3.3 Let C be a plane convex curve with perimeter P and λ a positive real number. Let μ(x) be the number of intersections of the circle of radius λ centered at x with C. Then  1 P= μ(x)d x, 4λ where the integration is with respect to Lebesgue measure over E2 . Proof of Theorem 4.3.2 First of all note that, it is enough to prove the case in which C is the boundary of a convex figure φ. For x ∈ E2 , let μ(x) be the number of intersections of the circle of radius λ centered at x with C. Then by Theorem 4.3.3, μ(x)d x = 4λP. For x inside the outer parallel set φλ of φ (see Section 2.4), but outside φ, the hypothesis implies that μ(x) ≥ 2 except on a set of measure zero. Since the area of φλ − φ is equal to λP + πλ2 , we obtain that 4λP ≥ 2(λP + πλ2 ), giving the desired result P ≥ πλ. In the case P = πλ, then 4λP = 2(λP + πλ2 ), implying that μ(x) = 0 for x in the interior of φ. From this fact, it follows that the width function w(φ, u) ≤ λ for every u ∈ S1 . But if w(φ, u 0 ) < λ for some u 0 ∈ S1 , then P < πλ, contrary to our hypothesis, thus proving that φ is a figure of constant width λ.  Regarding these ideas, Bezdek and Connelly [116] succeeded in proving that given any figure  of constant width 1, every closed curve C of perimeter 2 can be covered by a translate of . Moreover, this property characterizes constant width in the sense that if K is any compact convex set with the property that every closed curve C of perimeter 2 can be covered by a translate of K , then the perimeter of K is equal to or larger than π, with equality if and only if K has constant width 1.

4.4 The Makai–Martini Characterization Let φ ⊂ En be a convex body of diameter 1. We say that φ has property (MM) if any n mutually orthogonal nondegenerate concurrent chords have total length at least 1. We shall prove that property (MM) implies constant width although, for n > 2, there is a body of constant width one that contains n mutually orthogonal nondegenerate concurrent chords having total length as small as desired. But for n = 2, property (MM) characterizes constant width. Theorem 4.4.1 Let φ ⊂ En be a convex body of diameter 1 with the property that any n mutually orthogonal nondegenerate concurrent chords of φ have total length at least 1. Then φ has constant width 1.

4.4 The Makai–Martini Characterization

81

Proof We denote by Hu the support hyperplane of φ with outer normal u ∈ Sn−1 . We shall first prove that if the dimension of Hu ∩ φ < n − 1, then Hu ∩ φ consists of a single point and the corresponding unique normal chord is a diameter. Take p ∈ Hu ∩ φ and let pq be a normal chord of φ in the direction u. Assume that the length of the chord pq is smaller than 1. Denote by Hu the translate of Hu by a distance of , translated inwards to φ. Then, the width of Hu ∩ φ tends to 0 whenever tends to 0. This implies that through the point Hu ∩ pq we can choose n − 1 mutually orthogonal chords of Hu ∩  of total length arbitrarily small provided is sufficiently small. These n − 1 chords together with the chord pq show that the property (MM) does not hold for φ unless the length of pq is equal to one. Let us now consider the set  of all unit vectors u ∈ Sn−1 for which the dimension of Hu ∩  equals n − 1. Since these directions correspond to pair-wise disjoint non-empty open sets of the boundary of φ, they constitute an at most countable set. From our previous arguments, it follows that w(φ, u) = 1 for every u ∈ Sn−1 \ , but this implies, by continuity of the width function, that φ has constant width 1.  Consider now the following n + 1 points in En : o = (0, . . . , 0), a1 = (2t, 0, . . . , 0), a2 = (0, 2t, 0, . . . , 0), . . . , an = (0, . . . , 0, 2t). We shall prove, following the ideas of Polyanskii [944], that if t > 0 is sufficiently small and n > 2, there is a body  of constant width 1 with the property that {o, a1 , . . . , an } ⊂ bd φ. To do so, let us consider the auxiliary points b1 = (t, x, . . . , x), b2 = (x, t, x, . . . , x), . . . , bn = (x, . . . , x, t),  2 , in such a way that, for 1 ≤ i ≤ n, |obi | = |ai bi | = 1. where x = 1−t n−1 Let T = {o, a1 , b1 , . . . , an , bn }. Suppose that for t > 0 sufficiently small, the diameter of T is equal to 1. Then, by the Theorem of Pál 7.2.2, there is a body  of constant width 1 that contains T . Furthermore, since for 1 ≤ i ≤ n, |obi | = |ai bi | = 1, we have that T ⊂ bd , and hence the chords {oa1 , . . . , oan } are a collection of mutually orthogonal nondegenerate concurrent chords having total length 2tn. Let us verify that the diameter of T is equal to 1. For 1 ≤ i < j ≤ n, n > 2 and t sufficiently small,   ai a j 2 = 8t 2 < 1   ai b j 2 = 1 − 4xt + 4t 2 < 1 2   bi b j 2 = 2(x − t)2 = 2 − 2t − 4xt + 2t 2 < 1. n−1   It is not difficult to see that for n = 2, bi b j  is not less than 1. Next, we present the following characterization of constant width figures first proved by Makai and Martini in [753]. Theorem 4.4.2 Let φ be a convex plane figure of diameter 1. Then, φ has constant width 1 if and only if any two orthogonal nondegenerate concurrent chords of φ have total length at least one. For the proof of this theorem, we need the following technical lemmas. Lemma 4.4.1 Let φ be a strictly convex plane figure and let p1 q1 , p2 q2 be mutually orthogonal chords of φ having a common point o. Then | p1 q 1 | + | p2 q 2 | attains its minimum if and only if the point o is in the boundary of φ and o is an endpoint of both chords p1 q 1 , p2 q 2 .

82

4 Figures of Constant Width

Proof Let us fix the chord p2 q2 and translate p1 q1 parallel to itself, the common point o traversing the whole segment p2 q2 . Then the length of p1 q1 is a strictly concave function of the position o on p2 q2 . The same holds for | p1 , q1 | + | p2 , q2 |, and hence the minimum is attained for o = p2 or o = q2 . Since φ is strictly convex, the minimum cannot be attained elsewhere, and o is also the endpoint of  p1 q1 and p2 q2 . Lemma 4.4.2 Let φ ⊂ E2 be a plane convex figureof constant width 1, and let p1 q1 , p2 q2 be orthogonal intersecting chords of φ. If | p1 q1 | = < 1 − 1 | p1 q 1 | + | p2 q 2 | ≥ + 2



3 , 2

then

1 √ − 3 + 2 + = 1 − o (1) 4

for → 0. Proof Let ab be the diameter of φ, orthogonal to p1 q1 . This diameter intersects p1 q1 , and φ contains, besides ab, the union E of two circular segments bounded by ab and circular arcs of radius 1 and with central angle π/3. This follows from the fact that φ is the intersection of the family of unit disks with centers in bd φ, see Theorem 6.1.4. Therefore p1 , q1 , p2 , q2 do not lie in the interior of E. Let p1 q1 be horizontal with p1 lying at the left-hand side, say, and let the center of E be higher than p1 q1 . Also, a should be the lower endpoint of ab. Suppose p1 is not farther from ab than q1 . Let r be any point of p1 q1 , and let sr be  that point of E lying above r such that |r sr | is maximal. Then∗ | p2 q2 | ≥ min{|r sr | : r ∈ p1 q1 } = q1 sq1 . Now let p1∗ q1∗ be   a translate  to the right of p1 q1 such that p1 q1 sq  ≥ q ∗ s ∗ , where s ∗ is the point of E lying lies on the left-handboundary arc of E. We have 1 1 1 1  above q1∗ such that q1∗ s1∗  is maximal. However, q1∗ s1∗ is a vertical chord of a centrally symmetric convex figure F, shaded in Figure 4.5, which is bounded by a subsegment of ab and a parallel segment lying to the right of ab, the distance of the respective parallel lines being , and the upper subarc of the right-hand boundary arc of E, lying in the above parallel strip, and the lower subarc of the arc, obtained by translating the left-hand boundary arc of E to the right by a distance , also lying in the above parallel strip. The shortest vertical chords of F are those lying on ab or on the parallel segment mentioned above, and these have length     q1 sq  = q ∗ s ∗  ≥ 1 + 1 1 1 2



1 √ − 3 + 2 . 4



We shall give the proof of Theorem 4.4.2. Proof of Theorem 4.4.2 (1) By Theorem 4.4.1, we only have to prove that the convex figure φ of constant width 1 satisfies the property (MM), with strict inequality for nondegenerate chords. Since a convex figure of constant width is strictly convex, by Lemma 4.4.1, we may assume that the common point of the chords p1 q1 and p2 q2 lies on the boundary of φ and is an endpoint of both chords, let us say, o = q1 = q2 . If either of the chords op1 , op2 is degenerate, the other is a diameter of φ and |op1 | + |op2 | = 1. From now on we shall suppose |op1 |, |op2 | > 0, where op1 ⊥ op2 . Also, we make the indirect hypothesis that |op1 | + |op2 | ≤ 1. (2) Let oo , p1 p1 , p2 p2 be diameters of φ, where o , p1 , p2 are points of the boundary of φ. We have       • oo  =  p1 p1  =  p2 p2  = 1, • diam{o, o , p1 , p2 , p1 , p2 } = 1, • |op1 | , |op2 | > 0, op1 ⊥ op2 .

4.4 The Makai–Martini Characterization

83

Figure 4.5

Conversely, by the Theorem of Pál 7.2.2, any such system of six points can be included in a convex figure φ of constant width 1, which has oo , p1 p1 , p2 p2 as its diameters. Hence, all the six points lie on bd φ, and op1 , op2 are orthogonal chords of φ with common endpoint o ∈ bd φ. This means that the problem of determining inf{|op1 | + |op2 |} for orthogonal chords op1 , op2 of some convex figure φ of constant width 1 is equivalent to the question of determining inf{|op1 | + |op2 | : diam{o, o , p1 , p2 , p1 , p2 } = 1,       oo  =  p1 p   =  p2 p   = 1, |op1 | , |op2 | > 0, op1 ⊥ op2 . 1 2 By the indirect hypothesis and Lemma 4.4.2, we may assume in the case inf{|op1 | + |op2 |} < 1 that |op1 | + |op2 | ≤ λ for some λ < 1, and thus min{|op1 | , |op2 |} ≥ for some > 0. However, in this case, the domain {(o, o , p1 , p1 , p2 , p2 ) : diam{o, o , p1 , p2 , p1 , p2 } = 1,       oo  =  p1 p   =  p2 p   = 1, |op1 | , |op2 | > 0, op1 ⊥ op2 , 1 2 becomes compact if we assume that, say, o is fixed. Thus the infimum is attained. If inf{|op1 | + |op2 |} = 1, the infimum is evidently attained as well. It remains to investigate how these extremal configurations attaining min{|op1 | + |op2 |} look like. (3) As is well known, any two diameters of {(o, o , p1 , p1 , p2 , p2 )} either have a common relatively interior point or have a common endpoint. Thus, the cyclic order of p1 , p1 , p2 , p2 on the boundary of the convex hull φ∗ of all these six points is p1 , p2 , p1 , p2 (in some orientation), where neighboring points may coincide. Similarly, the cycle order of o, o , p1 , p1 , p2 , p2 on bd φ is (again in some orientation) i) ii) iii) iv)

p1 , p2 , o, p1 , p2 , o or p1 , p2 , o , p1 , p2 , o or p1 , o, p2 , p1 , o , p2 or o, p2 , p1 , o , p2 , p1 ,

84

4 Figures of Constant Width

Figure 4.6

where neighboring points (possibly more than two) may coincide. We shall show that each of these four cases leads to a contradiction.   (3.1) In the first case, we have π/2 = ∠ p2 op1 = ∠ p2 oo with | p2 o| > 0, oo  > 0, and hence    p2 o  > 1, a contradiction. Interchanging the role of the indices, we see that the second case is contradictory, too. (3.2) For the third case, we introduce a coordinate system x y (see Figure 4.6), with o = (0, 0), p1 = (a, 0), p2 = (0, b), where, in accordance with the indirect hypothesis, a, b > 0, a + b = |op1 | + |op2 | ≤ 1. The points p1 , p2 lie in the intersection of the unit circles about o, p1 , p2 , on the boundaries of the unit circles about p1 , p2 , respectively, and by the hypothesis about the order of the points on bd φ∗ , both lie in the quadrant x, y ≥ 0.     Thus, the point p1 is from the unit circle S( p1 , 1) about p1 , and also op1  (=  p1 p1 ); hence p1 lies on the arc of S( p1 , 1) lying to the left of its endpoints 

a2  , 1− , 2 4

a

a 2

 ,− 1 −

a2  . 4

Similarly, p2 lies on the arc of S( p2 , 1) lying below its endpoints  

b2 b  1− , , 4 2



 −

1−

b2 b  , . 4 2

Also both points p1 and p2 lie in the quadrant x, y ≥ 0. Let us now introduce another orthonormal coordinate system ξ, η, having the same orientation as the x, y-system, by letting its origin be the midpoint of p1 p2 and letting     1 1 2 p2 = 0, p1 = 0, − a + b2 , a 2 + b2 2 2 in this ξ, η-system. From now on, the coordinates of points will be given in the x, y-system, unless otherwise specified.

4.4 The Makai–Martini Characterization

85

     √ 2 By the last paragraph, p1 lies on the shorter arc of S( p1 , 1), between 0, 1 − a 2 and a2 , 1 − a4 .    2 We claim that this arc lies in the open quadrant ξ, η > 0. In fact, the point a2 , 1 − a4 lies above  √ 2 ( a2 , b2 ) (because of b2 < 1 − a4 ) and the point (0, 1 − a 2 ) lies above (0, b) (by a, b > 0, a + b ≤ 1 and hence a 2 + b2 < 1). Therefore, these two points lie in the quadrant ξ, η > 0. Since p1 lies on the negative half of the η-axis, the whole considered arc lies in the quadrant  ξ, η> 0. Thus, this arc (and P1 , too) lies on the shorter subarc J1 of S( p1 , 1) bounded by a2 , 1 − a4 and the point obtained by projecting  p radially from p1 to S( p1 , 1). Similarly, p2 lies on the shorter subarc J2 of S( p2 , 1)   2 2 1 − b4 , b2 and the point obtained by projecting p1 radially from p2 to S( p2 , 1). We bounded by   claim that if p  , p  are allowed to vary in the whole arcs J1 , J2 , the minimum of  p  p   is attained for 2

1

2

1 2



a2  , 1− , 2 4   b2 b  p2 = 1− , . 4 2       2 2 In fact, suppose, e.g., that p1 = a2 , 1 − a4 . Replacing p1 by a2 , 1 − a4 , the distance of    2 p1 to p2 decreases, since the orthogonal bisector of p1 and a2 , 1 − a4 passes through p1 and the    2 open half plane nearer to a2 , 1 − a4 contains the whole arc J2 . In an analogous manner, we can        2 2 1 − b4 , b2 . Therefore,  p1 p2  is actually at least the distance of a2 , 1 − a4 and replace p2 by        2 2 2 1 − b4 , b2 . Now we claim that the distance of a2 , 1 − a4 and 1 − b4 , b2 is greater than 1. In fact, since both these points lie on S(o, 1), this is equivalent to the fact that the angle subtended by  p1

=

a

them at o is greater than π/3. In other words, the sum of the angles subtended by p1 and    2 respectively, by a2 , 1 − a4 and p2 , at o, is smaller than π/6, which is equivalent to

1−

b2 b , 4 2

,

b a arcsin( ) + arcsin( ) < π/6. 2 2 However, the convexity of the arcsine function implies that a b b a + arcsin( ) + arcsin( ) < arcsin(0) + arcsin 2 2 2 2 ≤ arcsin(1/2) = π/6 by the indirect hypothesis. This gives

    2       a b2 b  p p  ≥  , 1 − a 1 − ,  > 1, 1 2  2 4 4 2 

and so we have a contradiction in the third case as well. (3.3) Finally, we have to settle the fourth case, where the order of the points on bd φ∗ is (in some orientation) o, p2 , p1 , o p2 , p1 , with the possibility that two or three neighboring points may coincide.

86

4 Figures of Constant Width

We shall make a distinction, first distinguishing according to the number of different points among our six points, and second distinguishing according to the diameter graph; i.e., the graph whose vertices are the different points among our given points, and whose edges are the diameters (i.e., the segments of length 1 among the vertices). From now on, we shall assume that our system of six points is that one at which the infimum of the reformulated problem in (2) is attained. (3.3.1) We have       oo  =  p1 p   =  p2 p   = 1, o = p1 , o = p2 , 1 2  0 < |op1 |2 + |op2 |2 = | p1 p2 | < |op1 | + |op2 | = 1, and hence also p1 = p2

and

|op1 | , |op2 | < 1.

  We suppose p2 = p1 . Then, we have | p1 p2 | =  p1 p1  = 1, a contradiction. Thus, p2 = p1 , and similarly p1 = p2 . Analogously, |op1 | < 1 implies o = p1 , o = p1 , and |op2 | < 1 implies o = p2 , o = p2 . Suppose now p1 = p2 . By the order of the points, as assumed, this implies p1 = o = p2 , contradicting the above statements. Therefore p1 , p2 , p1 , p2 are all different. If o coincides with one of them, then this one must be either p1 or p2 . For o = p1 , by the order of the points, as assumed, we have o = p2 = p1 , a contradiction. Thus o = p1 , and similarly o = p2 . If o coincides with one of p1 , p2 , p1 , p2 , then this one must be either p1 or p2 . For o = p1 , by the order of the points we have o = p2 = p1 , a contradiction. Thus o = p1 , and similarly o = p2 . Hence, we have shown that all six points are different. (3.3.2) Now, we shall discuss all the possible diameter graphs on six different points o, o , p1 , p1 , p2 , p2 , where we know that this graph contains the edges oo , p1 p1 , p2 p2 (without using the order of the points on bd φ∗ or the result of (3.3.1)). We know from Section 6.3 that the number of edges of the diameter graph is at most the number of vertices, in our case six, and the diameter graph does not contain even circles. Thus any circle can only be a triangle or a pentagon. If there is a triangle, it is a regular one of edge length 1, in which case we let φ be the Reuleaux triangle with the same vertices. We have φ∗ ⊂ φ, and each diameter of φ∗ is a diameter of φ as well. Hence, all six points lie on bd φ. Therefore op1 , op2 are two nondegenerate orthogonal chords of a Reuleaux triangle, and in such a case Schmitz [1030] already proved that |op1 | + |op2 | > 1, a contradiction. Thus, a cycle in the diameter graph can be only a pentagon. Now suppose the diameter graph is not connected. Then, the points o, o lie in a component other than the one in which, say, p1 , p1 lie. That is, there is no edge from the set {o, o , p2 , p2 } to { p1 , p1 }, or else from {o, o } to { p1 , p1 , p2 , p2 }. In the first case we may translate { p1 , p1 } a bit, keeping ∠ p1 op2 = π/2 and decreasing |op1 |, contradicting the fact that our six points realized the infimum of |op1 | + |op2 |. Similarly, in the second case we may translate {o, o } a bit, keeping ∠ p1 pp2 = π/2 and decreasing |op1 | + |op2 |, which is again a contradiction. Thus, the diameter graph has to be connected. Since oo , p1 p1 , p2 p2 are edges of the diameter graph, there are no isolated points. Hence, all possible diameter graphs are the following ones. (A) There is a pentagon in the graph, and the sixth vertex is joined by an edge to a vertex of the pentagon. So we already have six edges, and hence there are no more edges. (B) The graph is a tree. In this case besides oo , p1 p1 , p2 p2 , there are two additional edges. One of them forms, together with the above three edges, a graph consisting of a path P of length 3 and of an edge e, as components. The second of these additional edges connect an endpoint or a non-endpoint of the path P to an endpoint of the edge e. Thus, we obtain either (B ) a path of length 5, or (B ) a tree having a vertex of valence 3, and two of the edges adjacent to this vertex each join to one more edge.

4.4 The Makai–Martini Characterization

87

Figure 4.7

(3.3.3) Now, we investigate the geometric realizations of the graphs of cases (A), (B ), (B ), which are the only possible cases in the fourth case, described in (3), taking into consideration the order of our points on bd φ∗ . In case (A), we have a star pentagon, and inside one of the angular domains of the vertices there is another diameter (see left of Figure 4.7). We have |op1 |, |op2 |, | p1 p2 | < 1, and o, p1 , p2 are the three non-neighborly vertices of φ∗ , which can be chosen in two ways. In both ways the triangle op1 p2 has an edge of length 1, a contradiction. In case (B ), we have an “open star polygon” (see right of Figure 4.7). Here again, choosing three non-neighborly vertices of φ∗ in either of the two possible ways, the triangle obtained in this way has an edge of length 1. This leads, as above, to a contradiction. In case (B ) (see Figure 4.8) one has an “open star polygon” of four edges and one additional diameter from the middle vertex, in the respective angular domain. If we choose three non-neighborly vertices of φ∗ , in one of the ways, the triangle so obtained has an edge of length 1, which is a contradiction, while in the other case there are no edges of length 1. In this second case, we have to investigate two essentially different subcases. let In the first subcase (left of Figure 4.8), keeping o, o , p1 , p1 fixed,   us move  p2 on the segment p2 o, toward o, and let us move p2 under the restriction  p2 p2  = o p2  = 1. This moving of p2 is possible because p2 p2 is not orthogonal to S(o , 1) at p2 ; since otherwise, p2 = o   would    hold, which has been excluded in (3.3.1), or else p2 would be the midpoint of o p2 , thus o p2  = 2 would hold, a contradiction. Then ∠ p1 op2 = π/2 remains valid, and |op1 | + |op2 | decreases, a contradiction. In the second case (right of Figure 4.8), keeping o, o , p1 , p1 fixed, let us move p2 and p2 as of p2 is possible because p2 p2 described in the first subcase, but replacing o p2 by p1 p2 . This  moving       is not orthogonal to S( p1 , 1) at p2 ; in view of p2 = p1 and p2 p1 = 2 (by the above argument). Then ∠ p1 op2 = π/2 remains valid, and |op1 | + |op2 | decreases, a contradiction. Thus both subcases of (B ) lead to a contradiction. So together with (A) and (B ) all subcases of the fourth case (cf. (3)) give a contradiction. By (3.1) and (3.2), the first, second and third cases (cf. (3))

Figure 4.8

88

4 Figures of Constant Width

were contradictory as well. This means that our indirect hypothesis |op1 | + |op2 | ≤ 1, |op1 |, |op2 | > 0, is not right, or, in other words, for |op1 |, |op2 | > 0 we have the inequality |op1 | + |op2 | > 1. By (1) and (2) this finishes the proof of Theorem 4.4.2. 

4.5 Intersection Properties of the Boundaries Now, we will examine what happens to the boundaries of two intersecting constant width figures. We have the following theorem. Theorem 4.5.1 The intersection of the boundaries of two intersecting figures of constant width h divides the boundaries into either an even or an infinite number of components. To prove this theorem, we shall require some definitions and a lemma. Let 1 and 2 be two figures with constant width h and let  be a component (necessarily an arc or a point) of bd 1 ∩ bd 2 . Suppose bd 1 ∩ bd 2 has a finite number of components. If this is so; we shall say that  is a crossing component if every neighborhood of  contains points of 2 \ 1 and of 1 \ 2 ; and  is a non-crossing component otherwise. It is obvious that there are two kinds of non-crossing components. One type has a neighborhood N with the property that N ∩ 1 ⊂ N ∩ 2 , while the other has a neighborhood N with the property that N ∩ 2 ⊂ N ∩ 1 . In the former case, we say that 1 ⊂ 2 locally; in the latter 2 ⊂ 1 locally. Lemma 4.5.1 Let  be a non-crossing component of bd 1 ∩ bd 2 in which 1 ⊂ 2 locally near , and let p be a point of . Then, every support line of 2 at p is also a support line of 1 at p. Proof Let N be a neighborhood of  in which N ∩ 1 ⊂ N ∩ 2 , and let x be an interior point of 1 and 2 . Without loss of generality we may assume that x is in N . Let L be a support line of 2 at p. If L is not a support line of 1 at p, then there exists a point q different from p in L ∩ 1 . This would imply that the triangle with vertices q, p and x contained in 1 contains a point of N not in 2 , which is a contradiction.  Proof of Theorem 4.5.1 Without loss of generality, suppose that the number of components of bd 1 ∩ bd 2 is finite. The fact that the number of crossing components of bd 1 ∩ bd 2 is even is obvious, even for arbitrary convex curves. It is therefore sufficient to prove that the number of non-crossing components of bd 1 ∩ bd 2 is even, showing a bijection between the set of non-crossing components for which 2 ⊂ 1 locally and 1 ⊂ 2 locally. If α ⊂ bd 1 , then we can define τ1 (α) = {q ∈ bd 1 | there exists a diameter whose ends are q and p, with p ∈ α}. Since every pair of diameters of 1 intersects, we have that if α is a connected arc of bd 1 , then so is τ1 (α). Let  be a non-crossing component of bd 1 ∩ bd 2 , where 2 ⊂ 1 locally; that is, there is a neighborhood N of  in which N ∩ 2 ⊂ N ∩ 1 ; and let p ∈ . Then, the circular arc τ1 { p} is contained in bd 1 ∩ bd 2 , because every diameter pq of 1 is a normal of 1 at p, and by Lemma 4.5.1, every pq is also a normal of 2 at p. By Theorem 3.1.3,

the chord pq is a binormal of 2 , which implies that q ∈ bd 1 ∩ bd 2 and therefore that τ1 { p} ⊂ bd 1 ∩ bd 2 . From this, it can be concluded that τ1 () is a subarc of the boundary of both 1 and 2 . We now prove that τ1 () is a non-crossing component of bd 1 ∩ bd 2 , where 1 ⊂ 2 locally. Suppose it is not; then there are points xi ∈ 2 \ T1 () such that xi ∈ bd 1 and xi approach T1 (). Note that, every diameter of 2 with endpoints in 1 is also a diameter of 1 , so T2 (xi ) ∩  = ∅, otherwise xi ∈ T1 (). Then, for i sufficiently large, T2 (xi ) ⊂ N ∩ bd 2 \  ⊂ int1 ,

4.5 Intersection Properties of the Boundaries

89

which is a contradiction because if q ∈ T2 (xi ), then q xi is a diameter of 2 with endpoints in 1 and hence a diameter of 1 . The bijection between the sets of non-crossing components for which 2 ⊂ 1 and 1 ⊂ 2 locally is therefore given by  → τ1 (). 

This concludes the proof of the theorem. Two corollaries follow immediately from Theorem 4.5.1.

Corollary 4.5.1 Let  be a figure of constant width h, and D be a disk of diameter h having the property that its interior intersects the interior of . Then, the intersection of the boundaries of  and D has an even or infinite number of components. Corollary 4.5.2 Let  be a figure of constant width h, and  be a congruent copy of  with the property that the interiors of  and  intersect. Then, the intersection of the boundaries of  and  has an even or infinite number of components. In 1972, Peterson [927] conjectured that each of the properties in Corollaries 4.5.1 and 4.5.2 completely characterizes constant width figures, see also [928]. That is, he conjectured first that if  is a convex figure of diameter h and the intersection of the boundaries of  and D has an even or infinite number of components for every disk D of diameter h, whose interior intersects the interior of , then  must be a constant width figure. He further conjectured that if  is a convex figure and the intersection of the boundaries of  and  has an even or infinite number of components, for every congruent copy  of  whose interior intersects the interior of , then  must be a constant width figure. The first conjecture was proved by Goodey in [438] in the smooth case, and later both conjectures were verified by Goodey and Woodcock [445] in 1978. In 1984, Goodey [440] proved a much more general result. Theorem 4.5.2 Let 1 and 2 be two convex figures. If the boundaries of 1 and 2 intersect to produce an even or infinite number of components for every translated copy 2 of 2 , where the interior of 2 intersects the interior of 1 , then the width of 1 is equal to the width of 2 in every direction. We finish the chapter including the proof of the converse which generalizes the ideas of the proof of Theorem 4.5.1. Theorem 4.5.3 The intersection of the boundaries of two intersecting figures having equal width in every direction divides the boundaries into either an even or an infinite number of components. Proof Suppose  and φ are convex figures having equal width in every direction and let φ be a translated copy of φ. If the number of connected components of bd  ∩ bd φ is finite, then the number of crossing components of bd  ∩ bd φ is even. By Exercise 4.10, bd( + B) ∩ bd(φ + B) has the same number of non-crossing components as bd  ∩ bd φ, where B is the unit disk. So, we may assume that  and φ have unique supporting lines at all points of their boundaries. If x, y ∈ bd , we denote by [x, y] the arc of bd  starting at x and ending at y following the clockwise orientation of bd . Now let I = [x, y] be a non-crossing component of bd  ∩ bd φ . Then  and φ share a support line L at x. Because the figures have the same width in every direction they will also share the parallel support line L  . We put L ∩ bd  = x z

90

and

4 Figures of Constant Width

L  ∩ bd  = x  z  .

We shall assume that, without loss of generality, an open neighborhood of I contains points of (intφ ) \  but not (int) \ φ . If a ∈ (L  ∩ bd φ) \ x  z  , we can find support lines of  arbitrarily close to L  which separate a from int. But φ would the have larger width than  in the corresponding directions. So we must have L  ∩ bd φ ⊂ x  z  . We let J be the component of bd  ∩ bd φ intersecting L  ∩ bd φ . If there were an open neighborhood of J containing points of (intφ ) \  then again φ would be wider than  in some directions. So we deduce that J is also a non-crossing component. In this way, we see that the non-crossing components occur in pairs, and so there must be an even number or infinite number of them. This concludes the proof of the theorem. 

Notes Aside from basic books and surveys (see [160], [1204], [238], and [527]), there are also various expository papers summarizing and nicely presenting geometric properties and characterizations of constant width figures; an example is [598]. We now want to collect further geometric properties and characterizations of figures of constant width in the plane, which are widespread in the literature. Characterizations and Properties In view of the importance of binormals (see Chapter 3), we start these notes with related characterizations of figures of constant width. As starting point, we should mention the papers [510], [511], [512], [501], [507], [508], and [509], where a thorough study of diametrical chords of constant width figures (also for Minkowski planes) is given. Theorem 3.1.6 (Theorem 4.1.1) goes back to Bückner [194], and based on this he observed also that a 3-dimensional convex body K has constant width if and only if the line perpendicular to every pair of nonparallel normals of K intersects K , see also Exercise 3.10. Extending another result of Bückner [194], Beretta and Maxia [99] proved that a strictly convex figure K is of constant width if and only if for all directions u the expression 2 f (u) − w(u)l(u) is constant; here l(u) and f (u) denote the length of the boundary as well as the area of a part of K on the same side of a diametrical chord of direction u, and w(u) is clearly the respective width of K , see Corollary 5.1.5. In [866], it is shown that a convex planar set is of constant width if and only if its boundary is the image of a continuous mapping of a circle, such that diameters are mapped to double normals. Kelly [617] proved the following: If C is a constant width figure in the plane, its pedal curve C  = p(C) with respect to an arbitrary point a has a as an equichordal point. Conversely, given a curve star-shaped with respect to its equichordal point a, its “negative pedal curve” C = p −1 (C  ) (easily obtained as an envelope of straight lines) has constant width if it is convex. The author studies non-convex curves obtained in this way which are inner parallel curves of constant width curves. In [778], the results from [617] are extended to higher dimensions. For example, it is shown that a smooth convex hypersurface with the origin as an equichordal point has an antipedal hypersurface which is a hedgehog of constant width (for hedgehogs we refer to the notes in Chapter 11). A condition is given ensuring that the pedal hypersurface of a given hedgehog of constant width is convex.

4.5 Intersection Properties of the Boundaries

91

Theorem 4.1.3 can also be formulated as follows (see [531]): A closed convex curve is of constant width if and only if each of its chords is the longest chord of one of the two arcs determined by its two endpoints. Inspired by the fact that each planar convex body has an inscribed square, Eggleston [314] constructed examples which limit generalizations of this result. He showed that there is a constant width figure in which no regular k-gon may be inscribed for k > 4 (in fact, the Reauleaux triangle is shown to have this property). The author of [250] derived an exact condition for the existence of a regular polygon circumscribing a strictly convex figure in the plane. His result yields a generalization of Pál’s old theorem (a plane figure of constant width has a circumscribed regular hexagon) in the following way: If the width function of a strictly convex figure K has period π/3, then there is a regular hexagon circumscribing K . In [1133], constant width curves are characterized by the property that, if three edges of a rhomb belong to lines of support of the curve, then also the fourth. If “rhomb” is replaced by a quadrilateral which is the union of two isosceles triangles, the condition characterizes circles. Let K be a convex body, and F be a set of pair-wise intersecting translates of K . It is known that if K is centrally symmetric, then for any F one can find three points which meet every member of F. The conjecture that this is also true if K is not centrally symmetric is proved in [242] for K being a constant width figure. This result follows from the next interesting theorem: If Q is a set of diameter 1 and K a figure of constant width not less than 0.9101, then Q is contained in the union of three translates of K . In [513], it is shown that a closed curve of length 2π can be covered by a rectangle of area 4, and if no smaller rectangle will do this, the covered curve has to be convex and of constant width 2 (see also [304]). More analytic characterizations of constant width curves are given in [409], [935], and [866]. In [878], planar curves are studied from which any point p has another point p  of constant distance, such that the oriented tangents at p and p  have opposite directions (and the spherical image of the curve should be closed). Such curves are shown to be either of constant width or to be so-called translation curves consisting of a set of congruent arcs. In [607], closed convex curves (and, in particular, polygons) are investigated all whose circumscribed rectangles are squares. Clearly, constant width curves occur as examples. Let C be a body of constant width h in Euclidean n-space. In [377], it is proved that the lengths of the segments intercepted by C on each of two parallel lines at most v distance apart differ by at most 2(2vh)1/2 , where this value depends on the width, but not on the shape of C. And in [621] the following is investigated: A convex body K in En has property (P), if for every direction u and every pair of parallel supporting hyperplanes parallel to u there is a chord of maximal length in the direction of u lying midway between the hyperplanes. Is any K satisfying (P) symmetric? (The converse is obvious.) An affirmative answer is given for general convex bodies if n ≥ 3, and for constant width figures if n = 2. Thus, for general convex bodies, the question is unsettled only in the planar case. There are also results characterizing special curves (e.g., disks) within the family of constant width figures, or describing at least respective properties. One old result in this direction is obtained in [511]: A constant width figure, whose boundary is divided by each diametrical chord into equal arcs is necessarily a circular disk. Assuming smoothness, this wasproved already by Hirakawa [541] and independently by Klamkin (see again [511] and also [104]). On the other hand, it was also proved in [511] (and also with smoothness assumption already in [541]) that if any diametrical chord of a constant width figure bisects its area, then it is again a circular disk. See Corollary 5.3.1 for both results. For more on these two characterization theorems and related circle characterizations, we also refer to [829] and to [407], where it is proved that the only Zindler curve of constant width is the circle, see Exercise 5.19. Based on X -rays, in [724] a sufficient condition for a constant width figure is given to be a disk. The author of [162] proved that, if C is a constant width curve, such that for each pair x, y of points from C there exists a nondegenerate rectangle R contained in C with x, y being also from R, then C must be a circle. The paper [777] contains results on hedgehogs, applications which yield characterizations of spheres among all convex hedgehogs of constant width. In [967], the following problem is raised: Put f (x, y) = 1/π(1 − x 2 − y 2 )1/2 for x 2 + y 2 < 1 and f (x, y) = 0 for x 2 + y 2 ≥ 1. Clearly, the integral of f (x, y) on every chord of the unit circle is 1. Is there any other curve of constant width for

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4 Figures of Constant Width

which there exists a function f (x, y), whose integral is constant on every chord? Baillif [63] derived a characterization of the circle among all C 2 curves of constant width via inscribed regular polygons. Mizel’s Conjecture The following notes refer to Section 4.2. In [1213], Zamfirescu proved an analogue of Theorem 4.2.1 for a Jordan curve (not convex a priori) and for a rectangle with the infinitesimal rectangular property satisfied by the following relation between its sides: |a/b| ≤ ε, ε > 0, where a and b are the side lengths of the used rectangle. The usual rectangular property is given by ε = 1. In 2006, M. Tkachuk extended this infinitesimal rectangular property to arbitrary compact planar sets, whose complements are not connected. Zamfirescu also showed that every analytic curve of constant width satisfying the infinitesimal rectangle property is a circle. The main result of the paper [1217], where Tkachuk is one of the authors, is to prove this theorem for a convex curve which is not analytic, and to discuss some further unsolved problems concerning Mizel’s problem. Wegner [1177] considered extensions to plane continua and higher dimensional manifolds. He showed that any planar continuum having the above rectangle property and which is not a Jordan arc must be a circle. Further on, any bounded subset of the plane with the rectangle property and containing a circle must in fact be a circle. For higher dimensions it is noted that there exist smooth closed curves in 3-space having the rectangle property, which is not circles. On the other hand, it is proved that a compact hypersurface immersed in n-space and having the rectangle property must be a sphere. Further extensions to curves and hypersurfaces in higher dimensions are given in [1178]. For example, it is proved that a C 2 simple closed curve in 3-space satisfying the infinitesimal rectangular property must be a transnormal curve (as generalization of the notion of constant width curves, see Section 16.2). If the curve satisfies the rectangle property then it is a transnormal, centrally symmetric, spherical curve. On the other hand, any C 1 transnormal, centrally symmetric, spherical curve satisfies the infinitesimal rectangular property. Several other related results, also referring to higher dimensions, are proved in [1178], assuming also weakened forms of the infinitesimal rectangular property. Alexander’s Conjecture Ault [49] verified Herda’s conjecture (see Section 4.3) extended to any real Hilbert space by using simply the law of cosines for a triad of triangles, a few elementary trigonometric formulae, and a generalized triangle inequality, thus making the proof rather easily understandable. Studying closed curves and actions of finite groups on differential-geometric manifolds in n-space, Aeppli [3] re-obtains results such as Theorem 4.3.2. His approach also allows to obtain a related characterization of spheres in 3-space. Lutwak [741] succeeded to use Falconer’s result from [329] for showing the following: Any closed curve of length 1 can be covered by a semicircular disk of radius 1/π. Moreover, this radius cannot be decreased if and only if the curve is of constant width 1/π. From this, it follows that any two planar domains, each bounded by a simple closed curve of length 1, may be placed inside a circle of radius 1/π, such that their interiors are disjoint. Two figures of constant width 1/π show that the radius of the circle cannot be decreased, and it is proved in [741] that there are no other cases of this kind.

Exercises 4.1. 4.2.

Prove the Chakerian generalization to Theorem 4.3.1, see [234]. In what sense does the Alexander conjecture generalize Chakerian’s characterization of constant width?

Exercises

4.3. 4.4. 4.5.

93

Prove that for the Reuleaux triangle of side length 1, any two mutually orthogonal nondegenerate concurrent chords have total length at least one. In the construction given by Polyanskii, prove that the lengths of the line segments Bi B j are not smaller than one when n = 2. In the context of the proof of Lemma 4.4.2, prove that      Q 1 SQ  =  Q ∗ S∗  ≥ 1 + 1 1 1 2

4.6.



1 √ − 3 + 2 . 4

In the context of the proof of Theorem 4.4.2, prove the statement at the end of (3.2). That is.     2       a b2 b  P P  ≥  , 1 − a 1 − ,  > 1. 1 2  2 4 4 2 

4.7. 4.8*.

4.9. 4.10. 4.11.

Prove that a diameter graph does not contain even circles. Let φ be a convex figure and φ a congruent copy of φ different from φ with the property that the interiors of φ and φ intersect. Suppose the intersection of the boundaries of φ and φ has two components. Is φ a disk? Let φ1 and φ2 be two convex sets. Prove that the number of crossing components of bd φ1 ∩ bd φ2 is even or infinity. Let φ1 and φ2 be two convex sets. Prove that bd(φ1 + B) ∩ (φ2 + B) has the same number of non-crossing components as bd φ1 ∩ φ2 , where B is the unit disk. Prove both Peterson’s conjectures using Theorem 4.5.2.

Chapter 5

Systems of Lines in the Plane

Geometry is knowledge of the eternally existent. Pythagoras

A system of lines consists of an assignment of a line in the plane for any direction. In this section, we will be interested in studying systems of lines, in particular those which are combined with a given convex set by a certain property. For example, the system of lines that leave a fixed proportion of area, or a fixed proportion of perimeter, in one side of the convex set for every direction. Consider, for instance, the collection of tangent lines of a given strictly convex set in the plane which varies continuously with respect an angle, or a system of diametral lines, a system of median lines, etc. The purpose of this chapter is to study some useful and important systems of lines. We begin in Section 5.1 by introducing the pedal function and the system of tangent lines of a strictly convex plane figure. This allows us to give a parametrization of its boundary from which we derive the Cauchy formula for its perimeter. In Section 5.2, we parametrize a system of lines by its pedal function and we introduce the notion of envelope. In particular, we study those systems of lines which intersect the interior of a convex figure leaving on the same side regions of the same area. Section 5.3 will be devoted to study systematically some other systems of lines, like the system of diametral and median lines or the system of lines splitting in two the area of a convex figure, all of the examples of systems of lines are called externally simple. This will allows us to derive the equation of the analytic curve of constant width due to Rabinowitz, and to construct the Euler curve of constant width as an evolute of the hypocycloid. Finally, in Section 5.4, we use all this machinery to study the famous floating body problem in dimension two.

5.1 The Pedal Function 5.1.1 The Parametrization in Terms of the Pedal Function Let θ be a real number. We let u(θ) ∈ E2 denote the unit vector u(θ) = (cos θ, sin θ).

© Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_5

95

96

5 Systems of Lines in the Plane

Therefore u(0) is the unit vector (1, 0) on the x-axis. In general, if θ ∈ [0, 2π), then u(θ) denotes the unit vector that makes an angle of θ with the unit vector u(0). Note that u(θ) = (cos θ, sin θ) has a meaning for any real number θ because u(θ) = u(θ + 2π). Taking the derivative with respect to θ gives u (θ) = (− sin θ, cos θ) , from which we have that u (θ) is perpendicular to u(θ) given by  π . u (θ) = u θ + 2 Note that for every real number θ we have u (θ + 2π) = u (θ), u (θ + π) = −u (θ),  π u θ + = −u(θ). 2

u(θ + 2π) = u(θ), u(θ + π) = −u(θ),  π u θ+ = u (θ), 2 Let ψ be a convex figure in E2 and let p(θ) = max{x, u(θ)|x ∈ ψ} ,

where, as before, ·, · denotes the scalar product. The real function p(θ) is called the pedal function of the figure ψ, and it is essentially the support function described in Section 2.4. Observe (Figure 5.1) that the line l(θ) = {x ∈ E2 |x, u(θ) = p(θ)}, which is a support line of ψ parallel to u (θ), is perpendicular to u(θ), and its distance from the origin in direction u(θ) is p(θ). The other support line of ψ parallel to u (θ) is l(θ + π), and its distance from the origin in direction u(θ + π) is p(θ + π). It follows that if w(θ) denotes as usual the width of ψ in direction u(θ), then w(θ) = p(θ) + p(θ + π). If ψ is strictly convex for every real number θ, there exists a unique real number f (θ) such that x(θ) = p(θ)u(θ) + f (θ)u (θ) ∈ l(θ) is a point in the boundary of ψ (see Figure 5.1). It is not difficult to prove (see [511]) that both p(θ) and f (θ) are periodic functions with period 2π, and that they are not only continuous, but also satisfy the Lipschitz condition. | p(θ1 ) − p(θ2 )| ≤ k, |θ1 − θ2 |

| f (θ1 ) − f (θ2 )| ≤ k, |θ1 − θ2 |

where k is a constant and θ1 , θ2 are any two reals. Thus the derivatives p  (θ) and f  (θ) exist for every θ except for a set with measure zero. From this point on, any claim we make about the derivative of a Lipschitz function will be for almost every θ, that is, for every θ except for a set of measure zero. Taking the derivative of x(θ) gives     x (θ) = p  (θ) − f (θ) u(θ) + p(θ) + f  (θ) u (θ),

5.1 The Pedal Function

97

Figure 5.1 Pedal function

but since x (θ) is perpendicular to u(θ), we have p  (θ) = f (θ). As θ moves forward, the support line of ψ parallel to u (θ) touches the boundary of ψ at the point x(θ), which implies that x (θ) is a vector parallel to u (θ) and in the same direction. That is: 

 p(θ) + p  (θ) ≥ 0,

  and hence p(θ) + p  (θ) turns out to be the radius of curvature of the boundary of ψ at point x(θ), as we will see later in Section 11.2.1. The function p(θ) is called the pedal function of the figure ψ, and the set of lines of the form l(θ) = {x ∈ E2 |x, u(θ) = p(θ)} is called the system of lines tangent to ψ. To summarize we have the following theorem: Theorem 5.1.1 Let ψ be a strictly convex figure. Then the pedal function p(θ) = max{x, u(θ)|x ∈ ψ} has the following properties: a) p(θ) is a periodic function with period 2π, b) p(θ) is a Lipschitz function like also its derivative p  (θ), c) the curve given by x(θ) = p(θ)u(θ) + p  (θ)u (θ) is a parametrization of the boundary of ψ, d) the derivative of x(θ) is given by the formula   x (θ) = p(θ) + p  (θ) u (θ),

 where

 p(θ) + p  (θ) ≥ 0,

e) the curvature of the boundary of ψ at the point x(θ) is p(θ) + p  (θ), f) the width function of ψ in direction u(θ) is given by w(θ) = p(θ) + p(θ + π).

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5 Systems of Lines in the Plane

And conversely we have: g) If p(θ) is a periodic Lipschitz function with period 2π and p(θ) + p  (θ) ≥ 0, then x(θ) = p(θ)u(θ) + p  (θ)u (θ) is a parametrization of a convex curve C. If in addition p(θ) + p(θ + π) = h, then C has constant width h.

5.1.2 Areas, Perimeters, and the Cauchy Formula We begin this section with the Cauchy formula for the perimeter. Theorem 5.1.2 Let ψ be a convex figure, p(θ) be its pedal function and w(θ) = p(θ) + p(θ + π) be its width function. Then the perimeter of ψ is 

π

P(ψ) =

w(θ)dθ.

0

Proof We begin by assuming that ψ is strictly convex. Let x(θ) be the parametrization of its boundary given in Theorem 5.1.1 c). Then 

P(ψ) =

   x (θ) dθ =

0



0





p(θ)dθ +

p  (θ)dθ

0 2π

p(θ)dθ + p  (2π) − p  (0)  π  π = ( p(θ) + p(θ + π))dθ = w(θ)dθ. =

0

0

0

The general result follows immediately from the fact that every convex figure can be adequately approximated by strictly convex figures.  The following corollary to Theorem 5.1.2 is known as the Theorem of Barbier [75]. Corollary 5.1.1 The perimeter of a figure of constant width h is hπ. Continuing the notation of the previous section, the diametral chord of ψ in direction u(θ) can be presented as the chord with endpoints x(θ) and x(θ + π). Making the appropriate substitutions we have that x(θ) − x(θ + π) = w(θ)u(θ) + w (θ)u (θ). It is easy to see from this formula that the diametral chord of ψ with respect to u(θ) is parallel to u(θ) for every θ if and only if ψ is of constant width, a result already obtained in Chapter 3. Let L θ be the directed line that contains the diametral chord of ψ in direction θ, that is, the chord that passes through x(θ) and x(θ + π). Observe that it divides the figure into two parts; denote the area of the region of ψ to the right of L θ going from x(θ) to x(θ + π) by A(θ), and the area of the region to the left we called by A(θ + π). Similarly, we call λ(θ) the length of the arc of the boundary of ψ to the right of L θ , and the length of the arc to the left λ(θ + π). Beretta and Maxia [99] proved the following result.

5.1 The Pedal Function

99

Figure 5.2

Theorem 5.1.3 Let ψ be a strictly convex figure. Then A(θ) and λ(θ) are Lipschitz functions. Moreover, 2 A (θ) − w(θ)λ (θ) = 0. Proof Let δ be a small positive real number. Consider the lines l(θ) and l(θ + π), the support lines of ψ perpendicular to u(θ), and the lines L θ and L θ+δ containing the diametral chords of ψ with respect to the directions u(θ) and u(θ + δ), respectively. We define {o} = L θ ∩ L θ+δ , { p} = L θ+δ ∩ l(θ), {q} = L θ+δ ∩ l(θ + π), {x} = {x(θ)} = L θ ∩ l(θ), {y} = {x(θ + π)} = L θ ∩ l(θ + π), s = x(θ + δ), t = x(θ + π + δ) and let α be the angle ∠ pxo as shown in Figure 5.2. To simplify the notation, we represent the length of an interval by the interval itself; for example, pq denotes both the interval with endpoints p and q and its length. We begin by observing that lim

δ→0

sin (π − α) px sin (∠x ps) = lim = = 1. δ→0 sin (∠xsp) sx sin (α)

(5.1)

qy = 1. ty

(5.2)

Similarly, lim

δ→0

By (5.1) and (5.2) we have that 1 area of psx px = lim ps sin(∠spx) = 0, δ→0 δ 2 δ→0 δ lim

because lim sin(∠spx) = sin (π − α), lim

δ→0

Analogously,

δ→0

  px sx = lim = x (θ) and lim ps = 0. δ→0 δ δ→0 δ

area of qt y = 0. δ→0 δ lim

(5.3)

(5.4)

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5 Systems of Lines in the Plane

Therefore, by (5.3) and (5.4), A(θ + δ) − A(θ) δ area of oyq area of ox p − lim = lim δ→0 δ→0 δ δ h 2 yq − h 1 x p = lim , δ→0 2δ

A (θ) = lim

δ→0

where h 1 is the distance between the point o and the line l(θ), and h 2 is the distance between the point o and line l(θ + π), such that h 1 + h 2 = w(θ). Moreover, by the similarity of the triangles pxo and qyo we obtain h 2 x p = h 1 yq, and hence yq − x p h 2 yq − h 1 x p (h 1 + h 2 ) = . δ δ This implies that

  2 A (θ) = |x (θ + π)| − |x (θ)| w(θ),

because by (5.1) and (5.2) |x (θ)| = lim

δ→0

xp xs yq yt = lim and |x (θ + π)| = lim = lim . δ→0 δ δ→0 δ δ→0 δ δ

Also, it is easy to observe that λ (θ) = lim

δ→0

arc (yt) − arc (xs) = |x (θ + π)| − |x (θ)|, δ

which implies that 2 A (θ) = λ (θ)w(θ).



Corollary 5.1.2 In a strictly convex figure ψ the following statements are equivalent: a) its diametral chords divide the area in half, b) its diametral chords divide the perimeter in half, c) ψ is centrally symmetric. Proof The diametral chords of the figure divide its area in half if and only if A (θ) = 0. By Theorem 5.1.3, A (θ) = 0 if and only if λ (θ) = 0; but this happens if and only if the diametral chords divide the perimeter in half. Clearly, the diametral chords of a centrally symmetric figure divide its area and its perimeter in half. Suppose now that in the convex figure ψ the diametral chords divide the perimeter in half, that is, suppose λ (θ) = 0. By Exercise 5.2, we know that λ (θ) = |x (θ + π)| − |x (θ)| = 0. Moreover, since x (θ + π) and x (θ) are parallel and opposite vectors, we have that x (θ + π) = −x (θ). Consider now m(θ) = (1/2)(x(θ + π) + x(θ)), the curve of the midpoints of the diametral chords. By the above m (θ) = 0, which implies that there exists a fixed point which is the midpoint of all the diametral chords of ψ. Then, by Exercise 5.3, it follows that ψ is a centrally symmetric convex figure.  Corollary 5.1.3 A binormal of a figure of constant width h divides the area in half if and only if it divides the perimeter in half.

5.1 The Pedal Function

101

Proof By Theorem 5.1.3 we have that 2 A (θ) = hλ (θ). Integrating from θ to θ + π we have that 2(A(θ + π) − A(θ)) = h(λ(θ + π) − λ(θ)). This implies that A(θ + π) = A(θ) if and only if λ(θ + p) = λ(θ).  The following corollary was proved by Hammer and Smith in [510], and it follows immediately from Corollary 5.1.2 and Theorem 3.5.1. Corollary 5.1.4 If all the binormals of a figure of constant width  divide its area, or its perimeter, in half then  is a disk. Corollary 5.1.5 A strictly convex figure ψ has constant width if and only if w(θ)λ(θ) − 2 A(θ) is a constant. Proof Let g(θ) = w(θ)λ(θ) − 2 A(θ). Then, by Theorem 5.1.3, g  (θ) = w  (θ)λ(θ), from which we can conclude (since λ(θ) is never zero) that w(θ)λ(θ) − 2 A(θ) is a constant if and only if ψ has constant width.  Corollary 5.1.6 Let  be a figure of constant width h and let θ0 be an angle. Then A() = 21 πh 2 − hλ(θ0 ) + 2 A(θ0 ). Proof Since the width w(θ) of  is the constant h, then hλ(θ) − 2 A(θ) is a constant K because, by Theorem 5.1.3, its derivative is zero. Our next purpose is to find the value of K . Suppose that hλ(θ)− 2 A(θ) = K = hλ(θ an angle θ0 and summing up, we have  +π) − 2 A(θ + π). Choosing  that h λ(θ0 ) + λ(θ0 + π) −2 A(θ0 + π) + A(θ0 ) = 2K , and therefore h(hπ) − 2 A() = 2K . It follows that hλ(θ0 ) − 2 A(θ0 ) = (1/2)πh 2 − A().



5.2 Systems of Lines and Their Envelopes 5.2.1 The Envelope Let U ⊂ E be an open set, and p : U → E be a continuous function. For every θ ∈ U consider the line L(θ) = {x|x, u(θ) = p(θ)}. (5.5) The collection of lines L = {L(θ)}θ∈U will be called a system of lines with pedal function p(θ). Define the envelope of L, E(L), as the set of points ρ for which there are sequences {θi } and {ϑi } (at least one of them nonconstant), both converging to θ, such that L(θi ) ∩ L(ϑi ) converges to ρ. Given θ, ϑ ∈ U , by Exercise 5.6, the coordinates (x, y) of the intersection point of the lines L(θ) and L(ϑ) are given by

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5 Systems of Lines in the Plane

p(θ) − p(ϑ) sin θ − sin ϑ + p(ϑ) θ−ϑ θ−ϑ x= , sin (θ − ϑ) θ−ϑ cos θ − cos ϑ p(θ) − p(ϑ) + p(ϑ) cos ϑ θ−ϑ θ−ϑ y= . sin (θ − ϑ) θ−ϑ − sin ϑ

For every θ ∈ U , let η(θ) = limh→0 (L(θ) ∩ L(θ + h)). Note that if p(θ) is a Lipschitz function, then the limit exists and is unique a.e. Indeed, calculating the coordinates of L(θ) ∩ L(θ + h), if p  (θ) exists, yields η(θ) = p(θ)u(θ) + p  (θ)u (θ). Consequently, for the case in which p(θ) is a Lipschitz function then the envelope E(L) is almost everywhere equal to the closed curve parametrized by η(θ) = p(θ)u(θ) + p  (θ)(θ)u (θ), to which almost every line in L is tangent, see Exercise 5.8. Clearly, this was the case in Section 5.1, when p(θ) is the pedal function of a convex figure and L is the system of its tangent lines.

5.2.2 Leaving on the Same Side Regions of the Same Area Let p(θ) be the pedal function associated to the system of lines L = {L(θ)}θ∈U . In this section we would like to characterize under what conditions a system of lines L, which intersect the interior of a convex figure φ, is leaving on the same side regions of the same area. To do so, we orient L(θ) in the direction of u (θ) and denote by A(θ) the area of the region of φ to the left of L(θ). For every θ consider the closed interval I (θ) = L(θ) ∩ φ and suppose {x(θ), y(θ)} = L(θ) ∩ bd φ in such a way that x(θ) − y(θ) has the same direction as u (θ). Let m(θ) be the midpoint of I (θ) (see Figure 5.3). If A(θ) is constant, it is not difficult to verify that the support lines of φ in x(θ) make angles with L(θ) which are uniformly bounded by 0 and π. This implies that there exists a constant k such that |x(θ + δ) − x(θ)| ≤ k |δ| for any real numbers θ and δ, such that θ, θ + δ ∈ U . Therefore x (θ) exists for almost every θ. Similarly, y(θ), m(θ) and the pedal function p(θ) are also Lipschitz functions (for further details see [511]). Let η(θ, δ) be the point on L(θ) where L(θ + δ) intersects L(θ) (see Figure 5.4).

Figure 5.3

5.2 Systems of Lines and Their Envelopes

103

Figure 5.4

By the previous section η(θ) = limδ→0 η(θ, δ) exists for those θ for which p  (θ) exists. In fact, η(θ) = p(θ)u(θ) + p  (θ)u (θ). Suppose now that we have L = {L(θ)}, a system of lines that intersect the interior of a convex figure φ, with the property that the pedal function p(θ) of L = {L(θ)} is a Lipschitz function, and let η(θ), m(θ), I (θ) and A(θ) be like in the paragraph above. Then the following lemma holds true. Lemma 5.2.1 The function A(θ) is a Lipschitz function. Moreover,   η(θ) − m(θ) = A (θ)/I (θ) u (θ). Proof Let δ be a small real number, θ ∈ U , and l(θ, δ) the directed line that passes through η(θ) parallel to L(θ + δ). Suppose that δ is so small that the line l(θ, δ) intersects the interior of . Let x(θ, δ) and y(θ, δ) be points in l(θ, δ) ∩ bd  with the property that x(θ, δ) − y(θ, δ) has the same direction as u (θ + δ). Let B(θ, δ) be the oriented area of  determined by the lines L(θ) and l(θ, δ). Let D(θ, δ) be the oriented area of  determined by the lines L(θ), L(θ + δ), and l(θ, δ), let E(θ, δ) be the oriented area of  determined by the lines −L(θ) and −L(θ + δ), and let F(θ, δ) be the oriented area of  determined by the lines −L(θ + δ), −L(θ), and −l(θ, δ), where −L denotes the directed line with the direction opposite to that of L, see Figure 5.4. Let us calculate the derivative of the function area A(θ): A(θ + δ) − A(θ) E(θ, δ) − (B(θ, δ) + D(θ, δ)) = lim δ→0 δ δ (F(θ, δ) + E(θ, δ)) − B(θ, δ) D(θ + δ) + F(θ, δ) − lim . δ→0 δ δ

A (θ) = limδ→0 = limδ→0

The proof consists of the following three steps: In the first step we shall prove that 2 limδ→0 = |y(θ) − η(θ)|2 − |η(θ) − x(θ)|2 . In the second step we shall prove that = 0. Finally, the theorem follows easily since

(F(θ,δ)+E(θ,δ))−B(θ,δ) δ limδ→0 D(θ,δ)+F(θ,δ) δ

|y(θ) − η(θ)|2 − |η(θ) − x(θ)|2 = 2I (θ)η(θ) − m(θ), u (θ), where I (θ) is the length of the chord L(θ) ∩ φ equal to x(θ) − y(θ) , see Exercise 5.4.

104

5 Systems of Lines in the Plane

For the first step let us fix θ, and for all τ ∈ [θ, θ + δ] ∪ [θ + π, θ + π + δ], let r (τ ) be the distance in the direction u(τ ) from η(θ) to the boundary of . Then    2[ F(θ, δ) + E(θ, δ) − B(θ, δ)] =

θ+δ θ

 r 2 (τ )dτ −

θ+π+δ θ+π

r 2 (τ )dτ ,

from which it follows that   F(θ, δ) + E(θ, δ) − B(θ, δ) = r 2 (θ) − r 2 (θ + π) 2 lim δ→0 δ = |y(θ) − η(θ)|2 − |η(θ) − x(θ)|2 . For the second step note that since  is bounded, there are constants K 1 and K 2 such that for a sufficiently small δ K 1 |η(θ) − η(θ, δ)| sin (δ) ≤ D(θ, δ) + F(θ, δ) ≤ K 2 |η(θ) − η(θ, δ)| sin (δ) holds, which implies that limδ→0

D(θ,δ)+F(θ,δ) δ

= 0.



From the above proof it follows that if η(θ) = m(θ), then lim

δ→0

1 D(θ, δ) + B(θ, δ) = I (θ)2 , δ 8

(5.6)

which will be used below. Theorem 5.2.1 A system of lines L = {L(θ)}, which intersect the interior of a convex figure φ, is leaving on the same side regions of the same area if and only if the curve m(θ) of midpoints of chords L(θ) ∩ φ is tangent to L(θ) at m(θ). That is, A(θ) is constant if and only if m (θ) is parallel to u (θ).  5.2.1, Proof The function A(θ) is constant if and only if A (θ) = 0.  By Lemma   A (θ) = 0 if and only     if m(θ) = η(θ). Furthermore, if this is so, m (θ) = η (θ) = p(θ) + p (θ) u (θ).

Theorem 5.2.2 Let L = {L(θ)}θ∈U be a system of lines that intersect the interior of convex figure φ. Suppose that for any open subset V ⊂ U the lines of {L(θ)}θ∈V are nonconcurrent. Then any two of the following properties of L imply the third property. i) All regions of φ on the same side of lines in L are equal in area. ii) All subarcs of the boundary of φ on the same side of the lines in L have the same perimeter. iii) All chords of φ determined by the lines in L have the same length. Proof Using the notation introduced previously and recalling that we identify an interval with its length for convenience, we have that x(θ) = m(θ) + (I (θ)/2)u (θ) and y(θ) = m(θ) − (I (θ)/2)u (θ). Suppose, moreover, that m (θ) = a(θ)u(θ) + b(θ)u (θ), such that     x (θ) = a(θ) − I (θ)/2 u(θ) + b(θ) + I  (θ)/2 u (θ) and

    y (θ) = a(θ) + I (θ)/2 u(θ) + b(θ) − I  (θ)/2 u (θ).

5.2 Systems of Lines and Their Envelopes

105

Consider these three conditions: 1) 2) 3)

a(θ)  b(θ)  0 and  = 0,   = x (θ) = y (θ), if and only if, a(θ)I (θ) = b(θ)I  (θ), I  (θ) = 0 and I (θ) = 0.

It is easy to verify that conditions 1), 2), and 3) are equivalent to conditions i), ii), and iii), respectively, and that any two of them imply the third condition.  Note that in the proofs of these theorems it is not necessary for the figure φ to be convex; it is sufficient that every chord I (θ), except its endpoints, is contained in the interior of φ and that the tangent rays of φ in x(θ) make angles with L(θ) uniformly bounded from 0 and π. This implies that there exists a constant K such that |x(θ + δ) − x(θ)| ≤ K |δ| for any real numbers θ and δ; therefore x(θ) and p(θ) are Lipschitz functions. For further details see [48].

5.3 Systems of Externally Simple Lines 5.3.1 Main Properties of Systems of Externally Simple Lines Following the ideas of Hammer and Sobczyk [511], we extract and examine in this section the main properties of some specific systems of lines in the plane that are widely used in convex geometry. Among others, these include systems of lines that contain diametral chords, median chords, and lines that divide the area or perimeter of a figure in half. We will use these properties to characterize central symmetry and, in particular, to prove that every figure of constant width of diameter h contains at least three different semicircles with diameter h. Let p(θ) be a Lipschitz function with the property that the equation p(θ + π) = − p(θ) is satisfied for every real number θ. Consider the line L(θ) = {x|x, u(θ) = p(θ)}.

(5.7)

It is easy to see that L(θ) = L(θ + π). The collection L of all lines of the form L(θ) will be called the system of externally simple lines with pedal function p(θ). As we shall see later, the name is due to the fact that one and only one line in L passes through each point in the plane outside a sufficiently large disk, including points at infinity. In fact, every point in the plane is covered by these lines. The following theorem characterizes systems of externally simple lines. Theorem 5.3.1 A collection L of lines in the plane is a system of externally simple lines if and only if L satisfies the following two properties: a) there is exactly one line in L in every direction and b) the set of all points where two different lines in L intersect is a bounded set. Proof Let L = {L(θ)} be a collection of lines with the property that there is one line L(θ) = L(θ + π) of L perpendicular to u(θ), and let p(θ) be its pedal function. Then the coordinates (x, y) of the intersection point of lines L(θ) and L(ϑ) are given by

106

5 Systems of Lines in the Plane

p(θ) − p(ϑ) sin θ − sin ϑ + p(ϑ) θ−ϑ θ−ϑ x= , sin (θ − ϑ) θ−ϑ cos θ − cos ϑ p(θ) − p(ϑ) + p(ϑ) cos ϑ θ−ϑ θ−ϑ y= . sin (θ − ϑ) θ−ϑ − sin ϑ

By analyzing these formulas carefully it is not difficult to see that p(θ) is a Lipschitz function if and only if the set of points (x, y), where two different lines in L intersect, is bounded, see Exercise 5.6. The following are some examples of systems of externally simple lines related to the chords of a convex figure ψ: lines that divide the area of ψ in half, lines that divide the perimeter of ψ in half, lines that contain diametral chords of ψ whenever ψ is strictly convex, the binormals of ψ where ψ is a figure of constant width, if p is a point of the boundary of ψ, the lines that contain the chords rq of ψ with the property that angle r pq is a right angle, and f) the median lines, i.e., those that pass exactly midway between two parallel support lines of ψ.

a) b) c) d) e)

With the exception of c) and d), it is quite simple (using Theorem 5.3.1) to prove that these systems of lines are externally simple. The proof of d) comes from Theorems 3.1.7 and 3.1.2. In all these cases the set of intersections is contained in the interior of ψ. For the proof that the diametral chords of a strictly convex figure form a system of externally simple lines, see Exercise 5.9. Theorem 5.3.2 Let L be a system of externally simple lines. Then the lines in L cover the plane. Indeed, one and only one line in L passes through each point of the plane outside a sufficiently large disk. Proof Let q be any point on the plane. We will prove that there is a line on L that passes through q. If q is not in L(θ) we may assume that, without loss of generality, q, u(θ) > p(θ), and therefore q, u(θ + π) < p(θ + π). It follows, by continuity, that there is a θ0 such that q, u(θ0 ) = p(θ0 ),  which implies that q is on L(θ0 ).

The Center of a System of Externally Simple Lines The set of points on the plane which are the intersections of two different lines of L is not necessarily a closed set. Let us denote by J (L) the set of points on the plane that are the intersections of two different lines of L together with its limit points. That is, the center of L, J (L), is the topological closure of the set of points that are the intersections of two different lines in L. Note that the set J (L) is closed and bounded, hence compact. Since three given lines are either concurrent or intersect in three noncollinear points, we have that J (L) is either a point, or it is not contained in a line. Furthermore, it follows from the definitions that the envelope E(L) ⊂ J (L). Let Ext(J (L)) be the set of extreme points of J (L), that is, those points q ∈ J (L) for which there is a support line L of J (L) through q such that L ∩ J (L) = {q}. Lemma 5.3.1 Let L be a system of externally simple lines.

5.3 Systems of Externally Simple Lines

107

a) If ρ ∈ Ext(J (L)) and there are two different lines such that ρ ∈ L(θ) ∩ L(ϑ), then there are θ1 < θ2 , such that ρ ∈ L(θ), if and only if θ ∈ [θ1 , θ2 ] ∪ [θ1 + π, θ2 + π]. b) Ext(J (L)) ⊂ E(L). Proof Suppose that ρ is an extreme point of J (L) and at least two different lines in L pass through ρ, say {ρ} = L(ϑ1 ) ∩ L(ϑ2 ). Let L be the support line of J (L) through ρ, such that L ∩ J (L) = {ρ}, and suppose u(ϑ0 ) is parallel to L. Without loss of generality we may assume that ϑ1 ≤ ϑ2 and ϑ0 is not between ϑ1 and ϑ2 . Then, since any other line in L always intersects the lines L(ϑ1 ) and L(ϑ2 ) on the same side of L, it is easy to see that ρ ∈ L(θ) for ϑ1 ≤ θ ≤ ϑ2 . This immediately implies a). Furthermore, if this is the case, by definition ρ ∈ E(L). Suppose now that two different lines in L never pass through ρ ∈ J (L). Then there is some sequence {ρi } that converges to ρ, and two sequences {θi } and {ϑi } such that L(θi ) ∩ L(ϑi ) = {ρi }. Without loss of generality we may assume that {θi } and {ϑi } converge to θ and ϑ, respectively, which implies that ρ is in L(θ) ∩ L(ϑ), therefore θ = ϑ. Hence, by definition, ρ ∈ E(L).  It is an interesting exercise to calculate the envelopes of some known systems of lines as in the following lemma. Lemma 5.3.2 Let ψ be a convex figure. Then a) the envelope of the system of lines L that divides the area of ψ in half is the set of midpoints of chords determined by the lines in L, b) the envelope of the system of median lines of ψ is the set of points of intersection of each median line with its corresponding diametral chords. Proof The proof of a) is basically contained in the proof of Lemma 5.2.1. Informally, if {θi } and {ϑi } both converge to θ, and L(θi ) ∩ L(ϑi ) = {ρ}i converges to ρ, then the areas of the sectors of ψ contained between the lines L(θi ) and L(ϑi ) that make an angle of |θi − ϑi | are the same. Since |θi − ϑi | approaches zero, the ratio in which ρi divides the chords of ψ determined by L(θi ) and L(ϑi ) approaches one, which implies that ρ is the midpoint of the chord of ψ determined by L(θ). Being similar, the proof of the converse of a) is left to the reader. Consider any two directions u(θ) and u(ϑ) and the parallelogram determined by the four support lines of ψ parallel to these directions. It is then easy to see that the diagonals of this parallelogram intersect precisely at the point where the median lines of ψ parallel to u(θ) and u(ϑ) intersect. If we now consider the system L of median lines of ψ and take {θi } and {ϑi }, both converging to θ, and L(θi ) ∩ L(ϑi ) = {ρi } converging to ρ, then ρi is on the diagonal of the parallelogram determined by the support lines of ψ parallel to u(θi ) and u(ϑi ). Letting i go to infinity we see that the point ρ is on the diametral chords of ψ corresponding to the direction u(θ). The converse of b) is easy to prove in a similar way.  V. Klee conjectured that every figure of constant width h contains a semicircle of diameter h. It was not much later that Besicovitch [106] proved that a certain class of figures of constant width does not contain merely one semicircle, but in fact three distinct ones. Falconer [330] proved the general case with the elegant proof presented below. Theorem 5.3.3 Let  be a figure of constant width h. Then there are three different semicircles of diameter h contained in . Proof A more general result will be proved. Remember that for a convex figure  we define (), the thickness of , as the minimum of the widths of  over all directions. We will prove that there are three different semicircles of diameter () contained in any strictly convex figure .

108

5 Systems of Lines in the Plane

Let L be the system of median lines of  and let ρ be an extreme point of J (L). By Lemmas 5.3.1 b) and 5.3.2 b) there are support lines of , L 1 , and L 2 parallel to u(θ), and points a ∈ L 1 ∩ , b ∈ L 2 ∩ , such that ρ is located at the intersection of the median line L(θ) with the diametral chord ab. As ρ is an extreme point of J (L), every median line intersects L(θ) on the same side of ρ, that is, there exists a ray R of L(θ) that starts at ρ and contains L(θ) ∩ J (L). Consider now the semicircle S with center ρ and diameter () bounded by the chord ab and on the same side of ab as R. Note that the diameter of S is contained in ab. Moreover, since every median line intersects L(θ) at R, the distance between ρ and the support lines of  parallel to some tangent of S is greater than ()/2, which implies that S is contained in . To complete the proof it is sufficient to note that either J (L) is a point, in which case there exists a circle of diameter () contained in , or J (L) is not contained in a line, in which case it has at least three different extreme points, and , therefore, contains three distinct semicircles of diameter ().  Many characterizations of the ball among constant width bodies are established by first proving that a convex body with a given property must be centrally symmetric. Following this spirit we will now make use of the developed concepts to give a characterization of central symmetry. Theorem 5.3.4 Consider the following three systems of externally simple lines associated with a strictly convex figure ψ: i) the system of median lines of ψ, ii) the system of lines that contain diametral chords of ψ, iii) the system of lines that divide the area of ψ in half. If any two of these systems coincide, then ψ is centrally symmetric. Proof First of all, let us think of a circle  as the closed interval [0, 2π] where we identify the angle θ with the angle θ + π. In other words, the circle  is the set of directions of the projective plane. Let f :→ be the following continuous function: given θ ∈ , let L 1 and L 2 be the support lines of ψ orthogonal to u(θ), and let xi ∈ L i , i = 1, 2. Define f (θ) as the unique direction in  such that the diametral chord x1 x2 is orthogonal to u( f (θ)). It is obvious that f (θ) is a homeomorphism without fixed points, see Exercise 5.10. Let L = {L(θ)} be the system of median lines of ψ, and assume that every diametral chord of ψ is contained in a line in L. Let ρ be an extreme point of J (L), and let L(θ) be the median line that passes through ρ. By Lemmas 5.3.1 b) and 5.3.2 b) there exists a diametral chord that intersects L(θ) precisely at ρ. As a consequence, at least two different lines in L pass through ρ. By Lemma 5.3.1 a) there are θ1 and θ2 such that L(θ) passes through ρ if and only if θ ∈ [θ1 , θ2 ] ∪ [θ1 + π, θ2 + π]. If L(θ) passes through ρ for every θ ∈ [0, 2π], then, by Theorem 2.12.7 the figure ψ is centrally symmetric. If not, then, we may assume that ρ is the origin and let p(θ) be the pedal function of ψ. The fact that every line L(θ) through the origin is a meridian line for θ ∈ [θ1 , θ2 ] ∪ [θ1 + π, θ2 + π] implies that p(θ) = p(θ + π) for θ ∈ [θ1 , θ2 ], and also that ρ ∈ L( f (θ)). This means that f [θ1 , θ2 ] ⊂ [θ1 , θ2 ]. Hence there is θ0 such that f (θ0 ) = θ0 , contradicting the fact that f (θ) is a homeomorphism without fixed points. Let (θ) be the system of lines that contain a diametral chord in such a way that for every θ ∈ , (θ) is orthogonal to u(θ). Assume that every median line L(θ) divides the area of ψ in half. By Lemmas 5.3.2 a) and b), if {x} = L(θ) ∩ ( f (θ)) then x is the midpoint of the chord L(θ) ∩ ψ. Let ρ be an extreme point of J (L) and let L(θ) be the median line passing through ρ. Then ρ is the midpoint of L(θ). Moreover, since ρ is an extreme point of J (L), every median line of ψ intersects L(θ) in the interval aρ, where a and b are the endpoints of the median chord L(θ) ∩ ψ.

5.3 Systems of Externally Simple Lines

109

Our next purpose is to prove that the chord ab = L(θ) ∩ ψ is the diametral chord (θ) and the corresponding meridian line L( f −1 (θ)) also passes through ρ. To do so, let L 1 be the support line of ψ at a, L 2 the support line of ψ parallel to L 1 , and assume that b is not in L 2 . The median line L( f −1 (θ)), parallel to L 1 and L 2 , would then intersect L(θ) in the half-closed interval ρb, which is a contradiction unless b ∈ L 2 and the median line L( f −1 (θ)) passes through ρ. As a consequence, at least two different meridian lines pass through ρ. Then, by Lemma 5.3.1 a), there exist θ1 and θ2 such that L(θ) passes through ρ if and only if θ is in the closed interval [θ1 , θ2 ] ∪ [θ1 + π, θ2 + π]. Summarizing, for every θ ∈ [θ1 , θ2 ] ⊂ , L(θ) = (θ) and ρ ∈ L( f −1 (θ)). This implies that f −1 [θ1 , θ2 ] ⊂ [θ1 , θ2 ]. Hence there is some θ0 such that f −1 (θ0 ) = θ0 , contradicting the fact that f (θ) is a homeomorphism without fixed points. Finally, if every diametral chord divides the area of ψ in half, then the theorem follows from Corollary 5.1.2.  As a corollary we obtain the following characterization of the circular disk due to Khassa [621]. Corollary 5.3.1 If every median line of a figure of constant width  is binormal, or every diametral (or median) line divides the area of  in half, then  is a circular disk.

5.3.2 A Parametrization of Figures of Constant Width The following theorem, whose proof is left to the reader, will be very useful. Theorem 5.3.5 Let L be a externally simple system of lines with pedal function p(θ). Suppose that all lines of L intersect the interior of the convex figure φ. Then there exists a Lipschitz function f (θ) of period 2π such that x(θ) = p(θ)u(θ) + f (θ)u (θ) parametrizes the boundary of φ (see Figure 5.5). Let  be a figure of constant width, L the family of all lines that contain a binormal of , and p(θ) its pedal function. By Theorem 5.3.5, there exists a Lipschitz function f (θ) such that x(θ) = p(θ)u(θ) + f (θ)u (θ) is a parametrization of the boundary of . The fact that  is figure of constant width and L is the family of all lines that contain a binormal of  implies that x (θ) is parallel to u(θ), but in opposite direction. Finding the derivative, we have     x (θ) = p  (θ) − f (θ) u(θ) + p(θ) + f  (θ) u (θ) . Consequently, − p(θ) = f  (θ). It follows that

and

x(θ) = − f  (θ)u(θ) + f (θ)u (θ)   x (θ) = − f (θ) + f  (θ) u(θ) ,

110

5 Systems of Lines in the Plane

Figure 5.5

where

f (θ) + f  (θ) ≥ 0.

Note that f (θ) + f  (θ) is the radius of curvature of the boundary of  at the point x(θ) because the tangent vector at x(θ) always has direction −u(θ), see Section 11.2.1. Note also that, since p(θ + π) = − p(θ), we have f  (θ + π) + f  (θ) = 0 , which implies that f (θ + π) + f (θ), the width of , is constant. Summarizing, we formulate Theorem 5.3.6 Let f (θ) be a Lipschitz function with period 2π such that its first derivative f  (θ) is also a Lipschitz function and satisfies the equations f  (θ) + f  (θ + π) = 0 and f (θ) + f  (θ) ≥ 0. Then the curve x(θ) = − f  (θ)u(θ) + f (θ)u (θ) is a parametrization of the boundary of a figure of constant width h = f (θ) + f (θ + π). The reverse is also true; every figure of constant width can be parametrized as above. Proof Since x (θ) = −( f (θ) + f  (θ))u(θ), the tangent vector at x(θ) always has direction −u(θ). Furthermore, since f (θ) + f  (θ) ≥ 0, the curve x(θ) is a parametrization of a convex figure . The support lines of φ at the points x(θ) and x(θ + π) are parallel to u(θ). The chord of  with endpoints x(θ) and x(θ + π) is perpendicular to these support lines, since x(θ) − x(θ + π) = ( f (θ) + f (θ + π))u (θ). As a consequence, the distance between two parallel support lines is constant since f  (θ) + f  (θ + π) = 0.  As we will see later, this theorem can be used to construct figures of constant width whose boundary is a smooth curve, or does not contain any arc of a circle. First, we would like to prove our first theorem on the curvature for figures of constant width. The corresponding theorems on curvature for higher dimensional bodies of constant width will be treated in Section 11.2.

5.3 Systems of Externally Simple Lines

111

Theorem 5.3.7 If  is a twice continuously differentiable figure of constant width h, the sum of the radii of curvature of the boundary of  at the endpoints of a binormal of  is h. Proof By Theorem 5.3.6, let x(θ) = − f  (θ)u(θ) + f (θ)u (θ), a parametrization of , where f  (θ) + f  (θ + π) = 0 and the radius of curvature ρ(θ) at the boundary of  in x(θ) is f (θ) + f  (θ). It follows that ρ(θ) + ρ(θ + π) = f (θ) + f (θ + π) = h. 

5.3.3 The Analytic Curve of Constant Width due to Rabinowitz Observe that by applying a rotation of π/2 degrees to u (θ) one obtains u(θ), and −u(θ) = u (θ) becomes u (θ). Then it is easy to conclude from Theorem 5.3.6 that if p(θ) is a periodic Lipschitz function with period 2π, where p(θ) + p(θ + π) is a constant h and p(θ) + p  (θ) ≥ 0, then x(θ) = p(θ)u(θ) + p  (θ)u (θ) is a parametrization of a curve of constant width h. Let us consider the function p(θ) = a cos2 (kθ/2) + b, where k is any odd positive integer and a, b ≥ 0. Clearly, p(θ) + p(θ + π) = 1. If k = 1, then x(θ) = p(θ)u(θ) + p  (θ)u (θ) is a parametrization of a circle. In general, the curve parametrized by x(θ) = p(θ)u(θ) + p  (θ)u (θ), where p(θ) = a cos2 (3θ/2) + b, is not always convex because p(θ) + p  (θ) is not always greater than or equal to zero. Choosing a = 2 and b = 8 gives the nice convex curve shown in Figure 5.6. We will show next that this is a polynomial curve of constant width. Following Rabinowitz, we find that in this case the parametric equations are equivalent to x = 9 cos(θ) + 2 cos(2θ) − cos(4θ), y = 9 sin(θ) − 2 sin(2θ) − sin(4θ), and the curve is traced as θ varies from 0 to 2π. To get an equation between x and y, we can proceed as follows. Expanding everything in terms of c = cos(θ) and s = sin(θ), we get x = − 3 + 9c + 12c2 − 8c4 , y = s(9 − 4c − 4c3 + 4cs 2 ).

Figure 5.6 Rabinowitz’s analytic curve of constant width

112

5 Systems of Lines in the Plane

Squaring the second equation and substituting s 2 = 1 − c2 give us two equations in the three unknowns x, y, and c. We can therefore eliminate c. Using Mathematica, we can find the following polynomial equation whose graph is our analytic curve of constant width: (x 2 + y 2 )4 − 45(x 2 + y 2 )3 − 41283(x 2 + y 2 )2 + 7950960(x 2 + y 2 ) + 16(x 2 − 3y 2 )3 − 48(x 2 + y 2 )(x 2 − 3y 2 )2   + (x 2 − 3y 2 )x 16(x 2 + y 2 )2 − 5544(x 2 + y 2 ) + 266382 = 7203 . The algebraic curve of Rabinowitz has also isolated points separate from the original curve. The authors of [914] present a modification such that a curve without isolated points can be created. In [76] a related question of Rabinowitz [956] on the lowest degree polynomial whose graph is a noncircular curve of constant width is solved. Computations of partial degrees of the defining polynomial of algebraic surfaces of constant width are also discussed there. Wegner [1171] proved that any curve of constant width can be approximated arbitrarily closely by analytic curves of constant width. A weaker result, where the analyticity is replaced by infinite differentiability, was previously treated by Tanno [1118] who showed in addition that if a convex curve of constant width has an axis of symmetry, then it can be approximated arbitrarily closely by an infinitely differentiable curve of constant width with an axis of symmetry. Moreover, there exists a curve of constant width whose set of singular points is everywhere dense. In higher dimensions not only this is also true. In addition, the family of n-dimensional bodies of constant width whose set of singular points is everywhere dense in the boundary arbitrarily closely approximates any n-dimensional body of constant width, see Falconer [331]. In the article [907] constant width curves generated in the sense of Rabinowitz [956] are used for distinguishing different curves of the same constant width by suitably defined shadow functions.

5.3.4 Evolutes and Euler’s Constant Width Curve Let E be a curve. Imagine wrapping a string with a pencil tied to one endpoint p around E. As we unwrap the string, maintaining it taut, the pencil traces the curve γ as in Figure 5.7. The point p in γ is the point corresponding to q in E that was drawn just when the string was tangent to E at q. If we trace a circular arc C with center in q and radius the length of the segment pq, then it can be observed–and of course also proven, see [147]–that the line which passes through the segment pq is normal to γ, and not only that the curves C and γ are tangents but also that C is the circular arc that best approximates γ at the point p. This implies that the curve E is precisely the curve formed by taking all the centers of the circles of curvature of γ. Note also that the tangent lines of E are precisely the normal lines of γ, and therefore that E is the envelope of the system of normal lines to γ. The curve E is called the evolute of the curve γ. By changing the length of the string, for every curve E an infinite family of parallel curves that have E as evolute can be traced. They all have the same system of normal lines, the envelope of which is the curve E. Consider again the system of lines L that pass through a binormal of the body of constant width , and let x(θ) = − f  (θ)u(θ) + f (θ)u (θ) be a parametrization of the boundary of , where f (θ) and f  (θ) are Lipschitz functions with period 2π and k(θ) = f (θ) + f  (θ) is the radius of curvature of  at the point x(θ). Recall that − f  (θ) is the pedal function of the system L of lines normal to  and that E(θ) = − f  (θ)u(θ) − f  (θ)u (θ) is the envelope of the system of lines L. Since E(θ) + k(θ)u (θ) = x(θ), we have that E(θ) is the curve of centers of curvature of x(θ) and is therefore the evolute of the boundary of . The system of curves parallel to the boundary of  that has E(θ) as evolute describes a family

5.3 Systems of Externally Simple Lines

113

Figure 5.7 The evolute

of figures of constant width which are identical to those obtained as Minkowski sum of  with circles of radius h, see Figure 5.8. This suggests another way to construct constant width curves. The evolutes of hypocycloids with an odd number of points studied by Euler [323] constitute an interesting example. One particularly noteworthy case is the curve with three sharp points constructed using the Steiner hypocycloid. The Steiner hypocycloid is obtained from the path of a point on a circle of radius h that rolls around the inside of a circle of radius 3h. The smallest member of the family of constant width curves that has this hypocycloid as evolute has width 16h/3 and touches the three tips of the hypocycloid. It is called the Euler curve of constant width, see Figure 1.12. For an analytical description of the curve, the pedal function p(θ) of the system of support lines of the Euler curve of width h is given by p(θ) =

 1 h 1 + cos 3θ , 2 8

Figure 5.8 A constant width curve with the hypocycloid as evolute

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from which it follows that a parametrization of the Euler curve is x(θ) = p(θ)u(θ) + p  (θ)u(θ). Therefore its radius of curvature is ρ = (1 − cos 3θ)/2. The Euler curve has six extremal points of curvature, located at θ = sπ/3, (s = 0, 1, . . . , 5), and no corner points. At those points where the curvature ρ is zero (s = 0, 2, 4) the curve resembles x = t 4 , y = t 3 , and t = 0. Its evolute E(θ), the Steiner hypocycloid, has the parametrization E(θ) = x(θ) − ρu(θ). Since the Euler curve is symmetric with respect to the line θ = 0, the surface obtained by rotating it around this line is a constant width body. Using polar coordinates for the direction u = (sin θ, sin ϑ cos θ, cos ϑ cos θ) gives the expression  1 h 1 + cos 3θ , p(ϑ, θ) = 2 8

0 ≤ ϑ < 2π,

0 ≤ θ < π,

for the pedal function of its support planes, from which the following parametrization in Cartesian coordinates is obtained, see Section 11.1: ∂ p(ϑ, θ) cos θ, ∂θ ∂ p(ϑ, θ) sin ϑ sin θ, y = p(ϑ, θ) sin ϑ cos θ − ∂θ ∂ p(ϑ, θ) cos ϑ sin θ. z = p(ϑ, θ) cos ϑ cos θ − ∂θ

x = p(ϑ, θ) sin θ +

This surface has no corner points, in fact, it may be observed, by substituting s = tan θ/2 and t = tan θ/2, that it is an algebraic surface. For θ = 0 and θ = 2π/3 one of the main curvatures (and therefore, the Gaussian curvature) approaches infinity. For more about analytic bodies of constant width see Section 8.5.

5.4 Figures Which Float in Equilibrium 5.4.1 Introduction Problem 19 of the Scottish Book [807] states: If a solid of uniform density ρ has the property of floating in equilibrium–without turning over or moving–in whatever position it is placed, must it be a sphere?

In this section, we will examine the 2-dimensional version of the problem, dealing with figures rather than solid objects. For a 2-dimensional interpretation of the problem, consider a right cylinder of uniform density (made out of the same material) which has—when its long axis is parallel to the ground—the property of floating in equilibrium without turning over in whatever position it is placed. Must the cylinder be a circular cylinder? This problem is related to several dynamical systems, for example, the tire track problem [1114], the problem of the existence of closed carrousels [173] and the problem of determining the trajectory of a charge moving in a perpendicular parabolic magnetic field [1182].

5.4 Figures Which Float in Equilibrium

115

Figure 5.9 Figures that float in equilibrium in every position for density 1/2.

In 1938, Auerbach [48] studied the case when the density is one half. In his work, he realized that when the cylinder has a radially symmetric cross section, it must be circular. However, he proved that, surprisingly, in general the cylinder is not necessarily circular. Two possible solutions are shown in Figure 5.9. Both figures admit a rigid line segment that rotates inside the figure and, in every position, divides its area and perimeter in half. In 1921 Zindler [1224] found figures ψ called Zindler curves with the curious property that all chords that divide the area of ψ in half are also of the same length, and also divide the perimeter in half. Strubecker (see [1102] and [1103]) showed that rotating a double normal of a curve of constant width about its midpoint by 90◦ , one gets the main chords of a corresponding closed Zindler curve. Many more basic properties of planar Zindler curves were studied in [407], [769], and [1201], spatial analogues with different applications were investigated in [1170], [554], [555], [953], and again [1201]. These special Zindler curves are analogously related to constant width curves or constructed from them, see also Chapters 11 and 16. More recent contributions, related to the floating body problem were published in [173] and [174] and are also discussed here in this book. For extensions of Zindler curves to Minkowski planes (see Chapter 10) we refer to [798] and [799]. In this section, we will prove that these Zindler figures are exactly the ones, that float in equilibrium in any position when their density is one half. We will also prove that figures of constant width are in a certain sense duals of Zindler curves, since, as we shall see, one can be obtained from the other by means of the following simple geometric procedure: Consider a figure of constant width, say . We know that there exists a binormal of  in every direction and that all these binormals have exactly the same length, say, one. We construct a figure, not necessarily convex, as follows: For every binormal I of , trace a segment I  of length one perpendicular to I such that the midpoints of I and I  coincide (see Figure 5.10). As the binormal I turns, taking on every direction, its endpoints move along the boundary of  while, at the same time, the endpoints of the segment I  trace the new Zindler curve ξ. Note that this process may be inverted in the following way: we start with the Zindler curve ξ, and for every chord I  that divides the area of ξ in half trace a segment I perpendicular to I  of the same length and such that the midpoints of I and I  coincide. As I  moves along the curve ξ, the segment I describes a figure of constant width  whose binormals are the chords I . Next, we follow the ideas of Auerbach from [48] to prove that every curve of constant width is “dual”, in the sense described above, to Zindler curves. Furthermore, we prove that Zindler curves give rise precisely to the figures that float in equilibrium in every position when the density is one half.

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Figure 5.10 A Zindler curve ξ obtained from the Reuleaux triangle

5.4.2 A Figure Floating Let F be a plane figure with area A, center of mass g and uniform density ρ, 0 < ρ < 1. Assume this figure is floating in the water at some position. Let L be the water line and suppose that the part of F which is under water has area ρA. Let us take any direction u(θ). Let L(θ) be the oriented line in direction u (θ) which has the property that the area of F to the left of L(θ) is precisely ρA. The region of F that is under water in direction u(θ), that is, the portion of F located to the left of L(θ) will be denoted by S(θ). Assume that F and ρ have been chosen in such a way that S(θ) consists of a single piece, that is, the directed chord I (θ) = L(θ) ∩ F is contained, except for its endpoints, in the interior of F. Denote the center of mass of S(θ) by g(θ) (Figure 5.11). The curve g(θ) of the centers of mass of the regions S(θ) will be called the curve of the centers of mass of F. We say that F is in equilibrium in the direction u(θ) following Archimedes’ Law, if the line that joins the center of mass g of F with the center of mass g(θ) of the underwater portion S(θ) of F is perpendicular to the waterline L(θ). That is, if g − g(θ) is a vector in the same direction as u(θ). In Figure 5.12, F is not in equilibrium.

Figure 5.11

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117

Figure 5.12

Finally, we say that F floats in equilibrium in any position if for every θ the figure F floats in equilibrium in direction u(θ).

The Curve of Centers of Mass We now give the main theorem on the curve of centers of mass. Theorem 5.4.1 The curve g(θ) of the centers of mass of a figure F with uniform density is differentiable. Moreover,  I (θ)3  g (θ) = − u (θ). 12ρA That is, the tangent to g(θ) at θ is parallel to the waterline L(θ), and its magnitude depends only on the length of I (θ). Before giving the proof of Theorem 5.4.1, we summarize what we need to know on centers of mass. Two Observations on Centers of Mass 1) Suppose that F is a figure made up of two parts, F1 and F2 , whose interiors do not intersect. Call the centers of mass of F, F1 , and F2 , respectively, g, g1 , and g2 . Then the point g is located on the line segment g1 g2 and, moreover, g1 g Area of F2 = . gg2 Area of F1 2) Let p, q, and r be the vertices of a triangle. Then the center of mass of the triangle pqr is the barycenter of the triangle and has coordinates ( p + q + r )/3. If the triangle pqr is isosceles, that is, if pq = pr , then the center of mass g of the triangle is located on pl, the height of the triangle; moreover, pg/ pl = 2/3 (Fig. 5.13). Proof of Theorem 5.4.1. Recall that by I (θ) we denote not only the flotation chord L(θ) ∩ F, but also its length. Let δ be a positive small number. Consider I (θ) and I (θ + δ) and let R(θ + δ) and T (θ + δ) be the regions between L(θ) and L(θ + δ) indicated in Figure 5.14 with centers of mass r (θ + δ) and t (θ + δ), respectively. Let q be the center of mass of S(θ) ∪ S(θ + δ).

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Figure 5.13

Clearly, the point q is located at the intersection of the segments g(θ + δ)r (θ + δ) and g(θ)t (θ + δ), since S(θ) ∪ S(θ + δ) = R(θ + δ) ∪ S(θ + δ) = S(θ) ∪ T (θ + δ). Moreover, since the figure F has uniform density, then the area of S(θ) is equal to the area of S(θ + δ) which is equal to ρA, and then the area of R(θ + δ) is equal to the area of T (θ + δ). Thus |g(θ + δ) − q| |g(θ) − q| = |q − r (θ + δ)| |q − t (θ + δ)| =

|g(θ + δ) − g(θ)| area of T (θ + δ) = . |t (θ + δ) − r (θ + δ)| ρA

(5.8)

It follows that the segment g(θ + δ)g(θ) is parallel to the segment r (θ + δ)t (θ + δ). As δ approaches zero, the segment r (θ + δ)t (θ + δ) comes closer and closer to being parallel to I (θ), and therefore the secant g(θ + δ)g(θ) also comes closer and closer to being parallel to I (θ); that is, g  (θ) is parallel to I (θ). To state it more formally, g(θ + δ) − g(θ) . g  (θ) = lim δ→0 δ In addition, by (5.8) this yields g  (θ) =



1 ρA

  area of T (θ + δ) lim r (θ + δ) − t (θ + δ) . δ→0 δ→0 δ lim

Figure 5.14

5.4 Figures Which Float in Equilibrium

119

Then by (Theorem 5.4.1) it can be proved that I (θ)2 area of T (θ + δ) = . δ→0 δ 8 lim

Also, from the formula of the center of mass of an isosceles triangle and Exercise 5.16 it can be seen that   2 lim r (θ + δ) − t (θ + δ) = − I (θ)u (θ). δ→0 3 Then it follows that g  (θ) = −



I (θ)3 u (θ). 12ρA



As a corollary of Theorem 5.4.1 we have Corollary 5.4.1 Let F be a figure not necessarily convex and suppose that the flotation chords L(θ) ∩ F are connected, where L(θ) is the waterline. Then F has at least two equilibrium positions. Proof Let θ1 and θ2 be the two directions for which the length of the segment gg(θ) takes its maximum and minimum values. It is then easy to see that g  (θi ) is perpendicular to the segment gg(θi ), which implies that I (θi ) is also perpendicular to gg(θi ) and, therefore, that F is in equilibrium in direction  θi , i = 1, 2. Theorem 5.4.2 Let F be a figure not necessarily convex, g its center of mass, and suppose that the flotation chords I (θ) = L(θ) ∩ F are connected, where L(θ) is the waterline. Then the following three statements are equivalent: 1) The figure F floats in equilibrium in any position. 2) The curve of centers of mass g(θ) of F is a circle with center in g. 3) All the flotation chords {I (θ)} have the same length. Proof Let us first prove that the figure F floats in equilibrium in any position if and only if the curve of centers of mass g(θ) of F is a circle with center in g. Let r (θ) = |g − g(θ)| and let f (θ) = r (θ)2 = g(θ) − g, g(θ) − g. Then f  (θ) = 2g(θ) − g, g  (θ). If the curve of centers of mass is a circle with center in g, then f  (θ) = 0, and hence g − g(θ) is perpendicular to g  (θ) which is, by Theorem 5.4.1, parallel to the waterline L(θ). Consequently, the figure F floats in equilibrium in any position. Conversely, if the figure F floats in equilibrium in any position, then g − g(θ) is perpendicular to L(θ), which is parallel to g  (θ). Therefore f  (θ) = 0, and hence r (θ) is constant; but if this is so, g(θ) is a circle with center in g. We shall now prove that the curve of centers of mass g(θ) of F is a circle if and only if all the flotation chords {I (θ)} have the same length. For that purpose note that the radius of curvature of g(θ) I (θ)3 , see Section 11.2.1. Consequently, the curve of centers of mass g(θ) is a circle if and only at θ is 12ρA if its radius of curvature is constant, if and only if all the flotation chords {I (θ)} have the same length.  Corollary 5.4.2 Let F be a figure that floats in equilibrium in any position, and let x(θ) and y(θ) be the endpoints of I (θ). Then |x (θ)| = |y (θ)|.

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Proof Let m(θ) be the midpoint of the flotation chord I (θ). By Theorem 5.2.1, m  (θ) = a(θ)u (θ). On the other hand, by Theorem 5.4.2 3), x(θ) = m(θ) + ku (θ) and y(θ) = m(θ) − ku (θ), where the  length of all flotation chords are 2k. Consequently |x (θ)|2 = a(θ)2 + k 2 = |y (θ)|2 . Remember that Zindler curves are those curves with the property that the chords that divide their area in half are of the same length and also divide the perimeter in half. The following theorem shows that Zindler curves are precisely the boundaries of the figures that float in equilibrium in any position with density one half. Theorem 5.4.3 The figure F floats in equilibrium in any position with density one half if and only if the boundary of F is a Zindler curve. Proof If F floats in equilibrium in any position with density one half, then the chords that divide the area of F in half are the floating chords, and by Theorem 5.4.2 3) all of them have the same length. Furthermore, by Corollary 5.4.2 a) the flotation chords L(θ) of F intersect the boundary of F in such a way that they leave on the same side subarcs of the same length. Conversely, if the boundary of F is a Zindler curve and F is floating with density one half, then by Theorem 5.4.2 3) the figure F floats in equilibrium. 

5.4.3 Floating with Density One Half and Constant Width It is time to prove that the figures that float in equilibrium in any position with density one half are the “duals” of constant width figures. Consider a figure of constant width . We know that there exists a binormal of  in every direction and that all these binormals have exactly the same length, say, one. Recall the following construction: For every binormal I of , trace a segment I  of length one perpendicular to I such that the midpoints of I and I  coincide (see Figure 5.10). As the binormal I turns, taking on every direction, its endpoints move along the boundary of  while, at the same time, the endpoints of the segment I  trace a new curve ξ. Our first goal is to prove that ξ is a simple closed curve and the relative interiors of all chords I  are in the open region bounded by ξ. Indeed, we can see by inspection on Figure 5.10 that exactly this happens if  is the Reuleaux triangle. To achieve our purpose, let L = {L(θ)} be the system of externally simple lines that contain a binormal I (θ) of , and let L = {L  (θ)} be the system of externally simple lines that contain the segment I  (θ + π/2), where I  (θ + π/2) has length one, and is perpendicular to the binormal I (θ) and the midpoints of I (θ) and I  (θ + π/2) coincide. Note first that the envelope E(L ) is the set of midpoints of all binormals I (θ) of . Assume, without loss of generality, for the rest of the proof that  is not a Reuleaux triangle. First we prove that the diameter of E(L ) is smaller than 1/2. This is so because it is not difficult to see, using elementary geometry, that if we take a convex quadrilateral with diameter h whose diagonals have length h, then the distance between the midpoints of these diagonals is less than h/2 and equals h/2 only in the degenerate case where the two diagonals are two sides of a equilateral triangle. But this cannot happen because, by Corollary 4.1.1,  should be a Reuleaux triangle contradicting our assumption. On the other hand, by Lemma 5.3.1 we have Ext(J (L )) ⊂ E(L ), and therefore, by the Krein–Milman Theorem 2.9.4, the diameter of J (L ) is smaller than 1/2. Now, if the curve ξ intersects itself or touches the relative interior of any of the intervals I  (θ), this is because there is some point of ξ that belongs to J (L ). In other words, there is θ0 such that the endpoints of the chord I  (θ0 ) are a and b, and furthermore a ∈ J (L ). This is a contradiction because belongs to the envelope of E(L ) which is contained in J (L ). Therefore the diameter the midpoint a+b 2  of J (L ) is 1/2.

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121

Finally, let us prove that ξ is a Zindler curve or, equivalently, that the region bounded by ξ floats in equilibrium in every position with density one half. To do this, remember that  is a figure of constant width. By Theorem 5.3.6, let x(θ) = − f  (θ)u(θ) + f (θ)u (θ) be a parametrization of the boundary of , where f (θ) and f  (θ) are Lipschitz functions of period 2π that satisfy the equations f  (θ) + f  (θ + π) = 0 and f (θ) + f  (θ) ≥ 0, where 1 = f (θ) + f (θ + π) is the width of . Note that m(θ) = − f  (θ)u(θ) + f˜(θ)u (θ), where f˜(θ) = ( f (θ) − f (θ + π))/2, the midpoint of the diametral chord in direction u(θ), is such that its derivative m (θ) = −( f  (θ) + f˜(θ))u(θ) is perpendicular to u (θ). As a consequence, the curve ξ is parametrized by ξ(θ + π/2) = f˜(θ)u(θ + π/2) +



1 f (θ) − u (θ + π/2), 2 

which implies that the chord I  (θ + π/2) with endpoints ξ(θ + π/2) and ξ(θ − π/2) has constant length 1, and its midpoint, η(θ + π/2), is precisely m(θ), because ξ(θ + π/2) − ξ(θ − π/2) = −u(θ + π/2). Furthermore, its derivative η  (θ + π/2) is parallel to u (θ + π/2). By Theorem 5.2.1, the chords I  (θ) of ξ are leaving on the same side regions of the same area. Furthermore, since all flotation chords I  (θ) have length one, by Theorem 5.4.2 c) the region bounded by ξ floats in equilibrium in every position. Conversely, if we take the flotation chord I  of a figure F that floats in equilibrium in any position with density one half and draw a segment I perpendicular to I  of the same length, such that the midpoints of I and I  coincide, then the proof that, as I  moves along F, the endpoints of the segment  describe a constant width curve uses Theorems 5.2.1, 5.3.6, and 5.4.2, and it is left to the reader. Conversely, let F be a figure that floats in equilibrium in any position with density one half, and let I  be a flotation chord in any position and trace a segment I perpendicular to I  of the same length, such that the midpoints of I and I  coincide. Using Theorems 5.2.1, 5.3.6, and 5.4.2 it is not hard to show that while I  moves along F, I describes a curve of constant width. (See Exercise 5.18.)

Notes Line Families Line families related to sets of constant width occur in different forms combined with the notions of outwardly simple line families (or systems of externally simple lines), pedal functions and pedal curves, envelopes, evolutes/involutes, hedgehogs, evolutoids, and ruled surfaces. In the following we want to collect references in which these notions are applied to sets of constant width, or which are combined with constant width sets. Since almost all these concepts are closely related to differential geometry, the reader is also referred to our Chapter 11. Hammer and Sobczyk (see [511] and [512]) studied the configurations of line families coming from diametrical chords of planar constant width sets. They called a family of straight lines in the plane an outwardly simple line family (or system of externally simple lines, see Section 5.3), if it covers the exterior of some circle simply (see also [220] for further results on such line families). Additionally they proved the fact that all orthogonal trajectories of an outwardly simple line family having a point outside of a sufficiently large circle are smooth constant width curves. Moreover, all such curves can be obtained in this way. Hammer [509] summarized this work nicely, referring also to extensions for normed planes. The papers [506], [507], [508], and [1072] are also related, containing negative results for n > 2. In [508], outwardly simple line families are explicitly used as a tool to get analytic expressions for all types of constant width curves; the author starts with the line families in the smooth case and then describes the eventually occurring corners via some limiting process. In this way one can also obtain nice characterization theorems, see [541], [104], [510], and [829]: e.g., if

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each diametral chord of a constant width curve bisects its circumference (its area) then this curve is a circle, see Corollary 5.1.4. Since diametral chords lie on double normals, the following results should also be mentioned here. Kuiper [663] has shown that the set of lengths of double normals of a convex body in En is of measure zero if n < 4, while for all higher dimensions there is a convex body not of constant width with the property that the range of its width function coincides with the set of lengths of its double normals. Heil (see [525] and [526]) proved that a 3-dimensional body of constant width either contains a point belonging to infinitely many normals, or an open set of points through each of which at least 10 normals pass. Let m(K , p) denote the number of normals of a convex body K passing through p ∈ K , and m(K ) be the average of m(K , p) with p running through K . Chakerian [236] could provide sharp bounds on m(K ) for planar constant width sets; the lower bound characterizes the circle, the upper bound the Reuleaux triangle. Also in [301] upper bounds for m(K ) are proved, and related results on n-dimensional bodies of constant width are derived. It is clear that the concept of pedal function, introduced in Section 5.1, and the headline of this chapter are closely related to the notion of pedal curve (a special case of envelopes). Namely, the pedal curve of a plane curve C with respect to a fixed point y is the locus of all points x, such that the line spanned by x and y is perpendicular to the support line of C passing through x. Considering support hyperplanes instead of lines, this notion can be carried over to hypersurfaces in En , continuing the considerations from the booklet [597] on parallel curves of constant width curves. Kelly [617] showed the following: If a closed convex plane curve has constant width, then its pedal curve with respect to an arbitrary point y has y as an equichordal point (i.e., all chords of it passing through y have the same length). Conversely, given a curve, star shaped with respect to y and having y as an equichordal point, its “negative pedal curve” (easily obtained as an envelope of straight lines by reversing the procedure) has constant width provided it is convex. In the same manner Kelly also considered inner parallel curves of curves of constant width. Given a plane convex body of area F with boundary curve C of class C 2 and fixed length, let A denote the area enclosed by the pedal curve of C with respect to the Steiner point of the given convex body. As shown in [273], the isoperimetric deficit of that body has the lower bound 3π(A − F). Based on this, an inequality from Groemer’s book (see Theorem 4.3.1 of [464]) is improved, holding for the special case that the given set is of constant width. The concept of pedal curves and Kelly’s result have natural analogues in higher dimensions. For example, a hedgehog is the envelope of a family of hyperplanes x, z = h(z), where h is a function of z ∈ S n−1 , the unit sphere. It is a hypersurface Hh = x h (S n−1 ) ⊂ En , and for a point x h (z) belonging to the smooth part of Hh the unit normal vector is z. The mapping x h can be interpreted as the inverse to the Gauss map of Hh . In [778], a higher dimensional analogue of Kelly’s result above is derived, referring via antipedal hypersurfaces to hedgehogs of constant width or ensuring (under smoothness assumptions) that the pedal hypersurface of a given hedgehog of constant width is convex. In [777], and [780] related characterizations of spheres among constant width sets are given. Further results on hedgehogs referring also to the concept of constant width can be found in [776], [779], [780], [781], [782], and [1042]. For example, there is a 6-normals theorem for hedgehogs of constant width (see [779]), and [782] refers to evolutes within this framework (see also below). Now we come to the concepts of evolutes and involutes. Already Euler [323] studied orbiforms as involutes of three-cuspid curves, and related representations can be found on pp. 313–316 of the basic book [731], see also Section 5.3.4. Essential in this direction are the papers [598] and [1028], and in the monograph [160] constant width curves are discussed as involutes of certain “k-cuspid” curves (see § 65 there). Based on this, also Gere and Zupnick [408] constructed constant width curves as involutes of such “star curves” consisting of an odd number of convex pieces of total curvature π. They derived necessary and sufficient conditions for such star curves to play that role in terms of inequalities satisfied by their “side-lengths”. Kurbanov [674] studied special curves of constant width being involutes of the hypocycloids with an odd number of cusps, and further related papers are [197], [1143], and [1144]. In [1144], the definition of constant width curves is generalized to include all closed curves with multiple points such that each normal is a normal to two points of the

5.4 Figures Which Float in Equilibrium

123

curve at constant distance. The author considered certain curvilinear polygons with an odd number of sides and constructed from them such multiple curves of constant width using involutes of these curvilinear polygons. For analogous considerations in normed planes we refer to Petty [929]. To verify an important result on the approximation of constant width curves by C ∞ -ovals of constant width, Tanno [1118] used and modified the related evolute–involute relation. A related higher dimensional result for rotational hypersurfaces of constant width is also given, and a generalization of the planar approximation result from [1118] is given in [1171]. Ostrowski [904] studied so-called finite ovals as curves with continuous, never vanishing curvature and an even number of vertices, and he gave also a precise description of their evolutes. He applied then these considerations to obtain extensions of some results on curves of constant width, due to Schilling [1028], under weaker assumptions. The authors of [446] studied evolutes of smooth closed convex curves in the plane, on the (generic) assumption that the evolute has only ordinary cusps and transverse crossings as singularities. In case that the evolute has exactly (the minimum number of) three cusps, the considered curve is of constant width, and the evolute has no self-crossings. Let a constant width curve be of class C+4 . It is known that such a curve has at least 6 vertices, and its interior contains either a point through which infinitely many normals pass or an open set of points through each of which pass at least 6 normals. If all its vertices are nondegenerate, then such a curve has exactly 6 vertices if its evolute is the boundary of a topological disk through each interior point of which pass at least 6 normals, see [782]. We mention here also the papers [1144], [1156], and [1157], in which properties of plane curves with possible self-intersections and a kind of double-normal property are investigated, e.g., also when they are closed. Related to the concept of evolutes, an evolutoid is a curve obtained as the envelope of lines making a fixed angle with the normal line at every point of a given curve. In [581], it is shown that a convex curve is of constant width if its evolutoid for a fixed angle is of constant width. These investigations have their continuation in [4], where also relations to the famous homothetic floating body problem are discussed and it is proved that a curve and any of its evolutoids have the same Steiner point. Moreover, relations between evolutoids and constant angle caustics are also shown, and also curves of constant width play a role in this paper. The paper [872] addresses secantoptics of a closed convex curve (i.e., isoptics of its evolutoids). It is proved that for constant width curves the distance between opposite points of the secantoptics remains constant, and that, conversely, under certain conditions this property characterizes curves of constant width. Isoptics of constant width curves are studied in [256], and for the more general rosettes of constant width we refer to [828], see also the first part of the notes to Chapter 11. Furthermore, ruled surfaces of constant width should be mentioned here since they are also related to line families. For example, Nádeník [884] investigated surfaces generated from the normal hyperplanes of a closed analytic curve as follows. Considering a closed convex (n − 2)-dimensional surface W in such a normal hyperplane, one can study the cylindrical surfaces having W as right section. Assuming that the envelope of these cylindrical surfaces is free of singularities and that W is of constant width, typical properties of constant width surfaces are proved. Wegner [1173] defined cylinders of constant width in 3-space as complete C ∞ -surfaces V of the topological type of a cylinder such that, for each p ∈ V , the normal line l at p intersects V again in a point p  , where l is also the normal line of V at p  . For example, it is proved how an important subclass of these surfaces can be generated by moving a planar curve of constant width along an appropriate space curve. And if V is a cylinder of constant width fibered by a one-parameter family of plane curves of constant width, then the preceding construction can also yield V . With the help of results from global kinematics in 3-space, Pottmann [952] constructed ruled surfaces, which generalize curves of constant width and Zindler curves, see Section 5.4.1. In [554], skew ruled surfaces in 3-space are considered whose generators have a constant angle with respect to a fixed 2-plane, and in which constant width projections of these surfaces suitably occur. For such surfaces a six-vertex theorem and some analogue of Barbier’s theorem are proved. Also in the survey [647] ruled surfaces of constant width are discussed (see § 5.3 there).

124

5 Systems of Lines in the Plane

Floating and Related Billiard Problems Floating problems and billiard problems related to them are for example discussed in the problem book [272], more precisely in the sections A 4 and A 6 there. Further references on billiards are collected in [497] (for the polygonal case) and in [1137] (for the smooth case). The paper [352] refers to rigid bodies on liquid surfaces subject to capillary (and not gravitational) forces. In equilibrium, the liquid surface must have constant mean curvature (law of Laplace), and at the “contact line”, the contact angle between it and the boundary of the floating body has to take a given value (Young’s law). The author shows that if this contact angle is π2 in three dimensions, and the floating body is smooth and strictly convex, then the spherical shape is obtained. But in two dimensions and again for contact angle π , any smooth constant width set admits a neutral equilibrium for any possible orientation. The notion 2 of constant width is only relevant for this isolated case π2 , and the used construction achieves its goal as the orthogonal trajectory of a line family, without using the constant length of the line segments cut out of the arising geometric figure. It is nice that the regular analytic curve of Rabinowitz [956] is also used in [352], see Section 5.3.3. In the related note [353] the connection of such floating body problems and billiard caustics is discussed, citing also the earlier works [351] and [498]. Gutkin [499] presented a detailed solution of problems on capillary floating in terms of billiard problems. Also here, in a special case, curves of constant width are essential. Quantum systems with time-reversal invariance and classically chaotic dynamics have, by common assumption, energy spectra distributed according to the Gaussian orthogonal ensemble type of statistics. Gutkin [496] introduced a class of smooth convex billiards of constant width whose dynamics are time reversible and “almost” chaotic. In [68], it is shown that any minimal closed billiard trajectory in a planar body of constant width is 2-periodic, see Theorem 17.2.3. The proofs are based on results from [108], with analogous results referring to disk polygons (i.e., 2-dimensional ball polytopes). We now come to caustics in relation to constant width (see again the notes in Chapter 17). In the beautiful paper [640] the geometry of caustics of plane billiard tables of constant width is studied; various nice figures are presented there. Using results on the so-called Wigner caustic, in [1225] an improved version of the isoperimetric inequality is given, yielding also a characterization of constant width curves. The Floating Body Problem Problem 19 of the Scottish Book [807] is still open in dimension three and only few special cases have been solved. In three dimensions, in the density limit ρ → 0 or 1, there are no solutions other than the sphere [842]. Independently, Falconer and Schneider proved that there are no nontrivial solutions among star-shaped objects with central symmetry for density ρ = 21 (see [332] and [1032]). Falconer’s proof, using spherical harmonics, will be presented in Section 13.3.1. On the other hand, F. Wenger has proposed a perturbation expansion scheme starting from the sphere for objects with central symmetry and ρ = 21 (cf. [1183]), as well as for bodies with arbitrary shape and ρ = 21 , see [1184]. His results point out the existence of many nontrivial solutions in these wider classes of shapes, even though the proofs are incomplete in that the convergence of the perturbation series has not been examined. Furthermore, no attempt to construct actual solutions of the problem in dimension three has been reported. Regarding the 2-dimensional floating body problem, when the density is not necessarily one half, F. Wenger gave a solution, showing the existence of noncircular figures that float in equilibrium in every position. If a figure F of density ρ floats in equilibrium in every position, then we know that the water surface divides the boundary of F in a constant ratio, say σ/1 − σ. We call σ the perimetral density of F. In [1181], F. Wenger was able to obtain noncircular solutions for ρ = 21 = σ, by a perturbative expansion around the circular solution. These figures have a p-fold rotational symmetry

5.4 Figures Which Float in Equilibrium

125

and have ( p − 2) different perimetral densities. On the other hand, Bracho, Montejano, and Oliveros [173] proved that if the perimetral density σ is 13 or 41 , then the solution is circular. Later, in [1182], F. Wegner was able to give nontrivial explicit solutions to this 2-dimensional version of the floating body problem.

Exercises 5.1. 5.2. 5.3. 5.4.

Let p(θ) be a Lipschitz function with period 2π such that p(θ) + p  (θ) ≥ 0. Prove that x(θ) = p(θ)u(θ) + p  (θ)u (θ)) is a parametrization of a convex curve.  the notation    given  at the beginning of the proof of Theorem 5.1.3, prove that λ (θ) = With  x (θ + π) − x (θ). Suppose ψ is a convex figure with the property that the midpoint of any of its diametral chords is the same fixed point. Prove that ψ is then centrally symmetric. With the notation given in the proof of Lemma 5.2.1, prove that |y(θ) − η(θ)|2 − |η(θ) − x(θ)|2 = 2I (θ)η(θ) − m(θ), u (θ).

5.5.

5.6.

Let  be a body of constant width and suppose p ∈  is an equichordal point, that is, a point p in the interior of  with the property that all chords of  through p have the same length. Prove that  is a ball. Let L be a collection of lines with the property that exactly one line L(θ) = L(θ + π) of L is parallel to each direction, and let p(θ) be its pedal function. Prove that the coordinates (x, y) of the intersection point of the lines L(θ) and L(ϑ) are given by sin θ − sin ϑ p(θ) − p(ϑ) + p(ϑ) θ − ϑ θ−ϑ , x= sin (θ − ϑ) θ−ϑ cos θ − cos ϑ p(θ) − p(ϑ) + p(ϑ) cos ϑ θ − ϑ θ−ϑ y= . sin (θ − ϑ) θ−ϑ − sin ϑ

5.7. 5.8.

5.9.

5.10.

Furthermore, prove that p(θ) is a Lipschitz function if and only if the set of points (x, y) where two different lines of L intersect is bounded. Let L be simple lines and let p(θ). Prove that for every θ, η(θ) =  a system of externally  limh→0 L(θ) ∩ L(θ + h) exists a.e. and is unique. Let L be a system of lines and let p(θ) be its Lipschitz pedal function. Prove that the envelope E(L) is almost everywhere the curve parametrized by η(θ) = p(θ)u(θ) + p  (θ)(θ)u (θ), which is tangent to almost every line in L. Let φ be a strictly convex body. Prove that given a direction, the chord with maximum length between all parallel chords of ψ is a diametral chord. Use this to prove that the system of lines containing a diametral chord is a system of externally simple lines. Let us think of a circle  as the closed interval [0, 2π] where we identify the angle θ with the angle θ + π. In other words, the circle  is the set of directions of the projective plane. Let f :→

126

5.11. 5.12*. 5.13*.

5.14*.

5 Systems of Lines in the Plane

be the following continuous function: given θ ∈ , let L 1 and L 2 be the support lines of ψ orthogonal to u(θ), and let xi ∈ L i , i = 1, 2. Define f (θ) as the unique direction in , such that the diametral chord x1 x2 is orthogonal to u( f (θ)). Prove that f (θ) is a homeomorphism without fixed points. Let p(θ) = a cos2 (kθ/2) + b, for k any odd positive integer and a, b ≥ 0. Prove that p(θ) + p(θ + π) = a + 2b. Let p(θ) = a cos2 (θ/2) + b, for a, b ≥ 0. Prove that x(θ) = p(θ)u(θ) + p  (θ)u (θ) is a parametrization of a circle. Let p(θ) = a cos2 (kθ/2) + b. Prove that x(θ) = p(θ)u(θ) + p  (θ)u (θ) is a parametrization of a convex curve, when a = 2, b = 3 and k = 3. Furthermore, find the values a, b, k for which x(θ) is not a parametrization of a convex curve. Let p(θ) = 2 cos2 (3θ/2) + 8. Prove that p(θ) can also by expressed in the form p(θ) = cos(3θ) + 9 and therefore the curve x(θ) = p(θ)u(θ) + p  (θ)u (θ) has the following parametric equations: x = 9 cos(θ) + 2 cos(2θ) − cos(4θ), y = 9 sin(θ) − 2 sin(2θ) − sin(4θ).

5.15. 5.16.

5.17*. 5.18.

5.19. 5.20.

5.21*.

Prove (5.6). Moreover, with the notation given at the beginning of the proof of Theorem 5.4.1 2 . proves that limδ→0 area ofδ T (θ+δ) = P(θ) 8  With the notation given at the beginning of the proof of Theorem 5.4.1 prove that lim δ→0 r (θ +  δ) − t (θ + δ) = − 23 I (θ)u (θ). Let φ ⊂ E3 be a centrally symmetric convex body that floats in equilibrium in every position with density one half. Prove that the surface of centers of mass is a sphere concentric with φ. Take a chord I  that divides in half the area of a Zindler curve F and trace a segment I perpendicular to I  of the same length, such that the midpoints of I and I  coincide. Prove that as I  moves along F, the endpoints of the segment I describe a constant width curve. Prove that the only Zindler curve of constant width is the circle. Suppose that two segments ab and cd of unit length in 3-dimensional space make an angle √ of θ ≤ π/2.√Prove that if the tetrahedron abcd has unit diameter then ac + bd ≤ 2 2 sin(θ/2) ≤ 2 sin(θ). Let  ⊂ E3 be a body of constant width 1 and let S∂(, v) be the shadow boundary of  in the direction v ∈ S2 (see Section √ 2.12.2). Prove that each polyline P inscribed in S∂(, v) has length not greater than 2π, thus proving √ that a shadow boundary of a 3-dimensional body of constant width 1 has length at most 2π (see [762]).

Chapter 6

Spindle Convexity

There is geometry in the humming of the strings, there is music in the spacing of the spheres. Pythagoras

6.1 Spindle h-Convexity Let h be a positive real number and let p and q be two points in Euclidean n-space En , no more than 2h apart. The h-interval determined by this pair of points is the intersection of all balls of radius h that contain p and q. We say that a set φ, with diameter less than or equal to 2h, is spindle h-convex if given a pair of points p and q in φ, the h-interval they determine is also in φ. Note that for h  ≥ h spindle h-convex sets are spindle h  -convex. We shall see that classic convexity may be interpreted as the limit of spindle h-convexity for h → ∞. Spindle convexity was first introduced in 1935 by Mayer [806], with the name Überkonvexität, and it was also studied by Bierberbach [124], Kupitz-Martini-Perles [671], and Bezdek et al. (see [118] and [120]). Analogous references for normed spaces or even gauges (with unit balls not necessarily centered at the origin) are [579] and [684], respectively. Our study of spindle convexity also shows that there are various relations to notions like ball polytopes, Reuleaux polytopes, ball hull and ball intersection, and ball convexity, occurring in different chapters of our book (examples are the Chapters 8 and 9). The simplest example of a spindle h-convex body is the ball of radius h. In fact, balls of radius h are the only spindle h-convex sets of diameter 2h. Since the intersection of a collection of spindle h-convex bodies is also spindle h-convex, the intersection of a collection of balls with radius h is a spindle h-convex set. Note, for example, that h-intervals are spindle h-convex bodies, and that every spindle h-convex body is strictly convex. In this chapter, unless otherwise stated, all our sets are contained in n-dimensional Euclidean space and have diameter smaller than or equal to 2h. We start by proving the following theorem. Theorem 6.1.1 Every set  of constant width h is a spindle h-convex body. Proof Let v ∈ Sn−1 be a unit vector and let pq be the binormal of  in the direction v. By Theorem 3.1.1 := B( p, h) and Bv := B(q, h) with radius h and centers p and q, iii), we know that the balls Bv  respectively, contain . Let φ = (Bv ∩ Bv ), where the intersection extends over all v ∈ Sn−1 . Clearly, since φ is the intersection of a family of balls of radius h, we have that φ is a spindle h-convex body which contains . We will prove that  = φ. Let p ∈ φ be a point that is not in . Then there exists a support hyperplane H of  that strictly separates p from . If u is the unit vector orthogonal to H , then Bu ∩ Bu does not contain the point p. It follows that p is not in φ and, therefore, that φ is equal to .  © Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_6

127

128

6

Spindle Convexity

We say that S(h) is an h-support sphere of a compact set A if S(h) intersects A and it is the boundary of a ball of radius h that contains A. For example, by Theorem 3.1.1 iii), for every boundary point p of a body  of constant width h, there is an h-support sphere of the body  at the point p. The following lemma shows that the same is true for any spindle h-convex body. Lemma 6.1.1 Let φ be a spindle h-convex body, p a boundary point of φ and H a support hyperplane of φ at p. Then φ is contained in a ball B(h) of radius h tangent to H at p. That is, the boundary of B(h) is an h-support sphere of φ at p. Proof We begin by noting that if p and q are two points of the spindle h-convex body φ and C is a circle with radius h that passes through p and q, then the shortest subarc of C with endpoints p and q is always contained in φ. Let B(h) be the ball of radius h, tangent to H at p, that is contained in the same half-space determined by H as the half-space containing φ. We shall prove that φ ⊂ B(h). Suppose that there is a point q of φ that is not in B(h). Let H  be the plane that contains p, q and the line that passes through p orthogonal to H , and let C be the circle with radius h that passes through p and q and is contained in H  . Since q is not in B(h), then H , the support hyperplane of φ, strictly separates φ from x, where x is a point in the shortest subarc of C with endpoints p and q. But this contradicts the assumption that φ is spindle h-convex (see Figure 6.1).  As corollaries, we have the following two theorems. Theorem 6.1.2 A spindle h-convex body is the intersection of a family of balls of radius h. In fact, it is the intersection of the convex hull of all its h-support spheres. Theorem 6.1.3 Let  be a spindle h-convex body of diameter h. The body  has constant width h if and only if the center of every ball of radius h that contains  belongs to . Proof Suppose  has constant width h, and let B be a ball of radius h containing  and with center x such that x ∈ / . Let H be a hyperplane separating x from . Then in the direction orthogonal to H , the width is smaller than h. On the other hand, assume that every ball of radius h that contains  has its center at , but  does not have constant width h. Then, there must be a unit vector v such that the width of  in the direction v is smaller than h. This is a contradiction because if we take H the support hyperplane of  orthogonal to v, by Lemma 6.1.1 the body  is contained in a ball of radius h tangent to H at  ∩ H whose center is not in . 

Figure 6.1

6.1 Spindle h-Convexity

129

Following Eggleston [316], we say that a convex body φ has the spherical intersection property if φ is the intersection of all balls of radius h with center at φ. Our next theorem states that convex bodies with the spherical intersection property are precisely the bodies of constant width h. Remember that B( p, h) denotes the ball of radius h and with center at p, and S( p, h) denotes the sphere which is the boundary of B( p, h). Theorem 6.1.4 Suppose  ⊂ En . We have =



B( p, h)

p∈bd 

if and only if  has constant width h.   Proof First of all, it is easy to verify that p∈ B( p, h) = p∈bd  B( p, h). Moreover, note that the h) is equivalent to the property that the diameter of  is smaller than or property  ⊂ p∈bd  B( p, equal to h, and the property p∈bd  B( p, h) ⊂  is equivalent to the spherical intersection property; that is, if a ball B( p, h) contains , then p ∈ . If  has constant width h, then the diameter of  is smaller than or equal to h, and by  Theorem 6.1.3 every ball of radius h that contains  has its center from . Suppose now that  = p∈bd  B( p, h). Then every ball of radius h that contains  has its center from , and furthermore, since  is the intersection of balls of radius h,  is spindle h-convex. Then, by Theorem 6.1.3,  has constant width.  Suppose now that  is a spindle h-convex body which is the intersection of balls of radius h and with centers from , then  is not necessarily a body of constant width. An example is the Reuleaux tetrahedron, constructed in a similar way as the Reuleaux triangle, as the intersection of four suitable balls centered at the vertices of a regular tetrahedron. As we will see in Section 8.2, it is possible to show that the Reuleaux tetrahedron is not of constant width because the diameter does exceed h. The following corollary shows that if in addition diam  ≤ h, then  is a body of constant width. Corollary 6.1.1 Suppose that  is a convex body of diameter h. If  is the intersection of balls of radius h and with centers in , then  has constant width.  Proof Let us consider φ = p∈ B( p, h). Since by hypothesis  is the intersection of balls of radius h and centers from , we havethat φ ⊂ . Moreover, the diameter of  is smaller than or equal  to h if and only if  ⊂ φ. Since p∈ B( p, h) = p∈bd  B( p, h), the corollary follows immediately from Theorem 6.1.4.  Let S be a sphere. We say that a subset X ⊂ S is spherically convex if X does not contain two antipodal points and, for a given a, b ∈ X , the shortest geodesic arc joining a and b in S is also contained in X . The following lemma will be important later. Lemma 6.1.2 Let φ be a spindle h-convex body and let S(h) be a sphere of radius h. Then φ ∩ S(h) is spherically convex in S(h) unless φ is a ball of radius h and S(h) is its boundary. Proof By Theorem 6.1.2, assume φ = ∩x∈X B(x, h) and let a, b ∈ φ ∩ S(h). We may assume that a and b are not antipodal points of S(h), otherwise φ is a ball of radius h and S(h) is its boundary. Consequently, let α be the shortest geodesic arc joining a and b in S(h). Since a, b ∈ B(x, h), for every x ∈ X , then also α ⊂ B(x, h) for every x ∈ X , and hence α ⊂  ∩ S(h). 

130

6

Spindle Convexity

The spindle h-convex hull of a set A contained in a ball of radius h is the intersection of all the spindle h-convex bodies that contain A; that is, the spindle h-convex hull of A is the smallest (regarding inclusion) of the spindle h-convex bodies that contain A. By Lemma 6.1.1, the spindle h-convex hull of A, denoted by cch (A), is the intersection of all balls of radius h that contain A. For example, the h-interval determined by two points is simply their spindle h-convex hull. Lemma 6.1.3 The spindle h-convex hull of a compact set A of diameter h also has diameter h. Proof Let cch (A) be the spindle h-convex hull of A and let p and q be any two points of cch (A). We wish to prove that the distance between p and q is less than or equal to h. Let x be any point of A, and B(x, h) be the ball of radius h with center x. As the diameter of A is h, B(x, h) contains A and therefore also cch (A). Thus, since p is in cch (A), the distance between every point x of A and p is less than or equal to h. Hence the ball B( p, h) with center at p contains A and therefore also cch (A). That is, q is in B( p, h), which implies that the distance between p and q is less than or equal to h.  Summarizing, we have Theorem 6.1.5 Let φ ⊂ En be a convex body. Then the following are equivalent: a) b) c) d)

φ is spindle h-convex, φ is the intersection of balls of radius h, cch (φ) = φ, and for every boundary point p of φ, there is an h-support sphere through p.

The following lemma and its proof will be used in the next section. Lemma 6.1.4 Let A be a compact set contained in a ball of radius h, and let cch (A) be its spindle h-convex hull. Then the following are equivalent: a) cch (A) is a body of constant width h, b) every ball of radius h that contains A has its center from cch (A), and c) there exists a unique body of constant width h that contains A. Proof We start by proving that (a) implies (b). Suppose that cch (A) is a body of constant width h and that A is contained in the ball B(h) of radius h. Clearly, cch (A) ⊂ B(h). By Theorem 6.1.3, the center of B(h) lies in cch (A). We next show that (b) implies (c). Let 1 and 2 be two distinct bodies of constant width h that contain A. Since 1 and 2 are spindle h-convex, so is their intersection, which implies that cch (A) ⊂ 1 ∩ 2 . Let x be a point of 1 that is not in 2 . Since the diameter of 1 is h, we have that A ⊂ 1 ⊂ B(x, h). Hence, by hypothesis, x ∈ cch (A), which contradicts the fact that cch (A) ⊂ 1 ∩ 2 . To prove that (c) implies (a), suppose that cch (A) is not of constant width h. Then there exist H1 and H2 , half-spaces supporting cch (A) such that cch (A) ⊂ H1 ∩ H2 , where the width of the strip H1 ∩ H2 is strictly smaller than h. By Lemma 6.1.1, there exist balls B(x1 , h) and B(x2 , h) such that x ∈ H1 \ H2 , x ∈ H2 \ H1 , and cch (A) ⊂ B(x1 , h) ∩ B(x2 , h). Note that the distance between x1 and x2 is greater than h. By Theorem 6.1.2, since the diameter of {xi } ∪ cch (A) is less than h, there are bodies 1 and 2 of constant width h such that {xi } ∪ cch (A) ⊂ i , i = 1, 2, which is a contradiction  since, as x1 is not in 2 , 1 is distinct from 2 . We now examine the boundary of the spindle h-convex hull of a plane figure. Lemma 6.1.5 The boundary of the spindle h-convex hull of a plane figure F of diameter less than or equal to h consists of points of the boundary of F and a collection of subarcs of circles, all of them of radius h.

6.1 Spindle h-Convexity

131

Figure 6.2

Proof Let cch (F) be the spindle h-convex hull of the plane figure F, and let p be a point of the boundary of cch (F) which is not a point of F. By Lemma 6.1.1, there exists an h-support sphere S of cch (F) that passes through p. To simplify the notation, suppose that the point p is the north pole of S and let us call the south pole q (see Figure 6.2). If the left semicircle determined by the points p and q does not contain any point of F, then the circle S could be moved to the right without uncovering F, which would imply that the point p was not in cch (F). Then there is a point x of F in the left semicircle of S, and similarly a point y of F in the right semicircle of S. Thus, since p, x, and y are points of cch (F), the distance between them is less than or equal to h, and hence the h-segment with endpoints x and y contains the circular subarc of S which contains p and whose endpoints are x and y. Since cch (F) is a spindle h-convex figure, we have that the point p is located on a subarc of the boundary of  cch (F) which is a circular arc of radius h. This concludes the proof of the lemma. From this section, we know that everybody of constant width h is the intersection of balls of radius h. It is not difficult to see that, in fact, everybody of constant width h is the intersection of countably many balls of radius h. At the same time, the boundary of Reuleaux polygons is obtained as a finite number of circles of radius h. We shall prove that in higher dimensions the strict analogue of the Reuleaux polygon does not exist. Indeed, we shall prove that the boundary of an n-dimensional body of constant width, n ≥ 3, is not the union of countably many closed subsets, each of them a subset of a sphere of radius h, and, consequently, that an n-dimensional body (n > 2) of constant width is not the intersection of finitely many balls of radius h. As before, Theorem 6.1.6 For n > 2, the boundary of an n-dimensional convex body of constant width h is not the union of countably many closed subsets, each of which being a subset of a sphere of radius h. Proof Suppose that  is a n-dimensional convex body of constant width h whose boundary is the union of countably many closed subsets, each of which is a subset of a sphere of radius h, n > 2. As before, for 1 ≤ i < ∞, let Si := Si (xi , h) be a sphere of radius h centered at xi and suppose that bd  = ∪∞ 1 i , does not cover bd . where i = Si ∩ bd . Furthermore, assume that any proper subfamily of {i }∞ 1 Note that, every i is a compact set, and if i = j, then int bd  i ∩ int bd   j = ∅. Our first objective  is to show that  there are i = j such that the dimension of i ∩  j is n − 2. Let  p1 ∈ 1 \ i =1 i and p2 ∈ 2 \ i =2 i . Thenp1 and p2 are in different components of bd  \ j =1 (1 ∩  j ). This implies that the dimension of j =1 (1 ∩  j ) = n − 2, and therefore we may assume, without loss of generality, that the dimension of 1 ∩ 2 is n − 2. The proof now consists of two steps. First, we show that there is a point p ∈ 1 ∩ 2 which is   , and second, we show that if  has constant width, then the boundary of not in the closure of ∞ i 3

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 contains an open subset which is an (n − 1)-dimensional surface of revolution, contradicting our assumption.  2 is contained in the closure of ∞ For the first step, suppose that 1 ∩ 3 i . Then, 1 ∩ 2 is equal   m to the closure of 3 1 ∩ 2 ∩ i . Then, the dimension of 1 ∩ 2 ∩ i is n − 2 for some i > 2. Since 1 ∩ 2 ∩ i ⊂ S1 ∩ S2 ∩ Si , the dimension of S1 ∩ S2 ∩ Si is n − 2, but this is impossible unless two of these three spheres coincide. For the second step, assume that   has constant width h and let U be an open neighborhood of p in bd  with the property that U ∩ ∞ 3 i = ∅. For every q ∈ U ∩ 1 ∩ 2 , a hyperplane H through q is a support hyperplane of  if and only if H is a support hyperplane of B1 (x1 , h) ∩ B2 (x2 , h), where Bi (xi , h) is the ball whose boundary is Si . For every q ∈ U ∩ 1 ∩ 2 , consider a line L through q and some point of the closed interval x1 x2 . It is clear that L is a normal line of B1 ∩ B2 through q, and therefore it is also a normal line of  through q. Since  has constant width, this implies that the arc of the circle αq of radius h centered at q between x1 and x2 lies in the boundary of . While q traverses U ∩ 1 ∩ 2 , the arc αq sweeps an open subset of the (n − 1)-dimensional surface of revolution obtained by rotating αq around the line through x1 and x2 in En . Noting that our surface of revolution is not a sphere of radius h, we are finished with the proof. This is so unless the distance between x1 and x2 is 2h, but in this case S1 ∩ S2 is a single point, which is impossible.  Corollary 6.1.2 For n > 2, an n-dimensional convex body of constant width h is not the intersection of finitely many balls of radius h.  Proof Suppose that our convex body  is such that  = m m, Bi (xi , h) 1 Bi (x i , h), where for 1 ≤ i ≤ is the ball of radius h centered at xi whose boundary is the sphere Si . Let us consider  = m 1 ( ∩ Si ). It is obvious that  ⊂ bd . Furthermore, if p ∈ bd , then p ∈ Si0 for some i 0 ; otherwise, if p ∈ int Bi (xi , h) for every 1 ≤ i ≤ m, then p should be an interior point of . This implies that  = bd , which is impossible by Theorem 6.1.6. 

6.2 Ball Polytopes The main goal of this section is to study the geometry of the intersection of finitely many congruent balls. Ball polytopes are natural objects for studying several important problems of discrete geometry, such as the Grünbaum–Heppes–Straszewics theorem on the maximal number of diameters of finite point sets in En [671], the Kneser–Poulsen conjecture [117], the proof of the Borsuk conjecture for finite point sets [120], and the analogue of Cauchy’s rigidity theorem for triangulated ball polytopes [120]. Basic references about ball polytopes are [671] and [118]. Since we want to study the geometry of the boundary of a ball polytope, it is important to establish and remember from Section 2.9 some terminology. Let  ⊂ En be a convex body. If more than one support hyperplane goes through a boundary point p of , we say that p is a singular point of the boundary of . If only one support hyperplane passes through it, the boundary point p is called regular. We denote by Sing() ⊂ bd  the set of singular points of . We say that a unit vector u ∈ Sn−1 is normal at a boundary point p if the hyperplane through p, orthogonal to u, is a support hyperplane of  and u points outside . As we know, p has at least one normal unit vector; but if it is singular it has more than one. Consider the set ( p) ⊂ Sn−1 of all normal unit vectors of  at p. The closed set ( p) has no antipodal pair of points. In fact, ( p) ⊂ Sn−1 is spherically convex because, given two vectors u, v ∈ (P), the shortest subarc of the circle between u and v is clearly contained in ( p). The dimension of ( p) enables us to classify the boundary points. For n = 3 the classification is the

6.2 Ball Polytopes

133

following: we say that p is a vertex singular point if dim( p) = 2, while p is called a 1-singular point if dim( p) = 1. Finally, p is a regular point if dim( p) = 0. Then, a ball polytope in En is the intersection of finitely many, but at least n balls of radius h > 0. Let us consider a ball polytope  B(v, T ), φ = B(T, h) = v∈T

for some finite set T ⊂ En . Assume first that φ is 3-dimensional and for any proper subset T  ⊂ T , we have that B(T  , h) = B(T, h). We wish to describe the boundary of the ball polytope . One can represent the boundary of a ball polytope in E3 as the union of vertices, edges and faces defined in a natural way as follows: a boundary point y ∈ bd φ is a vertex if it belongs to at least three of the spheres defining the ball polytope whose centers are not in a circle with center at y. A face is the intersection of one of the generating spheres with the ball polytope, and the intersection of two faces, if non-empty, is the union of (possibly degenerate) circular arcs. All points in the relative interior of these nondegenerate arcs (called edges) are 1-singular points. Clearly, the empty set, vertices, edges, faces, and the ball polytope are partially ordered by inclusion forming the structure of the ball polytope. This structure is not necessarily a lattice (see [671]), unless the intersection of any two faces is either empty, a vertex, or an edge. To be more precise, let Vert(φ) be the set of vertex singular points of bd φ; i.e., the set of points y ∈ E3 such that T ⊂ B(y, h) and S(y, h) ∩ T is not contained in any great circle of S(y, h). The set of 1-singular points of bd φ is given by the set of points y ∈ E3 such that T ⊂ B(y, h) and S(y, h) ∩ T contains at least two points and is contained in some great circle of S(y, h). Lastly, the set of regular points of the boundary of φ is given by the points y ∈ E3 such that T ⊂ B(y, h) and |S(y, h) ∩ T | = h. Consequently, the set of singular points of the boundary of φ, Sing(φ), consists of an embedding of a graph G φ whose vertices are Vert(φ) and whose edges are subarcs of circles, each of them joining a pair of points of Vert(φ) and contained in S(vi , h) ∩ S(v j , h), for some vi = v j in T . Given adjacent  vertices a, b ∈ Vert(φ), we denote the corresponding edge by ab. The complement of the graph G φ = Sing(φ) in the boundary of φ consists of the regular points of φ, whose components are spherically convex open subsets of a sphere of radius h. The closure of each component is precisely a face that has the form: S(x, h) ∩ φ, where x ∈ T . A 3-dimensional ball polytope or polyhedra φ is standard ball polyhedra if the intersection of two faces is either empty, a vertex of G φ , or a single edge of G φ . In fact, the following is proved in Proposition 6.2 of [671]. Theorem 6.2.1 The graph G φ of the 3-dimensional ball polytope φ is a 2-connected planar graph. Furthermore, if φ is a standard ball polyhedra, then G φ is a simple 3-connected planar graph. As a consequence, we have the Euler–Poincare formula v − e + f = 2 for any 3-dimensional ball polytope with v vertices, e edges, and f faces. For more about the face structure of ball polytopes and polyhedra, see Sections 5 and 6 of [671]. Now, let φ ⊂ En be a ball polytope. Suppose φ = ∩x∈T B(x, h), but for any proper subset T  ⊂ T , φ = ∩x∈T  B(x, h). A support sphere Sl is a sphere of dimension l, where 0 ≤ l ≤ n − 1, which can be obtained as intersection of some of the spheres {S(x, h)}x∈T . Then, as before, we say that an n-dimensional ball polytope φ is standard ball polytope if for an l-dimensional support sphere Sl , the intersection φ ∩ Sl is spherically convex in Sl . If this is so, we call σ a face of φ if σ = φ ∩ Sl , for some l-dimensional support sphere Sl of φ. Bezdek et al. [118] proved the following theorem.

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Theorem 6.2.2 Let φ ⊂ En be a standard ball polytope. Then the faces of φ form the closed cells of a finite CW decomposition of the boundary of φ. Furthermore, we have the following Euler–Poincare formula: n  (−1)i f i (φ), 1 + (−1)n+1 = i=0

where f i (φ) denotes the number of i-dimensional faces of φ.

6.3 The Vázsonyi Problem Let T be a finite set of points of diameter h in Euclidean n-space. This section is devoted to characterize finite sets T for which the diameter is attained a maximal number of times as a segment of length h with both endpoints in T . For n = 3, a basic result is due to Grünbaum [484], Heppes [529], and Straszewicz [1000], which they independently proved in the late 1950s by means of ball polytopes, see also the survey [670]. In the plane, a finite set T of size m and diameter h has at most m diameters, and the diameter is attained m times if and only if T ⊂ bd B(T, h) and every singular point of the boundary of B(T, h) belongs to T . To be more precise, remember that a Reuleaux polygon is a plane convex figure of constant width h whose boundary consists of a finite (necessarily odd) number of circular arcs of radius h. Suppose the set T of size m and diameter h has the property that the diameter is attained m times. Then there is a Reuleaux k-gon P (3 ≤ k ≤ m, k odd) such that the diameters of T form a self-intersecting k-circuit (the main diagonals of P) plus m − k “dangling edges”. Each dangling edge connects a vertex of P with an interior point (“dangling vertex”) of its opposite arc. See Figure 6.3, where k = 5 (a Reuleaux pentagon) and m = 9. In 3-space, Vázsonyi conjectured that a finite set T of size m and diameter h has at most 2m − 2 diameters. Indeed, Grünbaum, Heppes, and Straszewicz proved the following theorem. Theorem 6.3.1 Let T ⊂ E3 be a set of size m and diameter h. Then T has at most 2m − 2 diameters, and the diameter of T is attained 2m − 2 times if and only if the points of T are singular points of bd B(T, h) and every vertex singular point of bd B(T, h) belongs to T . Furthermore, the face structure of the boundary complex of bd B(T, h) is self-dual in the strong sense. For a definition of self-dual polyhedra, we refer to [1056]. For a finite set T ⊂ E3 , the diameter graph D(T ) of T consists of a graph whose set of vertices is T and whose edges are pairs {x, y} ⊂ T such that the segment x y ⊂ E3 is a diameter of T .

Figure 6.3

6.3 The Vázsonyi Problem

135

Let us start with a description of the simplest example of a set T ⊂ E3 of diameter 1 and size m ≥ 4, for which the diameter is attained exactly 2m − 2 times. Let T = {x1 , x2 , x3 , x4 } be the set of four vertices of a regular tetrahedron of edge length 1, say. Then the diameter 1 is attained exactly six times and D(T ) = K 4 . The ball polytope B(T, 1) is the Reuleaux tetrahedron, which is a strictly convex body whose boundary has the same face structure as a regular tetrahedron, but with curved faces and edges (see Section 8.2). It has four “spherical triangles” intersecting at six “circular arcs”, where the center of each spherical portion is at the opposite vertex of the tetrahedron and the center of each circular arc is at the middle point of the opposite side of the tetrahedron. The self-dual graph G B(T,1) of singular points of the boundary of the ball polytope B(T, 1) is K 4 . √ 2 . The circle C is the intersection of the Denote by C the circle of radius 23 and with center at x1 +x 2 spheres S(x1 , 1) ∩ S(x2 , 1), so x3 , x4 ∈ C. Take as x5 any point in the relative interior of the shortest subarc of C between x3 and x4 . For this new example, take T = {x1 . . . x5 }. The diameter 1 is attained six times at the edges of the tetrahedron with vertices x1 , x2 , x3 , x4 , and two more times for the segments x1 x5 and x2 x5 . In this case, the graph G B(T,1) of singular points of the boundary of the ball polytope B(T, 1) is shown in Figure 6.4. Consequently, the vertices of the boundary of the ball polytope B(T, 1) are {x1 , x2 , x3 , x4 }, and although x5 is not a vertex point of B(T, 1), it is certainly a singular point of the boundary. Note that the graph G B(T,1) of singular points of the boundary of the ball polytope B(T, 1) is a planar involutive self-dual graph metrically embedded in E3 in such a way that the distance between two vertices of T is less than or equal to 1, and it is exactly 1 if and only if one point is in the dual cell of the other. We give now a brief description of the proof of the Grünbaum–Heppes–Starszewicz Theorem. Let T be a set of m points in E3 . Denote by e(T ) the number of pairs {x, y} such that the length of the segment x y is the diameter of T , and define e(m) to be the maximum of e(T ) over all sets T of m points in E3 . Let us first prove that e(m) ≥ 2m − 2 by showing a set T in which the diameter is attained 2m − 2 times. The example is the same as the example shown in Figure 6.4 except that we take m − 4 points instead of one point x5 in the relative interior of the shortest subarc of C between x3 and x4 . So the diameter 1 is attained six times at the edges of the tetrahedron {x1 , x2 , x3 , x4 } and twice more for every point from T = {x5 . . . xm }. For the proof that e(m) ≤ 2m − 2, we proceed by induction on m. The case m = 4 is obvious. If |T | = m + 1 and a point x ∈ T is incident with at most two diameters (that is, at most two points of T are at distance diam T from x), then we apply the induction hypothesis to T \ {x} and find that

Figure 6.4

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e(T ) ≤ (2m − 2) + 2 = 2(m + 1) − 2. Thus the only case that needs some extra consideration is when each point x ∈ T is at distance diam T from at least three other points of T . This extra consideration consists of the following: we establish a relation between the spherical face structure of the ball polytope B(T, h) and the diameter graph D(T ). Applying Euler’s polyhedral formula for the face structure to the ball polytope B(T, h), we conclude that e(T ) ≤ 2m − 2. In fact, when e(T ) = 2m − 2, the vertex points of the boundary of B(T, h) are exactly the points of T (this motivates the main definition of the next section) and, furthermore, the face structure of B(T, h) is strongly self-dual. That is, it admits a self-duality of order 2 (involution) which is strong in the following sense: it is fixed-point-free when it acts as an automorphism of the first barycentric subdivision of the boundary of the cell complex B(T, h). For n ≥ 4, the quantity e(m) has quadratic growth with m. More precisely, if n = 2d or n = 2d + 1 2 and m → ∞, then e(m) is asymptotically (1 − d1 ) m2 . We refer to [1111] for a discussion of the Vázsonyi problem in higher dimensions, see also [670].

6.4 Reuleaux Polytopes Following the ideas of Sallee [998], we define a Reuleaux polytope as a convex body  ⊂ En satisfying the following three properties:  • there is a finite set T ⊂ E3 with  = B(T, h) = x∈T B(x, h), •  is a standard ball polytope, and • the set Vert() of singular vertex points of bd  is T . In dimension 2, Reuleaux polytopes are exactly the Reuleaux polygons. It is clear, by Corollary 6.1.2, that Reuleaux polytopes, except in dimension 2, are not bodies of constant width. The simplest example of Reuleaux polytope is the Reuleaux tetrahedron which is the analogue of the Reuleaux triangle in dimension 3. It is, as already described, the intersection of 4 balls of radius h centered at the vertices of a regular tetrahedron of side length h, see also Section 8.2. The corresponding graph of singular points for the Reuleaux tetrahedron is the complete graph K 4 with 4 vertices. In dimension 3, as Sallee expected it in [998] and it was anticipated by Kupitz, Martini, and Perles in Remark 1.2 of [671], Reuleaux polytopes are the key to construct constant width bodies, see again Chapter 8. Theorem 6.4.1 Let  ⊂ E3 be a Reuleaux polytope. Then, G  is a self-dual graph, where the automorphism τ is given by τ (x) = S(x, h) ∩ , for every x ∈ T . Furthermore, τ is an involution; that is, a vertex x belongs to the cell τ (y) if and only if the vertex y belongs to the cell τ (x).  Proof Let  = x∈T B(x, h) and suppose that T is the set of vertex boundary points or 0-singular points of bd . Taking x, y ∈ T , the corresponding dual faces are τ (x) = S(x, h) ∩  and τ (y) =  of G  , where S(y, h) ∩ . Assume first that the faces τ (x) and τ (y) intersect in the edge ab  is the shortest arc joining x to y in the circle S(x, h) ∩ S(y, h). This implies that a, b ∈ T and ab |x − a| = |x − b| = |y − a| = |y − b| = h, and therefore that x, y are both vertices of the dual faces τ (a) ∩ τ (b). Since  is a standard ball polytope, then τ (a) = S(a, h) ∩  and τ (b) = S(b, h) ∩  intersect in the edge x y. This proves that if τ (x) ∩ τ (y) = ∅, then {x, y} is an edge of G  . The proof of the converse is completely analogous. Furthermore, if the vertex x belongs to the dual face τ (y) = S(y, h) ∩ , then |x − y| = h, and therefore the vertex y belongs to the dual face τ (x) = S(x, h) ∩ .  An important property of the embedding of the graph G  in E3 is that, for every pair of points of x, y ∈ T ,

6.4 Reuleaux Polytopes

137

Figure 6.5

|x − y| ≤ h and |x − y| = h iff x is in the dual face of y.

(6.1)

A 3-connected planar self-dual graph G that admits an automorphism τ which is an involution (x ∈ τ (y) iff y ∈ τ (x)) and is such that for every vertex x of G we have x ∈ / τ (x), will be called an involutive self-dual graph. A metric embedding of an involutive self-dual graph G in E3 is an embedding of the vertices T of G in E3 as a geometric graph in such a way that (6.1) holds true. Examples of metric embeddings of involutive self-dual graphs are K 4 with the vertices of the equilateral tetrahedron and the following two concrete examples of Figure 6.5. To characterize the situation when an involutive self-dual graph admits a metric embedding is an interesting problem. The spherically self-dual polyhedra constructed by Lovász in [733] in connection with the following problem, stated by Erdös and Graham in [321], are examples of metric embeddings of involutive self-dual graphs. Let n ≥ 2 be an integer and 0 < α < 2. Construct the graph G(n, α) on the points of Sn−1 by connecting two of them if and only if their distance is at least α. The graph G(n, α) obtained this way has chromatic number at least n + 1. This fact is equivalent to the Theorem of Borsuk [168] (see Section 15.3.1). Motivated by this fact, consider the graph H (n, α) on the points of Sn−1 by connecting two of them if and only if their distance is exactly α. The distance problem consists in proving that the chromatic number of H (n, α) is at least n and therefore tends to infinity with n, as Erdös and Graham conjectured. Lovász derived this result by the fact that the graph formed by the principal diagonals of their spherically self-dual polyhedra in En has chromatic number n + 1. 3 Theorem 6.4.2 Let  T ⊂ E be the vertices of a metric embedding of the involutive self-dual graph G. Then B(T, h) = x∈T B(x, h) is a Reuleaux polytope. Furthermore, its polyhedral face structure is lattice isomorphic to the polyhedral structure (points, edges, faces) of the planar graph G.

Proof It will be enough to prove that the vertex boundary points of B(T, h) are precisely the points of T . If this is case, both lattices are isomorphic, because for every point of T , the dual face (as a set of points) of the face structure of the boundary of B(T, h) and the dual face of the abstract polyhedron determined by the planar graph G coincide. To show this, let us prove first that for the number of diameters of T we have e(T ) = 2m − 2, where |T | = m. We start by proving that, for every x ∈ T , deg(x, G) = deg(x, D(T )). The degree of x is equal to the number of faces of G containing x, and since G is an involutive self-dual graph, this number is the same as the number of vertices of the dual face of x, which is the degree of x in D(T ). This proves that the number of edges of G and of D(T ) coincide and, since G is a self-dual graph, by the Euler formula the number of edges of G is 2m − 2; therefore the number of edges of D(G) is also 2m − 2.

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Figure 6.6 A Reuleaux polytope

By the Grünbaum–Heppes–Straszewicz Theorem 6.3.1, every vertex point of the boundary of B(T, h) is a point of T . On the other hand, since every x ∈ G has degree greater than 2, then it follows from the proof of the Grünbaum–Heppes–Straszewicz Theorem (see [671]) that every point x of T is a boundary point of B(T, h).  Figures 6.6 and 6.7 show the corresponding Reuleaux polytope for the self-dual graphs of Figure 6.5.

Figure 6.7 A Reuleaux polytope

6.4 Reuleaux Polytopes

139

Suppose G  ⊂ bd  is a metric embedding of the self-dual graph G  of the singular points of  the Reuleaux polytope  = B(T, h) = x∈T B(x, h), where T coincides with the set of vertices x y ∈ E(G  ) there is an edge of Vert(G  ). Then, by the proof of Theorem 6.4.2, for every edge  ∈ E(G  ) such that ab • |x − a| = |x − b| = |y − a| = |y − b| = h, • the edge x y is contained in S(a, h) ∩ S(b, h); that is, x y is the subarc of the circle with center at a+b , between x and y, contained in the plane orthogonal to ab, and 2  is contained in S(x, h) ∩ S(y, h); that is, ab  is the subarc of the circle • similarly, the edge ab with center at x+y , between a and b, contained in the plane orthogonal to x y. 2  is the dual edge of If this is so, we say that ab x y. Let L be the line between the midpoint of the interval ab and the midpoint of the interval x y. Then it is not very difficult to verify that L is a binormal line of the Reuleaux polytope . Indeed, if  and q lies at the middle of the arc { p, q} = L ∩ bd , then p lies at the middle of the arc ab, x y. A chord of a Reuleaux polytope  connecting the midpoint of two dual edges is called a dual binormal.  Lemma 6.4.1 Let  = B(T, h) = x∈T B(x, h) be a Reuleaux polytope and let J be a binormal of . Then either the length of J is h or J is a dual binormal of length greater than h. Proof Let J = pq be a binormal of , where p, q ∈ bd . If either p or q is regular, then the other point is a vertex, and therefore the length of J is h. The same holds if either p or q is a vertex point. So, we may assume that both p and q lie in edges of G  . But if this is the case, the fact that J is a binormal implies that J is a dual binormal.  and q is in its dual edge Suppose J = pq is a dual binormal, where p ∈ ab x y. Since

|x − a| = − |x − b| = |y − a| = |y − b| = h, then |a − b| = |x − y| = t. Furthermore, | a+b 2

2 2 and therefore the length of J is equal to 2 h 2 − t2 − h 2 − t4 > h.

x+y | 2

=

h2 −

t2 , 4



As a corollary we have Corollary 6.4.1 Let  = B(T, h) = of  is h.

 x∈T

B(x, h) be a Reuleaux polytope. Then the thickness ()

Notes Ball Convexity and Spindle Convexity Replacing intersections of half-spaces by intersections of balls, one gets possibilities to extend the classical convexity notion in a natural way to generalized convexity notions. It is the spherical intersection property of constant width sets that motivates also the study of notions like ball convexity, spindle convexity, and ball polytopes. A subset S of En is called ball convex if it coincides with its ball hull, which is the intersection of all balls of fixed radius containing S. On the other hand, S is said to be spindle convex if, for all x, y ∈ S, S covers the whole spindle of x and y, i.e., the intersection of all balls of fixed radius containing x, y. Spindle convex sets are not necessarily closed, but ball convex sets are. In En , closed sets are spindle convex iff they are ball convex, see Exercise 10.16. Since these notions are also important for Banach space theory, we remark that this coincidence does not hold in normed spaces (see Corollary 3.4 of [118], Corollaries 3.13 and 3.15 of [684], and also [579] for situations that are even more general). To avoid confusion by homonymous usage of names, we shortly mention another concept which is not discussed in this book. Namely, a set

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6

Spindle Convexity

is (also) called ball convex if with any finite number of points it contains the intersection of all balls of arbitrary radii containing the points; the ball convex hull of a set S is then the intersection of all ball convex sets containing S (see e.g., [688]). It is obvious that in the Euclidean case this type of ball convexity coincides with linear (i.e., usual) convexity. With the name “Überkonvexität” (overconvexity), the notion of spindle convexity was introduced by Mayer [804], and different older contributions (partially also referring to notions like ball convexity in the sense of ball hulls) are [192], [1150], [265], [921], [126], [128], [1017], and [124]. One starting point with such notions in discrete geometry was given by Fejes Tóth [346]. All these concepts naturally fit into the framework of generalized convexity notions as they are discussed in § 9 of the famous survey [278]. We discuss now some more recent results on spindle convex sets. First, we note that different names like K convexity, r -convexity, R-convexity, hyperconvexity, overconvexity (and perhaps more) are also used in the literature. In [114], results of Schramm on the illumination number of bodies of constant width (=smallest number of smaller homothets needed to cover such a body) are extended to spindle convex sets. The paper [371] contains analogues of important inequalities (like the Blaschke–Santaló inequality and Ball’s reverse isoperimetric inequality) for spindle convexity. Dowker’s theorem from the theory of packings and coverings says that the maximum area of m-gons inscribed in a convex disk K is a concave function of m, while the minimum area of m-gons circumscribed about K is a convex function of m. In [340], some Dowker-type theorems for spindle convex disks are proved. In [96], results about the Gaussian curvature of spindle convex sets are obtained, and we also refer to [122], where the usual notion of starshapedness is extended to spindle starshapedness via a spindle convex variant of visibility. The authors then establish respective analogues of the well-known theorems by Krasnosel’ski˘i, Carathéodory, and Klee, as well as spindle convex variants of some more recent results, including also a version of the art gallery problem from computational geometry. The papers [118] and [670] are basic for the study of spindle convexity in connection with the related notion of ball polytopes. Further interesting results on spindly convexity were obtained in [112] and [114], see also [916]. Also Chapters 7 and 8 of the monograph [115] refer to spindle convexity and ball polytopes. Ball Polytopes A ball polytope is a nontrivial intersection of finitely many unit balls (or balls of equal radii) in En . There are many applications and relations between the notions of spindle convexity, ball convexity, ball polytopes, ball hulls, and the spherical intersection property, see again the two papers [118] and [671]. For example, in [118] analogues of the classical theorems of Kirchberger, Carathéodory, and Steinitz for spindle convex sets are proved, and the face structure of ball polytopes is studied. This is continued in [671] in a very detailed way. Since diagonal faces occur and two faces may intersect in more than one edge, the respective graph of 3-dimensional ball polytopes is 2-, but not necessarily 3-connected. The authors of [671] derive also results on characterizations of finite sets V of diameter 1, say, in which diameters (now meant as segments of length 1 in a geometric graph with vertices from V ) occur a maximal number of times. This is related to topics like the Vázsonyi problem and the Grünbaum–Heppes–Straszewicz theorem, see Section 6.3. In addition, the results of Sallee [998] on Reuleaux polytopes are discussed again in [671], in the light of the framework of ball polytopes. Further properties of ball polytopes refer to rigidity (see [121]) and to the Kneser–Poulsen conjecture on volumes of unions and intersections of finite systems of balls (see, for example, [112]). There are more papers on ball polytopes and the above- generalized convexity notions (most of them cited in the references listed here), but since their contents are too distant from the concept of constant width, we stop here.

6.4 Reuleaux Polytopes

141

Circular Intersection Properties The planar version of the spherical intersection property is usually called circular intersection property. Closely related is the so-called weak circular intersection property introduced in [668]. A set S of diameter 1 has this property, if the intersection of all unit circles, whose centers belong to S, is of constant width. A finite example is the vertex set of a Reuleaux polygon. The authors of [668] determine the smallest number of points necessary to complete a finite planar set of given diameter such that the resulting set (obtained as the described intersection) has the weak circular intersection property. This number is given in terms of the diameter graph of the set. In [210], the fact is recalled that, given a set of points, the Minkowski sum of the intersection of all disks of fixed radius centered at these points and the intersection of all disks of this radius, which contain the points, is a constant width set. This is used to prove a nice conjecture comparing the two areas of intersections of two families of equal disks. For the spherical intersection property in higher dimensions and its numerous relations to other notions (like completeness, constructions, etc.) discussed in this book, we refer to the Sections 4.5, 7.6, and 10.7, where also related references are collected.

Exercises 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.10*.

Prove that h-intervals are spindle h-convex sets and that every spindle h-convex body is strictly convex. Prove that the h-interval determined by the points p and q is the union of all short circular arcs of radius ρ, h ≤ ρ ≤ ∞, through p and q. Prove that for h  ≥ h spindle h-convex sets are h  -convex. Prove that the intersection of a family of spindle h-convex sets is spindle h-convex. Prove that balls of radius h are the only spindle h-convex sets of diameter h. Let  be a spindle h-convex body and suppose p ∈ / . Prove that there is a ball B(h) of radius h with the property that  ⊂  B(h), but p ∈ / B(h).  Let  be a convex  body. Prove that p∈ B( p, h) = p∈bd  B( p, h). Prove that  ⊂ p∈bd  B( p, h) if and only if the diameter of  is smaller than or equal to h.  Prove that the property p∈bd  B( p, h) ⊂  is equivalent to the spherical intersection property; that is, if a ball B( p, h) contains , then p ∈ . Let L a and L b be two parallel lines at distance h apart and consider points a ∈ L a , b ∈ L b such that the segment ab is orthogonal to these two lines. Consider a curve C between L a

Figure 6.8

142

6.11. 6.12. 6.13.

6.14. 6.15.

6

and L b , to the right of ab tangent to L a at a, tangent to L b at b (see Figure 6.8) and such that through every point c ∈ C there is a support h-circle to α of c. Complete the curve α to a closed curve α˜ by adding the centers of all these support circles. Is the curve α˜ a curve of constant width h? Prove that a convex body of constant width h is the intersection of countably many balls of radius h. Let S ⊂ En be a set of diameter smaller than h. Then the diameter of cch (S) is equal to the diameter of S. Prove the Vázsonyi problem in the plane. That is, a finite set V of cardinality m and diameter h has at most m diameters, and the diameter is attained m times if and only if V ⊂ bd B(V, h) and every singular point of the boundary of B(V, h) belongs to V . Prove that a 2-dimensional Reuleaux polytope is a Reuleaux polygon.  and q is in its dual edge Prove that if J = pq is a dual where p ∈ ab x y, then

binormal,

2

6.16.

Spindle Convexity

2

the length of J is equal to 2 h 2 − t2 − h 2 − t4 , where |x − a| = |x − b| = |y − a| = |y − b| = h. Let n ≥ 2 be an integer and 0 < α < 2. Construct the graph G(n, α) on the points of Sn−1 by connecting two of them if and only if their distance is at least α. The graph G(n, α) obtained this way has chromatic number at least n + 1. Prove that this fact is equivalent to the Theorem of Borsuk [168], see Section 15.3.1.

Chapter 7

Complete and Reduced Convex Bodies

Geometry is the most complete science. David Hilbert

7.1 Introduction We say that a compact set in En is complete (or diametrically complete) if, adding any point to it, its diameter will increase. If we take the partially ordered set h of all compact sets of diameter h in n-dimensional Euclidean space ordered by inclusion, complete bodies are precisely the maximal elements of h . That is, a compact set A in h is a maximal element of h , or a complete body, if A is equal to B whenever A is contained in B, for B in h . The two main results of this chapter are that complete bodies are precisely bodies of constant width h, and that every element of h is contained in a maximal body; that is, that it can be completed to a body of constant width. These results are known as the Theorems of Meissner and Pál, respectively. Section 7.4 will be devoted to the study of reduced convex bodies, a notion somehow “dual” to completeness, and in Section 7.5 we complete convex bodies preserving some of their original characteristics, such as symmetries. The following lemma states that every body of constant width h is a complete body. Lemma 7.1.1 Let  be a body of constant width h and suppose that  is contained in a compact set φ with diameter h. Then  is equal to φ. Proof Let p be any point of φ \ . As the diameter of φ is h, the ball B( p, h) with center p and radius h contains φ and therefore also . Consider the hyperplane H through p that does not intersect  together with the hyperplane H  tangent to B( p, h) and parallel to H . It is clear that  is properly contained in the strip between H and H  , which is a contradiction because the distance between H  and H  is precisely h. We will prove that if φ is a compact set of diameter h that is not of constant width, then φ is properly contained in a body  of constant width h, proving that every complete body is a body of constant width. To do so, we shall use the notion of spindle convexity discussed in Chapter 6.

© Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_7

143

144

7 Complete and Reduced Convex Bodies

7.2 The Theorems of Meissner and Pál Theorem 7.2.1 (Theorem of Meissner) The maximal bodies of h , or complete bodies, are precisely the bodies of constant width h. Proof Since every set of constant width h is a complete body, it suffices to prove that every compact set φ of diameter h that is not of constant width is properly contained in a compact set of diameter h. If φ is not spindle h-convex, then, by Lemma 6.1.3, φ is properly contained in its spindle h-convex hull, which has diameter h. Suppose now that φ is a spindle h-convex body. Since φ does not have constant width h, there are parallel support hyperplanes of φ, H , and H  , with the property that the distance between them is less than h. By Lemma 6.1.1, there exists an h-support sphere of φ tangent to H whose center q is strictly separated from φ by H  . The spindle h-convex hull  of the compact set formed by the union of the point q and the set φ is a spindle h-convex body that properly contains φ and whose diameter, by Lemma 6.1.3, is h since the distance between q and every point of φ is less than or equal to h.  The Theorem of Pál, like the Theorem of Meissner, is considered as a classical theorem. Theorem 7.2.2 (Theorem of Pál) If φ is a body of diameter h, then there exists a body  of constant width h which contains φ. Proof In the proof of the Theorem of Meissner we showed that if φ is a body of diameter h and v is any unit vector in Sn−1 , then there exists a body φ of diameter h which contains φ and has the property that the width of φ in direction v is h. Let vi , i = 1, . . . , ∞, be a dense collection of unit vectors in Sn−1 . · · of diameter h, Hence there exists a nested sequence of convex bodies φ = φ0 ⊂ φ1 ⊂ · · · ⊂ φi ⊂ · in the direction v is h. Let  be the closure of φi ; that  is,  is with the property that the width of φ i i  φi plus its boundary points. It is easy to see that the diameter of  is h since the diameter of φi is also h. What is more, as φi is contained in , the width of  in the direction vi is h for every i. Since the collection of unit vectors vi is dense in Sn−1 and the width function is a continuous function, the width of φ in every direction is h.  The Theorem of Pál enables us to prove the following characterization of bodies of constant width due to Reidemeister [964]. Theorem 7.2.3 (Theorem of Reidemester) A convex body φ has constant width if and only if every boundary point of φ is the endpoint of a diameter. Proof Suppose that every boundary point of φ is the endpoint of a diameter. Let  be a body of constant width that contains φ by the Theorem of Pál. By hypothesis, every boundary point of φ is a boundary point of . This immediately implies that φ = .  If X is a set of diameter h, we can find a body  of constant width containing X , by adjoining to X all possible points that do not increase the diameter of the set. This leads to a complete set containing X . We describe a new method of constructing such a set by intersecting balls of radius h until a set of constant width is obtained. This method follows the ideas of Sallee [998]. Let D = {xm } be a countable dense set in En . We will construct our set inductively. For each integer m, we associate the convex body Fm to a finite set X m ⊂ D defined inductively as follows: F1 = B(x1 , h) and X 1 = {x1 }. Let Fm = Fm−1 ∩ B(xm , h) and X m = X m−1 ∪ {xm }, if xm ∈ Fm−1 , Fm = Fm−1

and and X m = X m−1 , if xm ∈ / Fm−1 .

7.2 The Theorems of Meissner and Pál

Note that for every m we have Fm = Define

145

 x∈X m

B(x, h). 

(D, h) =

Fm

1≤m 2, the boundary of an n-dimensional convex body of diameter h which has either constant minimum width h or is reduced is not the union of countably many closed subsets, each of them being a subset of a sphere of radius h. Proof Suppose φ is reduced and is the union of countably many closed subsets, each of them being a subset of a sphere of radius h. In this case it is clear that φ is almost spherically convex and, by Theorem 7.4.6, has constant width. But this contradicts Theorem 6.1.6. Finally, assume that φ has constant minimum width h and is the union of countably many closed subsets, each of them being a subset of a sphere of radius h. Then clearly φ is strictly convex, and therefore, by Theorem 7.4.3, a reduced body, contradicting our previous conclusion.  Corollary 7.4.3 For n > 2, an n-dimensional convex body of diameter h which has either constant minimum width h or is reduced is not a ball polytope.  Proof Suppose the convex body φ is such that φ = m B(xi , h) is the 1 Bi , where for 1 ≤ i ≤ m, Bi = ball of radius h centered at xi whose boundary is the sphere Si . Let us consider  = m 1 (φ ∩ Si ). It is obvious that  ⊂ bd φ. Furthermore, if p ∈ bd φ, then p ∈ Si0 for some i 0 ; otherwise, if p ∈ int Bi for every 1 ≤ i ≤ m, then p should be an interior point of φ. This implies that  = bd φ, which is impossible by Theorem 7.4.7. 

154

7 Complete and Reduced Convex Bodies

7.5 Regular Constant Width Hulls We know from Section 7.2 that every body with diameter h is contained in a body of constant width h. We may wonder, however, how similar these two bodies must be. For example, if a body is symmetric with respect to a hyperplane, does the body of constant width that contains it necessarily have the same property? In this section we prove that given a convex body φ, there exists a body  of constant width containing φ such that i) every symmetry of φ is a symmetry of  and ii) every corner point of  is a corner point of φ for which the set of antipodal points of  yields the circular convex hull of the points that are already antipodal points of φ. This problem was first posed by Danzer and Grünbaum in 1961, and it was solved by Schulte [1048] in 1981. We conclude this section by giving, based on this, a partial solution to the problem of Borsuk. For this famous partition problem the reader can also consult Section 15.3, as well as Section 10.6 and the Notes of Chapter 10 (the latter two Sections referring to normed spaces). Recall that we denote the ball with center in x and radius r by B(x, r ) and the sphere, which is its boundary, by S(x, r ). For convenience, in this section the line passing through x and y will be denoted by l(x, y); the ray with apex at x and passing through y by r [x, y), and the interval between x and y by x y. From this point on, φ will be a compact, convex set with diameter 2. A symmetry of this body is an isometry (a function that preserves distances) of the Euclidean space that fixes φ. The set of endpoints of all the diameters of φ will be denoted by D(φ). For x in φ we define ˜ φ) := sup{t|y ∈ φ and t = |x − y|}, d(x, and in the case that x belongs to D(φ) we write (x, φ) := φ ∩ S(x, 2) and S ing(x, φ) := cc{



r [x, y)|y ∈ (x, φ)} ∩ S(x, 2).

We differ in this section from the definitions of singular and regular points given in Section 2.9 and in Bonnesen–Fenchel [160]. During this section a singular point is a corner point x of φ such that x is an endpoint of some diameter of φ for which S ing(x, φ) consists of more than one point. If a boundary point is not singular, then it will be termed regular. Note that for a body of constant width every corner point is singular, but that in general a corner point can be even a regular point. We now enclose the convex body φ in a body of constant width, but without adding additional singular boundary points. Lemma 7.5.1 Let  be a convex body of constant width 2 in En , and let x be a singular point of . Let y, y1 , y2 ∈ (x, ) be three points in the boundary such that y is different from both y1 and y2 and, moreover,   r [x, y) ⊂ cc r [x, y1 ) ∪ r [x, y2 ) . Then y is a regular point of . Proof Since every body of constant width 2 is a 2-convex body, we know that the 2-interval with endpoints y1 and y2 is contained in the body . It is easy to see that y is at the same time a regular

7.5 Regular Constant Width Hulls

155

boundary point of  and of the 2-interval with endpoints y1 and y2 , but in the latter set, the only corner  points are the endpoints y1 and y2 . We now describe an iterative procedure—which exhausts φ—by which a body  of constant width 2 can be obtained. The body  contains φ, and its singular points are already the endpoints of the diameters of φ. We will need the following inductive definition: a) For every x in φ and i ∈ N let ρ0 (x, φ) := 0, and ρi+1 (x, φ) := ρi (x, φ) + (1/2)[2 − sup{t|y ∈ φ, where t = ρi (x, φ) + |x − y| + ρi (y, φ)}]. b) For every x in φ let [i → ∞]. ρ(x, φ) := lim ρi (x, φ) c) Let (φ) :=

 {B(x, ρi (x, φ))|x ∈ φ}.

Immediately it can be verified that for every x and y in φ and for every i ∈ N, ρi (x, φ) + |x − y| + ρi (y, φ) ≤ 2. It follows that the sequence {ρi (x, φ)}i is monotone for every x in φ, and also that the limit defined in b) exists. Transferring this inequality to ρ(x, φ), it follows that the diameter of (φ) is 2. We also have, for x and y in φ and λ, 0 ≤ λ ≤ 1, that   λρ(x, φ) + (1 − λ)ρ(y, φ) ≤ ρ λx + (1 − λ)y, φ ; so the convexity of (φ) follows from the convexity of φ. The inequality |ρi (x, φ) − ρi (y, φ)| ≤

(2i − 1) |x − y| , 2i

where the slow rate of convergence plays an important role, follows easily. The following lemma is essential. Lemma 7.5.2 For x in φ, i) there exists a point y in φ such that ρ(x, φ) + |x − y| + ρ(y, φ) = 2; ii) ρ(x, φ) = 0 if and only if x is the endpoint of a diameter of φ. Proof Since the function ρi (x, φ) is continuous in φ, there exists yi in φ for every i in N such that 2(ρi+1 (x, φ) − ρi (x, φ)) = 2 − sup{t|z ∈ φ and t = ρi (x, φ) + |x − z| + ρi (z, φ)} = 2 − [ρi (x, φ) + |x − yi | + ρi (yi , φ)]. Without loss of generality we may assume that the sequence {yi }i converges to a point y, from which i |x − y| and the convergence of the it follows, together with the inequality |ρi (x, φ) − ρi (y, φ)| ≤ 2 2−1 i sequence {ρi (x, φ)}i , that

156

7 Complete and Reduced Convex Bodies

ρ(x, φ)+ |x − y| + ρ(y, φ) = lim{ρi (x, φ) + |x − yi | + ρi (yi , φ)} = 2,

[i → ∞],

which proves Lemma 7.5.2 i). The proof of Lemma 7.5.2 ii) follows directly from the definitions of  ρ1 (x, φ) and ρ(x, φ). This concludes the preliminary results needed for the proof of the following theorem. Theorem 7.5.1 Let φ be a convex body with diameter 2 in En . Then φ is contained in a body  of constant width 2 such that i) any boundary point of  that also belongs to φ is an endpoint of some diameter of φ, ii) any singular boundary point of  is an endpoint of some diameter of φ, and iii) every symmetry of φ is a symmetry of . Proof The body  will be constructed with the help of the constructions described above. That is, let 0 := φ,

i+1 := (i ) [i ∈ N]

and

 := cl

 {i }i∈N .

To begin, it follows from Lemma 7.5.2(i) that  is a body of constant width 2. Part i) of the theorem follows from Lemma 7.5.2(ii). Since the group of symmetries of each body i , i ∈ N, contains the group of symmetries of φ, it follows that iii) is true. It now remains to prove i). To do so, we show that every boundary point of  outside of 0 is regular. Let x be a boundary point of  that is not in 0 . We then have two cases. Case 1. There exists some i ∈ N such that x ∈ i . Let i 0 be the minimum integer with this property. Since x is not in 0 , i 0 ≥ 1. There is some y in i0 −1 for this x for which ρ(y, i0 −1 ) > 0, and such that   x ∈ S y, ρ(y, i0 −1 ) ⊂ .   Then x is at the same time a boundary point of B y, ρ(y, i0 −1 ) and of , which implies that x is a regular boundary point of . Case 2. The point x is not in i for all i ∈ N. Assume the contrary to be true; that is, that x is a singular boundary point of . Since  is a body of constant width 2, there are distinct points y1 , y2 ∈ (x, ). It follows that  contains the 2-interval with endpoints y1 and y2 . At the midpoint y3 of y1 y2 , which is an interior point of , there is some i for which y3 ∈ i . Let i 0 be the smallest i with this property. Now, using Lemma 7.5.2 i) and the fact that B(y3 , 2 − |x − y3 |) is contained in the 2-interval with endpoints y1 and y2 , it follows immediately that ρ(y3 , i0 ) = 2 − |x − y3 | . Let y4 denote the point different from x where the boundary of  intersects the line l(x, y3 ). It follows from the above that y4 is in i0 +1 , and that, moreover, d(y4 , i0 +1 ) = 2 . ˜ 4 , i0 +1 ) < 2, then making use of Lemma 7.5.2 ii), ρ(y4 , i0 +1 ) > 0 and This we have since if d(y therefore y4 would not be in the boundary of i0 +1 , which, however, is not possible.

7.5 Regular Constant Width Hulls

Let x1 ∈ i0 +1 be such that

157

|y4 − x1 | = d(y4 , i0 +1 ) = 2.

Since, by Lemma 7.5.1, y4 is a regular point of , the interval x1 y4 is perpendicular to the support hyperplane of  at y4 , which implies that x1 = x and, therefore, that x ∈ bd i0 +1 , which contradicts the assumption. It follows that the assumption about x is false and, therefore, that x is a regular point of . Every singular boundary point of  is also in φ, which implies, by i) of the theorem, that any singular boundary point of  is also the endpoint of some diameter of φ, proving the theorem.  It is likely that only a finite number of iterations is needed in the above proof. It has only been possible to prove this, however, and with considerable effort, for convex figures. In fact, it is known that 2 = 1 for convex figures. This result is interesting in its own right since it implicitly means that every figure φ of diameter 2 can be placed inside a figure of constant width 2 which is the union of circles with unit radius whose centers are in φ. The claim cannot be generalized to 3-dimensional space, as illustrated by the example of the regular tetrahedron with sides of length 2. Theorem 7.5.1 guarantees that we can avoid adding any more singular boundary points to those which φ already has. Nevertheless, the following question has not been answered, yet: how to avoid that the set of diameters increases at a singular point? It could conceivably happen since in the above iteration process it is not clear that regularization of the set of endpoints of the diameters follows from the construction process. Therefore, the following theorem is necessary. Theorem 7.5.2 Let φ be a convex body with diameter 2. Then φ is contained in a body  of constant width 2 with the following properties: i) Any endpoint of a diameter of  is an endpoint of a diameter of φ. ii) If x is an endpoint of a diameter of  and H is a support hyperplane of  at x, then the line perpendicular to H has a point in common with the singular set S ing(x, φ) of x in φ. iii) If x is an endpoint of a diameter of  and  is a convex set of diameter 2 which contains φ, then S ing(x, ) = S ing(x, φ). iv) Every symmetry of φ is a symmetry of . Proof The proof is by construction. We start by defining, for every x in D(φ) and every a in S(0, 1), ˜ + σa, φ) ≤ 2}m , σ(x, a) := sup{σ|d(x and for every i = 0, 1, 2,

pi (x, a) := x +

σ(x, a) a. 2i

Moreover, let φ0 := φ ∪

 { p2 (σ, a)|x ∈ D(φ), a ∈ S(0, 1)}

and

 := cc(φ0 ).

From the construction of  it can be seen immediately that any symmetry of φ is also a symmetry of . The continuity of σ(·, ·) in both variables guarantees the compactness of φ0 and also of .

158

7 Complete and Reduced Convex Bodies

Now let x, y ∈ D(φ), a, b ∈ S(0, 1) and σ(x, a) > 0. It then follows from the definitions of σ(·, ·) and pi (·, ·) that | p2 (x, a) − p2 (y, b)| = 21 (x + p1 (x, a)) − 21 (y + p1 (y, b)) ≤ ≤

1 |(x + p1 (x, a))| − 21 |(y + p1 (y, b))| 2 1 max{|x − y| , |x − p0 (y, b)|} 2 + 21 max{|x − y| , |y − p0 (x, a)|} ≤ 2

and | p2 (x, a) − y| ≤ max{|x − y| , | p0 (x, a) − y|} ≤ 2. Together this proves that φ0 retains diameter 2 and that the value of 2 is only reached between two points of φ. Both properties are transferred to , which suffices to prove part i) of the theorem. Again, let x ∈ D(φ), and if y is a point of S ing(x, φ), denote the support half-space at x corresponding to y and perpendicular to the line l(x, y) by H (x, y). Let C(x) := ∩{H (x, y)|y ∈ S ing(x, φ)}. The support hyperplanes of C(x) at x are the only ones permitted in part iii) of the theorem. What remains to be proved in iii) is that C(x) is the support cone of  at x. We have that for every a ∈ S(0, 1), such that x + a is contained in the interior of C(x), σ(x, a) > 0. Let one such a be given. For i ∈ N, let xi := x + (1/2i )a and let Hi be the closed half-space that contains x and is orthogonal to the line l(x, x + a) at the point xi . Let H be the closed half-space that does not contain x + a and that is orthogonal to the line l(x, a) at the point x. Suppose that σ(x, a) = 0. Then for every i ∈ N there is a point yi in φ with |xi − yi | > 2, from which it follows that yi ∈ Hi+1 . Without loss of generality suppose that the sequence {yi }i converges to y. Then we have that y ∈ H , but also that |x − y| = lim{|xi − yi |}i→∞ ≥ 2 , and therefore that y ∈ S ing(x, φ), which contradicts that x + a is in the interior of C(x). Therefore C(x) is the support cone of  in x. From the above, it also follows—even in the case where the interior of φ is empty—that  is a convex body. Finally, let x ∈ D(), and let  be a convex set containing φ, with diameter 2 and y ∈ S ing(x, ). It follows from the fact that the interval x y is a diameter of  that the support hyperplane of  orthogonal to l(x, y) at x is also a hyperplane orthogonal to  at x. From part (ii) of the theorem it follows that y ∈ S ing(x, φ), which concludes the proof.



The combination of Theorems 7.5.1 and 7.5.2 leads to the proof of the claim given at the beginning of this section, which can now be stated as a theorem.

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Theorem 7.5.3 Let φ be a convex body with diameter 2. Then φ is contained in a body  of constant width 2 having the following properties: i) Every boundary point of  that belongs to φ is the endpoint of a diameter of φ. ii) Every singular boundary point x of  is a singular boundary point of φ and, moreover, (x, ) = S ing(x, φ). iii) Every symmetry of φ is also a symmetry of . Proof Let  be the convex body of diameter 2 given by Theorem 7.5.2 for the set φ, and let  be the body of constant width 2 given by Theorem 7.5.1 for the convex body . Any boundary point of  which is in φ is, by Theorem 7.5.1(i), the endpoint of a diameter of  and also, by Theorem 7.5.2, the endpoint of a diameter of . Any singular boundary point of  is, by Theorem 7.5.1(ii), the endpoint of a diameter of  and by Theorem 7.5.2 also the endpoint of a diameter of φ. By Theorem 7.5.1(ii) and Theorem 7.5.2(iii), we have for every singular point x of  (x, ) = S ing(x, ) = S ing(x, ) = S ing(x, φ), which proves (ii). Part (iii) of the theorem follows directly from Theorems 7.5.1(iii) and 7.5.2(iv). This proves the theorem.  Theorem 7.5.3 may be used to give a partial solution to the conjecture of Borsuk, since, by Hadwiger’s Theorem 15.3.1, the conjecture is true for constant width bodies without corner points. Theorem 7.5.4 Let φ be an n-dimensional convex body with diameter h. If the singularity S ing(x, φ) of every endpoint x of a diameter of φ consists of a unique point, then φ may be decomposed into n + 1 pieces each having diameter less than h. Proof By Theorem 7.5.3 above, φ can be enclosed in a smooth convex body of constant width h. Then Theorem 15.3.1 of Hadwiger can be applied to complete the proof. 

Notes Completeness and Completions A bounded subset of En is called complete (or diametrically complete) if it cannot be enlarged without increasing its diameter. One can define this type of sets with the same formulation within the class of convex bodies and will get the same family of sets. It turns out that in Euclidean space the classes of complete sets and of bodies of constant width coincide (this is the so-called Meissner Theorem), see Theorem 7.2.1. For a convex body in En , any complete set containing it and having the same diameter is called a completion of that body. Furthermore, each convex body has at least one completion, which is the Theorem 7.2.2 of Pál. For the history of and early contributions to these notions we refer to [160, Section 64]. Meissner’s Theorem was derived in [817] for n = 2 and n = 3, and for arbitrary dimension in [586]. It can be used to show that a convex body is of constant diameter if it is of constant width, see [964]. And Pál’s Theorem was derived in [909] for n = 2 (see also [966]), and in general by Lebesgue [708]. Based on ideas from [966], also Bückner [192] proved Pál’s Theorem (cf. also [193] and [99]), and Sallee [998] gave another proof in view of the spherical.5 intersection property. Namely, he showed that the completion of the convex body K is an intersection of balls of the same diameter whose centers

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are taken from K and some countable, everywhere dense set in En . See the proofs of Theorems 7.2.4 and 10.3.2. This construction he used already in [997] to obtain new planar sets of constant width from old ones, replacing old boundary parts by certain circular arcs. Closely related to the concept of completeness in the plane, Rademacher and Toeplitz [957, § 20b], Mayer [806], and Bückner [192] found exact criteria for planar arcs to be boundary parts of 2-dimensional sets of constant width, and in [192] also 3-dimensional analogues have been verified. Related are also the papers [1048] and [389], the latter again only discussing the planar case. For Minkowski spaces, such investigations were done by Sallee in [1003], see Section 10.3 and the Notes of Chapter 10 below. Constructing a completion of K , say, by an increasing (regarding inclusion) sequence of sets, Scott [1052] showed that all of these sets must lie inside the circumsphere of K , yielding the result that any convex body K has a completion with the same circumradius as K , see Section 14.4. This containment result was also obtained by Vre´cica and even extended to normed spaces, see [1159]. Considering not only the diameter of K in this framework, but also other quantities (such as minimum width, in- and circumradius, etc.), Groemer developed in [456] a general concept to look for related extremal bodies, regarding minima and maxima. He derived a general existence theorem for corresponding maximal and minimal sets and applied this also to extremal sets with respect to packing and covering densities. Regarding minimal sets, we have to mention that this concept inspired Heil to define the notion of reducedness (cf. [524]), motivated also by the isodiametric problem (asking for volume-maximizers for given diameter). Namely, he asked for the “dual” of this problem, i.e., for the convex bodies which are volume-minimizers for fixed given minimum width. Such sets have to be reduced, see our representation in this chapter and also Heil’s problems posed in the problem collection [481], as well as [272, A 18]. In [457] Groemer studied volume-maximizers in the family of all completions of a convex body, and he investigated also completions in Minkowski spaces. Replacing volume by mean projection measures, he further proved the existence and uniqueness of maximal completions having prescribed symmetry properties, thus extending related results of Schulte [1048] and Schulte and Vre´cica [1049] (see also [1050]) to normed spaces. Another contribution to the preservation of symmetry properties when completing a convex body is [982]. It is shown there that any bounded n-dimensional set having a symmetry group G and diameter 1, say, is contained in a body of constant width 1 having the same symmetry group G, see also again [1048] and Section 7.3. In [747], Maehara introduced a completion operation for convex bodies in En (see Section 7.5 and the Notes of Chapter 10), and in [748] he showed that by this procedure several known results on convex bodies can be reobtained, but in a way that their derivation does not depend on Blaschke’s Selection Theorem, see also the part “Pairs of constant width” in these notes. By Falconer [331] a point p of a set of diameter 1 is said to be singular if there exist two distinct points of this set having distance 1 to p. Falconer proved that every compact subset S of En of diameter at most 1 is contained in a set X of constant width 1 such that S and X have the same set of singularities. This yields a related approximation procedure (taking into consideration the structure of these singularities) within the family of sets of constant width; similar results were derived by Schulte in [1048]. A nice characterization of plane sets with unique completion was found by Boltyanski [146]: This property holds if the respective sets cannot be covered by two sets of smaller diameter each, thus giving a basic result on Borsuk partitions in two dimensions. An extensive discussion of the role that sets with unique completion and constant width sets play for results around Borsuk’s partition problem is given in Chapter 5 of [151], see also the related research problems in Chapter 8 there. Parts of the representation of Borsuk’s problem in normed spaces given in [151] are strongly based on the papers [485], [316], and [155], see also [488], [272, D 14], and [154]. Although mainly referring to normed spaces, the paper [890] contains also results on the Euclidean situation, e.g., about sets with unique extension to bodies of constant width, see the Notes of Chapter 10. Further results in this direction are given in [651] and [652]. For example, in the second paper it is shown that a planar convex body has unique completion if for any non-diametral chord of it there exists a diametral chord not meeting the interior of this chord; further results refer to Borsuk numbers (see Section 15.3) of convex bodies all whose diametral chords are in special positions. Still regarding Borsuk’s problem we refer also

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to A 27 in [272], where the following problem is posed: What is the largest possible diameter of the set M of midpoints of all diametral chords of a set of unit constant width in En , taken over all such sets? In particular, is there a simplex containing M and contained in K ? A positive answer would imply an affirmative answer of Borsuk’s problem. In view of approximation procedures, Stefani (see [1097], [1091], and [1093]) studied measure-theoretic questions related to complete sets. And in [592] (continuing investigations from [591]) completions of regular simplices in En are characterized via Minkowskian measures of asymmetry. Following Eggleston [310], we say that a convex body K in En of diameter h has the spherical intersection property if K is the intersection of all balls B(x, h) of radius h with center x ∈ K . It is easy to see that any body of constant width h has this property, and via completeness (see Chapter 7) also the converse is trivially shown. Hence a convex body K has the spherical intersection property if it is of constant width (we note that this equivalence no longer holds in Minkowski spaces, see Chapter 10). In papers like [806], [194], [127], [81], and [671] the relationships between constant width sets and sets which are convex in the sense of certain generalized convexity notions (like over-, super-, or hyperconvexity), the so-called adjoint transform, and the concept of ball polytopes is investigated. Going further to notions like ball hulls and ball intersections, one sees that these concepts are especially interesting in Minkowski geometry, see Sections 7.3 and 10.4. The same holds true for the concept of pairs of constant width, see [1001] and [747]. Here we have to refer also to the Notes of Chapter 4. Pairs of Constant Width Maehara [747] defined pairs of constant width as pairings (X, Y ) of compact, convex sets whose sum h(X, u) + h(Y, −u) of support functions for all directions u equals a constant r > 0, thus yielding that X + (−Y ) is a ball. He proved that (X, Y ) is such a pair if the intersection of all balls of radius r with center in X equals Y and, vice versa, X is obtained analogously from Y (examples of such pairs are constructed, too). In addition, Maehara [747] showed that the insphere of X and the circumsphere of Y are concentric (as they are in the usual constant width situation X = Y ). We also refer to Section 7.3.2 and also to the part “completeness” of Section 10.7. Earlier investigations in this direction are due to Valette [1141]. He also obtained (as a corollary) that for any compact, convex set X with twice continuously differentiable support function one can find a convex body Y of constant width such that X + Y is centrally symmetric. Sallee [1002] extended the notion of pairs of constant width to that of S-pairs (useful also for Minkowski spaces), meaning that h(X, u) + h(Y, −u) yields the support function of a prescribed centrally symmetric, convex body S (instead of a usual ball). Among other results he showed that X is a Minkowski summand of S if there is a set Y such that (X, Y ) is an S-pair. The same author continued these considerations in [1001], involving also circumradii, inradii, and the spherical intersection property. The results of [1002] were extended in [167] replacing S by a set which, in general, is nonsymmetric (thus referring to constant width sets for gauges, which are more general than norms). Let S be a nonempty compact, convex set in the plane (not a singleton), and for each integer k let Fn be the set of points each having at least k farthest points in S. Among other things, the authors of [292] proved that T3 is countable and T2 is contractible to the circumcenter of S. For r > 0 let Cr be the set of points whose distance to a farthest point in S is r ; Cr is a strictly convex curve in case r is larger than the circumradius of S. The authors of [292] were unaware of the fact that the boundary of S and the curve Cr form the boundaries of a pair of constant width as defined in [747]. Also unaware of [747], the authors of [253] characterized pairs of plane constant width sets in terms of the Fourier coefficients of their radii of curvature. Various results for such pairs are proved, and also a generalization of Barbier’s Theorem due to Maehara [747] is reproved in [253]. Aitchison [6] calls two convex bodies equivalent if the ratio of their widths for equal directions is constant over all directions. He applies this notion to suitable parallel pairs of sections of strictly convex bodies in 3-space, concluding that the boundary of these bodies has to consist of

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finitely many boundary pieces of ellipsoids. (Also a characterization of ellipsoids due to Blaschke is generalized in this paper, but this is more related to our Section 2.12.) Reduced Bodies The definition of reduced bodies, going back to Heil [524], can somehow be considered as a dualization of that of complete sets. Recall that the latter notion can be defined as follows: A convex body in En is said to be complete (i.e., in En of constant width) if any proper superset of it has larger diameter. Replacing superset by subset and diameter (= maximal width) by thickness (=minimal width), we have on the other hand: a convex body in En is called reduced if any proper convex subset of it has smaller minimal width. But the family of reduced bodies forms a proper superset of that of complete bodies (see below). It turns out that this is no longer the case in Minkowski spaces, see [797], [180], and our Chapter 10. Groemer [456] generalized the concept of reducedness calling a convex body K ⊂ En f -minimal if, when f denotes an extended real-valued function on the family of convex bodies in En , any convex body K  properly contained in K satisfies f (K  ) < f (K ). In the cases when f denotes circumradius and inradius, all f -minimal bodies are determined in [456], and for smooth convex bodies and f standing for minimal width, f -minimality is shown to be equivalent to constant width. Other ways of generalization are given by leaving the concept of convex bodies, but staying with minimal width. For example, in [2] a connected arc of minimal width 1, say, is called reduced if any connected subarc of it has smaller minimal width. The authors want to determine, among all rectifiable planar arcs of minimum width 1, that arc having the least length. For example, the union of two adjacent sides of an equilateral triangle of altitude 1 has minimal width 1 and length 2.309...; it is proved in [2] that the required minimum length is approximately 2.2783, achieved by a caliper-shaped arc described in the paper. The relation to Buffon’s needle problem is clarified, and the authors ask for the shortest connected union of arcs of minimum width 1, also in higher dimensions. We will survey now results on reduced bodies in Euclidean space. A wider representation is given in [701]. Since in En any constant width body is reduced and there are reduced sets not being of constant width, the superset statement should be shown by some examples of reduced sets which are not of constant width (see also [677] and [701], § 1), starting with the planar case. For example, every regular odd-gon is reduced, and there are also non-regular reduced odd-gons with at least five vertices. Further on, if we take, for polar coordinates φ and r and each fixed k ∈ [ 21 , 1], the convex hull of points )/2 and r ∈ {1 − k, k}, we get a planar reduced body. In particular, such that |ϕ| ≤ (π − arc cos 1−k k for k = 21 and k = 1 we obtain the unit disk and the quarter of a disk as extreme examples. Both these examples generalize also to En , the latter as intersection of an n-ball, centered at the origin, and a closed orthant. If a planar reduced body has an axis of symmetry, we can rotate it about this axis to get a 3-dimensional reduced body of revolution, and this rotational procedure can be suitably generalized for the step from En−1 to En , see [701], § 2. The only centrally symmetric reduced bodies are the balls, see Claim 2 of [700], Remark 5.3 of [58], and also Theorem 7.4.2. A surprising result was obtained by Lassak [693]: Different to the planar situation, there are reduced bodies in En , n > 2 having fixed minimal width, but arbitrarily large finite diameter! It turns out that the family of reduced bodies in En has an interesting richness of forms, inspiring geometric research on it! Regarding applications (e.g., various types of extremal and containment problems related to the notion of thickness (= minimal width); see, e.g., [46] and [451]), the following fact, immediately following from Zorn’s lemma, is important: Every convex body of minimal width h contains a reduced body of the same minimal width h. By the following, still open problem an attractive example is shown (we repeat that this problem inspired Heil [524] to introduce the notion of reducedness, see also the discussion A 18 in [272]): Which convex bodies of fixed minimal width in En , n > 2, have minimal volume? In the spirit of our “dualization remark” above, this can be seen somehow as a dualization of the isodiametric problem which, in Euclidean like also Minkowski spaces, has the ball as solution (see [820]). (It asks for the point set of fixed diameter that has the largest volume.) The

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solution to Heil’s question in E2 is the regular triangle, see [910]. And for higher dimensions this problem was already discussed in [160], Section 44, and [481], pp. 260–261; a good candidate for n = 3 is discussed in [524] and [481], and it is depicted in [524] and [143], Fig. 2.13. Oudet [906] introduced new numerical methods to solve optimization problems for convex bodies involving also minimum width as geometrical constraint. Besides discussing the possible optimality of Meissner’s bodies, the developed algorithms are used to approximate the optimal solution that Heil proposed for n = 3. Further related investigations and results are presented in [356], [456], [209], [53], and [56], see also the discussion in [272, Problem A 18]. For example, in [209] it is shown that among all bodies of revolution in E3 the compact cone obtained as rotated equilateral triangle has, for fixed minimal width, minimal volume. We remind the reader here that even the restriction of this problem to bodies of constant width in En , n > 2, is still open (with the Meissner bodies as conjectured extremal bodies), see also Section 8.3. The restriction to rotational bodies of constant width yields the rotated Reuleaux triangle as the only extremal set for n = 3, see again [209] and also Theorems 14.2.2 and 14.2.3. Replacing volume by quermassintegrals Wi , i = 0, 1, ..., n − 1 (see [1039]), one can generalize the “dual isodiametric problem” staying within our framework, since the extremal bodies still have to be reduced (see [701], § 7]). For i = 1 we get the original problem, and for i = n − 1 the solution is trivially given by the family of constant width bodies. For i = 1 it was shown in [637] that for n > 2 the balls are the only convex bodies of given minimal width having smallest surface area. It would be essential to get a complete geometric picture how the boundary of a reduced body in En has to look like, e.g., also regarding symmetry conditions. Unfortunately, we are far away from this (except for the case n = 2). Helpful statements, which are proved in this direction, are the following ones: Through every extreme point of a reduced body R in En , one support hyperplane of a thickness strip passes, and every extreme point of R is one endpoint of a thickness chord of R. From these and many related observations (especially in [677] many of them are derived) one can get results of the following type: Every smooth reduced body in En is of constant width (claimed in [524] and proved by Groemer [456]), and every strictly convex reduced body in the plane is also of constant width (cf. [282]), see also Theorem 7.4.4. It is not known whether for n > 2 every strictly convex, reduced body in En is of constant width (unfortunately, the approach of Nikonorov [893] is erroneous). Let H be a support hyperplane through a boundary point x of a convex body K in En , and let B be a ball which is tangent to H at x; let N be a neighborhood of x such that the intersection of N and K belongs to B. Then B is said to support K , and if K is supported in this way almost everywhere in its boundary, it is called almost spherically convex. Dekster [283] proved that any reduced, almost spherically convex body in En is of constant width, see Theorem 7.4.6. We come now to results on reduced bodies in E2 . A very detailed study of their boundary properties is given in [677], see also [701], § 3. We call the union of two opposite sides of a nondegenerate convex quadrangle Q and the diagonals of Q a butterfly B F, and these two opposite sides the arms of B F. Based on this notion, in [677] (see also [328]) the following was established: the boundary of any reduced body R of minimal width h in the Euclidean plane consists of at most countably many pairs of “opposite” segments (namely occurring as pairs of arms of one butterfly, in each case) and of at most countably many pairs of opposite pieces of curves of constant width h. Thus R is in fact the convex hull of the endpoints of all its thickness chords. This inspired the following generalization of Blaschke’s result (see [131]) on the approximation of planar constant width bodies by constant width bodies whose boundaries consist only of circular arcs. Namely, Lassak [697] showed that for any planar reduced body R of minimal width h and an arbitrary ε > 0 there exists a reduced body R  whose boundary consists only of circular arcs of radius h and of arms of butterflies of R, such that the Hausdorff distance between R and R  is at most ε. Now we will discuss some metrical problems yielding also extremal reduced bodies. Since we are in a superset of the family of constant width sets, extremal sets from there (like the Reuleaux triangle) need no longer be extremal. In the following, we write d, A, and p for the diameter, area, and perimeter of a reduced body R of minimal width h. Already in [677] it was shown

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√ that the two inequalities d/ h ≤ 2 and p/ h ≤ 2 + 21 π hold, in both cases with equality only for the quarter of a disk. On the other hand, the first ratio has the sharp lower bound 1 satisfied only if R is of constant width, and the second ratio has the sharp lower bound π with equality only if R is a disk. Due to the surprising result on arbitrarily large finite diameters in En , n > 2, derived in [697], there are no higher dimensional analogues of these results. Since the regular triangle is reduced, by [910] it √ follows that the area of a reduced body R of minimal width h satisfies 13 3 · h 2 ≤ A, and conversely in [695] the upper bound h 2 for A was confirmed. A sharpening for reduced polygons is also given in [695], yielding the interesting problem whether each planar reduced body R of minimal width h satisfies A ≤ 14 πh 2 , with equality only for R a disk and a quarter of a disk. By Jung’s Theorem, any √ body of constant width h in the plane is contained in a disk of radius 13 3h. Lassak [697] showed that √ any planar reduced body R of minimal width h is contained in a disk of radius 21 2h, which is sharp for a quarter of a disk. It is well known that for each boundary point x of a set  of constant width h the disk of radius h and centered at x covers . Clearly, this does not hold for reduced sets in E2 , but one can ask for special positions of disks doing this. This yields the open question whether any planar reduced body R of minimal width h can be covered by a disk of radius h whose center is from the boundary of R. For reduced polygons this is confirmed in [327]. On the other hand, since a related result of Blaschke [128] refers to the regular triangle, we have from this that any planar reduced body of minimal width h contains a disk of radius h/3, characterizing this figure. Results on annuli formed by two concentric disks and containing the boundary of planar reduced sets are derived in [697]. Now we will discuss the geometry of reduced polygons. Their vertex number is odd, and under this supposition a polygon P is reduced with minimal width h if the orthogonal projection y of any vertex x onto the affine hull of the opposite side is an interior point of that side such that the distance between x and y equals h; moreover, x and√y always halve the √ perimeter of P (cf. [677]). For reduced polygons P also the inequalities d/ h ≤ 23 3 and p/ h ≤ 2 3 hold, both sharp only for the regular triangle. √ This implies that every reduced polygon is contained in a disk of radius 23 3 · h, sharp only for the regular triangle, and it yields also (see again [677]) that among all reduced m-gons of minimal width h and with m not larger than n, only the regular n-gon has minimal diameter and minimal perimeter. We note that the statement on the perimeter was extended by [46] to arbitrary convex polygons. For the area of reduced polygons the inequality A < 41 π · h 2 holds, which in general cannot be improved. We continue with results on reduced n-polytopes, n > 2. The intriguing question whether there exist reduced n-polytopes in En , n > 2, was posed in [677], see also [690]. It was shown in [792] that there are no reduced n-simplices (see [796] for n = 3, and for a large class of n-pyramids [61]). This is connected with the paper [128], where Blaschke erroneously assumed that the minimal width of a regular n-simplex is attained in the normal directions of its facets. (This statement, true only for n = 2, was corrected by Steinhagen [1090].) A deeper study of this problem was done in [60], where generalized antipodality notions were introduced as tools, and also the extension to Minkowski spaces is considered. For the Euclidean subcase it was shown there that any n-dimensional polytope (n > 2) having r facets and s vertices is not reduced in the cases r = n + 2, s = n + 2, and s > r (therefore, no simple n-polytope is reduced). And in [700] centrally symmetric polytopes were excluded. After proving a new necessary condition for n-polytopes, n > 2, to be reduced, the authors of [437] surprisingly succeeded in constructing a 3-dimensional Euclidean reduced polytope!

Exercises 7.1. 7.2.

 Prove that x∈V B(x, h) consists precisely of the centers of all balls of radius h that contain V . Prove that V ⊂ B(V, h) if and only if the diameter of V is smaller than or equal to h.

Exercises

7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10. 7.11. 7.12. 7.13*. 7.14. 7.15. 7.16*. 7.17 7.18*.

7.19. 7.20*. 7.21. 7.22.

165

Let  be a convex body. Prove that B(bd , h) ⊂  if and only if  has the following property: if a ball B( p, h) contains , then its center p belongs to . that B(B(B(V, h))) = B(V, h). Let V ⊂ En be a compact set. Prove  For sets X, T ⊂ En prove that t∈T (X − t) = {y ∈ En | y + T ⊂ X }. For convex bodies , φ, and , prove that φ +  = φ +  implies  = . Prove that X ∼ (X ∼ T ) is the intersection of all possible translates of X that contain T . Prove that X ∼ (X ∼ (X ∼ T )) = (X ∼ T ). Prove that if X = B(0, h), then the strongly X -convex sets are the spindle h-convex sets and cc X (T ) = cch (T ). Prove that if (, ) is a pair of constant width h, then  +  is a body of constant width 2h. Prove that if (, ) is a pair of constant width h, then any insphere of  is concentric to the circumsphere of  and the sum of their radii is equal to h. Prove that if (, ) is a pair of constant width h in the plane E2 , then the sum of the perimeters of  and  is equal to 2πh. Describe a centrally symmetric convex body G and a strong G-convex set which is not a summand of G. Use Zorn’s lemma to prove that every convex body K contains a reduced body  of the same thickness. Prove that an equilateral triangle is a reduced body, but a square is not. Prove that a tetrahedron, and in general an equilateral n-simplex in En , is not reduced. Prove that the common part between the unit ball and the first octant is reduced. Let L be an (n − 1)-plane which is a hyperplane of symmetry of the reduced n-dimensional body φ. Is the (n + 1)-dimensional body of revolution obtained by rotating φ along L a reduced body? Prove that every reduced convex figure has constant minimum width. Prove or disprove: every strictly convex body contains a strictly convex reduced body of the same thickness. Prove that every φ-strip through a regular point p of the boundary of a strictly convex reduced body φ is a thickness strip. In the context of the proof of Lemma 7.4.1, prove that limsi →0 si =  is a support hyperplane of φ.

Chapter 8

Examples and Constructions

It is the enjoyment of shape in a higher sense that makes the geometer. Alfred Clebsch

This chapter is dedicated to concrete examples of constant width sets and procedures on how to construct them. The most notorious convex body of constant width is undoubtedly the Reuleaux triangle of width h which is the intersection of three disks of radius h and whose boundary consists of three congruent circular arcs of radius h. In Section 8.1, we will see that the Reuleaux triangle can be generalized to plane convex figures of constant width h whose boundary consists of a finite number of circular arcs of radius h. They are called Reuleaux polygons. The plan for the rest of the chapter is the following: In Section 8.2, we will study the 3-dimensional analogue of the Reuleaux triangle, and in Section 8.3, we will construct Meissner’s mysterious bodies from it. In fact, in this section, we will use the concepts of ball polytope and Reuleaux polytope to construct 3-dimensional bodies of constant width with the help of special embeddings of self-dual graphs. In Section 8.4, we will give a procedure of finitely many steps to construct 3-dimensional constant width bodies from Reuleaux polygons, and in Section 8.5, we will construct constant width bodies with analytic boundaries.

8.1 Reuleaux Polygons The Reuleaux triangle of width h can be generalized to Reuleaux polygons. A Reuleaux polygon is a plane convex figure of constant width h whose boundary consists of a finite (necessarily odd) number of circular arcs of radius h. Let us present a procedure to construct all Reuleaux polygons. Let P be a plane convex polygon of diameter h with an odd number of vertices, i.e., with vertex set V = {x1 , . . . , x2n , x2n+1 }, n = 1, 2, . . . . Suppose opposite chords are diameters of P, i.e., suppose that for every k = 1, . . . , 2n + 1 modulo 2n + 1, we have |xk − xn+k | = h = |xk − xn+k+1 |.  Then, the “ball polygon” φ = 2n+1 B(xk , h) is a convex figure of constant width h in which the 1 vertices of the polygon P coincide with the singular points of the boundary of φ, and this boundary consists of the union of 2n + 1 circular arcs. Indeed, for every k = 1, . . . , 2n + 1, the shortest circular arc, centered at xk and between xn+k and xn+k+1 (mod(2n + 1)), belongs to the boundary of φ, see

© Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_8

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8 Examples and Constructions

Figure 8.1

Figure 8.1. We leave it as an exercise for the reader to prove that this procedure gives rise to the construction of all Reuleuax polygons. Note that with the exception of a finite number of points, the curvature of every point of the boundary of a Reuleaux polygon of width h is 1/ h. So it is surprising that every plane figure of constant width can be approximated in an arbitrarily close sense by Reuleaux polygons, as shown in the next theorem. Theorem 8.1.1 If  is any plane figure of constant width h, then a Reuleaux polygon can be constructed which is, regarding the Hausdorff metric, arbitrarily close to φ. Proof For the proof we follow closely [1204]. Let us consider a polygon x1 x2 . . . x2n with the following properties: 1) Opposite sides xk xk+1 and xn+k xn+k+1 , 1 ≤ k ≤ n, are parallel and at distance h apart, 2) for every 1 ≤ k ≤ 2n, there is a point yk in the side xk xk+1 with the property that yk yn+k is orthogonal to xk xk+1 and xn+k xn+k+1 , and consequently the distance between yk and yn+k is h. We allow the possibility in which xk = yk = xk+1 and x2n+1 = x1 . Under these circumstances, we are going to construct a Reuleaux polygon which is inscribed in the polygon y1 y2 . . . y2n and circumscribed about the polygon x1 x2 . . . x2n . For that purpose, choose a circular arc of radius h between yk and yk+1 and a curve composed of two circular arcs joining yn+k and yn+k+1 in such a way that the new curve has constant width h and remains inscribed in the polygon y1 y2 . . . y2n . Let z k be a point whose distance from yk and yk+1 is equal to h. Furthermore, choose z k in such a way that it lies in the convex hull of the following four points: {yk yn+k ∩ yk+1 yn+k+1 , yn+k , xn+k+1 , yn+k+1 }. Now, for every 1 ≤ k ≤ n, draw a circular arc of radius h about z k that joins yk and yk+1 , a circular arc of radius h about yk that joins yn+k and z k , and a circular arc of radius h about yk+1 that joins z k and yn+k+1 , see Figure 8.1. It is evident that the new curve φ obtained this way is a curve of constant width h. In fact, if one of two parallel support lines L and L  of φ touches the arc yn+k z k , then the other passes through the vertex yk : if L goes through z k , then L  is tangent to the circular arc yk yk+1 ; if L touches the circular arc z k yn+k+1 , then L  passes through the vertex yk+1 . Therefore, by Theorem 3.1.3, φ has constant width h. Furthermore, φ remains inscribed in the polygon y1 y2 . . . y2n and circumscribed about the polygon x1 x2 . . . x2n . If  is a figure of constant width, let us choose a polygon with vertices y1 , y2 , . . . y2n in the boundary of  and with the property that the distance between opposite vertices yk and yn+k , 1 ≤ k ≤ n, is h. Let L 1 , L 2 , . . . L 2n be support lines to  at y1 , y2 , . . . y2n , respectively, in such a way that opposite support lines L k and L n+k , 1 ≤ k ≤ n, are parallel. Of course, we must allow the possibility that yk = yk+1 . Finally, let {xk } = L k−1 ∩ L k .

8.1 Reuleaux Polygons

169

Figure 8.2 The Reuleaux tetrahedron T

First note that,  is inscribed in the polygon y1 y2 . . . y2n and circumscribed about x1 x2 . . . x2n . Moreover, by the first part of the proof, it is possible to construct a Reuleaux polygon φ which is also inscribed in the polygon y1 y2 . . . y2n and circumscribed about x1 x2 . . . x2n . If, in addition, we choose the polygon x1 x2 . . . x2n in such a way that the distance between yk and yk+1 (1 ≤ k ≤ 2n), y2n and y1 , is smaller than , then the Hausdorff distance between  and φ tends to zero when  tends to zero. 

8.2 The Reuleaux Tetrahedron Let us start by considering a regular tetrahedron with sides of length h, say, in E3 . As in the case of the Reuleaux triangle in the plane, take the four balls of radius h and with centers at the vertices of the tetrahedron. The intersection of these four balls is known as the Reuleaux tetrahedron and will be denoted by T . As we will observe, it turns out that the Reuleaux tetrahedron T is a strictly convex body whose boundary has the same face structure as a regular tetrahedron, but with curved faces and edges. It has four “spherical triangles” intersecting at six “circular arcs”, where the center of each spherical portion is at the opposite vertex of the tetrahedron and the center of each circular arc is at the middle point of opposite sides of the tetrahedron (see Figure 8.2). To be more precise, let {a, b, c, d} be four equidistant points in Euclidean 3-space whose pairwise distance is h, and consider their convex hull, which is a tetrahedron abcd. Let B(a, h), B(b, h), B(c, h), and B(d, h) be the balls of radius h with centers at a, b, c, and d, respectively, and their boundaries be the spheres S(a, h), S(b, h), S(c, h), and S(d, h). Hence, the Reuleaux tetrahedron T is defined as T = B(a, h) ∩ B(b, h) ∩ B(c, h) ∩ B(d, h). describe the boundary of T . Note that S(a, h) ∩ S(b, h) is the circle of radius √ Our first purpose is to 3h/2 with center at a+b and contained in the plane orthogonal to the side ab. Clearly, c and d, the 2

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8 Examples and Constructions

Figure 8.3

 the shortest subarc of other two vertices of the tetrahedron, lie in this circle; hence we denote by cd this circle, between c and d (cf. Figure 8.3). Analogously, we obtain six circular arcs, each one for every side of the tetrahedron abcd, and all of them contained in the boundary of T . The union of ˜ ∪ cd,  ∪ ac  ∪ bc  ∪ bd  consists of the singular points of the boundary these six circular arcs, ab  ∪ ad of T . Besides these singular points and the vertex points {a, b, c, d} all other points of the boundary of T are regular points and lie in four spherical triangles bounded, each of them, by three of these     circular arcs. Each of these four spherical triangles S(a, h), S(b, h), S(c, h), and S(d, h) is contained in S(a, h), S(b, h), S(c, h), and S(d, h), respectively. Next, let H be the plane through the face abc of the tetrahedron. Then H ∩ T is a Reuleaux triangle,  h) the circular arc of radius h between b and c with center see Figure 8.4. Denote by  (bc)a ⊂ H ∩ S(a, at a, which is part of the boundary of the Reuleaux triangle H ∩ T . Similarly, we may define the other twelve circular arcs, with the other three planes through the corresponding faces of the tetrahedron abcd. Let us denote by (bc) the region of the boundary of T between the circular arc  (bc)a and the   circular arc (bc)d , observing that bc lies precisely in the middle of this region. The other six regions are defined analogously. Our next purpose is to describe the normal chords at each point of the boundary of T and to detect which of them are binormals. We may start by considering a vertex of T , say a, and observe that the normals of T at a are precisely the lines through a and any point of the face bcd. In order to see this, consider the planes through a orthogonal to ab, ac, and ad, and the three half-spaces determined by them that contain T . The intersection of these half-spaces is a tangent cone of T , with vertex a as apex and normal lines at a, which are precisely the lines through a and any point of the face bcd. Further on, we shall observe the behavior of a regular point r of T ; this point r lies in one of these  cd,  and spherical triangles. Assume, for example, r ∈  Sa := S(a, h) ∩ T , the surface bounded by bc,  db. Consequently, the normal chord of T at r is the unique chord Jra = ra. If, in addition, the chord Jra passes through a point in the face bcd of the tetrahedron, this chord becomes a binormal. If in contrast Jra does not intersect the face bcd, then the chord Jra is normal at r , but not at a, and therefore it is not a binormal. In fact, the corresponding diametral chord of T , with one extreme point being the regular point r , has the other extreme point in one of the circular arcs of singular points that arrive at a.  different from c and d, a normal Finally, if we assume that p is a point of the circular arc, say cd, chord of T at p is any line through p and some point x ∈ ab. Thus, there are as many normal lines

8.2 The Reuleaux Tetrahedron

171

Figure 8.4

as points x in ab. Now, let us note that except for the case of the chord through the middle point in  and the middle point of the arc ab,  the corresponding normals are not binormals. If L is the the arc cd √ √ c+d and , then L ∩ T is a binormal of length ( 3 − 2/2)h, which is approximately line through a+b 2 2 √ √ (1.02)h > h. We clearly have three of these binormals, each of length ( 3 − 2/2)h. If we call a point q abnormal in the boundary of T when the normal chord at q is not a binormal of length h, by the previous discussion we can conclude that every point in the interior of the surfaces (ab), (ac), (ad), (bc), (bd), and (cd) is abnormal, see the red patches in Figure 8.4. We know that the diameters and the thickness chords √ of√a convex body are binormals. Thus, our previous discussion proves that the diameter of T is ( 3 − 2/2)h and the thickness is h. It is not very difficult to calculate the surface area of T . In fact, Weisstein [1196] was able to use the Gauss–Bonnet formula to obtain, for h = 1, S(T ) = 8π − 18 cos−1 (1/3) = 2.975471... , and for the volume V (T ) =

√  1 √ 3 2 − 49π + 162 tan−1 2 = 0.42215773... . 12

8.3 Meissner Bodies 8.3.1 Original Meissner Bodies The first nonspherical 3-dimensional body of constant width can be generated by rotating the Reuleaux triangle around one of its axes of symmetry, see Figure 8.5.

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Figure 8.5 Rotated Reuleaux triangle

In 1911, this body of revolution appeared for the first time in a catalogue of mathematical models produced by Martin Schilling [1029]. There he showed also the first non-rotational body of constant width defined by Ernst Meissner. It is worth mentioning that inspired by this Meissner body the famous painter Man Ray came up with his work “Hamlet”, see “Man Ray-Human equations: A Journey from Mathematics and Shakespeare” and Chapter 18 for more information. As in the case of the Reuleaux triangle, the Reuleaux tetrahedron is the intersection of four balls centered at the vertices of a regular tetrahedron. This time the obtained convex body does not have constant width. Meissner and Schilling [818] showed how to modify the Reuleaux tetrahedron to obtain a convex body of constant width. They replaced three of its edge arcs by curved patches formed as the surface of rotation of a circular arc, see Figure 8.6. Now we will describe this procedure more detailed.

Figure 8.6 Schilling’s plaster model of a Meissner solid (University of Toronto Libraries)

8.3 Meissner Bodies

173

Figure 8.7

8.3.2 Performing Surgery to the Reuleaux Tetrahedron As we have noticed in the previous section, T is not a convex body of constant width. In this section, we will perform a series of trimmings of T to eliminate the abnormal points. We will denote by W (b, c) the wedge along the segment bc, the surface of revolution with axis  d . Thus, the wedge W (b, c) is the union of all  a and the circular arc bc bc between the circular arc bc  see Figure 8.7. circular arcs of radius h between b and c with centers at points e of the circular arc ad, Let us modify the boundary of T , namely replacing (bc) by the wedge W (b, c). The convex hull of the surface obtained by this change is denoted by Tbc . Consequently, we say that we obtain Tbc by performing surgery on T along (bc). Note that Tbc is a convex body contained in T . We will now analyze Tbc under the following aspects: 1) Detect the singular points of the boundary of Tbc . 2) Describe for every point p of the boundary of Tbc its normal chords. 3) Characterize the region in which the abnormal points of the boundary of Tbc lie. 1) Being part of a surface of revolution, the points at the interior of W (b, c) are regular points of Tbc . Furthermore, the wedge W (b, c) fits so well with bdT \ int(bc) that the points of the circular arcs  d are also regular points of Tbc . Therefore, the boundary of Tbc has only four vertex points  a and bc bc a, b, c, d at its boundary, and five circular arcs of singular points.  then the normal chords of Tbc at e are binormals of length h 2) If e is a point of the circular arc ad, because the second extreme points of these binormals describe the circular arc of radius h between b and c with center at e. Hence e is no longer an abnormal point of Tbc . At the same time, if p is a point of the wedge W (b, c), then the normal chord of Tbc at p is a binormal pe of length h, where e is a  So, again, in this case p is not an abnormal point of Tbc . In the following, point at the circular arc ad. we shall study the normal chords at vertex points of Tbc . The normal chords of Tbc at a are precisely the intersections with Tbc of a line through a and any point of the face bcd. The analogous situation we have with the normal chords of Tbc at d. The situation, however, is different with the vertex points  and the plane H orthogonal to be at b. Since H is a support plane of c and b. Consider a point e ∈ da  a along the axis bc, then H is also a support plane of Tbc the body of revolution of the circular arc bc at b. This implies that the chord bx is a normal chord of Tbc at b if and only if x lies in the triangular  ac  b . Therefore, the regular points of  b , and cd spherical region of S b bounded by the circular arcs da,  ac  b are  b , and cd Tbc contained in the triangular spherical region of Sb bounded by the circular arcs da, not abnormal points of Tbc .

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8 Examples and Constructions

3) As noted above, the abnormal points of Tbc lie in the interior of the following region: (ab) ∪ (bd) ∪ (dc) ∪ (ca). The following Figure 8.8 from the classical book “Convex Figures” of Yaglom and Boltyanski (see [1204]) shows the surgery procedure.

8.3.3 A Description of Meissner Bodies The first Meissner body is obtained from the Reuleaux tetrahedron by performing the surgery along three concurrent edges. To be more precise, let M1 be the convex body obtained by performing the surgery to T along the surfaces (ab), (ac), and (ad). Due to the discussion in the subsection above, the boundary of the convex body M1 has four vertex points a, b, c, d at its boundary, and the  cd,  other singular points of the boundary of M1 lie in the circular triangle, whose circular sides are bc,  Since M1 does not have abnormal points, by Theorem 3.1.4, M1 is a body of constant width and db. h contained in T , see Figures 8.6 and 8.9. The second Meissner body M2 is obtained from the Reuleaux tetrahedron by performing surgery to T along the edges bc, cd, and dc. As per the discussion in the former section, the boundary of the convex body M2 has four vertex points a, b, c, d at its boundary, and the other singular points of the  ac  Since M2 does not have abnormal boundary of M2 lie at the concurrent circular arcs ab,  and ad. points, again by Theorem 3.1.3, M2 is a body of constant width h contained in T . In contrast to our geometrical proof, Meissner confirmed the constant width of his bodies using Fourier series. Like Hurwitz [566], he originally studied convex-closed curves inscribed in a regular polygon, which remain tangent to all the sides of the polygon during rotations of the curve. These curves are called rotors, see Section 17.1. He characterized their support function, which can be expanded in a Fourier series. With the analogous technique in three dimensions, he was able to determine the rotors of the cube as bodies of constant width. He proved that nonspherical surface rotors of the cube, the regular tetrahedron, and the octahedron exist. In addition, it turned out that only spherical rotors exist for the regular dodecahedron and the icosahedron, see [819]. Both Meissner bodies have identical volume V and surface area S. When h = 1, their volume is given by

Figure 8.8 Figure from the classical book [1204] showing the surgery procedure described in 8.3.2

8.3 Meissner Bodies

175

Figure 8.9 A Meissner solid (University of Toronto Libraries)

V (Mi ) =

2 3

√  3 cos−1 (1/3) π = 0.419860 . . . , 4

which is considerably smaller than the volume of the rotated Reuleaux triangle R given by V (R) =

2 3

π π = 0.449461... . 6

In fact, it was conjectured in [160] that the Meissner bodies minimize volume among all 3-dimensional bodies of given constant width, see also Section 14.2 regarding Isoperimetric Inequalities and the Blaschke–Lebesgue conjecture. The surface areas of both Meissner bodies are also identical. We will see in Chapter 12 that for all 3-dimensional convex bodies  of constant width h, the equality 3h S() − 6V () = 2πh 3 holds. If h = 1, then √   3 cos−1 (1/3) π = 2.934115 . . . . S(Mi ) = 2 − 2 Furthermore, by the lower bound of the surface area of a body  of constant width h, obtained by Chakerian (Theorem 14.1.2), we have √ S() ≥ 2πh 2 ( 6) = h 2 (2.7928 . . . ). The Minwoski sum M=

1 1 M1 + M2 2 2

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8 Examples and Constructions

is a body of constant width with tetrahedral symmetry. Its volume, however, is larger than that of the Meissner bodies M1 and M2 , due to the Brunn–Minkowski inequality discussed in Section 2.7. Since the boundaries of Meissner bodies are unions of spherical caps of radius 1, or surfaces of revolution of circular arcs of the same radius, then we have that the smooth components of the boundaries of Meissner bodies have their smaller principal curvature constantly equal to 1.

8.3.4 Meissner Polyhedra A Meissner polyhedron is a body of constant width constructed from a Reuleaux polytope P ⊂ E3 by performing surgery in one of the edges of each pair of dual edges of an involutive self-dual graph G P , see Section 6.4. Let G P ⊂ bdP be the metric embedding of the  involutive self-dual graph G P performed by the singular points of the Reuleaux polyhedron P = x∈X B(x, h), where X is the set of vertices of the  ∈ E(G P ) such that x y ∈ E(G P ). There is dual edge ab graph G P . Let us concentrate on an edge  • |x − a| = |x − b| = |y − a| = |y − b| = h, • the edge  x y is contained in S(a, h) ∩ S(b, h), that is,  x y is the subarc of the circle with center at a+b , between x and y, and contained in the plane orthogonal to ab, 2  is contained in S(x, h) ∩ S(y, h), that is, ab  is the subarc of the circle with • similarly, the edge ab center at x+y , between a and b, and contained in the plane orthogonal to x y. 2 Denote by τ (x) the dual face of the vertex x in P. That is, τ (x) = S(x, h) ∩ P. By Lemma 6.1.2, τ (x) is a spherically convex-closed subset of the sphere S(x, h). As a subset of the boundary of P, the face τ (x) of the ball polytope P is bounded by a finite number of subarcs of  see Sections 6.2 and 6.4. circles, each one being an edge of G P . One of these edges is ab, We now follow closely the procedure described in Section 8.3 when we performed surgery in one of the edges of the Reuleaux tetrahedron. Let a ⊂ S(a, h) be the shortest geodesic joining x and y. By convexity of the faces of P, a ⊂ τ (a). Similarly, let b ⊂ τ (b) be the shortest geodesic joining x and y in S(b, h). Denote by (x y) the region x y is contained in (x y), and of the boundary of P between the arcs a and b . Note that the edge  with the exception of these points, all points of the boundary of P contained in (x y) are regular and belong either to the sphere S(a, h), or to the sphere S(b, h). This is so because by definition P is a standard ball polyhedron.  ∈ E(G P ) dual to The above implies, in particular, that if p is a point in the interior of an edge ab  x y ∈ E(G P ) and if pq is a normal chord of P at p, then either q is a vertex of P and the length of pq is h, or q belongs to (x y). We denote by W (x, y) the wedge along the segment x y, the surface of revolution with axis being the line containing the segment x y between the circular arcs a and b . Thus, the wedge W (x, y) is the union of all circular arcs of radius h between x and y with centers at points of the circular arc  contained in the boundary of P. We leave it as an exercise for the reader to prove that the wedge ab W (x, y) is contained in P. Let us modify the boundary of P by replacing (x y) with the wedge W (x, y). We shall denote by (P) the surface obtained from the boundary of P by performing surgery on one edge of each pair of dual edges of the involutive self-dual graph G P . The following theorem is due to Montejano and Roldan [853].

8.3 Meissner Bodies

177

Figure 8.10 Meissner polyhedra

Theorem 8.3.1 Let P ⊂ E3 be a Reuleaux polyhedron. The surface (P), obtained from the boundary of P by performing surgery on one edge of each pair of dual edges of the involutive self-dual graph G P , is the boundary of a constant width body. Proof The proof consists of two steps. In the first step, we prove that for every point p in the surface (P), there is a point q in the surface (P) such that | p − q| = h. In the second step, we prove that the diameter of the surface (P) is equal to h. If this is so, then by Theorem of Pál, 7.2.2, there is a body of constant width containing the surface (P) which, of course, has (P) as its boundary. For the first part of the proof, note that a point p in the surface (P) belongs either to a face τ (x), x y of G P . In the first case, for some vertex x of G P , or belongs to a wedge W (x, y) for some edge   of  | p − x| = h, and in the second case, | p − q| = h for some point q in the dual edge ab x y. For the second part of the proof, suppose pq is a diameter of (P). Then, the planes orthogonal to pq at either p or q are support planes of (P). Furthermore, since the surface (P) is contained in P, by the strict convexity of P, all the points strictly between p and q are interior points of P. At this point, we need a classification of the points of (P). First, we have the vertex points and  ∈ E(G P ). All second, we have the points of (P) which are in the relative interior of some edge ab other points of (P) are regular points. From this class, we have those which are in the interior of some face τ (x) of P and therefore belong to S(x, h) for some vertex x of P, those which are in the interior of a wedge W (x, y), and finally those regular points of (P) which are in the intersection of a wedge W (x, y) and a face τ (a). Suppose first that p is a regular point of (P). If p belongs to a face of P, then the chord pq is a normal chord of P at p and therefore there is a vertex z of P between p and q. Since z ∈ bdP and since every point strictly between p and q is an interior point of P, then z = q, and therefore the length of pq is h. Similarly, if p is a point of the wedge W (x, y), then the chord pq is normal to W (x, y) at  ∈ E(G P ) dual to  x y ∈ E(G P ) which is between p and p, and hence there is a point z in the edge ab q. As before, since z ∈ bdP, then z = q, and therefore the length of pq is h. Suppose now neither p nor q are regular points of (P). If both p and q are vertices of P, then the  length of pq is smaller than or equal to h; so we may assume that p is in the interior of the edge ab. If this is so, since pq is normal to the surface (P) at p, then it is also normal to P at p. Since q is either a vertex of P or lies in some edge of P, we have that pq is a chord of P; in fact, it is a normal chord P. Hence either q is a vertex and then the length of pq is h, or q belongs to (x y), which is impossible by construction. This completes the proof.  As we mentioned before, a body of constant width that can be obtained from a Reuleaux polyhedron P by performing surgery in one of the edges of each pair of dual edges of the involutive self-dual graph G P is called a Meissner polyhedron (see Figure 8.10).

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Since Sallee proved in [998] that every 3-dimensional smooth convex body of constant width h can be closely approximated by Reuleaux polyhedra with arbitrarily small edges, we have the following Theorem 8.3.2 Every 3-dimensional body of constant width can be closely approximated by Meissner polyhedra.

8.3.5 Meissner Bodies A Meissner body (or Meissner solid)  ⊂ E3 is a body of constant width with the property that the smooth components of the boundary of  have their smaller principal curvature constant. Clearly, every Meissner polyhedron is a Meissner body. The Blaschke–Lebesgue problem (see our Section 14.2) consists of minimizing the volume in the class of convex bodies of fixed constant width. Anciaux and Guilfoyle [31] proved the following Theorem 8.3.3 A minimizer of the Blaschke–Lebesgue problem is always a Meissner body. On the other hand, Shiohama and Takagi [1069] proved that a nonspherical surface with one constant principal curvature must be a canal surface, that is, the envelope of a one-parameter family of spheres or, equivalently, a tube over a curve (i.e., the set of points which lie at a fixed distance from this curve). They are made of spherical caps of radius h and surfaces of revolution over a circle of radius h, exactly like the Meissner polyhedra.

8.4 More Constructions 8.4.1 The Bull; A Concrete Example In the following two sections we follow closely the ideas of [853]. Let us consider the Reuleaux polygon P shown in Figure 8.11 whose boundary consists of five √ 3 arcs of a circle of radius 2. We will assume that P ⊂ E2 × {0} √⊂ E . The vertices {1, 2, 3, 4, 5} are such √ 2) = d(2, 4) = d(4, 3) = d(3.5) = d(5, 1) = 2 and d(1, 3) = d(3, 2) = 1. So 5 that d(1, B(i, 2). P = i=1

Figure 8.11 A Reuleaux polygon P with five vertices

8.4 More Constructions

179

Figure 8.12 The Voronoi diagram of P

Between the vertices of P, there are not 4 in a circle, but triples of vertices (x, y, z) with the property that the circle through {x, y, z} contains P completely inside it. We have: (3, 2, 5), (4, 3, 5) and (1, 3, 4). In fact, the circle through {4, 3, 5} is the circumcircle of P. Let us consider the Voronoi diagram for the farthest point. In other words, for a vertex i of P we introduce iˆ = {x ∈ P | d(x, i) ≥ {d(x, j) for 1 ≤ j ≤ 5} . Figure 8.12 shows this Voronoi diagram, where A is the center of the circle through {1, 3, 5} of radius ra , B is the center of the circle through {2, 3, 5} of radius rb , and C is the center of the circle through {1, 3, 4} of radius rc . The corresponding Delaunay triangulation with the three triangles 325, 435, and 134 is shown in Figure 8.13. √ √ E3 be the√3-dimensional ball of radius 2 and with center at i. For 1 ≤ j ≤ 5, let B 3 (i, 2) ⊂  = 51 B 3 (i, 2). We want to describe the geometry of the boundary Consider the ball polyhedron P  see Section 6.2. of P, √ √ Let V = {y ∈ E3 | {1, 2, 3, 4, 5} ⊂ B 3 (y, 2) and | S(y, 2) |≥ 3} be the set of vertex singu3   lar points of the boundary √ of P. The set √ of 1-singular points of bd P is given by the set {y ∈ E | 3  are the {1, 2, 3, 4, 5} ⊂ B (y, 2) and | S(y, 2) |= 2}, and the regular points of the boundary of P

Figure 8.13 The Delaunay triangulation of P

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8 Examples and Constructions

√ √ √ points {y ∈ E3 | {1,√2, 3, 4, 5} ⊂ B 3 (y, 2) and | S(y, 2) |= 1}, where S(y, 2) is the spherical boundary of B 3 (y, 2).  consists of an embedding of a graph Consequently, the set of singular points S of the boundary of P G P whose vertex set is V and whose edges are subarcs of circles, each of them joining a pair of points  consists of the regular points of P,  of V . The complement of this graph G P = S in the boundary of P whose open components are spherically convex opensubsets of a sphere of radius h.   In our case, let A = (A, 2 − ra2 ) ∈ E3 , B = (B, 2 − rb2 ) ∈ E3 and C = (C, 2 − rc2 ) ∈ E3 , and let A , B and C be their reflections with respect to the {x y}-plane, respectively. Then, the vertices of  C   AB,  A  A B , G P are the points {A, B, C, A , B , C }, and the edges of G P are the nine arcs AC, √      , B4B this ball polyhedron has five faces: 1˜ ⊂ B 3 (1, 2), 2˜ ⊂ B1B √ , C5C , C2C√, and A3A . Of course, √ √ B 3 (2, 2), 3˜ ⊂ B 3 (3, 2), 4˜ ⊂ B 3 (4, 2), and 5˜ ⊂ B 3 (5, 2). Figure 8.12 can also be interpreted  from the top, where each face has different color. as a view of P Define  ∩ {z ≥ 0} , P+ = P and for every 1 ≤ i ≤ 5, let σi = ˜i ∩ {z ≥ 0} be the five spherical faces of the boundary of P + . Lemma 8.4.1 The convex body P + has the following properties: √ 1) The diameter of P + is 2, 2) for every point x ∈ bdP√+ ∩ {z > 0}, there is a point y in the boundary of the Reuleaux polygon P such that d(x, y) = 2. Proof The second statement is obvious. For the proof of the first statement, let x y be a diameter of P + . Then we may assume, without loss of generality, that x belongs to the boundary of√the polygon P and a singular y ∈ bdP + ∩ {z > 0}. If y is a regular point of the boundary of P + , then d(x, y) = 2. If y is√  point of P + , then, since x is in the boundary of P, x is a vertex of P and hence d(x, y) = 2. √ By Theorem 7.2.2 of Pál, there is a body ϒ of constant width 2 containing P + . As we will see soon, it is unique and will be called the bull. Furthermore, by the above, bdP + ∩ {z > 0} is contained in the boundary of ϒ. We know that ϒ ∩ {z ≥ 0} = P + . We want to describe ϒ ∩ {z ≤ 0}, the bottom of ϒ. It turns out that the bottom of ϒ is determined by the binormals at singular points of the boundary for every point x ∈ bd ϒ ∩ {z ≤ 0} there is a singular point y ∈ bdP + ∩ {z > 0} such of P + . That is, √ that d(x, y) = 2. First note that the points belonging to a normal line of P + at A form a set equal to the cone A with apex the point A though the triangle 135, the points belonging to a normal line of P + at B unify to a set equal to the cone B with apex the point B through the triangle 235, and the points belonging to a a set equal to the cone C with√apex the point C through the triangle normal line of P + at C form √ √ 134. Therefore σA = A ∩ S(A, 2) ⊂ bd, σB = B ∩ S(B, 2) ⊂ bd, and σC = C ∩ S(C, 2) ⊂ bd.  then the points belonging to a normal line of P + at x form the cone with Note now that if x ∈ AB,  is the intersection between the face σ3 apex x through the segment 35. This is so because the arc AB and the face σ5 of bdP + . This implies that in the boundary of ϒ, the face σA is connected with the face σB through a surface of revolution with axis the line through the segment 35. Similarly, in the boundary of ϒ, the face σA is connected with the face σC through a surface of revolution with axis the line through the segment 34. Note also that for the same reason, in the boundary of ϒ, the face σA is connected with the face σ3 through a surface of revolution with axis the line through the segment 45, the face σB is connected with the face σ4 through a surface of revolution with axis the line through the segment 23, the face σB

8.4 More Constructions

181

Figure 8.14

is connected with the face σ1 through a surface of revolution with axis the line through the segment 25, the face σC is connected with the face σ5 through a surface of revolution with axis the line through the segment 13 and, finally, the face σC is connected with the face σ2 through a surface of revolution with axis the line through the segment 14. Figure 8.14 shows ϒ from the bottom. So the boundary of ϒ consists of eight spherical faces or caps σA , σB , σC , σ1 , σ2 , σ3 , σ4 , and σ5 and seven surfaces of revolution connecting them. This completely describes the boundary of . A picture of  is shown in Figure 8.15. Let us consider the ball polyhedron P=

5

B 3 (i,

2) ∩ B 3 (A,

2) ∩ B3 (B,

2) ∩ B3 (C,

2).

1

From the above discussion it is easy to see that {1, 2, 3, 4, 5, A, B, C} are the vertices of P. So P is a Reuleaux polyhedron whose involutive self-dual graph G P is shown in Figure 8.16. Thus, actually {1, 2, 3, 4, 5, A, B, C} ⊂ E3 is a metric embedding of G P , and of course, the bull ϒ can be obtained from the Reuleaux polyhedron P by performing surgery in one edge of every dual pair of edges of P, as shown in Section 8.3.4.

Figure 8.15

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Figure 8.16 A metric embedding of the self-dual graph G P

8.4.2 Constructing Bodies of Constant Width from Reuleaux Polygons As in the previous section, let P be a Reuleaux npolygon in the plane with vertex set { p1 , p2 , . . . , pn }. B( pi , 1). Of course, n is an odd integer. Suppose P = i=1 Let us consider the farthest point Voronoi diagram of P, see [189]. In other words, for a vertex pi of P denote by i the set of all point x ∈ P with the property that d(x, i) ≥ {d(x, j) | 1 ≤ j ≤ n}. As in the case of the previous section, this gives a cell decomposition of P into n convex cells, each of them containing the original arc of S1 ( pi , 1) ∩ P. The boundary between any two of these convex cells is a straight line edge, and the collection of these edges gives rise to the embedding of a tree with straight line edges and vertices denoted by V˜ (P), see Figure 8.12. The farthest point Delaunay triangulation of P is the planar dual of the farthest point Voronoi diagram of P, see [320]. It gives rise to a family F of subsets of { p1 , . . . , pn } whose convex hulls divide cc({ p1 , . . . , pn }), see Figure 8.13. It turns out that A ∈ F if and only if | A |≥ 3, and there is a disk B(xa , ra ) with center at xa and radius ra containing P whose boundary intersects P exactly at A. Furthermore, the collection {x A | A ∈ F} is precisely V˜ (P) \ { p1 , . . . , pn }. Now consider the 3-dimensional ball polyhedron =

n

B 3 ( pi , 1) ,

i=1

and consider the graph G  of singular points of the boundary of . As in the previous section, let Vert() be the singular vertex points of the boundary of . That is, Vert() = {y ∈ E3 | {1, . . . , n} ⊂ B 3 (y, 1), and | S(y, 1) |≥ 3} be the set of singular vertex points of the boundary of . The set of 1-singular point of bd is given by the set {y ∈ E3 | {1, . . . , n} ⊂ B 3 (y, 1) and | S(y, 1) |= 2}, and the regular points of the boundary of  are the points {y ∈ E3 | {1, . . . , n} ⊂ B 3 (y, 1) and | S(y, 1)|= 1}, where S(y, 1) is the spherical boundary of B 3 (y, 1). It is easy to verify that Vert() = {(x A , ± 1 − r A2 ) | A ∈ F} and that the orthogonal projection of the cell decomposition of the boundary of  coincides precisely with the cell decomposition of the Voronoi diagram for the farthest point discussed above.

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183

Writing {z ≥ 0} for the closed upper half-space, we define P + =  ∩ {z ≥ 0} , and for every 1 ≤ i ≤ n, let σi = S2 ( pi , 1) ∩ P + be the spherical face of the boundary of P + . Then, following the proof of Lemma 8.4.1, the convex body P + has the following properties: • the diameter of P + is 1, • for every point x ∈ bdP + ∩ {z > 0}, there is a point y in the boundary of the Reuleaux polygon P such that |x − y| = 1. By the Theorem 7.2.2, which is due to Pál, there is a unique body  of constant width 1 containing P + . Furthermore, by the above argumentation, bdP + ∩ {z > 0} is contained in the boundary of  and  ∩ {z ≥ 0} = P + . We want to describe the bottom of , namely  ∩ {z ≤ 0}. It turns out that the bottom of  is determined by the binormal chords at singular points of the boundary of P + . That is, for every point x ∈ bd  ∩ {z ≤ 0}, there is a singular point y ∈ bdP + ∩ {z > 0} such that |x − y| = 1. Recall that F is the farthest pointDelaunay triangulation of P, and let A ∈ F. The set of normal

lines of P + at the vertex vA = (zA , 1 − rA2 ) is equal to the cone  A with apex the point vA through ccA; therefore σ A =  A ∩ S(vA , 1) ⊂ bd is a spherical cap face of . Our next purpose is to show that as in the above section, two spherical cap faces of , σ A and σ A  , are connected by a surface of revolution whose axis is the line through the line segment pi p j , where 1 ≤ i < j ≤ n and pi p j is an edge of the Delaunay triangulation F. Assume that v, v  are two vertices  . We wish to adjacent in the tree defined by the Voronoi diagram of P. Let x be a point on the edge vv determine the set of normal lines of P + at x. There are two possibilities. If v = c A and v = c A for some A, A ∈ F, then A and a  share a side of the form pi p j . Thus, the set of normal lines of P + at x is equal to the cone with apex x through the segment pi p j . This implies that, in the boundary of , the face σa is connected with the face σ A through a surface of revolution whose axis is the line through the segment pi p j . If v = c A for some A ∈ F and v  = pi for some i, we may reason similarly to conclude that, in the boundary of , the face σ A is connected with the face σi through a surface of revolution with axis the line through the segment of the form p j pk opposite to pi . These surfaces of revolution are illustrated in Figure 8.14 for the case of the Reuleaux pentagon. So the boundary of  consists of the spherical caps σ1 , . . . , σn , and σ A with A ∈ F and surfaces of revolution on the bottom part of  for every edge of the Delaunay triangulation of P. From the above discussion, it is easy to see that the vertex set of  is V = { pi | 1 ≤ i ≤ n} ∪ {vs | A ∈ F}. Thus φ=

B 3 (v, 1)

v∈V

is a Reuleaux polytope and therefore induces an involutive self-dual graph G φ . Actually, V ⊂ E3 is a metric embedding of G φ and, of course,  is a Meissner polyhedron because it can be obtained from the Reuleaux polytope φ by performing surgery on one edge of each pair of dual edges of φ, as shown in Section 8.3.4. As a second example, we show in Figure 8.17 the Voronoi diagram and Delaunay triangulation of a Reuleaux polygon with 7 vertices. We can also interpret these two figures as the top and bottom view of the corresponding Meissner body.

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8 Examples and Constructions

Figure 8.17 A Meissner polyhedron constructed from a Reuleaux heptagon

8.4.3 The General Case In this section, we will consider the Euclidean n-space En to be contained in the Euclidean (n + 1)space En+1 as the set of points whose last coordinate is equal to zero. We shall denote by En+1 + the upper n+1 consists of all those points of E with last closed half-space of En+1 bounded by En . That is, En+1 + coordinate greater than or equal to zero. Furthermore, E denotes the last coordinate axis of En+1 , and n+1 \ En+1 if A ⊂ En+1 , we shall denote by A+ , A− and At , respectively, the sets: A ∩ En+1 + , A ∩ (E + ), and A ∩ {xn+1 = t}. Let  ⊂ En be a body of constant width 1. By Theorem 6.1.4, for every p ∈ bd, the n-dimensional ball B n ( p, h) of radius 1, centered at p, contains  and =

B n ( p, 1).

p∈bd 

For every ball B n ( p, 1) ⊂ En , let us consider the (n + 1)-dimensional ball B n+1 ( p, 1) ⊂ En+1 centered at p. Consider + =

B n+1 ( p, 1)+ .

p∈bd 

Lemma 8.4.2 The diameter of + is 1. Furthermore, for every point q of bd + \  there is a point p ∈ bd  such that the length of pq is 1.

8.4 More Constructions

185

Figure 8.18 B n+1 ( p, 1)+

Proof Let x y be a diameter of + . Since the orthogonal projection of + onto En is , then one of the endpoints of the diameter x y, say x, lies in En . Furthermore, since x y is a binormal of + , then x ∈ bd. The fact that + ⊂ B n+1 (x, 1) implies that the length of x y is 1. Let q be a point of bd+ \ . It will be enough to prove that there is a point p ∈ bd such that  ⊂ B n+1 ( p, 1) and q ∈ bdB n+1 ( p, 1). Let {qi } be a sequence of points in En+1 , outside + , converging to q. For every i > 0 / B n+1 ( pi , 1). Choose a subsequence such that pi j → p and there is a point pi ∈ bd such that qi ∈ n+1 n+1  B ( pi j , 1) → B ( p, 1). Clearly, q ∈ bdB n+1 ( p, 1), because by definition q ∈ B n+1 ( p, 1). Lemma 8.4.3 Let  be a convex body of constant width 1 containing + . Then the following properties hold. 1) 2) 3) 4)

bd + \  ⊂ bd . If pq is a diameter of  and p ∈ bd − , then q is a singular point of bd + . Every point of bd − is a regular point of bd . If q ∈ bd  has (0, 0, ...., 1) as an outer unit normal, then q is a singular point.

Proof 1) follows because by Lemma 8.4.2 for every p ∈ bd+ \ , there is a point q in + such the length of pq is 1. For the proof of 2), suppose that pq is a diameter of  and p ∈ bd \ bd + = bd− . We shall show first that q ∈ En+1 + . Suppose not. Let L be the line containing the diameter pq, and let x be the point in which L cuts En (x may be at infinity). Then it is well known that there is a diameter line of  through x. This implies that there is a diameter J of  which does not intersect pq but lies with pq in a common 2-dimensional plane. This is a contradiction because in this case the diameter of J ∪ pq is greater than 1. Now, suppose q is a regular point of bd+ . Then, by Lemma 8.4.2, p should be a point of the boundary of  contradicting our hypothesis. This completes the proof of 2). For the proof of 3), consider the following subset of En+1 . Let , θ > 0 be two positive real numbers and let  = {(cos t, sin t, 0, . . . , 0) | − ≤ t ≤ } ∪ {(1 − cos t, sin t cos θ, sin t sin θ, 0, . . . , 0) | 0 ≤ t < }. Then it is an exercise to prove that the diameter of  is larger than 1. Consider p ∈ bd \ bd + = bd − and suppose that p is a singular point of bd. Then, there is a subarc α of a circle of radius 1 centered at p such that α ⊂ bd. By 2) every point of α is a singular point of the boundary of , but hence the above claim contradicts the fact that the diameter of  is 1. Finally, for the proof of 4), assume that q is a regular point and let qp be the unique diameter of  at p. Hence the interval qp is parallel to (0, 0, ...., 1), and by Lemma 7.4.1, p must be in bd, which is a contradiction. 

186

8 Examples and Constructions

As a consequence of the above lemma, we have the following theorem. ˜ of constant width 1 that contains + . Theorem 8.4.1 There is a unique (n + 1)-dimensional body  Proof By the Theorem 7.2.2, due to Pál, and Lemma 8.4.2, there is a body of constant width 1  containing + . The uniqueness follows immediately from Lemma 8.4.3, 1) and 2). The unique (n + 1)-dimensional body of constant width 1 constructed from the n-dimensional body ˜ This construction is closely related to the construction  of constant width 1 will be denoted by . given by Lachand and Oudet in [677]. ˜ is the Reuleaux triangle. If  ⊂ En is a ball, then  ˜ is the Note that if  ⊂ E1 is an interval, then  body of revolution obtained from the Reuleaux triangle. If P is the Reuleaux pentagon as described ˜ is the Meissner in Section 8.4.1, then P˜ = ϒ. Furthermore, if  is the Reuleaux triangle, then  body M1 . Summarizing, we denote the described correspondence of the given n-dimensional with the unique (n + 1)-dimensional body of constant width 1 by 

˜ .

˜ we have the following description for ˜ For the upper part  ˜ + of , Next, we would like to describe . ˜ the  sections t orthogonal to the last coordinate axis E, in terms of the adjoint transform B(V, h) := x∈V B(x, h), see Section 7.3. Let 0 ≤ t ≤ R, where R > 0 is the circumradius of  and c ∈ En is the circumcenter. Then  ˜ t = B(, 1 − t 2 ) + (0, . . . , 0, t).  √ ˜ 0 = , for t = R,  ˜ R is the point {(c, 1 − R 2 )}, and for t > R,  ˜ t = ∅. Hence, for t = 0,  √ Furthermore, by 4) in Lemma 8.4.3 the point (c, 1 − R 2 ) is a vertex point of bd with external normal vector (0, . . . , 0, 1). By 3) in Lemma 8.4.3 the lower part − of  consists of regular points and depends completely on ˜ − , then, by 2) in the singular points of the boundary of the upper part. This holds because if p ∈ bd Lemma 8.4.3, p is a regular point of the boundary and the corresponding unique binormal pq through ˜ +. p is such that q is a singular point of the boundary of  ˜ Finally, we leave it as an exercise to the reader to prove that the group of symmetries of  and  ˜ if and only if there is a symmetry f : En → En coincide. That is, F : En+1 → En+1 is a symmetry of  of  such that F = f × I d. This implies that if  ⊂ En has constant width and is a body of revolution ˜ ⊂ En+1 . with respect to the axis L, then L × E is an axis of revolution for  On the same lines as in this section, Lachand and Oudet [677] describe a more general method to obtain constant width bodies.

8.5 Algebraic Constant Width Bodies In this section,  ⊂ E3 is a convex body whose boundary is C ∞ -differentiable The inverse Gauss map γ : S2 → bd 

8.5 Algebraic Constant Width Bodies

187

Figure 8.19 An algebraic body of constant width;  = 1/2

is the diffeomorphism that assigns to every unit vector u ∈ S2 the point γ(u) in the boundary of  for which u is an outward unit normal vector (see also Section 2.9). Furthermore, for every u ∈ S2

γ(u), u = P (u), where P (u) : S2 → E is the support function of . See also Sections 11.1 and 11.5. Note that the differential of γ at u ∈ S2 has as its image the subspace parallel to the tangent plane to  at γ(u), which is perpendicular to u. These two facts enable us to determine γ in terms of P . One has the following vector equation: γ(u) = P · u + ∇P (u) , where ∇P is the gradient vector field as a function on S2 . The above parametrization of the boundary of  is called the support parametrization, see Theorem 11.1.1. If we consider two parallel support planes of  orthogonal to ±u, they are the support planes of  at γ(u) and γ(−u), and the distance between these two planes, P (u) + P (−u), is the width of  in the direction u. Thus,  has constant width 2h if and only if its support function P satisfies P (u) + P (−u) = 2h. In particular, the function P0 = P − h must be an odd function on S2 . That is, P0 (−u) = −P0 (u), for all u ∈ S2 . Conversely, if P0 : S2 → R is an odd function then, by Theorem 11.5.1, P = P0 + h is the support function of a convex body of constant width 2h, provided h > 0 is sufficiently large. Alternatively, for all sufficiently small  > 0, the function P = 1 + P0 is the support function of a convex body  of constant width 2, see Exercises 8.18. In this section, we present the examples of algebraic constant width bodies found by Robert Bryant. For more see http://mathoverflow.net/questions/222159/algebraic-surface-of-constant-width/222161 No. 222161. To construct an algebraic constant width body, it suffices to take P0 to be an odd algebraic function on S2 . For example, if we take P0 , the restriction of a linear function in E3 to S2 will only give a convex body which is a translate of the unit ball B 3 (0, 1). Thus we need a slightly more complicated odd function.

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8 Examples and Constructions

Figure 8.20 An algebraic body of constant width;  = 3/4

Let P0 = x yz, where x, y, z are the standard coordinates of E3 . If  < 1, the convex body  is a smooth convex body of constant width 2, which is algebraic because it is parametrized by algebraic functions on the algebraic surface S2 . It is, therefore, the zero locus of an irreducible polynomial function f  (x, y, z) on E3 . A calculation using elimination theory (carried out by MAPLE) shows that f  has degree 20 and, for most values of , has more than 1000 terms. The two Figures 8.19 and 8.20 show the corresponding bodies for  = 1/2 and  = 3/4. If  = 1, then 1 is still an algebraic body of constant width 2, but it is no longer smooth. It is easy to show that 1 has the symmetry group of the regular tetrahedron. Since we have an explicit parametrization of 1 , it is easy to draw pictures using graphics software. Figure 8.21 shows how 1 looks like. For a parametrization of the boundary of 1 , see Section 11.1. If  > 1, then 1 +  p0 is not the support function of a convex body.

Figure 8.21 An algebraic body of constant width;  = 1

8.5 Algebraic Constant Width Bodies

189

More generally, by taking P = 1 + P0 , where P0 = λ1 λ2 λ3 , and each λi is a linear function of x, y, and z, one can produce a 7-parameter family of distinct algebraic bodies of constant width 2. Furthermore, we have the following approximation result due to Schneider [1038] Theorem 8.5.1 Every convex body  ⊂ En of constant width h can be arbitrarily closely approximated (in the Hausdorff metric) by bodies of constant width h with analytic boundaries (in fact, with algebraic support functions).

Notes Reuleaux Triangles The most popular noncircular set of constant width in the plane is the Reuleaux triangle, whose first mechanical application is ascribed by Reuleaux himself (see [968], § 155) to Hornblower, the inventor of the compound steam engine. We refer also to our Chapter 18 for many related results. Replacing boundary parts of planar constant width sets by arcs of circles, new sets of constant width may be obtained. This was used in [997] to show that if there is a set of maximal constant width whose interior misses a locally finite family of convex sets, then it is a Reuleaux polygon with at least one contact point on each edge. Analogously for lattices, a unique maximal set exists and forms a Reuleaux triangle. Schmitz [1030] showed that the total length of two orthogonal intersecting chords of a Reuleaux triangle of width 1 is larger than 1. This observation motivated Martini and Makai to conjecture and prove Theorem 4.4.2 (see [753]). For the corresponding investigations in normed planes, see [22], [25], and the notes to Chapter 10. Stability results on the Hausdorff distance of sets of constant width to Reuleaux triangles (and circles) are derived in [458], and Weissbach [1195] obtained the following characterization of Reuleaux triangles: A convex set K is called a covering set of a class Q of planar sets if any member of Q can be covered by a congruent copy of K ; K is particularly a minimal covering set if it cannot be replaced by a proper subset. It is shown in [1195] that there are minimal covering sets for the class of all planar sets of constant width, except for Reuleaux triangles. Chakerian [234] proved that if a convex body in the plane can be covered by a translate of a Reuleaux triangle, then it can be covered by a translate of any set of the same constant width, see Theorem 15.1.2. In [1094], it was shown that the sum of the diameters of three sets, each of diameter smaller than h, which together cover a set of constant width h, is greater than 2h. The main proof part depends on the special case of a Reuleaux triangle; namely, the perimeter of a triangle whose vertices belong to distinct arcs of a Reuleaux triangle of width h must be larger than 2h. For a planar, C 2 -smooth set  of constant width 1, say, and the subset of  consisting of all points lying on at least three diameters of , Makeev [759] showed that the area of this subset has an upper bound which is attained if and only if  is a Reuleaux triangle. In [516], the so-called m-diameter of a set S is defined as the supremum of the geometric mean of all Euclidean distances among m points from S (thus, m = 2 yields the ordinary diameter of S). It is proved that for constant width sets in the plane, circular disks have the smallest and Reuleaux triangles the largest 3-diameter. The Reuleaux triangle also occurs in extremum problems from distance geometry in metric spaces, see [936]. A type of “generalized Reuleaux triangle” is introduced in [1202]. Reuleaux Polygons and Related Notions Reuleaux polygons are defined as those planar sets of constant width h whose boundary consists of a finite, odd number of circular arcs of radius h; their centers are called the vertices of the Reuleaux polygon, and they also form the vertex sets of a star polygon, whose edges are the extremal segments of the diameter graph of the considered set (see [160], pp. 130–131). Reuleaux polygons were introduced

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8 Examples and Constructions

by Reuleaux himself (cf. [968] and also [1028]). Blaschke ([130] and [131]) showed that arbitrary sets of constant width h can be approximated arbitrarily closely by Reuleaux polygons of width h, see also [192], and for a quantitative version relating the Hausdorff distance to the odd number of sides, [603]. We also refer to Section 8.1. In [417], regular Reuleaux polygons are characterized among all constant width curves by an extremal property of the perimeter of circumscribed equiangular polygons. On the other hand, Zalgaller [1210] proved extremal properties of certain convex m-gons with the help of Reulaux polygons. In [868], Reuleaux polygons are discussed in connection with isodiametric-like problems for convex polygons, and in [672] a simple characterization of those geometric graphs is given which are diameter graphs of the vertex set of Reuleaux polygons. From this, a linear-time construction of Reuleaux polygons was obtained, clarifying the basis for traditional constructions of Reuleaux polygons involving the set of diameters. We mention here some further related algorithmical results. Continuing [227], the authors of [228] presented two linear-time ruler and compass constructions of figures of constant width h circumscribed about a given simple polygon whose diameter is h. Their approach is closely related to algorithmical work of Hershberger and Suri on the notion of circular hull, see [538]. Since circular intersections, circular hulls and their higher dimensional analogues carry over to normed planes and spaces and are important there (see e.g., the subsection on intersection properties in the Notes of Chapter 10), we mention here also the papers [773], in which the results from [538] are extended to normed planes, and [775], where the same is done also for nonsymmetric convex distance functions. A geometric graph all whose edges intersect is an intersector. In [669], close connections between intersectors (with segments of equal lengths) and curves of constant width are discussed, in particular also referring to Reuleaux polygons. Sgheri [1058] studied the deformation of regular Reuleaux polygons into non-regular ones. In [317], it is shown that among the largest equilateral triangles, which may be inscribed in different curves of constant width, that one inscribed in a “Reuleaux pentagon” is the smallest. In [331], a point x of a set of diameter 1 is called a “singularity” if there exist two different points in this set whose distance to x equals 1. By a clever construction, the author showed that every compact subset S of En with diameter ≤ 1 can be covered by a set Y of constant width 1 such that S and Y have the same set of singularities and that these are of the same dimension. By a related construction, in [927] constant width sets with a given number of singularities are generated. Hammer [502] presented a generalization of Reuleaux polygons for normed planes whose Euclidean subcase yields constant width sets formed by a finite number of circular arcs having different radii, see also [958], p. 167, and [398]. The study of constant width curves representable as involutes of hypocycloids or further types of curves goes even back to Euler, see [160], pp. 131–132, [408], [197], [674], and [1144] for related results, and see also Section 5.3.4. Silverman [1064] investigated indecomposability (in the sense of Minkowski addition) for certain classes of convex sets, and from her results it follows that Reuleaux polygons are indecomposable within the family of constant width sets in the plane, see also [603]. In many situations, Reuleaux polygons are also popular geometric objects for demonstrating geometric facts in a suggestive way. For example, in [1126] they are used as examples for observations around Holditch’s theorem. They play also a role for number-theoretic Favard-type problems concerning algebraic integers (see [680]): Depending on the notion of transfinite diameter and its relation to that of usual diameter, certain subsets of the complex plane suggest some interesting conjectures of a purely geometric nature, concerning also the transfinite diameter of Reuleaux polygons. For related results, we refer also to the papers [678], [679], and [681]. The concept of m-diameter (see above, where we refer to Reuleaux triangles) yields also results on Reuleaux pentagons. Namely, in [570] it is proved that the Reuleaux pentagons have the largest 5-diameter among all sets of given constant width.

8.5 Algebraic Constant Width Bodies

191

Constructions of Curves of Constant Width Let U be an open set in the plane, and let α be a real number, 0 < α ≤ π. Saroldi [1023] introduced the angle property of opening α for U : this property is fulfilled if every point on the boundary of U is the vertex of an angle of measure α whose interior does not intersect U . Thus, the angle property is a natural generalization of convexity, and it is equivalent to usual convexity when α = π (if U is connected). The main result in [1023] is that if U is such a set of diameter h, then the perimeter of U is less than or equal to πh/ sin 2( α2 ). This is the best upper bound when π2 ≤ α ≤ π. For α = π, any convex region of constant width h will have perimeter πh. If α < π, then one cannot achieve the upper bound for the perimeter, but one can approach it arbitrarily closely (at least for α ≥ π2 ) by a construction whose starting point is a region U of constant width h. In [1144], the notion of curve of constant width is extended in a constructive way, namely to all closed curves with multiple points such that each normal is a double normal, being normal to the curve at two points of it having constant distance. The author considers certain curvilinear polygons with an odd number of sides, with a cusp of the first type at each vertex, but such that each side is a simple arc without further singularities. Then multiple curves of constant width are constructed by describing an involute of such a polygon, and relations involving the total curvatures of both families of curves are given, see also our Chapter 11. In [408], constructions of constant width curves as involutes of certain curvilinear “star” arrangements of convex arcs (continuing [160], p. 132) are discussed. Geometrical constructions of curves of constant width are also used in computational geometry to derive time complexities of other problems, like polygon simplicity testing etc., see, e.g., [578]. Constructive approaches to rational ovals and rosettes of constant width formed by piecewise rational Pythagorean–Hodograph curves are presented in [8], see again also Chapter 11. A nice construction of a curve of constant width consisting of four arcs is given in [1203], where an isosceles trapezoid is the starting point and the two obtainable borderline cases are the disk and the Reuleaux triangle. Similarly, [786] showed how a planar constant width curve can be continuously constructed from a Reuleaux triangle while preserving the constant width property in all intermediate steps of this process. In [389], a necessary and sufficient condition for a convex arc of class C 2 is given to be extendable to a closed convex curve of constant width. Based on Eggleston [316], where it is proved that a compact set X of diameter h has constant width h iff X is equal to B(X ) = ∩{B(x, h) : x ∈ X }, Sallee [998] used the spherical intersection property to construct sets K (Y, D) of constant width h containing a prescribed set Y of diameter h and  determined B(xn , h) or B(Y ) and F = F by a sequence D = {xn } which is dense in En . Defining F0 = n n−1  / Fn−1 and then setting K (Y, D) = {Fn : n = 0, 1, 2, . . . }), he employs Fn−1 as xn ∈ Fn−1 or xn ∈ this construction (which was already used in [997] and [999] in a modified way) to define Reuleaux polytopes and to show that for the 3-dimensional case this class is dense in the family of constant width bodies regarding the Hausdorff metric, see Sections 7.2 and 10.3. In view of completing convex bodies (see Chapter 7), Schulte [1048] presented the following nice construction: For a convex body K of diameter 2 in En , there exists a body  of constant width 2 containing K such that every symmetry of K is one of  and every singular boundary point of  is a boundary point of K for which the set of antipodes in  is the convex hull of the antipodes in K . (As a consequence, the author proved Borsuk’s conjecture for convex bodies having no point as endpoint of more than one diameter.) In [1049] and [1050], these investigations are continued, see also Section 7.5 and the Notes to our Chapter 7. Bavaud [81] provided an exposition of the properties of the adjoint transform, associating to a set the intersection of all disks of given radius centered at the set. The relationships between this transform, the double adjoint transform, basic notions from convexity (like also completion of sets and the concept of constant width) are studied, see also Section 7.3. Further constructions of orbiforms from polygons are investigated in [228].

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Constructions in Higher Dimensions A century ago, Meissner introduced special convex bodies of constant width which, constructed from a regular tetrahedron, are now named after him. It is a well known and still unsolved conjecture that these Meissner bodies have the smallest volume among all 3-dimensional convex bodies of the same constant width, see also Sections 8.3 and 14.2. A well-written and nicely illustrated survey describing the Meissner bodies, presenting their history, and explaining some recent observations regarding that extremal problem is [612]. A short biography of Ernst Meissner is also there. In [545], it is asked whether it is possible to confirm this conjecture with the help of computers. However, besides Meissner bodies the “easiest” type of concrete 3-dimensional bodies of constant width are simply obtainable via rotation of axially symmetric 2-dimensional ones, taking an axis of symmetry as rotation axis. In [1136], volume and surface area of the body obtained in such a way from the Reuleaux triangle are computed and compared with the analogous quantities of Meissner bodies. The starting point of the Meissner construction, the intersection of four congruent balls called Reuleaux tetrahedron (which is clearly not of constant width), shows the necessity of rounding off curvilinear edges of such ball polytopes, see Section 8.2. Weissbach [1191] investigated the n-dimensional analogue, computing its diameter. This process of rounding off curvilinear edges was described by many authors, see, e.g., [160], pp. 135–136, [1204], § 7, and [151], pp. 378–379. Besides the tetrahedral case, this was studied by Gray [448] also for other types of pyramids. An older related paper is [192], where so-called “elementary bodies” in E3 as analogues of Reuleaux polygons are constructed, namely as intersections of balls of radius h, whose centers were chosen at the members of a finite point set or at points of circular arcs of radius h. It turns out that bodies of this type can approximate any 3-dimensional constant width body as precise as one wishes. These investigations were continued by Sallee [998] with another construction, but yielding the same approximation property. Note that constructions of this type, yielding Reuleaux polytopes, are also discussed in [853], see also Chapter 6. In [677], various characteristic properties of constant width bodies in En are derived, and a construction of several types of n-dimensional constant width bodies is presented, which has a given (n − 1)-dimensional constant width body as an orthogonal projection (one of the bodies obtained in this way is a classical Meissner body). For a similar construction see Section 8.4.3. However, in contrast to the constructions in [853], which give rise to finite procedures (see Sections 8.4.1 and 8.4.2), the procedures in [677] and Section 8.4.3 are not finite because they still need to intersect an infinite collection of balls. Danzer [276] constructed a 3-dimensional body of constant width h such that the minimum width of each of its 2-dimensional sections are smaller than h, see Section 9.2. It is also natural to ask for symmetry properties of bodies of constant width in higher dimensions. Usually, explicit higher dimensional constant width bodies presented until now have some symmetry properties, like rotational or tetrahedral symmetry. In [350], it is shown that there are analytic hypersurfaces of constant width in En whose symmetry group is trivial, and that, on the other hand, there are also ndimensional constant width bodies with analytic boundary, having the symmetry group of the regular n-simplex. (A proof gap in [350] can be filled via constructions presented in [982] and [1048].) For n = 3 it is also proved which subgroups of the rotational groups occur as symmetry groups of surfaces of constant width. Related to this and motivated by Borsuk’s problem, Rogers [982] proved that for any set S of diameter 1 and invariant under a group G of congruences, leaving a point invariant, there is a set of constant width 1 containing S and being invariant under G. Surprisingly, it is shown that in E8 a set of constant width 1, invariant under the symmetry group of a regular 8-simplex, exists that has no cover, invariant under this group, by nine sets of diameter smaller than 1. Coming back to Meissner bodies, we mention that they play also an important role in view of various geometric inequalities; see e.g., [591] and our Chapter 14.

8.5 Algebraic Constant Width Bodies

193

Analytic Representations Explicit representations of planar constant width sets in terms of polar coordinates are presented by Hammer [502], Kearsley [614], and Tennison [1122]; in the latter case also an analytic representation of smooth curves of constant width in terms of their curvature is given. Rabinowitz [956] applied the classical parametric representation of a plane convex curve in terms of its support function to derive curves of constant width. Using Mathematica, one of the simpler examples is converted to a polynomial equation for the curve, see Section 5.3.3. The question for the lowest degree polynomial whose graph is a noncircular curve of constant width is raised. Expressing it as an algebraic curve, isolated points not from the original curve are obtained; in [914] it is shown how to avoid this problem, i.e., how to construct a constant width curve in this way that has no isolated points. The authors of [86] present a complete analytic parametrization of 3-dimensional constant width bodies. For parametrizations of 3-dimensional constant width bodies see also Sections 8.5, 5.3.4 and 11.1. Approximation Results Recall that as starting point for the discussion of approximation results on constant width sets, already Blaschke ([130] and [131]) showed that they can be approximated by Reuleaux polygons in an arbitrarily good manner. Some other results on approximations are also contained in the discussion of 3-dimensional constructions, in particular of Meissner bodies (see Section 8.3). Bodies of constant width occur in the framework of approximation results in the surveys [470, subsection 2.4] and [476, Section 7]. Schneider [1038] showed that each convex body in En can be approximated arbitrarily closely by convex bodies having an algebraic support function and everywhere positive radii of curvature, and the approximating bodies can be chosen in such a way that they have at least the same group of symmetries as the approximated body. Special care is taken for the concept of constant width, and it is shown that for constant width bodies one may choose approximating bodies of constant width in this way, see also Theorem 8.5.1. These results nicely continue related investigations and results from [1118] and [1164]. Aumann [50] gave approximations of constant width sets in the plane with the help of Bézier splines, and he also proposed algorithms for determining quantities like their width, inradius, and circumradius. In [393], the evolution of strictly convex curves with a representation under some special flow (different to the known and usual curve shortening flow) is studied. With demands like smoothness and curvature conditions on the initial curve, this flow tends to curves of constant width, and thus also leaves curves of constant width invariant. Kharazishvili [620] showed that if a plane curve of constant width is sufficiently smooth, then it can be approximated by algebraic curves of constant width. Having uniform approximation of periodic functions in mind, the authors of [727] studied the approximation of convex curves by piecewise circular curves, discussing also constant width curves. Falconer [331] proved that any n-dimensional body of constant width h can be approximated by another body of the same constant width h having a dense set of singularities (= points of that body having at least two other points from it at distance h). The authors of [1097] derived results on discontinuities of measures approximating the 1-dimensional Hausdorff measure for planar sets of constant width; they also gave some sufficient condition for having no such discontinuities. The necessity of these conditions was then confirmed in [1092]. See also [1091], [1093], and [1094], where propositions about these continuity results on approximation measures of constant width sets are added. Further similar results in this direction are presented in [1095] and [1096]. For a given integer m ≥ n + 1, in [451] estimates on the minimal width of convex polytopes are given, where these polytopes have at most m vertices and are inscribed to bodies of given constant width. In special cases, even explicit values are obtained. Via the spherical intersection property, the notion of spindle convexity is close to that of constant width, see our Chapter 6. Two papers on spindle convexity are closely related to the approximation concepts discussed here. A compact spindle-convex set in En is the intersection of a finite family of closed unit balls, and a convex disk-polygon is the 2-dimensional variant thereof. In [372], the authors study the approximation of spindle convex sets with twice continuously differentiable

194

8 Examples and Constructions

boundaries by circum- and inscribed convex disk-polygons with at most m sides regarding Hausdorff metric, area deviation, and perimeter deviation. And in [370] famous results of Rényi and Sulanke are carried over from usual convexity to spindle-convex sets: Given a compact, spindle-convex set in the plane satisfying certain boundary conditions (e.g., smoothness), the authors take m independent uniform random points in this set and derive asymptotic formulae for the mean number of vertices, area, and perimeter deviation regarding their spindle-convex hull. The papers [192] and [998] contain also related approximation results.

Exercises 8.1.

8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8.

8.9. 8.10. 8.11.

8.12*. 8.13.

 Prove that in the procedure described at the beginning of Section 8.1,  = 2n+1 B(xi , h) 1 is a convex figure of constant width h in which the vertices of the polygon P coincide with the singular points of the boundary of  and this boundary consists of the union of 2n + 1 circular arcs. That is, for every i = 1, . . . , 2n + 1, the shortest circular arc, centered at xi between xn+i and xn+i (mod(2n+1)), belongs to the boundary of . Prove that every Reuleaux polygon can be constructed with the procedure described at the beginning of Section 8.1. Prove that for a Reuleaux polygon of width h, the length of every of its circular arcs is smaller than or equal to πh/3. Prove that if the length of a circular arc of a Reuleaux polygon is πh/3, then it must be a Reuleaux triangle. Construct a figure of constant width h whose boundary consists of countably many circular arcs of radius h. Prove that if abcd is a tetrahedron of side length 1, then S(a, 1) ∩ S(b, 1) is a circle of radius √ 3/2 with center at a+b that contains c and d. 2 Prove that if H is the plane through the face abc of the Reuleaux tetrahedron T , then H ∩ T is the corresponding Reuleaux triangle.  different to c and d, in the Reuleaux tetrahedron T Let p be a point of the circular arc cd, with vertices a, b, c, d. Prove that the normals of T through p are precisely the lines through p and some point x ∈ ab. √ √ Prove that the Reuleaux tetrahedron T has exactly three diameters of length ( 3 − 2/2). Prove that the thickness chords of the Reuleaux tetrahedron T are precisely the intersections of T with a line through a vertex and a point in the opposite face of the tetrahedron. Let L be a line through a point p in the relative interior of Sa and a which does not cut the face bcd. Prove that the chord L ∩ T is normal at p but not at a. Prove that the diametral chord of T with one extreme point p has the other extreme point in one of the circular arcs of singular points that arrive to a. Calculate the volume and the surface area of the Reuleaux tetrahedron. Let  < 0 and π/2 ≤ θ ≤ 3π/2 be two positive real numbers and let  = {(cos t, sin t, 0, . . . , 0) | 0 ≤ t ≤ } ∪ {(1 − cos t, sin t cos θ, sin t sin θ, 0, . . . , 0) | 0 ≤ t < }.

8.14.

Prove that the diameter of  is larger than 1. Use the above exercise to give conditions under which both extremes of a diameter of a body of constant width can be singular points?

Exercises

8.15. 8.16. 8.17. 8.18*.

8.19.

8.20. 8.21*.

195

Prove that the Reuleaux simplex constructed via a regular simplex n of side h, n > 2, has thickness h and diameter larger than h. Prove that the thickness of a Reuleaux polyhedron  = ∩x∈X B(x, h) is h. Suppose  is a Reuleaux polyhedron and  x y is an edge of G  . Prove that the wedge W (x, y) is contained in . Prove that if P0 : S2 → R is an odd function, then P = P0 + h is the support function of a convex body of constant width 2h, provided h > 0 is sufficiently large. Alternatively, prove that for all sufficiently small  > 0, the function P = 1 + P0 is the support function of a convex body  of constant width 2. Prove that if P0 is the restriction of a linear function in E3 to S2 and h > 0 is sufficiently large, then P0 + h is the support function of a convex body which is a translation of the unit ball B 3 (0, 1). Let p = 1 +  p0 , where p0 : S2 → R is the restriction of a linear function in E3 to S2 . Prove that p is the support function of a translation of a unit ball B 3 . Prove that the convex body whose support function is the restriction of 1 + x yz to S2 has the symmetry of a regular tetrahedron.

Chapter 9

Sections of Bodies of Constant Width

Mighty is geometry; joined with art, resistless. Euripides

In Chapter 3 it was proven that the property of constant width is inherited under orthogonal projection but not under sections. The proof of this fact was not a constructive one, that is, no nonconstant width section of a body of constant width was actually exhibited. In fact, it was proven that if all sections of a convex body have constant width, then the body is a ball. Since there are bodies of constant width other than the ball, it was concluded that they must all have at least one section that is not of constant width. To show this could, however, be tricky, even in cases as simple as the body produced by rotating the Reuleaux triangle around one of its axes of symmetry. In fact, in Section 9.1 we will prove that for a nonspherical body of constant width and any point, there is a section through the point which is not of constant width. What is more, in Section 9.2, we will construct bodies of constant width h with the property that none of their sections with diameter h is of constant width. After this discussion, a simple question arises: what are the sections of an n-dimensional body of constant width like? Section 9.3 is devoted to studying the sections of bodies of constant width h; it is proven that every spindle convex body is a section of some body of constant width.

9.1 Concurrent Constant Width Sections The main goal of this section is to prove that the ball is the only convex body with the property that all its sections through a point are bodies of constant width. We say that every k-section of a convex body φ through p0 is a body of constant width if for every k-plane (plane of dimension k) H through p0 , φ ∩ H is either empty, a single isolated point, or a k-dimensional body of constant width. Theorem 9.1.1 Let φ be a convex n-dimensional body, 2 ≤ k < n, and let p0 be a point with the property that every k-section of φ through p0 is a body of constant width. Then φ is a ball. Theorem 9.1.1 was proved by Süss [1107] using differentiability conditions for φ when n = 3 and p0 ∈ int φ. Later Wegner [1168] proved the same result without the differentiability conditions, and finally Montejano [845] proved it in the general case. If φ is a convex n-dimensional body with the property that all its k-sections through p0 are balls, 2 ≤ k < n, then it is known that φ must be a ball (see, for example, [844]). As a consequence, for the proof of Theorem 9.1.1 it will be enough to consider the case in which φ is a convex (k + 1)-dimensional body. © Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_9

197

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9 Sections of Bodies of Constant Width

Throughout this section φ is a convex (k + 1)-dimensional body, 2 ≤ k, and p0 is a point with the property that every k-section of φ through p0 has constant width. The first step in the proof of Theorem 9.1.1 is to show that φ has constant width (Lemma 9.1.4). Next, one has to prove that all the binormals of φ are concurrent. To do so, Lemma 9.1.2 will be proven to show that there is a fixed binormal B0 of φ with the property that any other binormal of φ intersects B0 at some point. The structure of the set of the binormals of φ that pass through x0 for x0 ∈ B0 will then be examined. If the set is non-empty, then its topological structure will be of such richness (Lemma 9.1.7) that connectivity arguments on the boundary of φ will allow the conclusion that any 2-plane through B0 intersects φ in a disk. In the final part of the proof, the argument will be / φ. different according to whether p ∈ int φ, p ∈ bd φ, or p0 ∈ Lemma 9.1.1 There is a line L 0 through p0 such that B0 = L 0 ∩ φ is a diameter of φ. Proof Let L be a line with the property that B = L ∩ φ is a diameter of φ. If p0 ∈ L, there is / L, let H be a k-plane that passes through p0 and L. By the hypothesis, nothing left to prove. If p0 ∈ H ∩ φ has constant width. Then there is a line L 0 in H passing through p0 with the property that B0 = L 0 ∩ (H ∩ φ) is a binormal and, therefore, a diameter of H ∩ φ. As B is also a binormal or a diameter of H ∩ φ, B and B0 must be of the same length, and therefore B0 = L 0 ∩ φ is a diameter of φ.  From this point on, let us fix the line L 0 through p0 with the property that B0 = L 0 ∩ φ is a diameter of φ of length h. Lemma 9.1.2 Let B be a binormal of φ. Then B ∩ B0 is non-empty. Proof If B and B0 are contained in some plane, then B ∩ B0 = ∅; otherwise one of the diagonals of the quadrangle created by B and B0 would have to have length greater than h. If B and B0 are not contained in a 2-plane, then there exists a k-plane H through L 0 that does not intersect the line L and that contains B. Then, since H ∩ φ has constant width h, there is a binormal of H ∩ φ, and therefore a diameter of φ, contained in H and parallel to B, which gives rise to a contradiction.  Lemma 9.1.3 If H is a k-plane through L 0 , then H ∩ φ has constant width h. Proof As B0 is a diameter of φ, it is also a diameter of H ∩ φ, and therefore the width of H ∩ φ is  equal to the length of B0 , that is, h. Lemma 9.1.4 The convex body φ has constant width h. Proof By Corollary (iv) of Theorem 3.1.1, it is sufficient to prove that φ has a diameter in every direction. Let H be a k-plane through L 0 . Then, by Lemma 9.1.3, H ∩ φ has constant width h. Thus, there are binormals of H ∩ φ, and hence diameters of φ parallel to every direction of H . But since H  is any k-plane through L 0 , φ has diameters parallel to every direction. Lemma 9.1.5 If  is a 2-plane through L 0 , then  ∩ φ has constant width h. Proof By Lemma 9.1.2, there is a diameter of  ∩ φ parallel to every direction contained in . Therefore  ∩ φ has constant width.  From this point on, assume that B0 is the only binormal of φ with the property that B0 and p0 are on the same line. This holds because, by Lemma 9.1.2, if B1 is another binormal of φ with the same property, then every binormal of φ that is not in the 2-plane determined by B1 and B0 must contain the point B1 ∩ B0 , and therefore φ must be a ball.

9.1 Concurrent Constant Width Sections

199

The following geometric figures occur in the next three lemmas: a) b) c) d) e) f)

a point q0 ∈ bd φ \ L 0 , a support k-plane  of φ through q0 , the binormal B = q0 a of φ orthogonal to , the line L 1 that passes through q0 and p0 , the 2-plane 0 determined by L 0 and L 1 , and the point b ∈ bd φ such that L 1 ∩ φ = q0 b.

Let H be any k-plane through L 1 and let H ⊥ be the line through q0 orthogonal to H . Then φ ∩ H is a body of constant width and, if H = , then  ∩ H is a support (k − 1)-plane of φ ∩ H through q0 . Note that if H ⊥ is not contained in 0 , then  = H ; otherwise B ⊂ H ⊥ , in which case, by Lemma 9.1.2, H ⊥ ⊂ 0 . Lemma 9.1.6 Let H be a k-plane through L 1 and assume that H ⊥ is not contained in 0 . Let q0 w be the binormal of φ ∩ H that is orthogonal to  ∩ H . Then there exists a binormal Bw of φ through w with the property that Bw ∩ B0 = B ∩ B0 . Proof Let  be the 2-plane orthogonal to  ∩ H at q0 . Then H ⊥ , B, and q0 w are contained in . Moreover,  ∩ L 0 = B ∩ B0 because of L 0 ⊂ . Then  = 0 , and as a consequence, H ⊥ ⊂ 0 . be the support (k − 1)-plane of φ ∩ H at w orthogonal to q0 w and, therefore, parallel Let L k−1 w to  ∩ H . Let (w) be the support k-plane of φ through L k−1 2 . Finally, let Bw be the binormal of φ  through w orthogonal to (w). Then Bw ⊂ , and therefore Bw ∩ B0 =  ∩ L 0 = B ∩ B0 . The strategy now is to “rotate” H around L 1 . Lemma 9.1.7 There exists a continuous function β : Sk−1 → bd φ that satisfies the following properties: a) for every w ∈ β(Sk−1 ) there exists a binormal Bw of φ through w with the property that Bw ∩ B0 = B ∩ B0 , b) β(Sk−1 ) ∩ 0 = {a, b}. Proof Assume next, that q0 is the origin of Ek+1 . Then Sk denotes the set of unit vectors based on q0 , the origin of Ek+1 . Moreover, we may assume that Sk−1 is the set of all unit vectors based on q0 and orthogonal to L 1 . We first describe a continuous function α : Sk−1 → Sk . Let us fix an orientation for the Euclidean space Ek+1 which contains φ, and an orientation for . Then for every v ∈ Sk−1 there exists an oriented = Hv ∩  is an oriented (k − 1)k-plane Hv passing through L 1 and orthogonal to v. Moreover, L k−1 v plane through q0 . Let α(v) be the unit vector of Sk contained in Hv with the property that α(v) is and they together generate the oriented k-plane Hv . orthogonal to L k−1 v Clearly, α : Sk−1 → Sk is a well-defined continuous function except in the case where L 1 ⊂ . In this case, α is well defined and is continuous at all points of Sk−1 except v0 , where v0 is the vector orthogonal to the oriented k-plane . Now let ψ : Sk → bd φ be the following continuous function. For every v ∈ Sk let ψ(v) ∈ bd φ be the point with the property that q0 ψ(v) = φ ∩ {tv|0 ≤ t}. Finally, if L 1 is not contained in , define β : Sk−1 → bd φ as the composition ψα : Sk−1 → bd φ. If L 1 ⊂ , we define β(v0 ) = q0 and β(v) = ψ(α(v)), for every v ∈ Sk−1 \ {v0 }. Then, since φ has constant width, β is a continuous function. / 0 , then L k−1 is a support Let w ∈ β(Sk−1 ). Then there is some v ∈ Sk−1 such that β(v) = w. If v ∈ v (k − 1)-plane of K v = K ∩ Hv , a convex body of constant width. Moreover, q0 w is the binormal of

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K v perpendicular to L k−1 v . As a consequence, by Lemma 9.1.6, B ∩ B0 = Bw ∩ B0 for some binormal Bw of φ through w. If v ∈ 0 , then (a) follows by continuity. Let w ∈ β(Sk−1 ) ∩ 0 and assume that w = a. Let v ∈ Sk−1 be such that β(v) = w. Then L k−1 v is perpendicular to both q0 w and q0 a. It follows that L k−1 is perpendicular to 0 since w = a. This v  implies that v ∈ 0 , but then w = b, thus concluding the proof. Definition Let q0 be a corner point of φ. By Lemma 9.1.5, there are two different points z 1 , z 2 ∈ B0 such that for every z ∈ z 1 z 2 , the interval q0 z is contained in a binormal of φ through q0 . We shall say that B is a proper binormal of φ if B ∩ B0 is contained in the interior of the segment z 1 z 2 . Lemma 9.1.8 Let q0 be a corner point of φ and suppose that B is a proper binormal of φ through q0 . Let β : Sk−1 → bd φ, as in Lemma 9.1.7. Then the points of β(Sk−1 ) \ {b} are not corner points of φ. Proof Continuing the notation of Lemma 9.1.7, let v ∈ Sk−1 . If v ∈ / 0 , then q0 is a corner point / 0 , the orthogonal projection of 0 onto v is a 2-plane that contains the of φv . Moreover, as v ∈ orthogonal projection of L 0 in v and also contains the binormal q0 w of φv . Since every binormal of φ through q0 is projected orthogonally in v onto a binormal of φv through q0 , we have that q0 has a neighborhood U in bd(φ ∩ ) which is an arc of the circle with center q0 . Assume now that w = β(v) is a corner point of φ. By the above argument, w is a corner point of φ ∩ 1 where 1 is the 2-plane in v which is the orthogonal projection of the 2-plane through L 0 and w. This is a contradiction because  1 = . Lemma 9.1.9 is an immediate consequence of Lemma 9.1.7. Lemma 9.1.9 Let L be a line through p0 and let qb = L ∩ φ. Let Bq be a binormal of φ through q. Then there exists a binormal Bb of φ through b such that Bq ∩ B0 = Bb ∩ B0 . Definitions For every y ∈ bd φ, let b(y) be the point in the boundary of φ with the property that yb(y) = L y ∩ φ, where L y is the line that passes through p0 and y. Moreover, for every point q ∈ bd φ \ L 0 , choose Bq = qa(q), a binormal of φ through q, in such a way that if q is a corner point of φ, then Bq is a proper binormal. Lemma 9.1.10 Let Bq0 = q0 a(q0 ) and Bq1 = q1 a(q1 ) be two binormals of φ contained in the 2-plane 0 through L 0 , q0 , q1 ∈ bd φ \ L 0 . Suppose that i) φ has no corner points in bd φ \ L 0 , or ii) the point q0 is a corner point of φ. If a(q0 ) and b(q0 ) separate a(q1 ) from b(q1 ) in bd(φ ∩ 0 ), then Bq0 ∩ B0 = Bq1 ∩ B0 . Proof Let β0 , β1 : Sk−1 → bd φ be the functions described in Lemma 9.1.7 for points q0 and q1 , respectively. By the hypothesis and Lemma 9.1.7(b), a(q1 ) and b(q1 ) are in different components / L 0. of bd φ \ β0 (Sk−1 ). Then w ∈ β0 (Sk−1 ) ∩ β1 (Sk−1 ) exists. Moreover, it is easy to see that w ∈ Therefore, by Lemma 9.1.8, w is not a corner point of φ and, as a consequence, by Lemma 9.1.7(a),  Bq0 ∩ B0 = Bw ∩ B0 = Bq1 ∩ B0 , where Bw is the only binormal of φ through w. Proof of Theorem 9.1.1 Case 1. p0 ∈ int φ. Suppose first that there are no corner points of φ in bd φ \ L 0 . We will show that every 2-plane through L 0 intersects φ in a disk. To do so, let 0 be a 2-plane through L 0 . It will suffice to prove that given a point q0 ∈ bd(φ ∩ 0 ), there exists a small neighborhood U of q0 in bd(φ ∩ 0 ) which is an arc of the circle with center Bq0 ∩ B0 . This can be done by proving that for every q1 ∈ U, Bq0 ∩ B0 = Bq1 ∩ B0 . As any two binormals of a convex 2-dimensional body of constant width intersect, and since p0 ∈ int φ, a(q0 ) and b(q0 ) separate a(q1 ) from b(q1 ) in bd(φ ∩ 0 ) for every q1 in a small neighborhood

9.1 Concurrent Constant Width Sections

201

U of q0 in bd(φ ∩ 0 ). Therefore, by Lemma 9.1.10, Bq0 ∩ B0 = Bq1 ∩ B0 for every q1 ∈ U. Note that a(q0 ) = b(q0 ), since it was assumed that B0 is the only binormal of φ through p0 . Suppose now that there is some point q0 ∈ bd φ \ L 0 which is a corner point of φ and, as always, let 0 be the 2-plane that passes through L 0 and q0 . By the above argument and Lemma 9.1.10, there is a small arc to the left or the right of q0 in bd(φ ∩ 0 ) that is an arc of the circle with center Bq0 ∩ B0 , which is a contradiction since it had been assumed that Bq0 is a proper binormal. Case 2. p0 ∈ bd φ. Since φ has constant width, the mapping β : Sk−1 → bd φ described in Lemma 9.1.7 has an antipodal mapping. That is, given q ∈ bd φ \ L 0 , there exists a continuous mapping ψq : Sk−1 → bd φ with the following conditions: a) for every w ∈ ψq (Sk−1 ) there is some binormal Bw of φ through w with the property that Bw ∩ B0 = Bq ∩ B0 , and b) ψq (Sk−1 ) ∩ 0 = {q, b( p0 )}. Suppose first that there are no corner points of φ in bd φ \ L 0 . We will prove that every 2-plane 0 through L 0 intersects φ in a disk. Let y ∈ (bd(φ ∩ 0 )) \ L 0 and let q0 ∈ bd(φ ∩ 0 ) such that a(q0 ) = y. Let b( p0 ) = q0 . Let γ be the subarc of bd(φ ∩ 0 ) that goes from y to p0 and does not contain b( p0 ). It suffices to prove that γ is an arc of the circle with center Bq0 ∩ B0 . Let q1 ∈ γ and let ψq1 : Sk−1 → bd φ. Let β : Sk−1 → bd φ as in Lemma 9.1.7 for the point q0 . By the proof of Lemma 9.1.10, β(Sk−1 ) ∩ ψq1 (Sk−1 ) is non-empty. This implies that Bq1 ∩ B0 = Bq0 ∩ B0 for every q1 ∈ γ and, therefore, that γ is an arc of the circle with center Bq0 ∩ B0 . Suppose now that q0 ∈ bd φ \ L 0 is a corner point of φ. Then we conclude, as in the first part of the proof, that γ is an arc of the circle with center Bq0 ∩ B0 , which is a contradiction, as Bq0 is a proper binormal. Case 3. p0 ∈ / K . Suppose first that there are no corner points of φ in bd φ \ L 0 . We will prove that every 2-plane 0 through L 0 intersects φ in a disk. Let q0 ∈ (bd(φ ∩ 0 )) \ L 0 . Suppose that the line L 1 that passes through q0 and p0 is a support line of φ ∩ 0 . We will show that there is a small neighborhood of q0 in bd(K ∩ 0 ) that is an arc of / L 0 ; otherwise this would contradict Lemma 9.1.9, the circle with center Bq0 ∩ B0 . Note that a(q0 ) ∈ as φ ∩ 0 has constant width. Since the line that passes through a(q0 ) and p0 is not a support line of φ ∩ 0 we conclude, using Lemma 9.1.10, that a small arc to the left or the right of q0 in bd(φ ∩ 0 ) is an arc of the circle with center Bq0 ∩ B0 , but by Lemma 9.1.9, a small neighborhood of q0 in bd(φ ∩ 0 ) is an arc of the circle with center Bq0 ∩ B0 . Let H be the closed half-plane of 0 determined by L 0 that contains q0 , and let L be a line that passes through p0 such that L ∩ φ = q1 q2 ⊂ H . Assume that the subarc of bd(φ ∩ 0 ) that goes from q1 to q2 and contains q0 is an arc of the circle with center Bq0 ∩ B0 , and assume that L is the closest line to L 0 with this property. We will show that L = L 0 , which will prove that φ ∩ 0 is a disk. Assume that L = L 0 and q1 ∈ ( p0 , q2 ). By Lemma 9.1.5, a(q1 )a(q2 ) is parallel to L and b(a(q1 )) is therefore in the interior of the subarc of bd(φ ∩ 0 ) that goes from q1 to a(q2 ) and does not contain q2 .   Therefore, by Lemma 9.1.9, a b(a(q1 )) is in the interior of the subarc of bd(φ ∩ 0 ) that goes from a(q1 ) to q2 and does not contain q1 . If y is sufficiently close to q1 , then the point b(y) is contained in the interior of the subarc of bd(φ ∩ 0 ) that goes from a b(a(q1 )) to a(q1 ) and contains q1 . Therefore, by Lemma 9.1.10, there is a small neighborhood of q1 in bd(φ ∩ 0 ) that is an arc of the circle with center in Bq0 ∩ B0 . By Lemma 9.1.9, this contradicts the choice of L. Suppose that q0 ∈ bd φ \ L 0 is a corner point of φ. Let L 1 be the line that passes through q0 and p0 , and let 0 be the 2-plane determined by L 0 and L 1 . If L 1 is a support line of φ ∩ 0 , then it can be argued, as in the first part of the proof, that there is a small neighborhood of q0 in bd(φ ∩ 0 ) that is an arc of the circle with center Bq0 ∩ B0 , which is a contradiction since q0 is a corner point of φ. If L 1 is not a support line of φ ∩ 0 , then φ ∩ L 1 = q0 q1 , where q0 = q1 . By Lemma 9.1.9, there is a binormal Bq1 = q1 a(q1 ) of φ through q1 such that Bq0 ∩ B0 = Bq1 ∩ B0 . We will first show that

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b(a(q1 )) = a(q0 ). If it were not so, we could use Lemma 9.1.10 to conclude that there is a small arc to the left or the right of a(q1 ) in bd(φ ∩ 0 ) that is an arc of the circle with center Bq0 ∩ B0 . By Lemmas 9.1.5 and 9.1.9, this is a contradiction since Bq0 is a proper binormal. Because q0 is a corner point of φ, we may choose another proper binormal B (q0 ) = q0 a (q0 ) of φ through q0 different from Bq0 . By Lemma 9.1.9, there is a binormal B (q1 ) = q1 a (q1 ) of φ through q1 with the property that B (q0 ) ∩ B0 = B (q1 ) ∩ B0 . We then also have that b(a (q1 )) = a (q0 ). Furthermore, it can be verified that if Bq0 ∩ B0 is very close to B (q0 ) ∩ B0 , then a(q0 )a(q1 ) ∩ a (q0 )a (q1 ) is contained in φ, since in this latter case, a(q0 ) and a(q1 ) separate a (q0 ) from a (q1 ) in  bd(φ ∩ 0 ). Therefore { p0 } = a(q0 )a(q1 ) ∩ a (q0 )a (q1 ) is a point of φ, which is a contradiction.

9.2 Thickness of a Body By using the notion of thickness, in this section we will prove that there is a body of constant width h with the property that none of its sections of diameter h is of constant width. Recall that the thickness of a body ψ, denoted by (ψ), is the minimum distance between two parallel support hyperplanes of ψ. For example, if ψ has constant width h, then its thickness is also h. Let us denote the maximum of the thicknesses of all sections (produced by hyperplanes) by s (ψ). Many years ago, Süss drew the attention to the following question. Which is the lower bound of the maxima of the thicknesses of sections of convex bodies of unit thickness, and for what convex body is it achieved? Next, we will construct an example of a 3-dimensional convex body of constant width  for which s () < (). We first note that if H is any plane whose section φ =  ∩ H is non-empty, where (φ) is the thickness of φ, and letting L 1 and L 2 be the corresponding parallel support lines of φ in H separated by a distance of (φ), then if we could find parallel support planes H1 and H2 of  that pass through L 1 and L 2 , respectively, () ≤ d(H1 , H2 ) ≤ d(L 1 , L 2 ) = (φ) ≤ s (), where d( , ) denotes the usual distance function. Therefore we would like to construct an example such that it is not possible to draw parallel support planes H1 and H2 of  passing through L 1 (φ) and L 2 (φ), respectively, for any section φ of  whose corresponding parallel support lines L 1 (φ) and L 2 (φ) produce the thickness (). Let φ0 =  ∩ H0 be a section of  for which (φ0 ) = s (), and let H1 and H2 be the support planes of  perpendicular to H0 , but parallel to the lines L 1 (φ0 ) and L 2 (φ0 ), where d(L 1 (φ0 ), L 2 (φ0 )) = (φ0 ). It follows that d(H1 , H2 ) > (φ0 ) = s (). If the body  is of constant width, then () = d(H1 , H2 ), from which we have that s () < (). It follows that a convex body  with the property that s () < () results from the following two properties: a)  has constant width. b) For every plane H that intersects the interior of , the section φ =  ∩ H has two parallel support lines L 1 (φ) and L 2 (φ), with d(L 1 (φ), L 2 (φ)) = (), such that it is not possible to draw parallel support planes H1 and H2 of  passing through L 1 (φ) and L 2 (φ), respectively.

9.2 Thickness of a Body

203

Figure 9.1

A body  with diameter h that simultaneously satisfies the above conditions has, of course, the property that none of its sections of diameter h is of constant width. Such a body was obtained by Danzer [276], by modifying a ball B of diameter h as follows. Let m be the center of the sphere C, and p1 , p2 , . . . , p8 be the vertices of a cube inscribed in C. The points pi are the centers of congruent circles Ci on the sphere such that each circle touches the three neighboring circles in such a way that six concave four-sided regions are left outside. We draw a smaller eccentric circle C j ( j = 9, 10, . . . , 14) inside each of these regions as shown in Figure 9.1. The centers of these concave quadrilaterals need not coincide with the centers of the circles and they must contain each circle C j in their interior, but the C j s are positioned so that their centers p j are not located in any of the planes of symmetry. In this way we also obtain six congruent circles whose centers are required to remain antipodal. Consider now a pair of antipodal points pk , pk (1 ≤ k, k ≤ 14) and substitute the regions of the sphere enclosed by the circles Ck and Ck for the two surfaces of revolution obtained by revolving the section indicated in Figure 9.2 (where ac and a c are arcs of the circle with center m 1 , cd, and c d are arcs of the circle with center m 2 , and the figure is symmetric with respect to the line that passes through pk and pk ). We do this for all seven pairs of antipodal circles (it makes no difference where the figure is flattened and restored). Exactly one support plane passes through each boundary point of the convex body  constructed by this procedure. Moreover, by Theorem 7.2.3, the figure has constant width h because each boundary point is the endpoint of exactly one diameter. A diameter of  that passes through m is either a diameter of the ball C or is contained in one of the lines that pass through pk and pk . Now assume that s () < () is not true for the body of constant width . Then s () = () = h. But this leads to a contradiction since we know that there is a section φ =  ∩ H of thickness h and therefore of constant width h. It follows that a diameter of  contained in H passes through every boundary point of φ. This means that this section is positioned in space in such a way that it is not completely contained in any of the substituted regions in the construction of , and φ contains at least one point of the unmodified portion of the sphere. This argument implies that H passes through the point m. Suppose, moreover, that there is some point r of φ in the interior of one of the circles Ck , but outside mpk . The diameter of  that passes through r intersects pk pk at a point other than m. The plane H contains two different points in the line pk pk , which implies that it contains the line entirely. Finally, to obtain s () = (), there must be a plane that passes through m without intersecting any of the circles Ck eccentrically. There is a plane that satisfies this condition for the circles C1 to C8 ,

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Figure 9.2

but it is not unique, because the condition is satisfied precisely by the planes of symmetry of the cube. Each of them, however, eccentrically intersects either two or four of the circles by construction. This proves that s () < (); that is, there exists a convex body whose thickness is greater than the maximum of the thicknesses of all its sections. In fact, this body is a body of constant width h with the property that none of its sections of diameter h is of constant width. The following question arrives naturally: Is there body of constant width with the property that none of its 2-dimensional sections is of constant width?

9.3 Which Bodies Are Sections? In this section it is proved that every spindle convex body is a section of some body of constant width h (see [845]), for h sufficiently large. Theorem 9.3.1 Let ψ be a spindle r -convex n-dimensional body with diameter d. Then for every h ≥ max{d, (4r 2 − d 2 )1/2 } there exists an (n + 1)-dimensional body  of constant width h such that ψ is a section of . Proof Let h ≥ max{d, (4r 2 − d 2 )1/2 }. Note that 0 < d ≤ 2r . By Lemma 6.1.1, there is an r -support sphere of ψ through every point of the boundary of ψ. Denote by C p the r -support sphere at p, and let x p be its center. Let T be the set of all points x p , where p varies over all the points in the boundary of ψ. We claim that the diameter of T is less than or equal to (4r 2 − d 2 )1/2 . For otherwise, if q and y were in the boundary of ψ and the length of xq x y would be greater than (4r 2 − d 2 )1/2 , then the diameter of Cq ∩ C y would be less than d, which is a contradiction since ψ ⊂ Cq ∩ C y and the diameter of ψ is d. Let us prove that h ≥ r . If d ≥ (4r 2 − d 2 )1/2 , then h ≥ d and d 2 ≥ 2r 2 , which implies that h ≥ r . On the other hand, if (4r 2 − d 2 )1/2 ≥ d, then h ≥ (4r 2 − d 2 )1/2 and 2r 2 ≥ d 2 , which implies once again that h ≥ r . Consider the Euclidean space En+1 as the Cartesian product of En with the real line. Then every element of En+1 can be written as a pair (m, t) where m is a point of En and t is a real number. Take the set T of all points of En+1 of the form y p = (x p , (h 2 − r 2 )1/2 ), where p varies over all points in the boundary of ψ. Note that the diameter of T is less than or equal to (4r 2 − d 2 )1/2 and that the length of the interval py p is h. Now let F = ψ ∪ T. We will prove that the diameter of F is h. Let x and y be two points in T. If both points are in T or both in ψ, then the distance between x and y is less than or equal to h because

9.3 Which Bodies Are Sections?

205

  of the choice of h. Suppose that x is in ψ and y = y p = x p , (h 2 − r 2 )1/2 for some p in the boundary of ψ. Then we have that     x − x p 2 + x p − y p 2 = |x − y|2 ,   2  and since x − x p  ≤ r , then since C p is a r -support sphere of ψ with center x p and x p − y p  = h 2 − r 2 , we have that the length of x y is less than or equal to h. Furthermore, for any p in the boundary of ψ the length of the interval py p is h. This proves that the diameter of F is h. By Theorem 7.2.2 of Pál, there exists a body  of constant width h that contains F. Let  =  ∩ En . Clearly, ψ ⊂ . Moreover, the boundary of ψ is contained in the boundary of , since for every point p in the boundary of ψ the interval py p is a diameter of . Since the points in the boundary of  are precisely the points of the boundary of  in En , the boundary of ψ is contained in the boundary of , and therefore, ψ = .  Corollary 9.3.1 Every spindle h-convex body is the section of a body of constant width 2h. Corollary 9.3.2 A convex body ψ is the section of a body of constant width if and only if ψ is a spindle h-convex body for some h. Proof Sufficiency follows from Corollary 9.3.1. Suppose that  is a body of constant width h and that ψ =  ∩ En . By Theorems 6.1.1 and 6.1.2,  is the intersection of a family of (n + 1)-dimensional balls with radius h. Each of these balls intersects En in an n-dimensional ball with radius at most h. The body ψ is therefore the intersection of a family of n-dimensional balls each with radius less than or equal to h. Since all of these balls are spindle h-convex, it follows that ψ is spindle h-convex.  Theorem 9.3.1 and its corollaries are of course true for linear sections that are not necessarily created by hyperplanes, but by lower dimensional flats.

Notes In [377], it is proved that the lengths of the segments, intercepted by a body of constant width h on 1 each of two parallel lines of distance at most d apart, differ by at most 2(2dh) 2 . This value depends on h, but not on the shape of that body. For parallel sections in higher dimensions, analogous results are obtained. In [847] the following is investigated: If K is a compact subset of En+k , what can we recognize about K from some of its n-dimensional linear sections. For example, topological conditions on a family X of n-plane sections are established which assure that K is a body of constant width if the size of X satisfies this topological condition and every section in X has constant width, see also Section 16.4. And it is shown in [845] that sections of convex bodies, which are themselves of constant width, can be used to characterize balls (see Section 9.1 and also [1168] for n = 3). Inspired by the paper [531], in [59] the following was shown: A convex body in a normed plane is of constant width if the following holds for any chord of it: one of the two parts, into which it is split by this chord, has diameter equal to the length of the chord. The only if part also extends to higher dimensions. Similarly, it was proved in [54] that if a body of constant width in a normed space is cut by a hyperplane, then one of the two parts has the same diameter as the original body. For two dimensions, this property even characterizes sets of constant width. A similar result was proved in [57]. Furthermore, if a body of constant width in a normed space is partitioned into a continuous family of hyperplane sections H (t), 0 ≤ t ≤ 1, then the diameter of H (t) is a unimodal function of t, and again in two (but not in higher) dimensions this property characterizes bodies of constant width (see again [54]). Constructing suitable bodies of constant width in 3-space, Danzer [276] answered a problem of Süss. Namely, he

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9 Sections of Bodies of Constant Width

showed that there are 3-dimensional convex bodies whose own thickness (i.e., minimal width) is larger than the maximum thickness of all their plane sections, see Section 9.1. Also, in Armstrong’s paper [37] planar sections of surfaces of constant width play a role. Results on sections and section functions related to spherical harmonics, Fourier series, and geometric tomography (see [461], [455], and [401]) will mainly be discussed in Chapter 13. Congruent Sections and Projections Let φ be 3-dimensional convex body with the following property: to every direction u continuously corresponds a plane Hu orthogonal to u such that all the sections Hu ∩ φ are congruent. By means of Brouwer’s fixed point theorem, Süss [1109] proved in 1947 that all these sections are disks; consequently, φ is a ball. By using spherical harmonics (see Theorem 13.1.7) it is possible to prove that a centrally symmetric convex n-dimensional body φ with the property that all its hyperplane sections through the center of symmetry are congruent is a ball. In 1979 Schneider [1037] proved this result without the hypothesis of central symmetry, and in 1990 Montejano [846] proved that if all the sections of a convex body through a point p0 are similar, then the convex body is also a ball, but the point p0 is not necessarily the center of the ball. For analogous problems and results referring to projections, see [848] and [675]. Consider the following related problem: Suppose φ is a convex (n + 1)-dimensional body with the property that all its sections through a fixed point p0 ∈ intφ are affinely equivalent. Is φ an ellipsoid? This problem is essentially reflected by the following conjecture of Banach. If 1 < k < n and V n is an n-dimensional (real or complex) Banach space with unit ball G and all the k-dimensional subspaces of V n are isometric (all the k-sections of G are affinely equivalent), then V n is an inner product space (G is an ellipsoid). Gromov proved in [466] that the conjecture is true if one of the following conditions holds: (1) k is even; (2) k is odd, V n is complex, n ≥ 2k; (3) k is odd, V n is real and n ≥ k + 2. Dvoretzky [305] derived the same conclusion under the hypothesis n = ∞. Furthermore, if additionally between any two sections there is an affine volume preserving isomorphism, then the conjecture is true [846]. Lately, some progress has been done by Montejano et al., in the solution of the problem when k = 5, 9. The topological technique behind these and other related results is the following. Let φ be a convex body in Rn . A field of convex bodies tangent to Sn and congruent to φ is a continuous function φ(u) defined for all u ∈ Sn , where φ(u) is a congruent copy of φ lying in the hyperplane tangent to Sn at u. If, additionally, φ(u) = φ(−u) for each u ∈ Sn , we say that φ(u) is a complete turning of φ in Rn+1 . Clearly, a convex (n + 1)-dimensional body with the property that all its hyperplane sections through a fixed point p0 are congruent to φ ⊂ Rn gives rise to a complete turning of φ in Rn+1 . Using the theory of fiber bundles, Mani [770] proved the following two results: (1) If n ≥ 2 is even, the only fields of convex bodies tangent to Sn are those congruent to balls. (2) If there is a field of bodies congruent to φ tangent to S n , where the group of symmetries of φ is finite, then Sn is parallelizable, that is, n = 1, 3, or 7. Furthermore, in [846] it is proved that if there is a complete turning of φ in Rn+1 , then φ is centrally symmetric. Regarding the symmetries of projections and sections of convex bodies, there is a very interesting and complete survey of Ryabogin [990] in which these and many other related problems are treated. In particular, the paper of Myroshnychenko and Ryabogin [880] is very interesting; it concerns polytopes with congruent projections or sections. Finally, for more about sections and projections of convex bodies see Rogers [980] and [1077], [1078], [584], [452], and [198]. For the problem of the false center of symmetry see Section 2.12.2 and the Notes of Chapter 2 regarding ellipsoids. Interesting problems and results concerning the reconstruction of convex bodies from their projections can be found in the papers of Golubyatnikov [433], [434], [432], and of Kuz’minykh [675]. See also Gardner’s book [401] and Section 16.4.

Exercises

207

Exercises 9.1. 9.2. 9.3. 9.4*. 9.5*. 9.6*. 9.7. 9.8. 9.9*. 9.10*. 9.11*.

Prove, without using Theorem 9.1.1, that if every section of a convex body φ through a point p0 ∈ En is a ball, then φ is a ball. In the proof of Theorem 9.1.1, prove that it is enough to consider the case n = k + 1. Prove that if every section of a convex body φ is an ellipsoid, then φ is an ellipsoid. If every section of a convex body φ is a polytope, is φ a polytope? If every section of a convex body φ is centrally symmetric, is φ an ellipsoid? If every section through a fixed point of a convex body φ is centrally symmetric, is φ centrally symmetric? Is φ an ellipsoid? If every hyperplane through a fixed point of a convex body φ is a hyperplane of orthogonal symmetry, is φ then a ball? If every hyperplane through a fixed point of a convex body φ is a hyperplane of affine symmetry, is φ then an ellipsoid? If a 3-dimensional convex body φ has two different axes of symmetry, is φ a ball? If φ is the intersection of an infinite family of balls, is φ then spindle h-convex for some h > 0? Is a strictly convex body the intersection of an infinite family of balls?

Chapter 10

Bodies of Constant Width in Minkowski Spaces

Some mathematician, I believe, has said that true pleasure lies not in the discovery of truth, but in the search for it. Leo Tolstoy

10.1 Introduction In Euclidean space, the length of a segment depends only on its magnitude, never on its direction. However, for certain geometrical problems the need arises to give a different definition for the length of a segment that depends on both the magnitude and the direction. An interesting example is the following. Let us suppose we are in a city “M” where half of the streets run vertically and the other half run horizontally as in Figure 10.1. A person wishes to go (walking or driving) from point a to point b in the city M. For practical purposes, what is the distance between these two points? Given the city map of M it is of course possible to draw the straight line segment ab on the map and measure its length. Nevertheless, such a notion of distance could be irrelevant here because in order to walk or drive in a straight line from point a to point b, a person would have to cross private property and even walk through the walls of houses. It is more realistic to consider the real distance between points a and b as the length of the line acb shown in Figure 10.2. There are also other feasible ways to go from a to b that are not longer than acb but no other feasible path that is shorter than acb. If we introduce a system of coordinates whose axis coincides with two perpendicular streets then it is clear that the real distance from point a = (x1 , y1 ) to point b = (x2 , y2 ) (that is, the length of the line segment ab in the geometry of the city) is equal to: d M (a, b) = |x2 − x1 | + |y2 − y1 | . Once we know how to calculate the distance between any two points in the city M, we would like to know the shape of the unit disk; that is, the set of all points whose distance from the origin o is less or equal to one. Since the origin has coordinates (0, 0), the distance from the origin to the point C(x, y) is given by the formula d M (o, c) = |x| + |y| . Consequently, in the geometry of the city M the unit disk is determined by the inequality |x| + |y| ≤ 1.

© Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_10

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10 Bodies of Constant Width in Minkowski Spaces

Figure 10.1

It is now clear that in the geometry of the city M, the unit disk has a square shape as shown in Figure 10.3. Now, using this unit disk it is possible to calculate the distance between any two points a and b in the city M by using the following procedure. First translate the unit disk so that the origin lies at a. Draw the straight line ray that starts at a, passes through b, and crosses the boundary of the unit disk at c. Then d M (a, b) = ab/ac. As we will define next, the basis of Minkowski geometry consists of using as a unit disk any centrally symmetric convex figure in the same manner as we use the square in the above example. Let G ⊂ En be a centrally symmetric convex body which is even centered at the origin. We will consider G as the unit ball of a Minkowski space M, whose points are the points of Euclidean n-space En , except that now we define the distance between any two points a and b of M by using the following procedure: Translate G until its center lies in a. Then draw the straight ray that starts in a, passes through b, and crosses the boundary of G at c. Then d M (a, b) = ab/ac. Note that if G is the unit disk of the Euclidean space En , then the distance d M (a, b) coincides with the Euclidean distance. Also note, since G is symmetric at the origin, that d M (a, b) = d M (b, a).

Figure 10.2

10.1 Introduction

211

Figure 10.3

Up to now we have not used the convexity of G. It turns out that the convexity is responsible for the triangle inequality. Triangle Inequality. Let a, b and c be three points in the Minkowski space M with unit disk G. Then d M (a, b) + d M (b, c) ≥ d M (a, c). Proof Let p, q be points in the boundary of G such that p = λ0 (b − a), and q = λ1 (c − b), λ0 , λ1 ≥ 0. Let n be a point in pq such that n = λ2 (c − a) and let m be a point in op such that q = λ3 (n − m), λ2 , λ3 ≥ 0. See Figure 10.4. Note that abc and omn are similar triangles. Also opq and mpn are similar triangles. Since G is convex we have that n ∈ G. Let n  in the boundary of G be such that n ∈ on  . From these similarities we have that op ab = ; ac on

bc mn = ; ac on

mn mp = . oq op

Figure 10.4 Triangle inequality

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By definition d M (a, b) =

ab op

and d M (b, c) =

d M (a, b) + d M (b, c) = ac

bc . oq

Hence

 ab bc  ac  om mp  ac + = + = . opac oqac on op op on

On the other hand, since G is convex we have that n ∈ G. Let n  in the boundary of G be such that n ∈ on  . Hence, d M (a, b) + d M (b, c) =

ac ac ≥  = d M (a, c). on on



A finite-dimensional Banach space is a finitely generated vector space M over the real numbers R equipped with a norm, which is a function that assigns a nonnegative length or size to each vector in M; that is,   : M → R satisfying the following properties: For all λ ∈ R and all x, y ∈ M, • x = 0 if and only if x is the zero vector in M, • λx = |λ| x, and • x + y ≤ x + y. Let G = {x ∈ M | x ≤ 1} be the unit ball of a Banach space M. Then it is easy to prove that G is a convex body centered at o ∈ M. Furthermore, if we use G to define the distance d M (x, y) between two points x and y in M as shown above (that is, M is a Minkowski space), then d M (x, y) = x − y. Consequently, a (normed or) Minkowski space M is a real finite-dimensional Banach space; in other words, it is the Euclidean space En endowed with a metric given by a unit ball which is a convex body G centered at the origin. With this unit ball we can measure the length of straight line segments according to their direction. Of course, the origin is always an interior point of G. Basic references regarding the geometry of Minkowski spaces are the book [1124] and the surveys [795] and [793], the latter containing a section on bodies of constant width in these spaces. For a broader audience, the field is geometrically introduced in [791]. For the remainder of this chapter, M will be a Minkowski space with a convex body G centered at the origin as unit ball. For every point x ∈ M, let G(x, r ) = {y ∈ M | d M (x, y) ≤ r }. Then G(x, r ) denotes the ball of radius r with center at x. Note that G(x, r ) is a homothetic copy of G; that is, G(x, r ) = x + r G. Let H ⊂ M be a hyperplane and let x ∈ / H . Then the distance between the hyperplane H and the point x, d M (x, H ), is the minimum of all possible distances between the points of H and x. Let r > 0 be such that G(x, r ) has H as a support hyperplane. Then it is clear that d M (x, H ) = r (see Figure 10.5). Thus, in the Minkowski geometry the distance from a point x outside a hyperplane H is not measured, generally speaking, by the perpendicular from the point x to the hyperplane H . In addition, it is possible that in the hyperplane H there is not just one point near x, but a convex set H ∩ G that consists of the nearest points to x in H . The distance d M (H1 , H2 ), between two parallel hyperplanes H1 and H2 or the G-width of the strip cc(H1 ∪ H2 ) is the minimum of all possible distances of points from H1 to H2 .

10.1 Introduction

213

Figure 10.5

A chord p1 p2 of a strip S is called a bridge if and only if p1 ∈ H1 , p2 ∈ H2 and d M ( p1 , p2 ) = d M (H1 , H2 ). Let L be a line of M which is not contained in a hyperplane H of M. We say that L is G-perpendicular to H , L ⊥G H , if and only if given a hyperplane H  parallel to H , the chord L ∩ cc(H ∪ H  ) is a bridge of cc(H ∪ H  ) if and only if d M (L ∩ H, L ∩ H  ) = d M (H, H  ). Of course, if L  is a line parallel to L and H  is a hyperplane parallel to H , then L ⊥G H if and only if L  ⊥G H  . Let r > 0 be such that the ball G(x, r ) with center at some point x has H1 and H2 as support hyperplanes. Then it is clear that d M (H1 , H2 ) = 2r , and p1 p2 is a bridge of the strip cc(H1 ∪ H2 ) if and only if there are points qi ∈ Hi ∩ G, i = 1, 2, such that q1 q2 is parallel to p1 p2 , see Exercise 10.1. Therefore, a line L is G-perpendicular to H if and only if the line L 0 through the origin, parallel to L, has the property that the hyperplanes parallel to H through L 0 ∩ bd G are support hyperplanes of G, see Exercise 10.2. This implies that given a line L, there is always a hyperplane H such that L ⊥G H . Such a hyperplane H may be not unique (up to translations) unless G is smooth. Similarly, given a hyperplane H there is always a line L such that L ⊥G H , and such a line L may be not unique (up to translations) unless G is strictly convex. Let  ⊂ M be a convex body, where M is the Minkowski space with the convex body G centered at the origin as unit ball. We say that  is a body of constant G-width h if and only if for every pair of parallel support hyperplanes H1 and H2 of , the G−width of the -strip determined by H1 and H2 is h. That is, if and only if d M (H1 , H2 ) = h.

10.2 Characterizations of Constant G-Width 10.2.1 G-Diameters Let M be the Minkowski space with the centrally symmetric convex body G as unit ball and let T be a compact subset of M. The G-diameter of T is the maximum of all possible G-distances between two points of T . Abusing the terminology, we call chords of a convex body that have maximum G-length G-diameters, and we will use the same name for their lengths. One of the essential characterizations of constant G-width bodies is that, like the unit ball G, they have a G-diameter in every direction. Indeed, it is the behavior of G-diameters that give bodies of constant G-width their basic properties.

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Theorem 10.2.1 Let  ⊂ M be a convex body of the Minkowski space M with unit ball G. Then the G-diameter of  is the maximum of the G-widths of all support -strips. In particular, a body of constant G-width h has G-diameter equal to h. Proof Let h be the maximum of the G-widths of all support -strips. Then the G-distance between any two points in a body  must always be less than or equal to h. This is because if there were two points p and q of  such that d M ( p, q) > h, then the body would have G-width greater than h in some direction. To see this, let H1 and H2 be the parallel support hyperplanes of  G-perpendicular to the segment pq and let x1 x2 be a bridge of the strip cc(H1 ∪ H2 ) parallel to pq. Then d M (H1 , H2 ) = d M (x1 , x2 ) ≥ d M ( p, q) > h.



Lemma 10.2.1 Let φ ⊂ M be a convex body of the Minkowski space M with unit ball G, and let pq be a G-diameter of φ. Suppose H is a hyperplane through p G-perpendicular to pq. Then H is a support hyperplane of φ at p. Moreover, the chord pq is a diametral chord of φ. Proof Suppose h is the G-diameter of φ. If pq is a G-diameter of φ, then φ ⊂ G(q, h) and p ∈ bd G(q, h). Furthermore, since pq is G-perpendicular to H , then H must be a support hyperplane of G(q, h) at p; consequently, H is a support hyperplane of φ at p. Similarly, the hyperplane H  through q and parallel to H is a support hyperplane of φ at q. Therefore, the chord pq is a diametral chord of φ.  Theorem 10.2.2 Let  ⊂ M be a convex body of the Minkowski space M with unit ball G. Then  is a convex body of constant G-width h if and only if the diametral chords of  are precisely the G-diameters of . Proof Suppose the diametral chords of  are the G-diameters of , and consider any hyperplane H . If a line L is G-perpendicular to H , by Exercise 5.9 there is a diametral chord pq of  parallel to L. Since pq is a G-diameter of , the width of the φ-strip parallel to H is equal to d M ( p, q), the G-diameter of . Consequently  has constant G-width. Suppose  has constant G-width, and let pq be a diametral chord of . Let H p and Hq be the corresponding parallel support hyperplanes of  at p and q, respectively. Therefore, since the G-diameter of  is h, d M ( p, q) ≤ h = d M (H p , Hq ). Also, d M ( p, q) = d M (H p , Hq ) = h, which implies that pq is a G-diameter of . Conversely, if pq is a G-diameter of , then, by Lemma 10.2.1, pq is a diametral chord of .  Corollary 10.2.1 A convex body  ⊂ M is a body of constant G-width h in the Minkowski space M if and only if all diametral chords have G-length h. Proof If  is a convex body of constant G-width h, then the diametral chords of  are the G-diameters of ; hence all of them have G-length h. Suppose all diametral chords have G-length h. By Lemma 10.2.1, the G-diameter of  is h. Therefore, all diametral chords of  are G-diameters. Consequently, by Lemma 10.2.1 and Theorem 10.2.2,  has constant G-width h.  Corollary 10.2.2 Let  ⊂ M be a body of constant G-width h in the Minkowski space M. Then, for each pair H1 and H2 of parallel support hyperplanes of , every diametral chord of  generated by H1 and H2 is G-perpendicular to H1 and H2 .

10.2 Characterizations of Constant G -Width

215

Proof Let pi ∈ Hi ∩ , i = 1, 2. Then, by Theorem 10.2.2, p1 p2 is a G-diameter of ; hence  d M ( p1 , p2 ) = h = d M (H1 , H2 ). Consequently, p1 p2 is G-perpendicular to H1 , H2 . Later we shall see that this property actually characterizes constant G-width. Theorem 10.2.3 A convex body  has constant G-width if and only if there is a G-diameter of  in every direction. Proof A convex body  of constant G-width has a G-diameter in every direction because it has a diametral chord in every direction. Suppose  has a G-diameter in every direction. Let h be the G-diameter of  and let H1 and H2 be two parallel support hyperplanes of . We shall prove that d M (H1 , H2 ) = h. Let qi ∈ Hi ∩ , i = 1, 2. Then h ≥ d M (q1 , q2 ) ≥ d M (H1 , H2 ) = d M ( p1 , p2 ), where p1 p2 is a bridge of the strip cc(H1 ∪ H2 ). By hypothesis there is a G-diameter x1 x2 of  parallel to p1 p2 . Therefore h ≥ d M (q1 , q2 ) ≥ d M (H1 , H2 ) = d M ( p1 , p2 ) ≥ d M (x1 , x2 ) = h. This implies that d M (H1 , H2 ) = h, as we wished.



10.2.2 The Homothety Theorem The following homothety theorem will be useful in this chapter. It was already used in Section 2.12 to give a characterization of central symmetry. The proof in the smooth case is an exercise and is delicate in all its generality. The interested reader may consult Theorem 16.7 of Busemann’s book [204]. Theorem 10.2.4 Let φ1 and φ2 be two convex bodies containing the origin as an interior point. Suppose that for every ray R starting at the origin there are parallel support hyperplanes H1 of φ1 at R ∩ bd φ1 and H2 of φ2 at R ∩ bd φ2 . Then φ1 is directly homothetic to φ2 ; that is, φ1 = λφ2 for some λ > 0. As an immediate consequence, we have the following corollary (see Exercise 3.7 and Remark 2.12.3). Corollary 10.2.3 Let φ ⊂ M be a convex body and suppose p0 is an interior point of φ. If for every point p ∈ bd φ there is a support hyperplane H of φ at p that is G-perpendicular to p0 p, then φ is directly homothetic to G. Corollary 10.2.4 Let φ be a convex body and suppose p0 is an interior point of φ. If for every chord pq through p0 there are parallel support hyperplanes of φ at p and q, then φ is centrally symmetric with respect to p0 . We have also the following characterizations of the unit ball G. Theorem 10.2.5 Let φ ⊂ M be a convex set and suppose that p0 is an interior point of φ. If every chord of φ through p0 is a G-diameter of φ, then φ is directly homothetic to G.

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Proof We may assume that p0 is the origin. Let R be any ray starting at the origin. Suppose H1 is a support hyperplane of G at R ∩ bd G. Let pq be a chord of φ through the origin, which, by hypothesis, is a G-diameter of φ, and suppose { p} = R ∩ bd φ. Let us consider the ball G(q, h), where h is the G-diameter of φ. Note that p ∈ bd G(q, h) and φ ⊂ G(q, h). Clearly, there is a support hyperplane H2 of G(q, h) at p parallel to H1 . Hence H2 is also a support hyperplane of φ at p. By Theorem 10.2.4, φ is directly homothetic to G.  Theorem 10.2.6 A convex body  ⊂ M is a body of constant G-width in the Minkowski space M if and only  + (−) is homothetic to the unit ball G. Proof Suppose  has constant G-width. Observe that, since the Minkowski sum of two bodies of constant G-width is again a body of constant G-width, it will be sufficient to prove that every centrally symmetric convex body  of constant G-width h is homothetic to the unit ball G. For this purpose we shall use Theorem 10.2.5. Let us suppose that the center of  is the origin, and let pq be a chord of  through the origin. We want to prove that pq is a G-diameter. By Theorem 10.2.3, there is a G-diameter r s of  parallel to pq. Since (−s)(−r ) is also a G-diameter of , by Exercise 10.11 it follows that pq is also a G-diameter . This implies that  is homothetic to the unit ball G. On the other hand, if  + (−) is homothetic to G, for every hyperplane H through the origin the G-width of the support -strip parallel to H is half the G-width of the support ( + (−))-strip parallel to H . Consequently the G-width of the support -strips is constant, independent of the direction.  Corollary 10.2.5 A centrally symmetric convex body of constant G-width is homothetic to G. Let φ ⊂ M be a convex body and let pq be a chord of φ. We say that pq is a G-normal of φ at p if there is a support hyperplane H of φ at p that is G-perpendicular to pq. Furthermore, we say that pq is a G-binormal of φ if there are parallel support hyperplanes of φ at p and q that are G-perpendicular to pq. We leave it as an exercise to the reader to prove that every G-diameter is a G-binormal. Theorem 10.2.7 A convex body  ⊂ M is a body of constant G-width in the Minkowski space M if and only if there is a G-binormal of  in every direction. Proof The proof of Theorem 10.2.7 closely follows the proof of Theorem 3.1.2. Suppose that  ⊂ M is a body of G-constant width. By Theorem 10.2.3 there is a G-diameter of  in every direction, but by Exercise 10.7, every G-diameter of  is a G-binormal. We will need the following construction to proceed with the proof of sufficiency. From any body  ⊂ M with the property that  has a G-binormal in every direction, we construct another body ∗ as follows. Let u be a direction. We define the G-width of  in the direction u as the G-length of the G-binormal parallel to u. Note that by Exercise 10.8, any two parallel G-binormals of  have the same G-length. Draw a ray from the origin o in direction u. Take a point pu on this ray such that d M (o, pu ) is precisely the G-width of  in direction u. As we vary u, the point pu describes a body ∗ with the property that if m is a point in the boundary of ∗ , then d M (o, m) is the G-width of  in direction om (Figure 3.2). The body ∗ is, of course, different from . In fact, if  is a constant width body, then ∗ is homothetic to G, since the G-width of  is the same in every direction. Conversely, if ∗ is G(o, h) for some h, then the G-width of  is the same in every direction, so  is a body of constant G-width h (see Exercise 10.9). Therefore, to show that  is a constant width body it is sufficient to prove that ∗ is homothetic to G. For this purpose, we will use Corollary 10.2.3. Take any point m in the boundary of ∗ . Let x y be the G-binormal of  parallel to om, and let H1 and H2 be parallel support hyperplanes of  at x and y, respectively, that are G-perpendicular to the

10.2 Characterizations of Constant G -Width

217

direction om. Let H be the hyperplane passing through m G-perpendicular to om and parallel to Hi , i = 1, 2. We wish to prove that H is a support hyperplane of ∗ . To do so, it will be sufficient to prove that for every direction u the segment opu does not cross H. If this is so, then we can be sure that ∗ is situated entirely on one side of H (see Figure 3.3). By identifying H with H1 –indeed, the two hyperplanes are parallel—that is, by superimposing  on ∗ in such a way as to make H1 coincident with H and the segment x y with om, we can see that H2 passes through o and that  is between H and H2 (see Figure 3.4). Consider any direction u. There is a G-binormal pq of  in direction u with the property that d M ( p, q) is of course the width of  in direction u. Moreover, the G-binormal pq is contained in  and therefore lies between H and H2 . As the segments pq and opu are parallel, since both are parallel to u and are of the same G-length, it is easy to see that this implies that opu does not cross the hyperplane H. But since this is true for every direction u, we have that the boundary of ∗ does not cross the hyperplane H. Therefore H, which is G-perpendicular to om at m, is a support hyperplane of ∗ . Therefore, ∗ is by Corollary 10.2.4 homothetic to G, and  is a body of constant G-width.  As a consequence we obtain the following two theorems, which correspond to statements (II ) and (III ) of [238] originally proved by Eggleston in [316]. Theorem 10.2.8 A convex body  ⊂ M is a body of constant G-width in the Minkowski space M if and only if for each pair H1 and H2 of parallel support hyperplanes of , and each direction u being G-perpendicular to H1 and H2 , there is a diametral chord of  generated by H1 and H2 being parallel to u. Proof Suppose  ⊂ M is a body of constant G-width in the Minkowski space M. By Theorem 10.2.3, there is a G-diameter pq of  parallel to u. Consequently, pq must be a bridge of the -strip bounded by H1 and H2 . For the converse, note that by hypothesis there is a G-binormal of  in the direction u. Consequently, by Theorem 10.2.7,  has constant width.  The following theorem states that constant width is characterized by the fact that the notions of binormal and diametral chord coincide. Theorem 10.2.9 A convex body  ⊂ M is a body of constant G-width in the Minkowski space M if and only for each pair H1 and H2 of parallel support hyperplanes of , every diametral chord of  generated by H1 and H2 is G-perpendicular to H1 and H2 . Proof By hypothesis, every diametral chord is a G-binormal. This implies, by Exercise 5.9, that there is a G-binormal in every direction and, consequently, by Theorem 10.2.7,  has G-constant width. The converse is Corollary 10.2.2.  Theorem 10.2.10 Suppose the unit ball G of the Minkowski space M is strictly convex. Let  ⊂ M be a convex body. Then the following are equivalent: 1) The convex body  has constant G-width; 2) any two parallel G-normals of  coincide; and 3) any chord pq of  which is a G-normal at p is G-normal at q. Proof Since G is strictly convex, we have that if two lines are G-perpendicular to a hyperplane, then they are parallel. Suppose  has constant G-width. By Corollary 10.2.2,  is strictly convex, and therefore by Theorem 10.2.2 every G-normal is a G-binormal. Conversely, if every G-normal is a G-binormal, then there is a G-binormal in every direction, and therefore, by Theorem 10.2.3,  has constant G-width.

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Suppose now that every two parallel G-normals of  coincide. Then  is strictly convex, and also any chord pq of  which is a G-normal at p is G-normal at q. Consequently, by the above,  has constant G-width. Conversely, if  has constant G-width, then  is strictly convex, and by Corollary 10.2.2, any two parallel G-normals of  coincide.  Let M be a Minkowski space with unit ball G. The G-inradius r (φ) of a convex body φ is the largest real number t such that φ contains some translate of tG. Analogously, the G-circumradius R(φ) of a convex body φ is the smallest real number t such that φ is contained in some translate of tG. These translates of r (φ)G and R(φ)G are said to be G-inspheres and G-circumspheres of φ, respectively. The following theorem was proved by Chakerian [231], see also [336]. Theorem 10.2.11 Let  ⊂ M be a convex body of constant G-width h. Then r () + R() = h. Moreover, the G-inspheres and G-circumspheres of  are concentric. Proof We start proving that if G(x, r ) ⊂ , then  ⊂ G(x, h − r ). First of all, since the G-diameter of  is h, then r < h. Suppose y ∈ G \ G(x, h − r ) and let L be the line through x and y. Take a point p ∈ L between x and y such that d M ( p, x) = h − r and take a point q ∈ L such that d M (q, x) = r but q is not between x and y. Then d M ( p, q) = h and hence d M (y, q) > h, contradicting the fact that the G-diameter of  is h. Next, we prove that if  ⊂ G(x, r ) and r ≤ h, then G(x, h − r ) ⊂ . Suppose y ∈ G(x, h − r ) \ , and let H be a hyperplane that strictly separates y from . Take a support hyperplane H1 of G(x, h − r ) parallel to H and a support hyperplane H2 of G(x, h − r ) such that the G-width of the strip cc(H1 ∪ H2 ) is h and also that H ∪  ⊂ cc(H1 ∪ H2 ). Therefore the G-width of  in the corresponding direction to H is smaller than h, contradicting the fact that  has constant G-width h. Suppose now G(x, R()) is a G-circumsphere of . Then R() ≤ h and by the above, G(x, h − R()) ⊂ . We claim that G(x, h − R()) is a G-insphere of . Suppose not, hence there is G(x  , r  ) ⊂ , with r  > h − R(). Therefore,  ⊂ G(x  , h − r  ) contradicting our assumption that R() is the G-circumradius, because h − r  < R(). Similarly, if G(x, r ()) is a G-insphere of , then G(x, h − r ()) is a G-circumsphere of . This proves that r () + R() = h and G-inspheres and G-circumspheres of  are concentric. We finish this section by stating a characterization of constant width in the Minkowski planes due to Averkov and Martini [59]. Theorem 10.2.12 Let  ⊂ M be a convex body in the Minkowski plane M with unit disk G. The convex body  has constant width if and only if every chord pq of  splits the body  into two compact sets such that one of them has G-diameter equal to the G-length of pq.

10.3 Complete Bodies in Minkowski Spaces Let M be a Minkowski space with unit ball G. We say that a compact subset of M is G-complete if adding any point to it increases its G-diameter. If we take the partially ordered set hM of all compact sets of G-diameter h in M, ordered by inclusion, G-complete bodies are precisely the maximal elements of hM . That is, a compact set  in hM is a maximal element of hM , or is G-complete, if  is equal to φ whenever  is contained in φ, for φ in hM .

10.3 Complete Bodies in Minkowski Spaces

219

The concept of completeness was introduced by Meissner in [817]. In an arbitrary Minkowski space, any body of constant G-width is G-complete, but the converse does not hold, even if the unit ball is smooth and strictly convex. The converse was believed for a long time until Eggleston [316] gave counter examples. However, in dimension two a set is G-complete if and only if it has constant G-width. For higher dimensions, Naszódi and Visy [890] proved that a smooth G-complete body has constant G-width. Let us fix h > 0. For every Y ⊂ M, define G(Y ) =



G(y, h).

y∈Y

The following theorem characterizes G-completeness. Theorem 10.3.1 Let φ ⊂ M be a convex body of G-diameter h in the Minkowski space M with unit ball G. Then φ is G-complete if and only if φ=



G(y, h).

y∈φ

Proof The point x ∈ G(φ) if and only if d M (x, y) ≤ h for every y ∈ φ. In particular, φ ⊂ G(φ). Note that x ∈ G(φ) \ φ if and only if φ is properly contained in φ ∪ {x} and the G-diameter of φ ∪ {x} is h. Consequently, G(φ) \ φ = ∅ if and only if φ is not G-complete.  As a corollary of the above characterization of completeness, we shall prove that every compact set of diameter h is contained in a G-complete set and, furthermore, that every body of constant G-width h is a G-complete body. Theorem 10.3.2 Let X be a compact set of diameter h. Then X is contained in a G-complete body of diameter h. Proof Let D = {xm } be a countable dense set in M. We will construct our set inductively. For each nonnegative integer m, we associate the convex body Fm to the finite set X m ⊂ D defined inductively as follows: F0 = G(X ) and X 0 = X . Let Fm = Fm−1 ∩ G(xm , h) and X m = X m−1 ∪ {xm }, if xm ∈ Fm−1 , Fm = Fm−1 Note that for every m, Fm = Define

 x∈X m

and and X m = X m−1 , if xm ∈ / Fm−1 .

G(x, h) and A ⊂ Fm .  Fm = 1≤m β(F). There are, however, cases for which βG (F) < β(F). Indeed, if G is a square and F is a circle, the reader might verify, as an exercise, that βG (F) = 2 while β(F) = 3. The Borsuk problem in the Minkowski plane was studied in 1957 by Grünbaum [485]. He proved the following theorem. Theorem 10.6.1 Let F ⊂ M be a compact set in the Minkowski plane M with unit ball G. Then βG (F) ≤ 4. Furthermore, equality holds when G is a parallelogram and cc(F) is directly homothetic to G. The case of the Minkowski plane in which the unit disk is a parallelogram was completely understood by Grünbaum [485]. The reader may consult [150] or [151] for proofs. Theorem 10.6.2 Let F ⊂ M be a compact set in the Minkowski plane M with unit ball the parallelogram G. Then βG (F) = 2 if and only if cc(F) does not contain an equilateral triangle (in the Minkowski sense) whose side-lengths equal the G-diameter of F. The Borsuk problem in Minkowski planes, in which the unit disk is not necessarily a parallelogram, was also studied by Boltyanski and Soltan [153]. For Minkowski spaces of dimension higher than 2 only one result obtained by Ivanov [575] is known. Using the properties of a spherical mapping of the 1  n p p . Gauss type, Ivanov studied the case when the unit ball G gives the norm x p = 1 |x i |

Notes The origins and basic definitions of Minkowski geometry are connected with names like Riemann, Minkowski, Banach, and Busemann. The field of Minkowski geometry can be located at the intersection of Finsler geometry, Banach space theory, and convex geometry, but it is also closely related to many other fields. An example is distance geometry (in the spirit of Menger and Blumenthal [141]), another one is combinatorial geometry (see, e.g., [151, Chapters II and V]). Minkowski geometry was also enriched by many results from applied disciplines such as operations research, optimization, theoretical computer science and location theory. For an excellent book covering mainly the analytic part of the theory see [1124], and for surveys covering hundreds of papers in Minkowski geometry widespread in very different fields one should consult [795] and [793]. Since it is very natural to carry over the concept of constant width to (finite dimensional) real Banach spaces, the number of publications in this little field is large. We try to give an overview which is as complete as possible. In former expositions, many of these publications are also cited, in particular in the three surveys [238], [527], and [793]. Many new results were published later. Summarizing, many contributions (also some older ones) cannot be found in the mentioned three surveys. Thus, we aim to give here an overview with update which is as complete as possible. Of course, we use the introduced notation as far as possible. Since in these notes here we always refer to Minkowski spaces (and also to more general Banach spaces), we omit the

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addendum G everywhere and we do not extra mention this. Note that sometimes in the literature the notion “constant relative width” occurs, meaning the same concept (but to avoid confusion, we should mention here also the related, but different concept of relative differential geometry; see the survey [80]). And when using the word orthogonality, we mean (if not otherwise said) the most common orthogonality concept for normed spaces, namely that of Birkhoff orthogonality, see [24]. This yields also that, when speaking about double normals, we naturally replace diameters, important for the Euclidean case, by affine diameters (see the excellent survey [1076] on this notion). Basic Geometric Properties of Constant Width Bodies The basic theorems from Section 3.1 in this book here and Section 2 in [238] about the ball shape of the difference body of constant width sets, the existence and length of diametral chords, their orthogonality to supporting strips, their coincidence in the parallel case as well as their double normal property (here for smooth and strictly convex balls) are derived in [316] and [238, Section 2], see also [336] and [509] and the comprehensive representation in [793, § 2.1]. For example, we have the following basic statements: A convex body in n-dimensional Minkowski space is of constant width h iff its difference body coincides with the unit ball, and iff all its diametral chords have length h. Thus, the only centrally symmetric constant width bodies are also in Minkowski spaces the balls; e.g., [58]. And if the unit ball is smooth and strictly convex, then a convex body is of constant width iff any two parallel normals of it coincide. Petty and Crotty [933] proved that a body of constant width having an equichordal point is a ball of the norm (an alternative approach to an important partial step in the proof was established in [509], see also [380] for n = 2 and n = 3). It was also shown in [933] that there are Minkowski spaces with convex bodies having exactly two equichordal points. In the mentioned paper [509] results on diametral chords of constant width sets, also derived in the papers [506], [511], [512], [508], and [1072], are extended to normed planes. Vre´cica [1159] confirmed that a convex body is of constant width iff for all points a, b from its interior, there is a set of constant width contained in this interior having a, b itself as boundary points. An important theorem for normed spaces is the monotonicity lemma, see subsection 3.5 of the basic expository paper [795] or [1124, Lemma 4.1.2]. Heppes [531] proved a theorem which can be interpreted as monotonicity lemma for sets of constant width in the Euclidean plane. One of the implications was extended to strictly convex normed planes by Grünbaum and Kelly [490], and in [59] the result from [531] was completely extended: Any hyperplane section S of an n-dimensional convex body of constant width divides that body into two compact, convex sets such that at least one of them has the same diameter as S. For n = 2 this even characterizes the sets of constant width. We refer also to the related papers [694] and [54]. In the latter paper it is shown that in higher dimensions at least the following holds: If the convex body is partitioned by a continuous family of hyperplane sections S(t), 0 ≤ t ≤ 1, then the diameter of S(t) is an unimodal function of t. Closely related is also [57]: In a strictly convex, smooth and n-dimensional Minkowski space a convex body with the same boundary restrictions is said to have property (P) if any manifold M0 homeomorphic to the (n − 2)-dimensional sphere and lying on the boundary of the body splits its boundary into two compact manifolds, one of them having the same Minkowskian diameter as M0 . It is shown that if a convex body has constant width, then (P) holds for it. Vice versa, if the body satisfies (P) and has at least two diametral chords, then it is of constant width. And still related is the paper [52]: a convex body in an arbitrary normed plane is of constant width iff it is splitted by every chord I of it into two compact convex sets K 1 and K 2 such that I is a Minkowskian double normal of K 1 or K 2 . In [55] the following result, which was announced by Hammer and Smith [510], is established for normed planes: If all binormal chords of a plane constant width set bisect its Minkowskian perimeter, then it is a ball. This paper contains also interesting results on normed planes whose unit circles are equiframed curves (i.e., centrally symmetric closed convex curves touched at each of their points by some circumscribed parallelogram of smallest area). Chakerian [231] (see also [336]) established the following well-known theorem for Minkowski spaces: The inradius r and the circumradius R of

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a convex body of constant width h satisfy r + R = h, and the corresponding in- and circumsphere are concentric. Sallee [1001] showed that this result still holds if “constant width” is replaced by “spherical intersection property”, related also to the notion of Minkowskian pairs of constant width. Barbier’s theorem says that any planar set of constant width h has the same perimeter hπ, and it holds analogously in normed planes. Most likely, the first proof of it was presented in [929], and further proofs and variants can be found in the papers [827], [785], [625], [270], and [697]. The latter paper contains even a generalization of Barbier’s theorem, and there it is also proved that for every ε > 0 and every convex body of constant width in a normed plane there exists a convex body of the same constant width whose boundary consists only of arcs of circles in the sense of the norm such that the Hausdorff distance between the two bodies is at most ε. As an approximation result, this generalizes the Euclidean case proved by Blaschke. Let M denote a normed plane with isoperimetrix I , and M  be the Minkowski plane with I as unit circle. In [625] nice results on curves of constant width in such pairings of normed planes are established. For example, if C has constant width h with respect to ) holds, where A denotes area and A(C, −C) the mixed area of M  , then A(C) + A(C, −C) = h A(I 2 C and −C. And if the perimeter L(C) of such C is measured in M, then L(C) = h A(I ). As already mentioned an analogue of Barbier’s theorem is derived, too. The paper [223] is based on the same concept with M  and I . It is shown there that if C is a smooth curve of constant width in M  each of whose diametral chords bisects the M  -perimeter (or the area), then C is homothetic to I . In [22], examples are provided which show that if one uses Birkhoff orthogonality, then the Makai–Martini characterization of curves of constant width (Theorem 4.4.1) cannot be extended to normed planes. The authors present also further results on intersecting orthogonal chords of unit circles of Minkowski planes, based on Birkhoff and also on James orthogonality. On the other hand, in [25] the new notion of affine orthogonality is introduced, and for this type of orthogonality the Makai–Martini characterization has some extension. The paper [25] contains also other characterizations of sets of constant width, e.g. via a double-normals property. Special Bodies of Constant Width There are different possibilities to define the Minkowskian analogue of the Reuleaux triangle. Analogous to the Euclidean situation, Ohmann [898], Chakerian [230] and Wernicke [1197] defined it as intersection of three circles, each of radius h and centered at one of the vertices of an equilateral triangle, and they described it as extremal figure regarding certain metrical problems (we refer to our subsection below on inequalities). Wernicke [1197] proved that the planes with parallelograms or affine regular hexagons as unit circles are the only ones in which Reuleaux triangles exist as Minkowski circles; this was reproved in [785]. Constructions of constant width figures and, in particular, Reuleaux polygons are presented in Chapter 4 of the monograph [1124], [672], and [667]. Reuleaux polygons are also investigated in [509] and [997]; Hammer [509] obtained analytical representations of special constant width curves, giving a specific formula (involving suitably properties of the unit ball) always leading to a curve of constant width. Petty [929] used the Minkowskian analogue of “evolute” to construct special curves of constant width. The authors of [270] defined notions like Minkowskian curvature, evolutes, and involutes for polygons of constant width. They confirmed that many properties analogous to those of the smooth case, which was studied before in [269], are preserved under corresponding iterative procedures. For example, the iteration of involutes of polygons of constant width generates a pair of sequences of polygons of constant width with respect to the given norm and its dual, respectively. It turns out that these sequences are converging to symmetric polygons with the same center, which can be regarded as a central point of the starting polygon. Further geometric properties of Minkowskian Reuleaux triangles are discussed in [785], [997] and [965].

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Completeness and Completions We recall that a bounded subset C of a Minkowski space is called complete or diametrically complete if it cannot be enlarged without increasing its diameter, and any complete set C having the same diameter as a convex body K with K ⊂ C is said to be a completion of K . Important early references on complete sets in Minkowski spaces are [316], [160], [457], see also [151, Chapter V]. We will survey now results on complete sets, completion procedures and applications of these notions in Minkowski spaces and in more general Banach spaces. For normed planes, the notions of completeness and constant width are equivalent, and in higher dimensions constant width still implies completeness, see [336, § 2] and [316] for an explicit proof, and also Theorem 10.3.3 and Corollary 10.3.1. The converse implication was confirmed for n = 3 by Meissner [817] and for higher dimensions by Kelly [616], but their proofs were erroneous. Eggleston [316] constructed an explicit example showing that for n > 2 the converse is no longer true, even for norms that are strictly convex and smooth, and he also constructed an example with polyhedral unit ball. The characterization of those Minkowski or Banach spaces in which completeness and constant width are equivalent properties became a famous research topic, see below and, e.g., [861] and [788]. The norms with this property are called perfect norms. This name is due to Karasev [606], and still no characterization of this important class of norms is known. Minkowski spaces with perfect norms are precisely those in which the family of complete sets is convex, or the family of completions of any given set is convex. Karasev also showed that if a norm is perfect and strictly convex, then the intersection of the unit ball with any translate of it is a summand of it. Polovinkin [940] defined an interesting notion which is closely related to perfect norms. He rediscovered an elegant completion procedure presented originally by Maehara [747] for the Euclidean case. This concept is also discussed in subsection “Related concepts” below, in these notes. According to Sallee [1002] and Polovinkin [940], this way of completion even works for a class of Minkowski spaces, namely those whose unit ball is a so-called generating set. A convex body is called a generating set if any intersection of translates of it is also a summand of it. Maehara [747] was the first who proved that Euclidean balls are generating sets, of course without using this name, and Polovinkin [940] introduced and extended this notion. The topological and algebraic properties of generating sets, the properties of related notions like M-strongly convex hulls and support functions of M-strongly convex sets (see below) were deeply studied in [64]. From results derived in [747] and [1002], Polovinkin could conclude that if the unit ball of a Minkowski space is a generating set, then the norm is perfect. (Note that at the end of this subsection we discuss types of generating sets which are known until now.) Based on [1002] and [64] it was also shown by Karasev [606] that in a reflexive Banach space the so-called Maehara set of a given compact convex set is of constant width. Since Maehara’s type of completion is not extendable to all Minkowski spaces, Moreno and Schneider [864] were motivated to look for further completion procedures extendable to all Minkowski spaces. Based on results of Bückner [192] (see also [966] for a 2-dimensional earlier version, and [677] for finding independently an important step in this approach), they succeeded with a constructive completion procedure in general Minkowski spaces and showed also that their procedure yields a locally Lipschitz continuous selection of the completion mapping. And also the elegant completion procedure of Maehara was studied in [864], also having locally Lipschitz continuous extensions. Modifying two well-known characteristic properties of bodies of constant width, Moreno and Schneider [862] presented a new characterization of complete sets in Minkowski spaces. More precisely, they proved that a convex body is complete iff the width of each regular supporting slab of it equals its diameter (equivalent to the assertion that the length of each regular diametral chord equals the diameter). Sharp estimates for circum- and inradius of a complete set of given diameter were derived, and it was also verified that in a generic Minkowski space of dimension > 2 the family of complete sets is not closed under the operation of adding a ball; also new results about perfect norms were obtained (see also [971]). In [861] properties of the family of all complete sets of diameter 2 in a given Minkowski space were studied. It was shown that this space (obtained when metrized by the

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Hausdorff distance) is generally not convex, and that it is not necessarily starshaped. A characterization of those spaces in which this set is starshaped was given, via some properties which were already obtained in [862]. Furthermore, it was proved that this space is contractible, and that for polytopal unit balls the space of corresponding translation classes is the union set of a finite polytopal complex. The authors of [863] also studied the mapping γ assigning to each compact convex set the family of all its completions and proved that in normed spaces with Jung constant less than 2, γ is locally Lipschitz continuous with respect to the Hausdorff metric induced by the respective norm (see also above and [856]). One motivation, interesting for us here, is the fact that γ is important for the study of bodies of constant width. It was proved that a Minkowski space has perfect norm if and only if γ is convex for every nonempty, compact and convex set (see also [606]). For a compact convex subset C of a Banach space, a closed convex set K containing C with the same diameter is called a tight cover of C. For finite dimensions, there is at least one tight cover of C with maximal volume, and this turns out to be a completion of C and is, for strictly convex norms, even unique (see also again [457]). Groemer [457] discussed also symmetry properties of such completions. The paper [858] is concerned with topological and stochastic properties of the family of all closed convex sets with unique completion. With a sharpened version of a lemma due to Groemer it was shown that, in strictly convex Minkowski spaces, this family is lower porous. This improved a previous result from [457] where this family was shown to be nowhere dense. In contrast to this, there is a stochastic construction procedure which provides a complete set with probability one. This generalizes an earlier result from [81] for the Euclidean plane. If in Minkowski spaces, also higher dimensional ones, a complete convex body is smooth and strictly convex, then it is also of constant width, see [890] and [336, Theorem 3.18]. In [890] also the first example of a norm was found such that the vector sum of a complete set and a ball is not complete. In the latter paper it is also proved that if there exist symmetric points p and − p such that every facet of the unit ball of the space contains one of them, then balls are the only sets of constant width in this space. Also in this paper, sets in Euclidean space are studied which have unique extensions to sets of constant width. Eggleston [316] proved that in any Minkowski space a compact set has the spherical intersection property (SIP) (see below) iff it is complete and iff each boundary point of it has its diameter as distance to at least one other point of it (an alternative proof was given in [336, § 3]). In [313] it was wrongly stated that the only unit balls for which any constant width set is a ball are the parallelotopes. To get the right answer, one has to look for all irreducible sets (cf., e.g., Shephard [1060]) i.e., for all convex bodies K centered at the origin for which the representation K = Q + (−Q) is only possible if Q is centrally symmetric. Yost [1208] showed that for dimension n > 2 most (in the Baire category sense) centered convex bodies are irreducible, and that most n-dimensional Minkowski spaces (having a smooth and strictly convex unit ball) have the property that all constant width sets are balls; hence they are complete (see also [1039, § 3.3]). On the other hand, any complete set is a ball iff the unit ball is a parallelotope, see [316] and [1074]. (The 2-dimensional case of the latter statement was already given in [509], and the infinite-dimensional situation was studied in [275] and [374].) Investigating 4-dimensional polyhedral unit balls in view of irreducibility, in [923] examples of Minkowski spaces important regarding the n-ball property and with respect to semi-M-ideals (basic notions from Banach space theory) were constructed. Uniqueness of completion is also important for Borsuk’s partition problem, see [151, § 33] and [788]. In addition, we refer to [77], where investigations from Groemer [457] are summarized and continued, and the relations of completeness and the spherical intersection property in the infinite-dimensional case are investigated. In [179] a one-to-one connection between the generalized Jung constant and the maximal Minkowski asymmetry of the complete bodies in an arbitrary Minkowski space is established. With this starting point, the authors generalize and unify also recent results on complete bodies. The paper [179] contains also results for nonsymmetric unit balls (i.e., for gauges or convex distance functions not necessarily symmetric), and the completeness of simplices plays an essential role for inequalities related to Bohnenblust’s inequality (see [144]) related to the Jung constant. Vre´cica [1159] observed that each bounded set has a completion contained in any circumball, which was sharpened in [336].

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Sallee [1003] described methods for generating complete sets which contain an arbitrary set, paying special attention to the problem of preassigning boundary points for such sets, and also the preservation of symmetry properties is discussed. Some of these results are reproved in [77], and in addition the extension to infinite dimensions is done there; also Vre´cica’s result carries over to infinite dimensions. In [518] all normed spaces are characterized that contain a two-point set with unique completion. They form a strict superfamily of all Banach spaces in which any complete set is a ball, and a strict subfamily of all Banach spaces in which each constant width set is a ball. On the other hand, in finite dimensions there is a completion C satisfying C ∩ S = K ∩ S, where K denotes the bounded set having S as circumsphere, and in infinite dimensions not. In [77] relations between the spherical intersection property and completeness in (finite and) infinite-dimensional Banach spaces are discussed, and it is shown that most, but not all constructions and related results of Sallee, Schulte, and Vre´cica also hold for infinite dimensions, where at least one completion always exists, too. In [787] results of the following type are shown: Let K 1 and K 2 be different complete sets. If there is no inclusion between the sets K 1 and K 2 , then their union cannot be complete; and if the three diameters of K 1 , K 2 , and their intersection are equal, then their intersection cannot be complete. On the other hand, a set K and its ball hull have the same completions. In [918] different construction methods are compared, some of which are connected with completions or uniqueness of completions. After a useful survey of known results on this topic, the authors of [920] present new methods to get completions in separable Banach spaces. They use techniques which exist for finite dimensions. The first one is inspired by Theorem 54 in the book [312]. For the second method, the wide and the tight spherical hull are used; the basic idea in finite dimensions is, independently, due to Bavaud [81] and to Lachand–Robert and Oudet [677]. Also in [519] these unions and intersections of completions (i.e., wide and tight spherical hulls) of compact sets are studied. For sets which are not complete, the authors derive boundary properties of their hulls which allow them to characterize diametral points of the original sets, and they also prove that the distance between such a set and the boundary of each of its hulls is always 0, though their intersection may be empty in infinite-dimensional spaces. In [919], several natural relations between balls, complete sets, and completions in Banach spaces are investigated, e.g., conditions for ball-shaped completions. Caspani and Papini [218] discuss various properties concerning complete and constant width sets; in particular, their radii, self radii, and the existence of centers and incenters are studied. Several nice examples, some of them rather pathological, give a fairly complete picture concerning different possible situations. In [219] it was shown that for closed, convex and bounded sets in a Hilbert space constant width and completeness are equivalent notions. If such a set has smooth boundary and for each boundary point there is another one at distance the diameter, then constant width is satisfied. In [767] it was proved that a special Banach space, which is a renorming of l2 , contains a complete set without interior points. The paper [768] is among the first to characterize the normal structure of spaces in terms of notions like constant width sets, diametrically complete sets, and sets with constant radius. It starts with a historical survey in that direction. Sets with constant radius and the related family of diametral sets are introduced via the notion of Chebyshev radius. In the previous work [859], Moreno, Papini and Phelps showed that in every normed space a complete set is a set with constant radius. These investigations were continued in [860] by the same authors. There they introduced additionally sets of constant difference and verified the inclusion sequence regarding the four families of sets of constant width, of constant difference, complete ones, and of constant radius. Each of the inclusions can be strict, but under certain conditions (which are also presented) equality holds. For example, the families of sets of constant radius and of complete sets coincide for finite-dimensional spaces; the families of sets of constant width and of constant difference coincide if the Banach space has the Mazur Intersection Property; and all four families coincide in Hilbert spaces. In [797] the authors construct examples showing that in Minkowski spaces a complete set is not necessarily reduced. Continuing such investigations, it was asked in [180] whether a convex body K being complete and reduced with respect to some gauge body (not necessarily centered at the origin) has to be of constant width. This

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implication is verified for the following large class: the implication holds if K possesses a smooth extreme point. In [917], Papini discussed the role of completeness in approximation theory. Now we cite results referring to the C(K ) spaces, i.e., of all real-valued continuous functions on a compact Hausdorff space K with the maximum norm  f  = maxt∈K {| f (t)|}. We denote by H the family of all closed, convex and bounded subsets of C(K ) furnished with the Hausdorff metric. In [855] the notions of completeness and completions were naturally introduced for C(K ) spaces, and a characterization of the family of all possible completions of sets from H was given. It was shown that if S is the family of all elements of H having a unique completion, then S is uniformly very porous in H if and only if K is not a singleton. Also following [859] and [860], the author of [856] studied the fundamental properties of the complete hull mapping in C(K ) spaces. As above, it associates to each C ∈ H the set of all its completions. It turns out that this mapping is Lipschitz continuous and has a Lipschitz selection, while it is convex-valued iff K is extremely disconnected. In Euclidean spaces, the mapping is always convex-valued (an example is provided to show that this is not true in a generic finite-dimensional space). In [857] it is shown that in C(K ) spaces the following holds: each C within the family of all sets which are intersections of closed balls has a best approximation within the family of completions of C, and within the family of all complete sets. Problems related to vector addition and completion procedures of convex bodies in C(K ) spaces are dealt with in [865], and geometric properties of convex bodies in C(K ) are shown to characterize the underlying compact Hausdorff space as a Stonean space. It was shown that there is a Maehara-like completion procedure (see [747]), and that the family of all complete sets of diameter h is starshaped with respect to any ball of radius h/2. As already announced, we finish this subsection with a little survey on what is known about generating sets and related notions (like results on Minkowski difference if directly related to generating sets, and strong convexity). Note that the system of generating sets is stable under linear transformations and direct sums. 2-dimensional convex bodies are generating (see [406]), like also the unit balls of Hilbert spaces (see [747] and [64]). Generating polytopes were investigated already in [811], together with other interesting polytope classes. In [165] these investigations were continued: the class of all 3-dimensional polyhedral generating sets coincides with the family of strongly monotypic 3polytopes from [811]. Except for the following class, all these examples are nonsymmetric. Namely, it was shown in [811] that the only centrally symmetric polytopal generating sets are direct sums of polygons and, for odd dimensions, segments. Besides this centrally symmetric case, from [811] the generating n-polytopes (n > 2) are only explicitly known if they have at most n + 3 facets. In [166] also non-polytopal generating sets are derived. Ivanov [577] derived a criterion for generating sets, yielding a few new (symmetric) examples of generating sets in Hilbert space. After a survey on results of the author, in [941] constructive approaches to completions of sets in reflexive Banach spaces with generating sets as unit balls are presented. Using the machinery of strongly convex analysis (see [939]), the author also established a criterion for unique completion to a constant width body and proposed algorithms for constructing all bodies of constant width containing a given set of the same diameter. Here we also mention [942]. And in [64] the notion of an M-strongly convex set, where M is generating, was introduced. The latter can be represented as the intersection of sets which are translates of the generating set M. The authors described various classes of generating sets and studied properties of corresponding M-strongly convex sets, also strengthening classical Carathéodory-type results and Krein–Milman theorems into this direction. The paper [549] contains interesting separation theorems where generating sets are basically involved. Closely related to the notions discussed here is the study of certain properties of Minkowski sum and difference, which are also important for pairs of constant width, see, e.g., [747], [1002], [167], [439], [849], and [851]. In the latter paper, several topological characterizations of Minkowski summands are given, also yielding a characterization of homothety of convex sets. Also two nice characterizations of the Euclidean n-ball in terms of homological (n − 2)-spheres or, respectively, in terms of connected components are derived there. We also mention the papers [603] and [604] of Kallay who gave a complete characterization of planar convex

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bodies which are indecomposable under Minkowski addition within the class of bodies of given width function. In particular, a planar set of constant width 1 is indecomposable with respect to the class of bodies of constant width iff its radius of curvature (which exists almost everywhere on the unit circle) is almost everywhere 0 or 1. There seem to be no analogous results in higher dimensions. Finally, we mention that also in our Sections 7.3 and 7.6 closely related topics are discussed. Reducedness It is natural to extend the concept of reducedness also to Minkowski spaces; the greatest part of recent knowledge in this direction is covered by the survey [702]. The first related investigations were done in [700], and the paper [328] contains basic results mainly for the planar case. Again, a convex body in a Minkowski space is said to be reduced if any convex proper subset of it has smaller minimal width, and every convex body contains also a reduced body of the same minimal width. Thus, the following open “dualization” of the isodiametric problem (completely solved for normed spaces by [820], with balls as the unique solution) refers to the much wider family of reduced bodies in normed spaces: Which convex bodies of fixed minimal width have, for any Minkowski space of dimension n > 1, minimal volume? (For the definition of volume in normed spaces we refer, for example, to [820] and [908].) There are certainly normed planes and spaces for which this can be easily answered, but we are not aware of a systematic investigation in this direction. However, interesting related results nevertheless exist; see, e.g., [53] and [56]. And in [180] it is asked whether a complete, reduced set is necessarily of constant width. This question is even posed for gauges, without the symmetry axiom, and it is answered affirmatively for a large class of norms/gauges. In normed planes the same implication as in E2 holds: each strictly convex reduced body is of constant width (see [328]). Also one might look for norms where the balls are the only reduced bodies. In [700], it was observed that again, for every norm, the only centrally symmetric reduced bodies are the balls (see also [58]). And the following also still holds: Every smooth reduced body is of constant width. Various examples of reduced bodies in normed spaces are collected, discussed, and depicted in the papers [700], [696], [328], [60], and [702]. For fixed minimal width, some of these examples can be “arbitrarily prolonged” like the Euclidean example from [693]. Referring to this, it is conjectured in [702] that in each normed space of dimension n > 2 there exist reduced bodies of fixed minimal width having arbitrarily large finite diameter. This was confirmed by Richter in [971]. We come now to reduced polytopes in normed spaces, again with an open problem: Do there exist normed spaces of dimension n > 2 in which no reduced polytopes exist? This question was posed in [702], but in view of [437] (where it is shown that in E3 such polytopes exist) perhaps an affirmative answer becomes less likely. Lassak [696] proved that a convex polytope P in a Minkowski space is reduced if and only if one supporting hyperplane of a minimal-width strip passes through any vertex of it, containing only this vertex of P. Various theorems on reduced polytopes in normed spaces having small vertex numbers were obtained in [60], using new generalized antipodality notions; see also [696] for reduced simplices. It should also be clarified in which normed spaces, for n > 2, there exist reduced simplices (in the planar situation this is always the case, even with an arbitrary direction of one side of a reduced triangle; see [700]). In [23], the geometry of reduced triangles is deeply investigated (see also [700]). For example, if the centroid and the incenter of a triangle coincide in the Euclidean plane, this implies equilaterality. In normed planes, in general a wider class of triangles is obtained in this way, but they are still reduced there. On the other hand, equilaterality of triangles in the antinorm is equivalent to their reducedness in the original norm. And equilaterality and reducedness of triangles coincide precisely for Radon norms. Also, reduced triangles can be applied successfully to study the Fermat–Torricelli problem and Steiner minimum trees in normed planes; cf. [237], [792], and [185]. Higher dimensional analogues for Minkowskian simplices are studied in [237], [51], and [721]. There are also nice results on reduced polygons in normed planes (cf. [696] and [60]): For every side of a reduced polygon P the minimal

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width is attained in the direction perpendicular to it (i.e., orthogonal, in the Euclidean background metric, to the corresponding minimal-width strip). This property has no analogue for n ≥ 3. If a convex polygon is reduced, then its central symmetral has to be a polygon circumscribed about a ball of the respective norm (i.e., each side of this polygon has to belong to some supporting line of that ball). Geometric properties of general reduced bodies in normed planes were mainly obtained in [328] and [697]. These are general boundary properties, and also inequalities, approximation results and further natural generalizations of the analogous results for E2 . Furthermore, in [971] Richter studied deeply the ratios of diameter and width of reduced and of complete convex bodies. Confirming a conjecture from [702], he showed that for n > 2 there exist reduced bodies of arbitrarily large ratio, whereas the ratio for complete bodies is bounded by (n + 1)/2. As a consequence of these results, every normed space of dimension at least 2 contains reduced bodies that are not complete. Staying with the idea of non-Euclidean geometries, we mention that research on spherical reduced bodies has also started. For example, in [698] the following results are derived: Every reduced spherical polygon is an odd-gon of thickness h at most π2 , and such a spherically convex odd-gon is reduced if and only if the projection of each of its vertices onto the great circle containing the opposite side belongs to the relative interior of that side, and the distance of this vertex from that side is h. An upper bound on the diameter of reduced spherical polygons is also given, with equality for regular spherical triangles. And in [699] it is shown that any smooth reduced spherical body is of constant spherical width. Intersection Properties An n-dimensional convex body K has the spherical intersection property (SIP) if it is the intersection of all balls whose center is from K and whose radius equals the diameter of K . In Euclidean space, the notions of (SIP) and constant width are equivalent; in Minkowski spaces this is no longer true, but (SIP) is still equivalent to completeness; see [316] and [238, p. 62]. If K is of constant width, let t (K ) denote the smallest number of balls whose intersection is K . Soltan [1075] studied the number t (K ), giving, e.g., necessary and sufficient criteria for t (K ) < ∞ and constructing unit balls for even values in the plane. Groemer [457] used the (SIP) to study sets with unique completions in Minkowski spaces. And we repeat once more that Sallee [1001] extended Chakerian’s inradius–circumradius theorem from sets of constant width (cf. [231]) to sets with the (SIP). In [668], [789] and [774], relations between the notions of ball intersection (the intersection of all unit balls with centers from the given set), completion, (SIP) and constant width are investigated. (Note that a set has the (SIP) if it coincides with its ball intersection.) In the first paper the related weak spherical intersection property (WSIP) is introduced (for the Euclidean plane): a planar set S of diameter 1, say, has the (WSIP) if the intersection of all unit circles with center in S is of constant width. A finite example is the vertex set of a Reuleaux polygon. The least number of points (given in terms of the diameter graph) to be added to a finite planar set of diameter 1, say, such that the resulting set has the (WSIP), is determined. It would be very interesting to study (WSIP) sets also in higher dimensions and Minkowski spaces. In [789], the following further results for normed planes are shown: A convex body of diameter 1 satisfies the (SIP) iff it is of constant width; if a convex body has constant width, then it satisfies the related spherical hull property (defined via the intersection of all unit balls containing the given set). If a convex body and its spherical intersection are both of diameter 1 and it satisfies the spherical hull property, then this convex body has constant width. Further results from [789] refer to Reuleaux polygons and (WSIP) sets in normed planes. The paper [774] refers to applications of these notions in n-dimensional Minkowski spaces to Chebyshev sets and Chebyshev centers of bounded sets, thus giving also links to approximation theory and location science. In [788] new insights into representations of complete sets and (pairs of) sets of constant width are obtained, e.g., as vector sums of suitable ball intersections and ball hulls, a strongly related notion. Baronti and Papini [77] proved several properties combining completeness and the (SIP), also for infinite dimensions.

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Curvature and Mixed Volumes It turns out that classical results on principal radii of curvature (see [160, p. 128]), the surface area function and mixed volumes (cf. [1039, § 4.2 and § 5.1]) and, coming from this, a nice characterization of bodies of constant width via their surface area function (= first curvature measure of second kind) at opposite Borel sets on the unit sphere S n−1 can be carried over to constant width sets in Minkowski spaces, using the tools of the so-called relative differential geometry discussed in [160, § 38] and [238, § 6], see also [522] and [523]. For a smooth convex body  ⊂ En let L(bd , x) denote the canonical linear mapping of its tangent space bd x at x ∈ bd  into the tangent space of Sn−1 at u ∈ Sn−1 , where x and u are connected by the usual Gauss map via parallel normals, with u as a unit outward normal of bd  at x. If for a smooth unit ball G in n-dimensional Minkowski space M, Ge denotes the tangent space of bd G at e having the same unit normal as bd  at x, then the linear map J : bd x → Ge can be defined by J = L(bd , x). L −1 (bd G, e) and is, as the canonical linear map of bd  into bd G via parallel normals, invertible with J −1 = L(bd G, e). L −1 (bd , x). Thus we may write J = J (u), where u is the outward unit normal vector of bd  at x, and of bd G at e. The relative principal radii of curvature R˜ 1 , . . . , R˜ n−1 of bd  at x are the reciprocals of the eigenvalues of J (u), and the corresponding relative principal directions are the eigenvectors of J (u). Following [160, p. 64], we write { R˜ 1 , . . . , R˜ ν } for the ν-th elementary symmetric function of the relative principal radii of curvature R˜ 1 , . . . , R˜ n−1 and we let Fν (, u) = { R˜ 1 , . . . , R˜ ν } at u be as above, see Section 11.2.1. With this notation, Chakerian [231] proved that if in addition  has constant width h, then R˜ i (u) + R˜ n−i = h, for every u ∈ Sn−1 , and in particular F1 (, u) + F1 (K , −u) = (n − 1)h, u ∈ Sn−1 . Furthermore, we may define the relative mixed surface area as S(, . . . , , G, . . . , G, E), where the convex body  appears r times, G appears n − r − 1 times, and E is a Borel set in Sn−1 , see Section 12.2. In the same paper Chakerian proved that if  has constant G-width then S(, G, . . . , G, E) + S(−, G, . . . , G, E) = 2S(G, E)

(10.2)

and, conversely, if the unit ball is smooth and (10.2) holds then  is of constant width. Hug [563] obtained further deep results in this direction, some of them also related to pairs of constant width in Minkowski spaces. In terms of difference bodies and for normed planes, the constancy of the sum of the radii of curvature of correspondingly opposite boundary points of a planar body of constant width was first observed by Vincensini [1152], and reproved by [1113, p. 31] and [929, Theorem (6.14)]. Also for normed planes, Chakerian [236] proved that an integral presented by the relative radius of curvature and the relative arc length element of a set K of constant width can be expressed by a double integral based on the number of diameters of K passing through the points of this set. For twice continuously differentiable convex curves in the plane, the four vertex theorem says that there are at least four vertices, i.e., points where the curvature has a stationary value. Heil [522] derived four vertex theorems for different definitions of curvature in normed planes, see also [523]. His results imply that in a normed plane any smooth curve of constant width has at least six vertices, see also [827]. Here we also mention the paper [66], in which all possible curvature types of smooth curves in normed planes are derived and classified. Based on this, several results (like Barbier’s theorem, or statements close to Theorem 11.3.1) are elegantly reproved, yielding also analogous statements for curves of constant anti-width (which refers to the width in the antinorm) if the anti-curvature radius (i.e., the inverse of the normal curvature) is used. A representation of k-th quermassintegrals of convex bodies of constant width in En in terms of mixed volumes due to Dinghas [296] was extended by Chakerian [235] to normed spaces. Finally we mention here that Guggenheimer (see [491, p. 327]) announced related results on sets of constant width, but more generally for gauges (where the unit ball is no longer centered at the origin).

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Inequalities The Blaschke–Lebesgue theorem for normed planes, saying that also there the Reuleaux triangle of width h has minimum area among all planar sets of constant width h, was proved by D. Ohmann and, independently, K. Günther (in both their dissertations, Marburg 1948); see also [898], [707], [929, pp. 15–16], [661], [230], and [410] for further approaches and modifications of this result. But in Minkowski planes Reuleaux triangles of given width h can have different areas. The particular type reaching the minimum (and obtained on independent, different ways in [898], [230], and [661]) can be constructed as follows: Let the Minkowskian unit vectors r1 , r2 , r3 add up to o. Then one can form a triangle a1 a2 a3 by suitable translates of the segments ori such that a1 = o and a2 , a3 are from the unit circle. Connecting a1 and a2 , as well as a2 and a3 , by corresponding boundary arcs of the unit ball, we get a figure of constant width having a1 , a2 , a3 as boundary points and called by Ohmann general Reuleaux triangle. The announced analogue of the Blaschke–Lebesgue theorem says that for Minkowski planes among all figures of constant width h > 0 a general Reuleaux triangle has minimum area. Ghandehari [410] gave an optimal control formulation of this statement. Here we mention also the paper [1063], in which (similar to [661]) inequalities between several quantities are established. More precisely, given two quantities, the extremum values of a third one are established and the corresponding extremal bodies are determined. For example, Reuleaux polygons play a role there. And Sallee announced in [999] that the Minkowskian analogue of the Firey–Sallee theorem, saying that among all Euclidean Reuleaux polygons of width h > 0 and having m ≥ 3 vertices, the regular one has maximum area, can be proved on the same lines as the Euclidean subcase. In [1197] and [785] it is shown that the ratio of the area of the unit ball to that of a Reuleaux triangle of width 1 lies between 4 and 6, and in [785] several further inequalities on areas of Reuleaux triangles and a new proof of Barbier’s theorem are given. Also the paper [965] refers to Reuleaux triangles as extremal figures. Castro Feitosa [336] used extremal properties of figures of constant width in normed planes to extend results of Scott [1052] from the Euclidean case to Minkowski planes. More precisely, he used that among all convex figures in normed planes with given diameter and circumradius the sets with largest thickness, perimeter, inradius, and area are of constant width. More inequalities between inradius, circumradius, diameter, and related quantities of convex bodies in Minkowski spaces yielding in some cases sets of constant width as extremal cases are discussed in [313], [1063], [144], [1082], [709], and [710]. The Rosenthal–Szasz inequality between perimeter and diameter of a planar convex figure was extended in [65] to Radon planes, and an analogue (considering the diameter in the antinorm) was also proved there. Here we mention also the papers [53], [718], [56], [222], and [626]. For example, in the latter one the authors give an upper bound on the area of a constant width figure in terms of the Minkowskian arc length of its pedal curve and further quantities. An analogous result in terms of its width, the minimal radius and the arc length of its pedal curve was derived in [224]. This bound is reached iff the constant width figure is homothetic to the pedal curve of the isoperimetrix of the normed plane under consideration. Ghandehari proved in [411] that in n-dimensional Minkowski space the volume of the polar reciprocal of a given convex body of constant width 2 is bounded from below by the volume of the dual of the unit ball; equality holds iff this constant width body is the unit ball. Another type of result is derived in [754]: For a finite point set X with n-dimensional convex hull P, the points xi , x j ∈ X are called antipodal if there are different parallel supporting hyperplanes H  , H  of P with xi ∈ H  , x j ∈ H  . If the considered space is Minkowskian, one might ask for the number of pairs in X whose Minkowski distance is maximal. This number is not larger than the number a(X ) of antipodal pairs in X , and if the polytope P is of constant Minkowski width, then equality holds. In [754] several upper bounds on a(X ) are derived. Projections We come now to Minkowskian analogues of results discussed in the notes to Chapter 13. In Minkowski spaces we say that a convex body K is of constant brightness if the brightness of K (measured

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as in the Euclidean case) is proportional to that of the unit ball. Chakerian [231] showed that if a 3-dimensional convex body K is of constant brightness and its unit ball have C 2 boundary with everywhere positive curvature, then K is a ball. More generally, Petty (see [930] and [931]) defined the brightness of K at u ∈ S n−1 in an n-dimensional Minkowski space as the minimal Minkowski cross-section area of the cylinder K + l, where l is a 1-subspace of direction u. In [930] and [931] he derived results on bodies of constant brightness and constant width, e.g., as extremal cases of certain inequalities. Chakerian [231] studied also sets of constant k-girth in Minkowski spaces, and Petty [931] studied bodies of constant Minkowskian curvature. Discrete Geometry Following Loomis [730], a set K three-covers the set L if L ⊆ {a1 , a2 , a3 } + K for some three points a1 , a2 , a3 . Having this notion, he showed that a Reuleaux triangle of width h three-covers any figure of constant width h, and that if the unit ball is a centrally symmetric octagon, then every figure of constant width three-covers every other set of the same constant width. Further on, the dissertation [730] contains also Helly-type theorems for sets of constant width in normed planes. Motivated by a question of Hammer, Sallee [997] investigated bodies of constant width in association with lattices. Defining Reuleaux polygons of width 1 as finite intersections of (properly chosen) translates of the unit ball and saying that a set S avoids another set X if the interior of S does not meet X , he proved the following statements for strictly convex Minkowski planes: Every set of maximal constant width avoiding a square unit lattice L is a Minkowskian Reuleaux triangle P where each of the three open “edges” of P contains at least one point from L. And if the lattice L is replaced by a locally finite family X of convex sets in an arbitrary normed plane, then the correspondingly maximal sets are Reuleaux polygons all whose open “edges” contain points from X . Our next theme is the famous partition problem of Borsuk. Surveys on this problem, also referring to Minkowski spaces, are [488], [151, Chapter V], and [960], see also [272, D 14], [150], [154]. This problem is closely related to bodies of constant width and completeness (see, e.g., [151, Chapters V and VIII]). A first investigation for n = 2 was done by Grünbaum [485]; he showed that if the unit ball is not a parallelogram, then any set of diameter 1 can be covered by three balls of smaller diameters. The so-called Borsuk number can be defined as follows. Let F be a bounded set in a Minkowski space having diameter h. What is the smallest integer k such that F is the union of k sets each having diameter < h? Denoting this smallest number by a B (F), Boltyanski and Soltan [155] proved for n = 2 that a B (F) ∈ {2, 3, 4}, where a B (F) = 4 iff the unit ball B is a parallelogram and the convex hull of F is homothetic to B. And a B (F) > 2 holds iff one of the following two conditions is satisfied: (A) There is a unique completion of F to a figure of constant width h. (B) For any two parallel supporting hyperplanes of this constant width completion at least one has nonempty intersection with F. In [575] some conditions are derived under which an n-dimensional body of constant width in the sense of the norm can be presented as union of n + 1 subsets of smaller diameter than K has. Also related to (unique) completions of compact sets, in [788] some new results on Borsuk numbers of sets of constant width in normed spaces are derived. We finish this subsection by some results on in- and circumscribing figures. It is well known that a hexagon (regular in the considered planar norm) can be inscribed in any Minkowski circle. Thompson [1124, Chapter 4] gives a nice discussion of related results, including a way how this theorem can be applied to construct curves of constant width, in particular also Reuleaux triangles. As a special case of a result of Doliwka [297] (conjectured by Lassak [691]), the following result on normed planes holds: any planar figure of constant width 1, say, has an inscribed pentagon whose vertices are in at least unit distance to each other. Related results can be found in [683] and [694]. Spirova [1086] extended a result of Chakerian to normed planes: if a planar convex body can be covered by a translate of a Reuleaux triangle, then it can be covered by a translate of any convex body of the same constant width. In [790] universal covers in Minkowski planes are investigated; it turns out that a related covering property of constant width sets is needed in the proofs.

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Related Concepts As we discussed it already in Section 7.3, Maehara [747] defined two convex bodies K 1 , K 2 in En to be a pair of constant width if K 1 + (−K 2 ) is a ball. Analogously, Sallee [1002] defined {K 1 , K 2 } to be a pair of constant width in Minkowski spaces if the sum of their support functions in the form h(K 1 , u) + h(K 2 , −u) is itself the support function of a Minkowskian ball. He proved that K 1 is a summand of the unit ball iff there is a K 2 such that {K 1 , K 2 } is a pair of constant width in Minkowskian sense. Also his generalization of Chakerian’s theorem on in- and circumradius of constant width sets, given in [1001], can be formulated in terms of pairs of constant width. The authors of [253] characterized pairs of planar sets of constant width in terms of the Fourier coefficients of their radii of curvature. Weakly related is also the paper [33]. Rodriguez Palacios [979] pointed out that results on summands of Banach spaces may be interpreted in terms of sets of constant width, see also [922]. In view of multiples with scalars, Minkowski sums, and suitable combinations thereof, the family of all convex bodies in En forms an abelian semigroup with scalar operators. Having such an algebraic structure and the Hausdorff metric in mind, Ewald and Shephard [326] introduced an equivalence class structure of the subclass of bodies of constant width, yielding an incomplete normed linear space, and they mentioned that this idea can (due to a suggestion of Grünbaum) easily be extended to bodies of constant width in Minkowski spaces. Sorokin [1083] gave such extensions, even for nonsymmetric unit balls (which, on the other hand, have to be smooth and strictly convex). Taking the minimum width of certain representatives as a metric (in the abovementioned space), Lewis [723] showed that then a conjugate Banach spaces with complete norm is obtained. Related is also the paper [324]. Finally in this subsection, we mention that relations between the concept of bodies of constant affine width (see Section 11.6) and Minkowski geometry were studied in [542] and [99]; the background is given by typical methods from relative differential geometry in the sense of [160] (see Section 38 there). Further papers of Hirakawa related to affine relative differential geometry and considering also curves of constant width within this framework are [543] and [544]. We continue with mentioning results about rotors in normed planes. Ghandehari and O’Neill [414] derived inequalities for the self-circumference of rotors in equilateral triangles and figures of constant width. Hereby these selfcircumferences are measured by taking these rotors or figures of constant width themselves as gauge figures or unit circles of Minkowski planes. The inequalities compare their areas and mixed areas (using also the polar reciprocal) with the self-circumference. Also higher dimensional results are given there. Sorokin [1083] investigated classes of convex bodies which can roll in Minkowskian spheres. We finish with a link to Zindler curves in normed planes. Our subsection 5.4.1 clearly shows that this type of curves is closely related to constant width curves in the Euclidean plane, and their analogues also in higher dimensions. Since this notion can also be carried over to normed planes (see [798] and [799]), it certainly yields an interesting field of further research. Constant Width in Other Non-Euclidean Geometries The concept of constant width was also extended to non-Euclidean geometries different to Banach space geometry. There exist results mainly for hyperbolic and spherical spaces, and also for the complete concept of spaces of constant curvature. In addition, also results on constant width sets in Riemannian manifolds and in further frameworks exist; we try to give an overview to results that are known. Again (as in Minkowski spaces) one can observe that already the definitions of notions give an interesting variety of possibilities. Hyperbolic Geometry Santaló [1015] defined geodesic convex curves of constant width in the hyperbolic plane by means of supporting and transversal geodesics, and based on that he derived already extremal properties of analogues of Reuleaux polygons and an analogue of Barbier’s theorem, see also [1020]. Continuing

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this, Fillmore [350] defined curves of constant width as horocycle convex curves having constant distance between every two supporting horocycles with the same center at infinity (where the distance is measured along certain geodesics). Fillmore showed that not all curves of the same constant width have the same length (as it is the case with the concept developed in [1015]); it is also shown how the Euclidean borderline case naturally leads to Barbier’s theorem. Using different methods, Leichtweiss [715] defined a closed convex curve to be of constant width h if its supporting strip region has the same width h for any direction (here a strip region is defined as a region, in the hyperbolic plane as well as on the sphere, bounded by two curves which have constant distance from a given geodesic line). Continuing [1015] and [350], Leichtweiss also derived a version of Barbier’s theorem, and based on approximation arguments for Reuleaux polygons and a careful analysis of their area he succeeded with a hyperbolic version of the Blaschke–Lebesgue theorem yielding analogues of Reuleaux triangles as extremal figures. Many of his results refer also to spherical space. Another way to extend Barbier’s theorem to complete, simply connected surfaces of constant curvature k is given in [34]. The author underlines that the case k > 0 is due to Blaschke [130] and k < 0 to Santaló [1015], but that his approach (using tools from differential geometry) is new and unifies these results. Coming from an isoperimetric inequality referring to horocycle convex curves of the hyperbolic plane (see also above), Pinkall [937] proved an inequality for curves of constant width, elegantly using an ingenious map of the horocycle convex curves on the centrally symmetric curves of the affine plane. He also reproved results from [1015] and [1020], such as Barbier’s theorem. Also [35] refers to such problems in the hyperbolic plane. A simple closed curve has constant width h if for each point from it the maximum hyperbolic distance to other curve points is h. Using a definition of Reuleaux triangle exactly as in the Euclidean case, the author of [35] confirmed that among all piecewise regular curves of the same constant width the Reuleaux triangle encloses the smallest area, and also this is obtained via new differential-geometric methods, giving an alternative proof without invoking the isoperimetric theorem that the circle encloses the largest area among all curves of the same hyperbolic constant width. This author continued in [36] with presenting a parametrization of curves of constant width in the hyperbolic plane, giving also examples for their usage. As a central result it is shown that each curve parametrized this way can be uniformly approximated by analytical curves of constant width, furthermore used for extending known results. In [582], Jerónimo-Castro and Jimenez-Lopez characterize the hyperbolic disk among constant width sets. It is well known that a planar convex body in E2 with the property that from every point outside of it the two respective tangent segments have equal length, has to be a Euclidean ball. The authors of [585] investigate various related problems and obtain a nice characterization of constant width sets in the hyperbolic plane working explicitly with the metric of the upper half-plane model. In [387] the relation between asymptotic values of the ratios area/length and diameter/length of a sequence of convex sets expanding over the whole hyperbolic plane is investigated. It is shown that diameter/length has its limit value between 0 and 21 , in strong contrast to the Euclidean situation where the lower bound is π1 , holding iff the convex sets have constant width. In [388], convex bodies in spaces of constant curvature are investigated, and several interesting and useful expressions for the measures of lines and planes intersecting such bodies are obtained. Since also here a natural definition of width comes into the game, bodies of constant width become interesting in this framework. Via a characterization of the ball by some inequality in terms of body volume, ball volume and ball area, finally a complete system of related inequalities for constant width bodies is obtained. Glasauer [420] derived a local Steiner formula and showed that this formula can be applied to bodies of constant width in Euclidean, spherical, and hyperbolic space. Dekster [289] introduced the concept of completeness for compact sets in hyperbolic, Euclidean, and spherical space, showing that in the first two cases the notions of completeness and constant width coincide. For getting this coincidence in the spherical case, convexity has to be assumed. Moreover, the author also constructed nice examples showing how strong complete sets of diameter larger than π2 can differ from bodies

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of the same constant width. Studying isoperimetric problems in hyperbolic, Euclidean, and spherical surfaces referring to regions between constant-curvature curves, Simonson [1066] showed that certain minimizing sets cannot occur in regions of constant width. As an extension from a ball-shaped basic region to arbitrary domains, the Apollonian metric is a generalization of the hyperbolic metric, and relations between these concepts were investigated in [567]. It turns out that if these domains are complements of sets of constant width, then the metric has special interesting properties, see [568], [569], and [165]. Spherical Space Before we discuss references referring only to spherical space, we underline that the above-cited papers [130], [34], [289], [420], [715], and [388] cover (besides results holding in hyperbolic geometry) also analogous results for the spherical case. Two early contributions are [130] and [1012]. Properties of constant width sets on the unit sphere are discussed, and an extremal property of the spherical Reuleaux triangle is proved. The paper [1014] continues Blaschke’s paper [130], characterizing spherical curves of constant width in two inequalities in terms of minimal width and diameter. The paper [291] combines transnormality and constant width sets in spherical n-space. Considering their inner and outer parallel sets (which are 2-transnormal topological (n − 1)-spheres), the authors prove that if C is a spherical body of constant width ( π2 ) + a, 0 < a < π2 , then C is smooth and contains a body C  of constant width ( π2 ) − a such that C is the union of the spherical caps of radius a whose centers are from C  . In analogy to quermassvectors of convex bodies in Euclidean space (see [1035]), Arnold [38] introduced quermassvectors for strictly convex subsets of the 2- and 3-dimensional spherical space. These spherical quermassvectors can, after renormalization, be interpreted as curvature centroids, and some properties of them are proved. In particular, vector-valued Steiner formulas for the quermassvectors of outer parallel bodies are derived. Also Lassak [699] studied n-dimensional spherical sets of diameter at most π2 and defined in a natural way spherical bodies of constant width and, in addition, also reduced sets. He showed that then all bodies of constant width are, like also any such spherical odd-gon for n = 2, reduced sets, and that any reduced and smooth spherical body is of constant width. Further results on spherical reduced sets are presented in [879]. Recently, Lassak and Musielak (see [698], [703], and [704]) obtained new results in reduced bodies in spherical geometry, see also [879]. Spherical versions of Reuleaux polygons of diameter < π2 , also extendable to the hyperbolic plane, are given in [673]. The paper [703] contains results of the following type, again referring to the n-dimensional spherical space: For a certain definition of width, the authors introduce bodies of constant width h, proving that their diameter is h and that for h < π2 these sets are strictly convex. The concepts of “constant width” and “constant diameter” in this framework are compared, leaving open the question whether any spherical body of constant diameter h < π2 is a body of constant width h. The concept of completeness in spherical spaces is investigated in [318], [289], and [1180]. Among other results, one can find there the theorem that for sets of diameter smaller than π2 the notions of completeness and constant width coincide. In [34] the following nice characterization of the circle within the family of constant width curves (for the Euclidean case due to Hammer and Smith [510]) is extended to the 2-dimensional spherical space: if each two points on a curve of constant width h, having spherical distance h from each other, divide the curve into two arcs of equal length, then the curve is a circle. Pottmann ([951] and [954]) showed that spherical curves of constant width π2 are interesting in view of spherical motions. Related to the problem of finding the maximal length of steepest descent spherical curves for quasi-convex functions satisfying suitable constraints, the spherical Reuleaux triangle occurs in [729] as extremal figure. The following result from [114] holds on the 3-sphere: the volume of any spherical convex body of constant width h is minimal among all bodies of constant width h iff the polar body of constant width π − h has minimal volume among all bodies of constant width π − h.

10.6 The Borsuk Conjecture in the Minkowski Plane

243

Representing results on systems of inequalities due to Minkowski one can use spherical geometry, and a related characterization of spherical sets of constant width π2 is given in [140]. Further on we refer in this subsection to [963]. On Manifolds and Further Non-Euclidean Concepts In a series of papers, Dekster worked successfully on the extension of the concept of constant width to manifolds, and one should also mention the earlier related papers [894] and [895]. In [285], Dekster investigated compact, geodesically convex subsets of an n-dimensional Riemannian manifold such that a geodesic segment from them never contains a pair of conjugate points. Such a set is called a body of constant width h if for each of its boundary points any normal geodesic chord emanating from it has length h and is a maximal geodesic chord of the set with this endpoint. The author confirmed that various standard properties of Euclidean sets of constant width also hold for their analogues in Riemannian manifolds, and others do not. In [284] he continued by showing that each incenter of such a constant width body is a circumcenter and, conversely, that there is a unique point equal to both, under certain curvature conditions. Results on in- and circumradii are given, too. In [286] the concept of width of a convex body in a Riemannian manifold is newly introduced (in other papers the author is not using the notion of “width” explicitly, but prefers some analogue of “constant diameter”), and it is confirmed that such a convex body is of constant width iff also this type of width is constant. Two extensions of the characterization via double normals are given in [288], and further possible characteristic properties of constant width sets in manifolds are discussed, see also [287]. In this last paragraph we discuss (somehow widespread) results on the concept of constant width in further non-Euclidean geometries. The paper [1] contains a deep study of many basic notions from convexity in complex vector spaces-among them also sets of constant width. We continue with a special Cayley–Klein geometry. Namely, Strubecker [1102] studied Zindler curves and constant width curves in isotropic geometry, transforming the framework suitably into isotropic space (thus using then a spatial approach to planar results). The papers [644] and [613] refer to spacelike curves of constant width in “the other Minkowski geometry”, i.e., in Minkowskian spacetime 4-space, whereas [900], [646], [1207], and [643] refer to the 3-dimensional analogue. More precisely, in [900] and [644] differential equations characterizing such curves are obtained, and in [613] Frenet-like formulae are developed. In [646] special types of such curves are investigated, and it is shown that the total torsion of a closed spacelike curve of constant width is zero. Properties of double normals of these curves are obtained in [644], in [643] these curves are investigated according to the Bishop frame, and [1207] contains a study of the correspondingly dual curves, sitting in the dual Lorentzian 3-space. In these publications also more related references can be found. We should also mention the paper [682] in which some concept of constant width in Möbius geometry is developed. In an abstract sense, tolerance classes can be introduced in a natural way in any metric space. In [658] relationships between two such classes and bodies of constant diameter in this model are established. Our last reference cited here refers to relations between lattice theory and metric spaces. Namely, in [615] lattices are “normed” such that they yield metric spaces with a well-defined distance function. The author establishes necessary and sufficient conditions for a general metric space to be a space associated with a normed lattice in this way, and it turns out, that sets of constant width (defined with the help of that distance function) are useful for such investigations.

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10 Bodies of Constant Width in Minkowski Spaces

Exercises 10.1.

10.2.

10.3. 10.4. 10.5. 10.6. 10.7. 10.8. 10.9. 10.10**. 10.11.

Let H1 and H2 be parallel support hyperplanes of the ball G(x, r ), and let x1 ∈ H1 and x2 ∈ H2 . Prove that x1 x2 is a bridge of the strip cc(H1 ∪ H2 ) if and only if there are points y1 ∈ H1 ∩ G(x, r ) and y2 ∈ H2 ∩ G(x, r ) such that x1 x2 is parallel to y1 y2 . Prove that a line L and a hyperplane H in the Minkowski space M are G-perpendicular if the line L 0 through the origin parallel to L has the property that the unit ball G has the support hyperplanes through L 0 ∩ bd G parallel to H . Suppose the unit ball G of the Minkowski space M is strictly convex. Prove that if two lines are G-perpendicular to a hyperplane, then they are parallel. Suppose the unit ball G of the Minkowski space M is smooth. Prove that if two hyperplanes are G-perpendicular to a line, then they are parallel. Suppose the unit ball G of the Minkowski space M is strictly convex, and let  ⊂ M be a body of constant G-width. Prove that  is strictly convex. Prove Theorem 10.2.4 assuming that φ1 and φ2 are smooth convex bodies. Prove that every G-diameter is a G-binormal. Prove that any two parallel G-binormals have the same G-length. Suppose that  has a G-binormal of G-length h in every direction. Prove that  has constant G-width h. Construct a 3-dimensional G-complete body which is not a body of constant G-width. Let φ be a centrally symmetric convex set in the Minkowski space M with unit ball G. Suppose the origin is the center of φ and the chord r s is a G-diameter of φ. Prove the following two statements. (a) The chord r (−r ) is a G-diameter of φ. (b) The chord pq through the center of φ parallel to r s is a G-diameter.

10.12.

10.13.

10.14. 10.15.

10.16.

For a fixed point p on the unit circle G of a Minkowski plane M and a variable point x, which is moving on bd G from p to − p, prove that the G-length of the chord px is not decreasing. Prove that if  ⊂ M has constant G-width in the Minkowski plane M with unit disk G, then every chord pq of  splits the body  into two compact sets such that one of them has G-diameter equal to the G-length of pq. A strip is called G-regular if the corresponding parallel support G-strip is regular. Prove that the G-diameter of  is the maximum of the G-widths of all G-regular support -strips. Let H1 and H2 be parallel hyperplanes of an n-dimensional Minkowski plane M with unit ball G and suppose d M (H1 , H2 ) = h. Let xi ∈ Hi , i = 1, 2. Prove that there exist parallel (n − 2)-dimensional affine planes L 1 , L 2 such that xi ∈ L 1 ⊂ Hi , i = 1, 2, and with the property that it is possible to parallel-rotate H1 , H2 in such a way that the G-width of the resulting strips is always smaller than or equal to h. Let G be a centrally symmetric convex body in Euclidean n-space En and let p and q be two points with the property that they both can be covered with a translated copy of G. The G-interval determined by this pair of points is the intersection of all translated copies of G that contain p and q. A closed set φ is called spindle G-convex if it has the property that, given a pair of points p and q in φ, the G-interval that they determine is also in φ. (a) Let φ be a strongly G-convex set. Prove that φ is spindle G-convex, (b) Is every spindle G-convex a strongly G-convex set? Under what conditions on G is this true?

Exercises

245

10.17*.

Prove that balls and simplices in every dimension are generating sets. Prove that all 2-dimensional convex sets are generating sets. Prove that the intersection of all spindle φ-convex sets containing the compact set V is φ ∼ (φ ∼ V ). For every direction u, let (φ, u) be the maximum length of a chord of the convex body φ in the direction u. Prove that (φ) = minu (φ, u). Give an example of a convex body which has constant G-minimum width but is not a body of constant G-width. Let F be a parallelogram. Prove that β F (F) = 4. Let G be a square, and F be a circle. Prove that βG (F) = 2.

10.18. 10.19. 10.20. 10.21. 10.22.

Chapter 11

Bodies of Constant Width in Differential Geometry

Poetry is the art of giving different names to the same thing; Mathematics is the art of giving the same name to different things. Henri Poincaré

11.1 The Support Parametrization in Terms of the Gauss Map Let φ ⊂ En be a strictly convex body whose boundary is twice differentiable and whose curvature never vanishes. Recall that the inverse Gauss map γ : Sn−1 → bd φ is a diffeomorphism that assigns to each unit vector u ∈ Sn−1 the point γ(u) in the boundary of φ for which u is the outward unit normal vector. If P φ : Sn−1 → E is the support function of φ, by definition γ(u), u = Pφ (u), for every u ∈ Sn−1 . Since the differential of γ at u ∈ Sn−1 has as its image the subspace u ⊥ , one expects the following vector equation. Theorem 11.1.1 Let φ ⊂ En be a strictly convex body whose boundary is twice differentiable and whose curvature never vanishes. Therefore, for every u ∈ Sn−1 , γ(u) = Pφ (u) · u + ∇Pφ (u),

(11.1)

where ∇Pφ is the gradient vector field of the support function Pφ as a function of Sn−1 . Proof By the definition of Pφ , we may write the decomposition γ(u) − Pφ (u) · u + t (u), for each u ∈ Sn−1 , where t (u) ∈ Tu Sn−1 u ⊥ . To show that t (u) is the gradient of Pφ we have to verify that   dPφ u (v) = t (u), v, © Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_11

247

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11 Bodies of Constant Width in Differential Geometry

n−1 for any u ∈ Sn−1 and v ∈ Tu Sn−1 . But u ⊥ , we always have t (u), v = γ(u), v. Now  Tu S  since we simply calculate the differential dPφ u (v) from the definition. We get

  dPφ u (v) = dγu (v), u + γ(u), du(v) = γ(u), v, where the last equality holds since dγu (v) ∈ Tγ(u) (bd φ) Tu Sn−1 u ⊥ and du is simply the derivative of the position vector.  The parametrization of the boundary of the convex body φ given by the vector equation (11.1) in terms of the support function is called the support parametrization of φ. Suppose U is an open subset of En−1 , and u : U → Sn−1 ˜ = Pφ (u(ϑ)) ˜ is the support function of the convex body is a C 1 -parametrization of the sphere and P(ϑ) n−1 ˜ φ in terms of the coordinates ϑ ∈ U ⊂ E . Since ∇Pφ , the gradient vector field of the support function Pφ as a function of Sn−1 is such ˜ = du | ˜ dP | ˜ . By Theorem 11.1.1 we obtain the following parametrization of the that ∇Pφ (u(ϑ)) ϑ ϑ boundary of the strictly convex body φ in terms of coordinates ϑ˜ ∈ U ⊂ En−1 : ˜ = P(ϑ)u( ˜ ϑ) ˜ + du | ˜ dP | ˜ . x(ϑ) ϑ ϑ

(11.2)

If n = 2, we obtain the parametrization x(ϑ) = P(ϑ)u(ϑ) + P (ϑ)u (ϑ). If the parametrization of S1 is given by u(ϑ) = (cos(ϑ), sin(ϑ)), then we obtain the parametrization given by Theorem 5.1.1, c): x(ϑ) = P(ϑ) cos(ϑ) − P (ϑ) sin(ϑ), y(ϑ) = P(ϑ) sin(ϑ) + P (ϑ) cos(ϑ). If n = 3, x(ϑ, θ) = P(ϑ, θ)u(ϑ, θ) +

∂P ∂u ∂P ∂u (ϑ, θ) (ϑ, θ) + (ϑ, θ) (ϑ, θ). ∂ϑ ∂ϑ ∂θ ∂θ

If we use polar coordinates   u(ϑ, θ) = cos(ϑ) cos(θ), sin(ϑ) cos(θ), sin(θ) ; 0 ≤ ϑ < 2π, 0 ≤ θ < π, we obtain the following parametrization in cartesian coordinates from (11.3): x(ϑ, θ) = cos(ϑ) cos(θ)P(ϑ, θ) − sin(ϑ) cos(θ)

∂P ∂P (ϑ, θ) − cos(ϑ) sin(θ) (ϑ, θ), ∂ϑ ∂θ

y(ϑ, θ) = sin(ϑ) cos(θ)P(ϑ, θ) + cos(ϑ) cos(θ)

∂P ∂P (ϑ, θ) − sin(ϑ) sin(θ) (ϑ, θ), ∂ϑ ∂θ

(11.3)

11.1 The Support Parametrization in Terms of the Gauss Map

z(ϑ, θ) = sin(θ)P(ϑ, θ) + cos(θ)

249

∂P (ϑ, θ). ∂θ

Recall from Section 8.1 that to construct an algebraic body of constant width 2 it is enough to consider the restriction of an odd function P0 : E3 → E and then to study the support function 1 + P0 , where  < 1, see Figures 8.19 and 8.20. In particular, if we consider the restriction of 1 + x yz to S2 , the resulting convex body is still of constant width 2 (see Figure 8.21), but it is no longer smooth. It has the symmetry group of a tetrahedron, and by (11.3) it has the following parametrization: x(ϑ, θ) = cos(θ)(cos(ϑ) + cos2 (θ) sin3 (ϑ) sin(θ) + 1/4 cos2 (ϑ) sin(ϑ)(5 sin(θ) − 3 sin(3θ)), y(ϑ, θ) = cos(θ)(sin(ϑ) + cos3 (ϑ) cos2 (θ) sin(θ) − cos(ϑ) cos2 (θ) sin2 (ϑ) sin(θ))+ cos(θ) sin(ϑ) sin(2ϑ) sin3 (θ)), z(ϑ, θ) = cos(ϑ) cos2 (θ) cos(2θ) sin(ϑ) + sin(θ).

11.2 Curvature 11.2.1

Introduction and Preliminaries

The Curvature of the Boundary of a Plane Convex Curve Let φ ⊂ E2 be a plane strictly convex figure whose boundary is twice differentiable and whose curvature never vanishes. See, for example, [1104] and [1087] for definitions. Let us fix an orientation of the boundary of φ, and according to this orientation let f : S1 → bdφ be the differentiable map that assigns to each u ∈ S1 the unique point p in the boundary of φ with the property that u is the unit tangent vector of bd φ at p. Although the derivative d f |u is a linear map between two 1-dimensional vector spaces, for convenience, we shall denote by d f |u also the absolute value of its determinant. Then the curvature κφ ( p) of bd φ at p, in this case, can be easily calculated by taking the inverse of the derivate of f at u. That is, κφ ( p) =

1 ∈ E. d f |u

If instead of a tangent vector we consider a unit normal vector, then the conclusion is obviously the same. So, let γ : S1 → bdφ be the inverse Gauss map. Then, κφ ( p) =

1 ∈ E. dγ |u

(11.4)

Intuitively, the reason is that dγ |u tells us, infinitesimally, how the length of the curve bdφ changes with respect to the length of S1 . The Curvature of the Boundary of a 3-Dimensional Convex Body Let φ ⊂ E3 be a a strictly convex body whose boundary is twice differentiable and whose curvature never vanishes. A preliminary non-rigorous definition of the curvature of bd φ at a point γ(u) = p is

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11 Bodies of Constant Width in Differential Geometry

κφ ( p) =

area(γ −1 (U )) , area(U )

where the limit is taken as the region U around p in bd φ becomes smaller and smaller. The derivative of the inverse Gauss map dγ |u : u ⊥ → u ⊥ , where u ⊥ is the subspace orthogonal to u, is a linear endomorphism called the inverse Weingarten map. The determinant of the inverse Weingarten linear endomorphism dγ |u : u ⊥ → u ⊥ tells us, infinitesimally, how the surface area of bdφ changes with respect the surface area of S2 . So, in this case, it is natural to calculate the Gaussian curvature of bdφ at a point p = γ(u) as 1 . Det (dγ |u )

κφ ( p) = Classic notation for this determinant is

Det (dγ |u ) = Fφ(2) (u). Surface Area For each Borel subset E of S2 , let γ(E) be the set of points of bdφ at which there is an outer unit normal vector in E. Intuitively we may define S(φ, E) as the surface area of γ(E) inside bdφ. Rigorously, since the determinant of the inverse Weingarten linear endomorphism dγ |u : u ⊥ → u ⊥ tells us, infinitesimally, how the surface area of bdφ changes with respect the surface area of S2 , we define, for each Borel set E ⊂ S2 ,  Fφ(2) (u)du. S(φ, E) = E

It is not difficult to see that S(φ, ∗) is a finite Borel measure in S2 called the surface area measure of bdφ. Note that S(φ, S2 ) is the surface area S(φ) of φ, that is:  Fφ(2) (u)du. S(φ) = S2

Principal Directions and Principal Curvatures In order to compute Fφ(2) (u), the determinant of dγ |u : u ⊥ → u ⊥ , it will be important to compute the eigenvalues and eigenvectors of this linear endomorphism, and to understand it geometrically. Consider a point p = γ(u) of bdφ with a unit normal vector u ∈ S2 . A normal plane H of φ at p is one that contains the normal vector u, and will therefore also contain a unique direction w ∈ S2 tangent to φ at p. Let  : E3 → H be the orthogonal projection in the direction w⊥ . Then the boundary of (φ) is a curve in H that contains p. These curves will in general have different curvatures for different normal planes. So, we will denote by ρw the radius of curvature of (φ) at p, where the normal plane H of φ at p is generated by u and w. Our next purpose is to show that ρw = dγ |u (w), w.

11.2 Curvature

251

This is indeed the case because γ : S2 ∩ H → H parametrizes the curve bd(φ). Furthermore, for every v ∈ S2 ∩ H , γ(v) ∈ H has as unit normal vector precisely the unit vector v. This implies, by (11.4), that the radius of curvature of the curve γ : S2 ∩ H → H at u is the derivative of this map, which by definition is dγ |u (w), w. The principal radii of curvature at p = γ(u), denoted by ρ1 and ρ2 , are the maximum and minimum values of all radii of curvatures ρw , with w a unit vector orthogonal to u. These two extreme curvatures are reached in two orthogonal directions, called the principal directions. In fact, we have the next formula, first proved by Blaschke in [132], pp. 117. ρw = ρ1 cos2 (θ) + ρ2 sin2 (θ),

(11.5)

where θ is the angle between w and the principal maximal direction. This formula follows because if v1 and v2 are the eigenvectors of dγ |u and λ1 , λ2 are the respective eigenvalues, then dγ |u (vi ) = λi vi , i = 1, 2, and so ρvi = dγ |u (vi ), vi  = λi . This implies that ρi = λi , and since θ is the angle between w and the principal maximal direction, then w = v1 cos(θ) + v2 sin(θ) and ρw = ρ1 cos(θ)v1 + ρ2 sin(θ)v2 , v1 cos(θ) + v2 sin(θ) = ρ1 cos2 (θ) + ρ2 sin2 (θ). Summarizing, we have that the principal radii of curvatures are the eigenvalues of the inverse linear Weingarten endomorphism. Furthermore, the curvature of bdφ at p = γ(u) is the product of the inverses of the principal radii of curvature. That is, Det (dγ |u ) = Fφ(2) (u) = ρ1 ρ2 , and hence κφ ( p) =

1 . ρ1 ρ2

Euler’s Theorem and the Gauss Map In this section, we are interested in the curvature of the normal sections of bdφ. For that purpose, let us consider the Gauss map  : bd φ → S2 , that assigns continuously to every point p ∈ bdφ its unit normal vector. Its derivative d | p : ( p)⊥ → ( p)⊥ is a linear endomorphism called the Weingarten map, where as usual ( p)⊥ is the subspace orthogonal to ( p). From the fact that  = γ −1 , we get immediately Theorem 11.2.1 The eigenvalues and eigenvectors of the Weingarten map D | p are the principal curvatures 1/ρ2 , 1/ρ1 and the principal directions of bdφ at p. Furthermore, the determinant of d | p is equal to the Gaussian curvature κφ ( p) of bdφ at p.

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11 Bodies of Constant Width in Differential Geometry

Figure 11.1 Sectional curvatures

Remember that a normal plane at p ∈ bdφ is one that contains the normal vector ( p), and will therefore also contain a unique vector w tangent to φ at p. This plane cuts the boundary of φ in a curve, called a normal section, see Figure 11.1. All these curves will in general have different curvatures for different normal planes. If κ1 and κ2 denote the maximum and minimum values of these curvatures, respectively, then we will show that these two extreme curvatures are reached precisely in the principal directions with κ1 = ρ12 , κ2 = ρ11 . It turns out that the curvature at every normal section is completely determined by the principal curvatures and the principal directions. For every unit tangent vector w of bd at p, let us denote by κw the curvature at p of the curve H ∩ bd φ, where H is the normal plane generated by ( p) and w. Theorem 11.2.2 (Euler’s Curvature Theorem). Let φ ⊂ E3 be a convex body whose boundary is twice differentiable. Let p be a point in bd φ. Then κw = κ1 cos2 (θ) + κ2 sin2 (θ), where θ is the angle between w and the principal maximal direction. Proof For any normal plane H in the direction w, let us consider the map g : bd φ ∩ H → S2 ∩ H (q) defined as follows: given a point q ∈ bd φ ∩ H , g(q) = |(q)| , where  : E3 → H is the orthogonal projection in the direction w⊥ . Note that the normal unit vector of the curve bd φ ∩ H at q is precisely g(q) ∈ H and hence, by (11.4), dg | p = κw . Furthermore, it is easily checked that

dg | p = d | p (w), w = κw . If v1 and v2 are the principal directions, then w = v1 cos θ + v2 sin θ, and therefore, by Theorem 11.2.1, we have that d | p (w) = v1 κ1 cos θ + v2 κ2 sin θ. Consequently, κw = κ1 cos2 (θ) +  κ2 sin2 (θ). This concludes the proof of the theorem. Corollary 11.2.1 Let φ ⊂ E3 be convex body whose boundary is twice differentiable, p a point of bd φ, κφ ( p) the Gaussian curvature of bd φ at p, and w a unit tangent vector to φ at p. Then

11.2 Curvature

253

Figure 11.2 An umbilic point

κw = κφ ( p), ρw⊥ where w⊥ is orthogonal to w in the tangent space of φ at p. Proof Let w be a tangent vector of φ at p. Then κw κ1 cos2 (θ) + κ2 sin2 (θ) = , κφ ( p) κ1 κ2 where θ is the angle between w and the principal maximal direction. Hence κw = ρ2 cos2 (θ) + ρ1 sin2 (θ) κφ ( p) = ρ1 cos2 (θ + π/2) + ρ2 sin2 (θ + π/2) = ρw⊥ .



A point p ∈ bdφ for which the two principal curvatures are equal is called an umbilical point of the boundary of φ. There is always an umbilical point (see Figure 11.2). Otherwise, choosing in every point of the boundary of φ the first principal direction, we contradict the classical topological theorem claiming that it is impossible to choose continuously a unit vector, tangent to every point of a sphere, see [833]. Remarks 11.2.1 Given a normal plane H at a non-umbilical point p (see Figure 11.2), consider the curvatures of the normal section H ∩ φ and of bd(φ), where  is the orthogonal projection onto H . Then these two curvatures coincide if and only if H is generated by ( p) and a principal direction, see Exercise 11.5.

11.2.2 Positive Constant Gaussian Curvature We start this section with a classical theorem of Liebmann (1900) about the curvature of closed surfaces [662].

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11 Bodies of Constant Width in Differential Geometry

Figure 11.3 A Surface with constant Gaussian curvature

Theorem 11.2.3 A convex body ψ ⊂ E3 with the property that all its boundary points have positive constant Gaussian curvature must be a ball. The hypothesis of convexity is not relevant here because it is clear that the theorem holds for every closed surface. See [662] for a proof. As we will see, the relevant hypothesis for the surface is to be closed, because there are classical examples of closed surfaces of revolution with constant Gauss curvature at regular points; see [98], p. 131. Theorem 11.2.4 There is a convex body of revolution ψ ⊂ E3 with only two singular points at the axis and with positive constant Gaussian curvature at any other boundary point. The convex body ψ will be a centrally symmetric body of revolution around the z-axis. We shall describe the 2-dimensional meridian section, lying in the {x, z}-plane. See the upper part of ψ in Figure 11.3. For each u ∈ S2 , let xu be the point on the boundary of body of revolution ψ, where the outer normal vector is u. Let u 1 , u 2 be the principal curvature directions corresponding with the principal radii of curvature ρ1 , ρ2 at xu . It follows that u 1 lies in the plane of u and the z-axis and u 1 is orthogonal to u. The direction u 2 is orthogonal to the plane spanned by u and u 1 . Then ρ2 (u) is the distance from xu to the z-axis measured along the line through xu parallel to u, see Exercise 11.6. The body ψ will be constructed so that the principal radii of curvature satisfy ρ1 (u)ρ2 (u) = c =

1 , κψ (u)

where c is a constant. We may concentrate on the convex figure φ, the slice of ψ lying in the {x, z}-plane. Denote by p(θ) the pedal function of φ, where −π/2 ≤ θ ≤ π/2 is the angle between u and the x-axis, see Section 5.1.

11.2 Curvature

255

Then xu = (r (θ), 0, h(θ)), where r (θ) = p(θ) cos(θ) − p (θ) sin(θ), and h(θ) = p(θ) sin(θ) + p (θ) cos(θ). Therefore ρ2 (u) =

r (θ) , cos θ

and by Theorem 5.1.1 e) we have ρ1 (u) = p(θ) + p

(θ) = −

r (θ) . sin θ

Let f (θ) = r (θ)2 , then, f (θ) = −2ρ1 (θ)ρ2 (θ) cos θ sin θ. Suppose now that f (θ) = c cos2 θ − b. Then ρ1 (θ)ρ2 (θ) = c =

− f (θ) , 2 cos θ sin θ

as we wished. Let 0 < b < c, and let 0 < θ0 < π/2 be such that cos2 (θ0 ) = bc . We shall then construct the meridian section φ in such a way that for −θ0 ≤ θ ≤ θ0 p(θ) cos(θ) − p (θ) sin(θ) =



c cos2 θ − b.

(11.6)

The last equation is a first-order linear differential equation. The solution   2 p(θ) = cos θ c cos θ − b + sin θ

θ

0

2 cos2 t c cos2 t − b

dt

(11.7)

may be found by means of an integrating factor followed by integration by parts. If this is so, then r (θ) =

  c cos2 θ − b and h(θ) = 0

θ

2 cos2 t c cos2 t − b

dt.

Therefore, the centrally symmetric convex body of revolution ψ has only two vertex points at θ cos2 t (0, ±a), where a = 0 0 √c2cos dt. At any other point of the boundary, the curvature is c. 2 t−b √ Moreover, ψ intersects the {x, y}-plane in the disk D := B(o, c − b) centered at the origin of √ radius c − b. At any point p ∈ bdD, the unit normal vector of ψ at p lies in the {x, y}-plane. Furthermore, the unit normal vectors of ψ at (0, ±a) are all those vectors whose angle from (0, 0, ±1), respectively, is smaller than or equal to (π/2 − θ0 ). Of special interest for Section 13.3.2 will be the case when c = 2, b = 1, and θ0 = π/4 (see Figure 11.4), because we will use this body to construct a body of constant brightness.

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11 Bodies of Constant Width in Differential Geometry

Figure 11.4 c = 2 and b = 1

11.3 The Curvature of a Body of Constant Width We shall start proving that for a figure  of constant width h, whose boundary is a twice continuously differentiable curve, the radii of curvature at the ends of a binormal of  sum up to h. Our approach will be different from the one given in the proof of Theorem 5.3.6 and will serve to introduce the ideas that allow us to prove the corresponding theorems on curvature for higher dimensional bodies of constant width. Remember from Chapter 5 that for a real number θ, u(θ) = (cos(θ), sin(θ)) and u (θ) = (− sin θ, cos θ). For the sake of simplicity, in this section our discussion will be restricted to curves whose curvature is continuous and never vanishes. Let C be a twice continuously differentiable convex curve, and for every θ let x(θ) ∈ C be such that the normal unit vector of C at x(θ) is u(θ). Let ρ(θ) be the radius of curvature of C at x(θ), where d x(θ) = ρ(θ)u (θ). dθ Now note that if C is a curve of constant width h, then x(θ + π) = x(θ) − hu(θ). So, d x(θ + π) d x(θ) = − hu (θ), dθ dθ but hence ρ(θ + π)u (θ + π) = ρ(θ)u (θ) − hu (θ),

11.3 The Curvature of a Body of Constant Width

257

and then ρ(θ + π) + ρ(θ) = h, thus proving that the radii of curvature at the ends of a binormal sum up to h. With this in mind, let us consider  ⊂ En as a smooth, strictly convex body whose boundary is twice continuously differentiable. It will be assumed that the principal curvatures are continuous and do not vanish. Let us consider now the inverse Gauss Map γ : Sn−1 → bd , where, for every s ∈ Sn−1 , the unit normal vector of bd at γ(s) is s. The fact that the inverse Gauss map is a differentiable map between differentiable manifolds of dimension (n − 1) implies that for every s ∈ Sn−1 the derivative dγ |s : s ⊥ → s ⊥ is a linear map, where, as usual, s ⊥ is the linear subspace of all vectors orthogonal to s. We define the Gaussian curvature of  at p = γ(s) by 1 , Det (dγ |s )

κ ( p) = and the mean curvature of  at p = γ(s) by H ( p) =

1 . T race(dγ |s )

Generalizing the concepts for surfaces in E3 , we must consider the eigenvalues and eigenvectors of the inverse Weingarten linear endomorphism dγ |s : s ⊥ → s ⊥ . The fact that the inverse Weingarten map is self-adjoint ensures us the existence of n − 1 linear independent eigenvectors. Let ρ1 (s) ≤ ρ2 (s), · · · ≤ ρn−1 (s) be the principal radii of curvature of bd at γ(s), and let {ξ1 (s), ξ2 (s), . . . , ξn−1 (s)} ⊂ s ⊥ be the corresponding principal directions of curvature of bd at γ(s). That is, for 1 ≤ i ≤ n − 1, dγ |s (ξi (s)) = ρi (s)ξi (s). Moreover, κ ( p) =

1 n−1 and H ( p) = . ρ1 (s)ρ2 (s)...ρn−1 (s) ρ1 (s) + ρ2 (s) + · · · + ρn−1 (s)

Suppose now that  has constant width h. Then γ(s) − sh = γ(−s). So dγ |s −dγ |−s = h I ds ⊥ , where I ds ⊥ : s ⊥ → s ⊥ is the identity linear map.

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11 Bodies of Constant Width in Differential Geometry

Consequently, for 1 ≤ i ≤ n − 1,   dγ |−s (ξi (s)) = h − ρi (s) ξi (s). Summarizing, we have the following Theorem on Curvature for bodies of constant width. Theorem 11.3.1 (Theorem on Curvature) Let  ⊂ En be a body of constant width h whose boundary is twice continuously differentiable. Then the following statements hold: 1) The principal directions of curvature at opposite endpoints of a diametral chord are parallel. 2) If pq is a diametral chord of  and ρ is a principal radius of curvature corresponding to a principal direction ξ of bd  at p, then h − ρ is the principal radius of curvature corresponding to the principal direction ξ of bd  at q. 3) The mean radii of curvature of  at opposite endpoints of a diametral chord sum up to h. 4) If n = 3 and pq is a diametral chord of , let κ ( p) and κ ( p) be the Gaussian curvature of bd  at p and q, respectively. Let H be a plane through pq in the direction w, and let κw ( p) and κw (q) be the curvature of the curve H ∩ bd  at p and q, respectively. Then κw ( p) κw (q) + = h. κ ( p) κ (q) Proof The fact that the differential map dγ |−s : s ⊥ → s ⊥ is such that dγ |−s (ξi (s)) = (h − ρi (s))ξi (s) at q, and hence that 1) and 2) holds. Furthermore, the mean implies that ξi (s) is a principal direction 1 n−1 ρi , whereas by 1) and 2), the mean radius of curvature of radius of curvature of  at γ(s) is n−1 1 1 n−1 1 n−1 1 n−1  at γ(−s) is n−1 (h − ρ ). So, ρi + n−1 i 1 1 1 (h − ρi ) = h. n−1 Let w be a tangent vector at p with the property that H is the plane through pq in the direction w. is the curvature at p of the orthogonal projection of  in the direction w⊥ . By Corollary 11.2.1, κκw(P) ( p)

By the same reason, κκw (q) is the curvature at q of the orthogonal projection of  in the direction w ⊥ . (q) Since the orthogonal projection of  is a figure of constant width h and pq is a diametral chord of this + κκw (q) = h.  projection, it follows by Theorem 5.3.6 that κκw (( p) p) (q)

In Minkowski spaces, Theorem 11.3.1 on curvature is proved by Balestro, Martini and Teixeira for the 3-dimensional case. See Theorem 4.1 of [67]. One interesting result in global differential geometry is the so called Six Vertex Theorem whose proof requires global, rather than purely local, arguments, see [903]. Theorem 11.3.2 The curvature function of any simple, closed, twice continuously differentiable curve in the plane, other than the circle, must have at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives from the convention in global differential geometry to call an extreme point regarding the curvature function a vertex. The first general proof of this theorem was given by Kneser in [639]. For curves of constant width we have the corresponding Six Vertex Theorem, an early reference of which appears in Bonnesen–Fenchel [160], p. 132. Theorem 11.3.3 The curvature function of every simple, closed, twice continuously differentiable curve of constant width, other than the circle, must have at least six local extrema. Furthermore, the number of local extrema, if finite, is of the form 4k + 2, k ≥ 1.

11.4 Baire Category and Curvature

259

11.4 Baire Category and Curvature A set in a topological space is called nowhere dense if its closure has empty interior. Any set which can be written as a countable union of nowhere dense sets is said to be of the first category. If the set is not of the first category, then it is of the second category. A topological space where each open set is of the second category is called a Baire space. Examples of Baire spaces are the following. • • • • • • •

The Euclidean space En . The space all of compact sets in En with the Hausdorff metric. The space of all compact convex bodies in En with the Hausdorff metric. The space of all convex bodies in En with the Hausdorff metric. The space of all convex surfaces in En with the Hausdorff metric. The space of curves of constant width h in the plane with the Hausdorff metric. A convex surface with its intrinsic metric.

We say that most elements of a Baire space enjoy a certain property, if those not enjoying it form a set of the first category. Likewise, we say that a typical element of a Baire space enjoys a certain property, if those elements not enjoying it form a set of the first category. Finally, properties of most elements of a Baire space, i.e., shared by all elements except those in a set of the first category, are called generic. For a survey on generic properties of convex sets see [1214]. Two further basic surveys on Baire category results in the spirit of convex geometry are [1212] and the Handbook article [474]. The purpose of this section is to prove that for almost all points in the boundary of most convex bodies of constant width h the radius of curvature is equal to h. Within this section, let C be a convex curve not necessarily differentiable, and let α(t) be a clockwise parametrization of C. Assume α(0) = α(2π) = p. Suppose that L is a support line of C at p. Define ρ(t) as the radius of the circle tangent to L at p through α(t). We define ρ+ s (L) = lim t→0 sup ρ(t) = inf t sup{ρ(s) | 0 < s < t}, ρi+ (L) = limt→0 inf ρ(t) = supt inf{ρ(s) | 0 < s < t}. + Note that if the limit of the line through p and α(t) is not L, then ρ+ s (L) = ρi (L) = 0. Likewise − + − we define ρs (L) and ρi (L), when the parametrization is counterclockwise. If ρ+ s (L) = ρi (L) = − − ρs (L) = ρi (L), then we say that the radius of curvature ρ(L) of C at L exists. If p is a regular point of a convex curve C, L is the support line of C at p, and the radius of curvature ρ(t) exists, then we define the curvature γ( p) of C at p as

γ( p) = 1/ρ(t). Note that this notion of curvature coincides with the one given in the above section, when the curve is twice continuously differentiable. Furthermore, we say that the curvature γ( p) of a vertex p of C is not finite. Without proof we give here an important and classical theorem of Aleksandrov [14]. Theorem 11.4.1 The curvature of a convex curve exists and is finite almost anywhere (in the sense of measure). Zamfirescu proved in [1215] the analogue of Theorem 5.3.6 for curves of constant width which are not necessarily differentiable.

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11 Bodies of Constant Width in Differential Geometry

Lemma 11.4.1 Let C be a curve of constant width h, and let L and L be parallel support lines of C. Then

ρi+ (L) + ρ+ s (L ) = h.

Proof Let L ∩ C = {q} and remember that L ∩ C = { p}. First of all, notice that ρ+ s (L ) ≤ h. Indeed,

if it were not so, there would be a point q ∈ C arbitrarily close to q and with the property that the distance between p and q is bigger than h, contradicting the fact that C has constant width. Hence the + +

inequality ρ+ s (L ) ≤ h − ρi (L) is verified when ρi (L) = 0. + In the case ρi (L) > 0, let r be the point of the diameter pq such that the distance between r and p is precisely ρi+ (L) and take r ∗ ∈ pr , different from p and r . Consider the concentric circles C , C

with center r ∗ and radii |r ∗ − p|, and |r ∗ − q|, respectively. Then a small arc α of C starting at p and lying on the right side of p is outside C except for the point p. Take an arbitrary point q

∈ C

collinear with r ∗ and some point of α − { p}. The orthogonal projection of C onto the line L through r ∗ and q

has length h and contains L ∩ C in its interior. So q

lies in its exterior, and therefore

∗ ∗ / C. This proves that ρ+ q

∈ s (L ) ≤ |r − q| and, since this is true for every r ∈ pr − { p, r }, then + +

ρs (L ) ≤ |r − q| = w − ρi (L). Thus,

ρi+ (L) + ρ+ s (L ) ≤ h.

If C includes an entire circular arc starting at p and lying on the right side of p, then C must also include a circular arc starting at q and lying on the right side of q. So, trivially

ρi+ (L) + ρ+ s (L ) = h.

If C includes no such circular arc and ρi+ (L) < h, then there is a sequence of points pn ∈ C converging to p from the right, such that the circle through pn tangent to L at p has its center rn ∈ rq \ {r, q}. Now consider the concentric circles Cn , Cn

with center rn ∈ rrn and radii |rn − p|, |rn − q| such that rn → r . Let αn be a circular subarc of Cn on the right side of p from p to { pn } = Cn ∩ rn pn . Then the open connected bounded set n with boundary rn p ∪ αn ∪ pn rn must intersect C. Thus any point of n ∩ C at minimum distance from rn is an endpoint of a diameter of C with the other endpoint outside Cn

. This proves that +

ρ+ s (L ) ≥ lim | r n − q | = |r − q| = h − ρi (L). n→∞

For ρi+ (L) = h, the inequality reduces to ρ+ s (L ) ≥ 0 and follows from the convexity of C. This concludes the proof. 

Corollary 11.4.1 Let C be a curve of constant width h, and let L and L be parallel support lines of C. Then + − −

ρ+ s (L) + ρi (L ) = ρi (L) + ρs (L ) = ρs (L) + ρi (L ) = h.

Theorem 11.4.2 Let C be a curve of constant width h, and let L and L be parallel support lines of C. If the radius of curvature ρ(L) at L exists, 0 ≤ ρ(L) ≤ h, then the radius of curvature ρ(L ) at L

exists and ρ(L) + ρ(L ) = h. Reuleaux triangles can be generalized to regular or arbitrary Reuleaux polygons. The latter are defined as plane convex bodies of constant width h whose boundary consist of a finite (necessarily

11.4 Baire Category and Curvature

261

odd) number of circular arcs of radius h, as it can be seen in Section 8.1. Any plane convex figure of constant width h can be approximated arbitrarily closely by Reuleaux polygons. This and the fact that, with the exception of a finite number, every point in the boundary of a Reuleaux polygon has curvature 1/ h suggest that for a typical curve of constant width h every point has curvature 1/ h, except for points forming a set of measure zero. Indeed, as T. Zamfirescu was able to prove in [1215], this is true. Theorem 11.4.3 Most curves C of constant width h have the following property: ρi+ (L) = 0 or ρ+ s (L) = h, for every support line L of C. Proof Let Dr (L) denote the half-disk with boundary line-segment of length 2r orthogonal to L at p, with p as a boundary point and with the circular boundary arc to the right side of p. Let n be the family of all convex curves C of constant width h with the property that there exist a point p ∈ C and a support line L of C through p such that D n1 (L) ⊂ C, α[0, t0 ]) ⊂ Dh− n1 (L), for some real number t0 > 0. The set n is nowhere dense in C. Indeed, it is clearly closed in the space of curves of constant width h of the plane with the Hausdorff metric. Moreover, by Theorem 8.1.1 the space of curves of constant width h of the plane can be approximated with Reuleaux polygons, and since every Reuleaux

 polygon is clearly not in n , the interior of n is empty. So ∞ n is of the first category in the space 1 of all convex curves of the plane

with constant width h. It is an exercise to prove that ∞ 1 n is precisely the set of all curves C of constant width h such that ρi+ (L) > 0

and

ρ+ s (L) < h, for some support line L of C.

Hence, most curves C of constant width h have the property that ρi+ (L) = 0

or

ρ+ s (L) = h

for a support line L of C.



Theorem 11.4.4 For most curves of constant width h, the curvature exists and equals 1/ h a.e. Proof By Theorem 11.4.1, the curvature of a convex curve exists and is finite a.e. Hence, by Theorem 11.4.3, those curves of constant width h for which the curvature is not 1/ h a.e. form a set of the first category.  For higher dimensions there should not be something similar. First of all, as we already mentioned, a classical theorem of Aleksandrov [14] shows that the curvature of a convex curve exists and is finite almost everywhere. In contrast, it is known that an n-dimensional convex body, n > 2, which is typical in the sense of the Baire category, shows a simple, but highly nonintuitive curvature behavior. At most points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which the curvatures are infinite, see Zamfirescu [1214] and Schneider [1041]. Moreover, by Theorem 6.1.6, in higher dimensions there is not a similar construction as that of Reuleaux polygons. So it is very surprising that Bárány and Schneider [73] were able to generalize Theorem 11.4.4 as follows: Theorem 11.4.5 For most n-dimensional convex bodies of constant width h, at almost all boundary points all curvatures are, in the sense of measure, equal to 1/ h. In contrast, note that a ball of width 1 has radius 1/2, and hence all its curvatures are equal to 2.

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11 Bodies of Constant Width in Differential Geometry

11.5 The Geometry of a Body of Constant Width 11.5.1 A Characterization of Constant Width in Terms of the Hessian This section is devoted to studying the differential geometry of the boundary of a convex body  of constant width h expressed in terms of the Hessian of η = P − h2 , where P : Sn−1 → E is the support function of the n-dimensional convex body. We follow closely Anciaux and Guilfoyle [31] and Bayern, Lachand–Robert, and Oudet [86]. Let  ⊂ En be a strictly convex body with support function P : Sn−1 → E. Then P admits first derivatives that are Lipschitz continuous (see, for example, [557]). By Rademacher’s Theorem it follows that the second derivatives are well defined almost everywhere and bounded. Consider the support parametrization (11.1) of  in terms of the inverse Gauss map γ : Sn−1 → bd ; namely γ(u) = P (u) · u + ∇P (u) , where ∇P is the gradient vector field. Consider also the mean of its support function  w=

Sn−1 

P (u)d A , Sn−1 d A

where d A denotes the canonical volume form on Sn−1 , and define η = P − w. Theorem 11.5.1 The convex body  ⊂ En has constant width 2w if and only if η = P − w is an odd function; i.e., η(u) + η(−u) = 0. Moreover, if η : Sn−1 → E is an odd function that admits second derivatives which are continuous, then for ω > 0 sufficiently large, η + ω is the support function of a body of constant width 2ω. Proof First note that a function f : Sn−1 → E is the support function of a body of constant width h if and only if for every 2-dimensional subspace H , the restriction f : Sn−1 ∩ H → E is the support function of a plane convex figure of constant width h. Next, note that for n = 2 the proof of the theorem follows immediately from Theorem 5.3.5. Now use the fact that the second derivatives of η are continuous to prove that for every 2-dimensional subspace H there is a constant K H depending continuously on H such that if ω ≥ K H , then η |Sn−1 ∩H +ω is the support function of a plane convex figure of constant width 2ω.  The Median Surface of a Body of Constant Width Suppose now that  is a body of constant width 2w, and let γ : Sn−1 → bd  be its inverse Gauss map. Then, for every u ∈ Sn−1 , γ(u) − γ(−u) = 2wu. Define M : Sn−1 → En as M (u) = γ(u) − wu = γ(−u) + wu , for every u ∈ Sn−1 .

11.5 The Geometry of a Body of Constant Width

263

The set of points M = {M (u) | u ∈ Sn−1 } is called the median surface of . Notice that M (u) = M (−u), and that the following is a parametrization of the surface M in terms of η: M (u) = η(u) · u + ∇η (u) . The following theorem, whose proof is left to the reader as an exercise, characterizes median surfaces of constant width bodies. Theorem 11.5.2 Let w > 0 be given, and let M : Sn−1 → En be a continuous map satisfying 1) M(−u) = M(u) for every u ∈ Sn−1 , and 2) M(u 1 ) − M(u 0 ), u 0  ≤ w2 |u 1 − u 0 |2 for every u 0 , u 1 ∈ Sn−1 . Then  = {M(u) + tu | u ∈ Sn−1 , t ∈ [0, w]} is a convex body of constant width 2w with median surface M. Conversely, any convex body of constant width 2w can be described as above.

11.5.2 The Local Geometry of the Boundary of a 3-Dimensional Convex Body From now on, we treat only the 3-dimensional case and follow closely the ideas of Anciaux and Guilfoyle in [31]. We start by expressing the local geometry of the boundary of our strictly convex body  in terms of w and η. Recall that the Hessian of η is the symmetric tensor defined by Hess(η)(X, Y ) = ∇ X ∇η, Y , where ∇ denotes the Levi–Civita connection of the round metric of S2 . The two invariants of Hess(η) are its trace, which is the well-known Laplace–Beltrami operator , and its determinant that will be denoted by H (η). The main theorem of this section is the following. Theorem 11.5.3 Let d A¯ be the area element of the boundary of a strictly convex body  ⊂ E3 . Let P be its support function, w the mean of P and η = P − w. Then   d A¯ = w 2 + αw + β d A, where α = 2η + η and β = η 2 + ηη + H (η). Proof Let (x, y)(u) be a coordinate chart from the dense subset of S2 given by the stereographic projection onto E2 , and denote the conformal factor by er ; i.e., er = |∂x | = ∂ y . Then the area element is given by d A = er d xd y. The coefficients of the Hessian of η in the coordinates (x, y) are: a =e−2r ∇∂x ∇η, ∂x , b =e−2r ∇∂x ∇η, ∂ y  = e−2r ∇∂ y ∇η, ∂x , c =e−2r ∇∂ y ∇η, ∂ y . In order to compute the first derivatives of the inverse Gauss map parametrization (11.1) of the boundary of , γ(u) = P (u) · u + ∇P (u) = P (u) · u + ∇η(u), we use the Gauss formula of the embedding of the sphere S2 in E3 , which relates the flat connection D of E3 to the Levi–Civita connection ∇ on the sphere: (D X Y )(u) = (∇ X Y )(u) − X, Y u.

264

11 Bodies of Constant Width in Differential Geometry

It follows that γx = P ∂x + ∇∂x ∇η = (P + a)∂x + b∂ y and γ y = P ∂ y + ∇∂ y ∇η = b∂x + (P + c)∂ y . The trace a + c = η and the determinant H (η) = ac − b2 of the Hessian matrix of η depend only on the metric of S2 and not on the choice of coordinates. Taking into account the coefficients of the first fundamental form of the immersion γ, we have   E =γx , γx  = (P + a)2 + b2 e2r ,   F =γx , γ y  = 2P + a + c be2r ,   G =γ y , γ y  = (P + c)2 + b2 e2r . It follows that √   21  E G − F 2 e−2r = ((P + a)2 + b2 )((P + c)2 + b2 ) − 4(P + a + P + c)2 b2     = P2 + P (a + c) + ac − b2 = w 2 + (2η + a + c)w + η 2 + (a + c)η + ac − b2 . From this we can deduce our theorem, because     d A¯ = E G − F 2 d xd y = E G − F 2 e−2r d A   = w 2 + (2η + η)w + η 2 + ηη + H (η) d A =(w 2 + αw + β)d A.



Theorem 11.5.4 Let  ⊂ E3 be a strictly convex body. Let P be its support function, w the mean of P and η = P − w. Then the principal curvatures κ1 and κ2 , whenever they exist, are κ1,2

 2w + α ± α2 − 4β , = 2(w 2 + αw + β)

where α = 2η + η and β = η 2 + ηη + H (η). Proof Let us calculate the coefficients for the second fundamental form. Since N (u) = u, we have l =∂x N (u), γx  = ∂x u, γx  = e2r (P + a), m =∂x N (u), γ y  = ∂x u, γ y  = e2r b, n =∂ y N (u), γ y  = ∂ y u, γ y  = e2r (P + c). Thus lG + n E − 2m F   = e4r (P + a)((P + c)2 + b2 ) + (P + c)((P + a)2 + b2 ) − 2b2 (2P + a + c)   = e4r (P + a)(P + c)(2P + a + c) − b2 (2P + a + c)   = e4r (w 2 + αw + β)(2w + α) ,

11.5 The Geometry of a Body of Constant Width

265

and     ln − m 2 = e4r (P + a)(P + c) − b2 = e4r w 2 + αw + β . At a point γ(u) where d A¯ does not vanish, the mean curvature and the Gaussian curvature of the boundary of  are given by 2w + α lG + n E − 2m F , = 2 EG − F2 w + αw + β ln − m 2 1 ; K = = 2 EG − F2 w + αw + β

2H =

thus H2 − K =

(2w + α)2 − 4(w 2 + αw + β) α2 − 4β = . 4(w 2 + αw + β)2 4(w 2 + αw + β)2

Hence the principal curvatures κ1 and κ2 of the immersion at the point γ(u) are   α2 − 4β 2w + α ± κ1,2 = H ± H 2 − K = . 2 2(w + αw + β)

The following technical lemma will be very useful in the next calculations. Lemma 11.5.1 Let  be a body of constant width 2w; then  S2

1 H (η)d A = 2

 S2

|∇η|2 d A,

where η = P − w. Proof Let us denote the complex structure on S2 by j. Then we have that j∂x = ∂ y

j∂ y = −∂x .

We shall use the following formula for the curvature tensor on the sphere: R(X ; Y )Z , W  = X, Z Y, W  − Y, Z X, W  with X = ∂x , Y = ∂ y , Z = ∇η, W = j∇η. On the other hand,   = U

(X, Z Y, W  − Y, Z X, W )d xd y U

(∂x , ∇η j∂x , j∇η − ∂ y , ∇η∂ y , j∇η)d xd y   2 2 |∇η|2 d A. = (ηx + η y )d xd y = U

S2



266

11 Bodies of Constant Width in Differential Geometry

On the other hand, using the fact that j is parallel, i.e., ∇ X jY = j∇ X Y , we have  R(∂x , ∂ y )∇η, j∇ηd xd y

 

U

 ∇∂ y ∇∂x ∇η, j∇η − ∇∂x ∇∂ y ∇η, j∇η d xd y =  U  − ∇∂x ∇η, ∇∂ y j∇η + ∇∂ y ∇η, ∇∂x j∇η d xd y = U   − ∇∂x ∇η, j∇∂ y ∇η + ∇∂ y ∇η, j∇∂x ∇η d xd y = U  = 2 ∇∂ y ∇η, j∇∂x ∇ηd xd y U  = 2 (b∂x + c∂ y ), (−b∂x + a∂ y )d xd y  U H (η)d A. = 2 (ac − b2 )e2r d xd y = 2 S2

U



For a function η : Sn−1 → E that admits first derivatives which are Lipschitz continuous and vanishing mean, define   1  |∇η|2 − η 2 d A. E(η) := Sn−1 n − 1 The Wirtinger inequality claims that E(η) ≥ 0. This inequality can be easily proved using the theory of spherical harmonics (see [483], p. 1288, or Section 4.4 of [464]). Theorem 11.5.5 Let  ⊂ E3 be a body of constant width 2w and let η = P − w. Then 4π 3 w − wE(η), 3 S() = 4πw 2 − E(η),

V () =

where V () and S() are the volume and surface area of , respectively. Proof In order to calculate the volume of , we use the divergence theorem. Recall that u is the unit outward normal vector of the smooth parts of the boundary of  and that γ(u) = P (u). u + ∇η(u). Then we have   1 1 ¯ V () = γ(u), ud A¯ = P (u)d A. 3 S2 3 S2 Hence, by Theorem 11.5.3,  1 (η + w)(w 2 + αw + β)d A V () = 3 S2      w2 w w3 3η 2 + 2ηη + H (η) d A dA + (3η + η)d A + = 3 S2 3 S2   3 S2  1 3 2 + η + η η + η H (η) d A. 3 S2

11.5 The Geometry of a Body of Constant Width

267

Since η has zero mean, the coefficient of w 2 vanishes. Moreover, the constant width condition implies that η is odd, and therefore that all the cubic expressions of η and its second derivatives are odd and hence have zero mean. Thus the constant term vanishes. Finally, using the divergence theorem and Lemma 11.5.1 we obtain     2 1 w3 η 2 + (− + ) |∇η|2 d A dA +w V () = 3 S2 3 6 S2   2  3 |∇η| 4πw = −w − η2 d A 2 3 2 S 4πw 3 = − wE(η). 3 The computation of the surface area of  again uses Lemma 11.5.1 and Theorem 11.5.3, yielding  S() =

bd 

   w 2 + (2η + η)w + (η 2 + ηη + H (η)) d A S2    1 |∇η|2 d A + |∇η|2 d A =w 2 S(B3 ) + η2 d A − 2 S2 S2 S2 =4πw 2 − E(η).

d A¯ =



As a corollary of Theorem 11.5.5 we obtain the famous classical Theorem of Blaschke that claims that for 3-dimensional bodies of fixed constant width h, the volume depends linearly on the surface area. The proof presented here is due to Bayern, Lachand–Robert and Oudet in [86]. See Section 12.1 for a second proof using mixed volumes. Corollary 11.5.1 Let  ⊂ E3 be a body of constant width h. Then V () =

πh 3 h S() − . 2 3

11.6 Constant Width in Affine Geometry A convex body  ⊂ En is called an affine constant width body if  is the affine image of a constant width body. The following characterization follows immediately from Theorem 3.5.2. Theorem 11.6.1 A convex body  ⊂ En is an affine body of constant width if and only if  + (−) is an ellipsoid. Given this result, it is natural to look for suitable characterizations of the ellipsoid. The following one, whose proof uses the technique of spherical harmonics (see Section 13.2) will be needed later. For more characterizations of ellipsoids, see Section 2.12. A rectangular parallelepiped will be called from now on a “box”. Theorem 11.6.2 A convex body  ⊂ En , n ≥ 3, is an ellipsoid if and only if all its circumscribed boxes have their vertices on a fixed sphere. Proof Suppose first that all circumscribed boxes of an n-dimensional convex body  have their vertices on a fixed (n − 1)-sphere. Performing, if necessary, a homothetic transformation, we may assume that

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11 Bodies of Constant Width in Differential Geometry

√ the sphere in this theorem is centered at the origin o and has radius n. Let P be the support function of . By hypothesis, for any n mutually orthogonal unit vectors u 1 , . . . u n , we have that P2 (u 1 ) + · · · + P2 (u n ) = n. Letting F(u) = P2 (u) − 1, we obtain that F(u 1 ) + · · · + F(u n ) = 0. Since n ≥ 3, this implies, by Lemma 13.2.1, that for every H ∈ Hmn   F, H  H (u 1 ) + · · · + H (u n ) = 0. Choosing a zonal harmonic Pmn (u 1 , u) = H ∈ Hmn , we obtain that   F(u), Pmn (u 1 , u) Pmn (1) + (n − 1)Pmn (0) = 0.

(11.8)

By the main properties of Legendre polynomials in Section 13.2, the second factor of Equation −1 if and only if, by (13.6), m = 2. Consequently for any m = 2 and (11.8) vanishes only if Pmn (0) = n−1 n−1 any fixed u 1 ∈ S , F(u), Pmn (u 1 , u) = 0. It is well known that any H ∈ Hmn can be written as a linear combination of Legendre polynomials of the form Pmn (x, u). It follows that for m = 2 and any H ∈ Hmn , F, H  = 0. Therefore, if F ∼ ∞ 0 Q m is the condensed expansion of F, then Q m = 0 for all m  = 2 and therefore F = Q 2 or, equivalently, P2 (u) = Q 2 + 1. But this equality implies that P (u) is the support function of an ellipsoid. For the converse, we may assume without loss of generality that E is a solid ellipsoid whose boundary points are determined by the equation n xi2 . ai2 1

Then, by Exercise 2.24, the support function of E is P E (u 1 , . . . , u n ) =

n 

a12 u i2

 21

,

1

where (u 1 , . . . , u n ) ∈ Sn−1 . Let Iu denote the circumscribed box with one of its facets orthogonal to u. If p is a vertex of Iu , then there are mutually orthogonal outer normal unit vectors w1 , . . . wn of the facet of Iu such that w 1 = ±u and

11.6 Constant Width in Affine Geometry

269

p=

n

P E (w j )w j .

j=1 j

j

j

Setting w j = (w1 , . . . , wn ) and noting that the matrix [wi ] is orthogonal, we obtain that  p2 =

n j=1

P E (w j )2 =

n n

j

ai2 (wi )2 =

j=1 i=1

n i=1

ai2

n n j (wi )2 = ai2 . j=1

Hence, the vertices of every tangential box of E lie on a sphere.

i=1



A convex body  ⊂ En has constant diagonal if the main diagonals of all its circumscribed boxes have constant length. For n > 2, Chakerian [229] proved that an n-dimensional convex body is the affine image of a body of constant width if and only if it has constant diagonal. For n = 2 this claim is not true. In [134], Blaschke gave examples of centrally symmetric convex curves with constant diagonal which are not ellipses. Such a curve could not be the affine image of a curve of constant width, since the only centrally symmetric curve of constant width is the circle. Therefore we would like to have a characterization of affine images of constant width curves. Theorem 11.6.3 (Chakerian). A convex body  ⊂ En , n ≥ 3, is an affine image of a body of constant width if and only if it has constant diagonal. Proof Let  ⊂ En be a convex body and let {u 1 , . . . , u n } be a basis of unit orthogonal vectors. Then the length of the diagonal of the circumscribing box of  in the directions {u 1 , . . . , u n } is n 

w(, u i )2

 21

,

1

where w(, u) is the width function of . Consequently,  has constant diagonal if and only if  + (−) has constant diagonal. On the other hand, for a centrally symmetric convex body , the midpoint of the diagonal of any box circumscribing  is the center of . Thus a centrally symmetric convex body has constant diagonal if and only if all its circumscribed boxes have their vertices on a fixed sphere if and only if it is an ellipsoid. This implies that  has constant diagonal if and only if  + (−) is an ellipsoid if and only if  is an affine image of a constant width body.  Theorem 11.6.4 An n-dimensional convex body, n ≥ 3, is an affine image of a body of constant width if and only if all its orthogonal projections onto hyperplanes are affine images of constant width bodies. Proof Let  ⊂ En be a convex body. For each unit vector u ∈ Sn−1 , let us denote by u the orthogonal projection of  in the direction u. If  is an affine image of a constant width body, then  + (−) is an ellipsoid. Hence [ + (−)]u = u + (−u ) is an ellipsoid, so u is an affine image of a constant width body. Conversely, if u is an affine image of a constant width body, for every u ∈ Sn−1 , then u + (−u ) = [ + (−)]u is an ellipsoid and therefore, by Theorem 2.12.5,  + (−) is an ellipsoid. Then, by Theorem 11.6.1,  is an affine image of a constant width body.  Two diametral chords p1 q1 and p2 q2 of a plane convex figure φ are said to be conjugate if there exist support lines through p1 and q1 parallel to p2 q2 and support lines through p2 and q2 parallel to

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11 Bodies of Constant Width in Differential Geometry

p1 q1 . The boundary of φ is a P-curve if each diametral chord of φ belongs to a conjugate pair. In other words, P-curves are those convex curves which have a simple infinity of inscribed quadrangles of maximal area, see [135]. For more results combining P-curves and affine and projective images of plane bodies of constant width we refer to the paper [947], which presents also a bridge to the affinely invariant concept discussed below. It is clear that the affine image of a P-curve is a P-curve. Furthermore, since every curve of constant width is a P-curve, the boundary of every affine image of a constant width figure is a P-curve. The converse is not true. Indeed, there are noncircular, centrally symmetric P-curves. Centrally symmetric P-curves are known as Radon curves (see [794]). In normed planes, they are essentially the boundaries of those unit disks for which the most common relation of perpendicularity (called Birkhoff orthogonality) is symmetric. That is, a centrally symmetric figure φ has as boundary a P-curve if and only if, whenever a line L is Birkhoff orthogonal to a line L , then L is Birkhoff orthogonal to L. A good reference for orthogonality types in normed spaces is [24]. The following characterization of n-dimensional affine images of constant width bodies, n ≥ 3, is due to Krautwald [655]. An n-dimensional convex body, n ≥ 3, is an affine image of a body of constant width if and only if all its orthogonal projections onto 2-dimensional planes are convex figures whose boundaries are P-curves.

Notes Many results in convexity refer to convex bodies with certain smoothness assumptions. Then, as it is natural, the used methods are often taken from differential geometry. This is done also in this chapter here, but other parts of chapters (e.g., Section 16.2) refer to that direction as well. Basic survey-like references closely related to this part of convexity are [160, Chapter 8, Chapter 17], [712], and [713]. In particular, curves and surfaces of constant width are also interesting from the viewpoint of differential geometry. Books from that field taking special care for constant width sets are [1147, Chapter I], [830, Chapter 3], [561, Chapter 2], and [18, Chapter 5] (see also [1104]). Regarding classical curve theory we also mention the impressive book [731, pp. 311–316]. This chapter here, together with its notes, is strongly related to our Chapter 5 about line families (topics like pedal curves, evolutes/involutes, envelopes, hedgehogs etc. are discussed there) and, via curvature notions, to the Chapter 12 on mixed volumes. Curves of Constant Width We start these notes with some elementary theory of curves (of constant width), later on coming also to higher dimensions and special topics (mainly that of curvatures and that of spatial curves of constant width). Braude [184] investigated cycloids of constant width curves, or used them as cycloids to generate new curves (including the case that a constant width curve rolls on a constant width curve). Santaló [1013] showed how to generate curves by an endpoint of a segment which moves (with the other endpoint) on a closed, arc-length parametrized curve. It turns out that a special case of this process yields curves of constant width. In [556], a cam mechanism with two rigidly connected disks is studied, leading to curves of constant distance sum of corresponding tangents. These can be interpreted as generalized constant width curves, and a kinematic generation additionally yields generalized Zindler curves, see Section 5.4. On these lines, the author obtains also a generalization of Barbier’s theorem. In [995], pairs of closed Bertrand curves, Frenet formulas, and an inequality of Fenchel are used to derive Barbier’s theorem. In [955] envelopes of lines and the global properties of some special one-parameter equiform motions are discussed, yielding also generalizations of Holditch’s and Barbier’s theorem. Using dual line coordinates, in [11] Bertrand offsets of ruled surfaces in view of their dual representation are

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271

examined. As a limit position of closed ruled surfaces, also results on closed space curves are derived. In this framework, Barbier’s theorem is reproved, and the results are illustrated by computer-aided examples. The paper [769] presents a deep study of Zindler curves, using constructively also their close relations to constant width curves; e.g., rotating the double normals (meant as chords) of a constant width curve about π2 and around their midpoints, one gets the main chords of a Zindler curve (see also [1102] and Section 5.4.3). This is also related to our notes on transnormality in Section 16.2. However, Wunderlich developed in [1201] a kinematic principle which is based on the family of tangent lines of the midpoint curve, allowing a spatial construction of Zindler curves without using spatial curves of constant width (and hence not done as in [555]). In [1125], osculating circles of smooth constant width curves (having the same width as these circles) are considered. It is shown that there exist at least three such osculating circles which cross the constant width curve exactly twice, both times tangentially, and that these circles are the usual osculating circles at each of the crossing points. Mozgawa [870] generalized Mellish’s theorem (see below under “curvatures”) about constant width curves in the following way: He introduced curves of constant α-width, defined with the help of a curve called the isoptic α-curve associated to the given curve C, with α ∈ (0, π). This isoptic α-curve is the curve consisting of points at which two tangents to C meet at an angle of π − α. The α-width of C is defined as the curvature of this associated isoptic α-curve. Clearly, in the limit (when α = π) the α-width coincides with the usual width. In fact, the concept of isoptics is much older: Nitsche [896] called a planar curve C an isoptic for a circle K if C subtends the same angle at all points of K , and he proved that C is a circle if it occurs as isoptic for two distinct concentric circles. In [257], results on rosettes (meaning plane C 2 curves with everywhere positive curvature) are derived, where certain pairs of points (such as antipodal points, orthodiameter points,...) in such a curve are introduced and studied in their relation to each other. It is shown that rosettes of constant width (as generalizations of constant width curves) have necessarily odd winding numbers, and that rosettes of constant width h and winding number k have perimeter πhk. More general results in this direction are presented in [1174], where the author also extended classical constructions of convex curves to rosettes of constant width. More general results on rosettes were obtained in [262]. In [828], the notion of isoptics is adjusted to that of rosettes, yielding again rosettes of constant width. Their length is computed, it is shown that the number of axes of symmetries for rosettes of constant width has to be odd, and that central symmetry is only possible for circles. Also in [255], [256], and [828] isoptics of plane closed convex curves are investigated, yielding again results about constant width curves in this framework. Isoptics as a topic are more than hundred years old; their generation is also of interest in the theory of mechanisms. For example, in [256] the preservation of constant width is investigated. And also in [828] various basic theorems on constant width curves are extended to rosettes of constant width. Porcu [947] proved several metric and affine properties of so-called P-curves (these convex curves, going back to Blaschke, have a simple infinity of inscribed quadrangles of maximal area) and curves of constant width and their affine and projective transforms. In [259], a six vertex theorem for equipower curves (as certain “counterpart” of constant width curves) is proved. Related also to [231], the authors of [871] study some geometric problems concerned with plane sets of constant width relative to a certain given oval. In particular, they prove a six vertex theorem for such curves. Also the paper [899] is concerned with vertex theorems of plane curves, yielding results on the perimeter of equiangular polygons circumscribed about constant width curves. Further results on the differential geometry of plane curves of constant width were derived in [735] and [995]. Motivated by CAD software applications, the authors of [8] used piecewise Pythagorean hodograph curves to construct curves and rosettes of constant width; the concept of involutes plays a role there. Related is the paper [7], in which the well-known concept of offset curves is extended to normed planes and where again (analogues of) Pythagorean hodograph curves are used, since their offset curves are rational and allow flexible interactive design. In this way the authors carry over tools and notions like Serret–Frenet equations, evolutes, and involutes to normed planes. Related, since also referring to

272

11 Bodies of Constant Width in Differential Geometry

constant width in normed planes, are the papers [269] and [270], where also the concepts of hedgehogs and involutes occur. Using smoothness of support functions, Kharazishvili [620] showed the following approximation result: if a closed convex curve of constant width has everywhere strictly positive curvature and a support function belonging to the class C 3 ([0, 2π]), then, for any ε > 0, the ε-neighborhood of this curve contains an algebraic curve of constant width. The paper [393] deals with an evolution procedure of strictly convex plane curves taking their tangential angle, width, and curvature radius suitably into consideration. After showing area-increasing and perimeter-preserving properties of this flow (under which constant width curves remain invariant), it is proved that under certain assumptions the limit curves are of constant width or (if the initial curve is centrally symmetric) even circles. Now we switch to the concept of space curves of constant width, see also Section 16.2. There are several definitions, also depending on the dimension. We will start with the following classical one which is perhaps also most common. A C 1 curve embedded in E3 is a space curve of constant width h if every normal plane of this curve intersects it in exactly two points which are antipodal and of distance h apart. Later on we will discuss also space curves in higher dimensions. It seems that Fujiwara started the topic with the paper [378], giving also the definition above. Answering a question of him, Blaschke [129] showed that such a curve has to lie in the boundary of a constant width body, see also [160, p. 139]. This was reproved by Bückner [192], who also showed that any space curve of constant width h has length at least πh (see [194]). General classes of space curves in E3 (with those of constant width as special cases) were investigated by [1004] and [878]. It is well known that the length of a space curve of constant width h in E3 satisfies l(C) ≥ πh. Using a result from [1167] (asserting that any such curve can be isotopically deformed through curves of constant width h to a curve on a sphere of radius h)√and Crofton’s formula for spheres, the author of [1123] derived the non-sharp upper bound l(C) < (3 2)πh. In [1167], the above mentioned deformation is shown to preserve length, integral curvature and integral torsion, whereas the variance of the geodesic curvature is minimized by the spherical curve. It is also proved that any nonplanar curve of constant width has at least 12 geodesic vertices. Also on these lines, Wegner [1169] proved that a spherical curve of constant width h lies between two parallel planes having distance √h3 , and that a curve of constant width is isotopic to the circle through curves of constant width. In [247], the author established some nice formulae relating the perimeter, curvature, and torsion of space curves of constant width in 3-space to their width. In [1004], several characterizations of such curves are obtained, via Gauss curvature, point-tangent distances, angles between “opposite tangents”, and further properties. Pottmann [951] investigated spherical motions associated to spherical curves in E3 and also in higher dimensions, and he found, within this framework, also applications to spherical curves of constant width. Further results about curves of constant width in E3 were obtained by Nádeník [881]. Armstrong [37] investigated certain types of space curves (e.g., geodesics, contact curves with circumscribed cylinders, lines of curvatures, planar sections, etc.) in surfaces of constant width in E3 and showed, which pairings of such properties imply all properties satisfied by these curves. Sezer [1057] characterized space curves of constant width via differential equations, and in [653] it is shown that the integral torsion of a space curve of constant width is an integer multiple of 2π. Staying with space curves of constant width, we go now to higher dimensions. After a historical survey, Nádeník studied in [882] space curves of constant width in even-dimensional Euclidean spaces, with constant ratios of curvatures. He then proved several theorems on such curves, e.g., on their lengths, relations between opposite points, centers of gravity, and particular types of such curves. In [142], pairs of closed curves in En having the same spherical images (obtained via normals) are studied. Using also the arc length of the spherical image, theorems on the difference or sum of their perimeters were obtained, generalizing known properties of constant width curves. Nádeník [883] presents eleven properties of space curves in E4 , again determined via such spherical images and guaranteeing constant ˘ width. Also in [645] and [749] constant width curves in 4-space are studied. Smakal [1071] derived n necessary and sufficient conditions for space constant width curves in E , n > 3, with symmetric

11.6 Constant Width in Affine Geometry

273

spherical images of their tangent vectors, yielding also a natural analogue of Barbier’s theorem and of Meissner’s theorem concerning the coincidence of the centroid and the Steiner center of curvature. The Steiner Point and Related Topics The center of mass of a uniform mass distribution on a convex curve C is the perimetral centroid, and the center of mass of a distribution on C, whose density at each point is equal to the curvature, is called the Steiner point; see Section 14.5, where the Steiner incircle will be used to generalize the isoperimetric inequality in the plane. It is known that if C has constant width, the perimetral centroid and the Steiner point coincide [882] (see also [169]; we refer to [882] for an analogue referring to space curves of constant width). Furthermore, the locus of the perimetral centroid of the outer parallel curves of a curve of constant width lies on a straight line. In [273] the areas of pedal curves with respect to Steiner points are used to get an inequality for constant width sets. Ganapathi [390] studied those curves whose outer parallel curves all have the same curvature axis (defined via mass distribution whose density is the curvature); he showed that all of them have at least six vertices, thus reproving the fact that each constant width curve has at last six vertices, see Theorem 11.3.3. Soloviev [1073] proved that the support function of any convex curve whose centroid coincides with its curvature centroid is representable by h(u) = s(u) + cg(u), where c is a constant, s(u) is the support function of a centrally symmetric curve, and g(u) is the support function of a parallel curve of a constant width curve. The following was shown in [1132]: A closed, connected, orientable surface W in E3 is called a surface of generalized constant width h if it satisfies the following conditions: (1) let n( p) be its inward unit normal at p ∈ W ; then the vector p = p + hn( p) lies on W ; (2) the map p → p is an involution. It is proved that if W is an analytic surface of generalized constant width h and the Gauss curvature of W at p equals that of p for every p ∈ W , then W is a sphere of diameter h. The author of [1145] studied differentiable hypersurfaces which have exactly two normals parallel to every direction. He confirmed properties known for bodies of constant width, but gave also more general statements. In [1035], Schneider generalized most of the classical real-valued functions associated to convex bodies to vector-valued functions. He proved that hereby many well-known relations either carry over exactly or slightly modified. As from the notion of volume in a natural way the notions of mixed volumes and quermassintegrals are obtainable, one can derive from the centroid a series of curvature centroids, among them the surface area centroid and the Steiner point. In the final section of [1035] it is shown that a whole class of linearly independent relations involving one of the vector-valued functions and the quermassintegrals holds for sets of constant width. For the spherical situation, similar results were derived in [38], and analogues for envelopes of cylindrical surfaces of constant width are given in [884]. Curvatures We now come to the important notion of curvature. Beretta and Maxia [99] showed that for curves of constant width the number of vertices (i.e., of points of extreme curvatures) is of the form 4n + 2, n > 0, when the radius of curvature is assumed to be continuous. The obvious fact of nowhere vanishing curvature was proved in [261]. Mellish [814] and Hsiao [560] proved several properties of principal directions and radii of curvature as well as mean curvatures at opposite points (defined via parallel normals) of curves and surfaces of constant width, see Theorem 11.3.1. Here we refer also to [261] and [66]. One can generalize Theorem 11.3.1, 3) to all convex bodies without smoothness restrictions by introducing the first curvature measure S1 (, E). See Section 12.2. For a body  of constant width h we have that  + (−) = h B. Therefore, by linearity and homogeneity of S1 (, E) as a function of , we obtain (11.9) S1 (, E) + S1 (−, E) = hγ(E),

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11 Bodies of Constant Width in Differential Geometry

where γ(E) is the spherical Lebesgue measure of the Borel set E ⊂ Sn−1 . Thus, if φ is a convex body satisfying (11.9) for all Borel sets E ⊂ Sn−1 , then S1 (φ + (−φ), E) = S1 (h B, E), and by the Aleksandrov–Fenchel–Jessen Uniqueness Theorem (see Schneider [1036]) we obtain that φ has constant width h. Thus (11.9) is a characteristic property of bodies of constant width h. For a very interesting treatment of the curvature of convex surfaces we may consult a paper by Mellish [814]. This paper, although it appeared in the Annals of Math., is not well known, maybe because the words “constant width” do not appear in the title. Mellish worked on a generalization of Barbier’s Theorem. He died at the age of 24 and had no mathematical publications during his lifetime. After his death, his colleagues at Brown University examined his notes on mathematics and prepared this paper based upon his notes. Formula (11.5), which clearly resembles Euler’s Sectional Curvature Theorem 11.2.2, was first proved by Blaschke in his book “Kreis und Kugel” (see [132], pp. 117). Mellish’s proof is different from that of Blaschke and was obtained independently. With respect to the curvature of a surface of constant width, Hsiao proved in [560] an interesting theorem involving the sectional curvature and the Gaussian curvature at the endpoints of a diameter (see Theorem 11.3.1 4). A general n-dimensional convex body shows a regular curvature behavior at almost all its boundary points. Due to a theorem by Aleksandrov [13] almost all boundary points of a convex body are, in the sense of measure, normal points. Hence all sectional curvatures exist, and they satisfy Euler’s Theorem. This is true even without differentiability assumptions. From the generic point of view the picture is as follows: while a typical convex body is, in the Baire category sense, strictly convex and smooth, it was proven by Zamfirescu [1214] that at almost all its boundary points the curvature is zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. In analogy to this result it is to be expected that for a typical convex body of constant width 1 the radii of curvature exhibit a preference for attaining the values 0 and 1. Indeed, it was also shown by Zamfirscu [1215] that for a typical convex figure of constant width 1 in the plane the radii of curvature attain only the values 0 and 1, see Theorem 11.4.3. A higher dimensional extension of this result was given by Bárány and Schneider [73]; they proved that the typical behavior of convex bodies of constant width 1 is that for almost all boundary points all curvatures are equal to 1. The proof of this theorem requires a more elaborate approximation procedure than Zamfirescu’s result. Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature. The Local Geometry of a 3-Dimensional Convex Body For the local geometry of the boundary of a 3-dimensional convex body (see Section 11.5.2) two papers are essential. The first is due to Bayern, Lachand–Robert and Oudet [86], and the second due to Anciaux and Guillfoyle [31]; they apply methods of variational analysis to study the Blaschke– Lebesgue Conjecture. If φ ⊂ E2 is a convex figure which has sufficiently smooth boundary, ρ(θ) denotes the radius of curvature of bd φ at the point with outward normal unit vector u(θ) and, in addition, φ has constant width h, then (11.9) becomes ρ(θ) + ρ(θ + π) = h , see Theorem 5.3.7. This equation is only a necessary condition on a given function ρ(θ) if we want to characterize the radius of curvature function of the boundary of a constant width function. Kallay [603] assigns a radius of curvature function to every figure  of constant width h and gives necessary and sufficient conditions for a nonnegative and measurable function ρ(θ) to be the radius of curvature of some figure of constant width. He uses this to characterize the “extreme bodies” among figures of constant width that are indecomposable with respect to Minkowski addition. Kallay proved that a

11.6 Constant Width in Affine Geometry

275

convex figure is extreme in this sense if and only if the radius of curvature takes the values 0 and 1 for almost all θ. For example, Releaux polygons (see Section 8.1) of width 1 are among the extreme bodies. A counterpart of Theorem 5.3.6 in Minkowski planes is given by Petty [929]. Vicensini [1152] gave an extensive treatment of related matters, including the analogue of Theorem 11.3.1 3) in Minkowski spaces. Affine Geometry Convex bodies of constant affine width were also studied from the viewpoint of affine differential geometry. This area was started by Süss [1108] in 1927. An excellent survey of the basic results in case n = 2 can by found in Beretta–Maxia [99], including the discussion of Hirakawa’s work [542] about the concept of affine width. Furthermore, for more related material on the affine geometry of convex curves see Heil [521]. An interesting affinely invariant concept was introduced in [1006]: For a plane convex curve C and each p ∈ C let t ( p) be the maximal area of all triangles inscribed in C having p as vertex. Then t ( p) is the triangular width of C at p, and C is said to be of constant triangular width if t ( p) is constant for all p ∈ C. In [1006] and [1007], such curves (different to ellipses) are constructed, and higher dimensional analogues are given. In [1008], a characterization of ellipses inside that class of planar curves is presented. Further on, in [1009] it is clarified which regular polygons can have this property. Furthermore, in [263] and [395] it was shown that 4-gons and 5-gons of constant triangular width must be affine images of regular polygons, but 6-gons and 7-gons not. And in [394] these considerations are continued, where some parallelism between sides and diagonals of polygons of constant triangular width is established as tool for possible further research on such notions. Using affine diameters of convex sets, Alonso and Spirova [25] introduced a new orthogonality type for normed planes, called affine orthogonality. This concept leads to characteristic properties of sets of constant width (with respect to the considered norm). Leichtweiss [710] considered inequalities referring to the ratio of the radii of smallest circumscribed and the largest inscribed spheres of n-dimensional convex bodies ranging over the class of all their images regarding affine transformations. If such a class contains a constant width set, the infimum case yields a characterization of constant width sets. An n-dimensional convex body is said to have constant diagonal if the main diagonals of all its circumscribed boxes (i.e., rectangular parallelotopes) have constant length. Chakerian [229] proved that for n > 2 this property characterizes all affine images of constant width bodies. And we have also the affine image of a constant width body iff all its orthogonal projections onto hyperplanes are affine images of a body of constant width within this hyperplane, see Section 11.6. For the planar case, the second condition is empty, and the first one is only necessary. Krautwald [655] obtained nice characterizations of affine images of n-dimensional constant width bodies. One of them says the following for a convex body K : let P be combinatorially equivalent to a cross-polytope and inscribed in K , having maximal volume among all such polytopes. Assume that all its vertices sit in the interiors of the facets of a suitable parallelotope circumscribed about K and having minimal volume among all parallelotopes circumscribed about K . Then K is the affine image of some constant width body. The paper [947] contains several results about generalized Radon curves and affine as well as projective images of (planar) constant width bodies. From results of Blaschke on ellipsoids it follows that the affine images of 3-dimensional constant width bodies are characterized by the property of having projections bounded by P-curves. See Exercises 11.17 and 11.18. Applying the Blaschke–Santaló inequality to the difference body of arbitrary convex bodies, Lutwak [742] derived an inequality involving n-means, diameters, and widths of them in a direction u, where equality holds for affine images of constant width bodies. Inspired by the coat of arms in the castle of Blois consisting

276

11 Bodies of Constant Width in Differential Geometry

of a packing of Reuleaux triangles, in [343] convex sets of direction-invariant packing density in the plane are investigated. Affine images of squares and constant width sets belong to this family. Let L denote a system of m lines in the plane. Depending on m, Alexander [19] studied homothety ratios for affine images of constant width sets in the plane such that their smaller homothets are not met by any line from L, whereas each line of L meets the original set. In [1005], pairs of affine equidistant curves in 3-space are introduced; when these pairs of curves coincide, spatial curves of constant width are obtained. Another related notion concerns the polar of a body of constant width. Let o be an interior point of the convex body  ⊂ En . We say that o is an equichordal point if all chords of  passing through o have the same length. The point o is a equireciprocal point if, for every chord pq of  through o, the quantity 1 1 + |op| |oq| is constant. Both concepts are wider discussed in the Notes of Chapter 13 below (see the part below the sub-headline “Constant section”). It is possible to prove that if a body of constant width  ⊂ En has the origin o as an interior point, then its polar body o has the origin o as an equireciprocal point. Furthermore, if S is the surface obtained from bd  by inversion with respect Sn−1 , then the origin o is a equichordal point of S. The above theorem is of interest with respect to the famous problem of Fujiwara asking whether there exists a plane convex figure with more than one equichordal point. Despite its elementary formulation, this problem remained unsolved for many years until it was finally proven in 1996 by Rychlik [993] who showed that such a figure with two equichordal points cannot exist. He used methods of advanced complex analysis and algebraic geometry. A discussion of this problem and related problems about equichordal points is given by Klee, see [631], [634] and also the notes to our Chapter 13.

Exercises 11.1.

  ˜ be the orthogonal projection of ∂Pφ (u(ϑ)), ˜ . . . , ∂Pφ (u(ϑ)) ˜ onto u ⊥ , and Let ∇Pφ (u(ϑ)) ∂u 1 ∂u n ˜ = Pφ (u(ϑ)) ˜ be the support function of the strictly convex body φ in terms of the P(ϑ) ˜ coordinates ϑ ∈ U ⊂ En−1 . Prove that ˜ = du | ˜ dP | ˜ . ∇Pφ (u(ϑ)) ϑ ϑ

11.2.

Let φ be a strictly convex body in the x z-plane with support function p(θ). Suppose that φ is symmetric with respect to the z-axis, that is, p(π/2 + θ) = − p(π/2 − θ). Then we can parametrize the boundary of φ by: x(θ) = p(θ) cos(θ) − p (θ) sin(θ) z(θ) = p(θ) sin(θ) + p (θ) sin(θ).   (a) Calculate the parametrization in cartesian coordinates x(ϑ, θ), y(ϑ, θ), z(ϑ, θ) of the surface of revolution obtained by rotating  φ around the z-axis.  (b) Use the polar coordinates u(ϑ, θ) = cos(ϑ) cos(θ), sin(ϑ) cos(θ), sin(θ) to obtain from 11.3 the same parametrization.

11.3.

Calculate the parametrization of the body of constant width 2, which has the restriction of 1 + x yz to S2 as support function.

Exercises

11.4.

11.5.

11.6.

11.7*.

11.8*. 11.9. 11.10.

11.11. 11.12*. 11.13. 11.14. 11.15*. 11.16.

11.17. 11.18.

277

Let p be a point in the twice differentiable boundary of a convex body  ⊂ E3 . Suppose u is a unit normal vector of  at p, and let H be a normal plane generated by u and w. Assume  : E3 → H is the orthogonal projection,  : bd  → S2 is the Gauss map and  : S2 → bd  is its inverse. a) Let g : S2 ∩ H → H be the curve defined by g(v) = γ(v). Prove that dg |u = dγ |u (w), w. (q) , for every boundary point b) Let g : bd  ∩ H → S2 ∩ H be defined by g(q) = |(q)| q ∈ bd  ∩ H . Prove that dg | p = d | p (w), w. Let  ⊂ E3 be a convex body whose boundary is twice differentiable, p a non-umbilical point in bd , and H a normal plane through p. Suppose that the curvatures at p of the normal section H ∩  and of bd(), where  is the orthogonal projection onto H , coincide. Prove that H is generated by a principal direction. Let  be a convex body of revolution around the z-axis in E3 . Suppose that the boundary of  is obtained by rotating a regular plane curve φ → (r (φ), h(φ)) in such a way that bd  admits a parametrization of the form f (φ, ψ) = (r (φ) cos ψ, r (φ) sin ψ, h(φ)). Furthermore, suppose that (r (φ), h(φ)) is a an arc length parametrization. a) Prove that the normal vector of bd at f (φ, ψ) is (−h (φ) cos ψ, −h (φ) sin ψ, r (φ)). b) Prove that u 1 = (r (φ) cos ψ, r (φ) sin ψ, h (φ)) is a principal direction of curvature of bd at f (φ, ψ). c) Prove that a principal direction of curvature of bd at f (φ, ψ) is u 2 = (−h (φ) cos ψ, −h (φ) sin ψ, r (φ)) × (r (φ) cos ψ, r (φ) sin ψ, h (φ)). d) Prove that the second radius of curvature is given by the distance from f (φ, ψ) to the zaxis measured along the line through f (φ, ψ) in the direction (−h (φ) cos ψ, −h (φ) sin ψ, r (φ)). c) Prove that the Gaussian curvature of bd at the point p = f (φ, ψ) is given by the formula r

(φ) . (11.10) κ ( p) = − r (φ) Let  be a figure of constant width h and let p be a point in the smooth part of the boundary of . Suppose the curvature of bd  at p is 1/ h. Must bd  be a circle of radius h at a small neighborhood of p in bd ? Prove the Six Vertex Theorem for curves of constant width. Prove Corollary 11.4.1.

In the context of the proof of Theorem 11.4.3, prove that ∞ 1 n is precisely the set of all (T ) < h, for some support line curves C of constant width h such that ρi+ (T ) > 0 and ρ+ s T of C. Complete the arguments of the proof of Theorem 11.5.1. Let p : S2 → E be an odd function. Prove that for  > 0 sufficiently small, P = 1 +  p is the support function of a body of constant width 2. Prove Theorem 11.5.2. Prove that  ⊂ En is the affine image of body of constant width if and only if  + (−) is an ellipsoid. Find a non-elliptical convex curve in the plane with the property that all its circumscribed boxes have their vertices on a fixed circle. For a unit vector u ∈ Sn−1 , let us denote by u the orthogonal projection of  in the direction u. Prove that if  and  are convex bodies, then ( + )u = u + u and (−)u = −u . Prove that the affine image of a P-curve is a P-curve. Prove that the boundary of a figure of constant width is a P-curve.

Chapter 12

Mixed Volumes

The art of doing mathematics consists in finding that special case which contains all the germs of generality. David Hilbert

12.1 Mixed Volumes The notion of mixed volumes represents a profound concept first discovered by Minkowski in 1900. In the letter [838] he wrote to Hilbert explaining his discoveries as interesting and quite enlightening. As we can see below, this concept will allow us to prove several classical theorems on the volume of constant width bodies in a somewhat unexpected way. For a more complete treatment see the book by Bonnesen and Fenchel [160], the excellent survey [1010], and Chapter 7 of Schneider’s book [1039]. Let Kn be the space of all convex bodies of dimension (less than or) equal to n with the Hausdorff metric. For every  ∈ Kn , denote the n-dimensional volume of  by V () and the (n − 1)-dimensional area of its surface by S(). Recall also that B denotes the ball with unit radius. A mixed volume is a nonnegative functional V : (Kn )n → E+ of convex bodies that satisfies the following properties: i) V is multilinear; that is, V(1 , . . . , λi , . . . , n )

=

λV(1 , . . . , i , . . . , n ),

V(1 , . . . , i + i , . . . , n )

=

V(1 , . . . , i , . . . , n ) +V(1 , . . . , i , . . . , n ).

λ ≥ 0,

ii) V is symmetric; that is, V(1 , . . . , i , . . . ,  j , . . . , n ) = V(1 , . . . ,  j , . . . , i , . . . , n ). iii) V(, . . . , ) is the n-dimensional volume V () of . The following results are immediate consequences of the definition.

© Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_12

279

280

12 Mixed Volumes

Lemma 12.1.1 For every mixed volume V, V(, . . . , , B) =

S() , n

where S() denotes the (n − 1)-dimensional area of the surface of . Proof V ( + r B) − V () r V( + r B, . . . ,  + r B) − V(, . . . , ) = lim . r →0 r

S() = lim

r →0

By multilinearity we have that    n S() = nV(, . . . , , B) lim r V(, . . . , B, B) + · · · r →0 2  + r n−2 nV(, B, . . . , B) + r n−1 V(B, . . . , B) = nV(, . . . , , B).



Theorem 12.1.1 Let 1 , . . . , r be convex bodies of dimension less than or equal to n, λ1 , . . . , λr positive reals and V a mixed volume. Then V (λ1 1 + · · · + λr r ) is a homogeneous polynomial of degree n in the variables λ1 , . . . , λr , where the coefficient of the term λi1 · · · λin is the mixed volume V(i1 , . . . , in ); that is, V (λ1 1 + · · · + λr r ) =

r r   i 1 =1 i 2 =1

···

r 

V(i1 , . . . , in )λi1 . . . λin .

i n =1

Proof The proof follows easily from the multilinearity of the mixed volume V and from the expression of the volume in terms of mixed volume.  Remark 12.1.1 In the homogeneous polynomial in this lemma, it is important to interpret the meaning of the coefficient of the term λi1 · · · λin correctly. For example, if r = 3, the coefficients of the terms λ1 λ1 λ2 , λ1 λ2 λ1 and λ2 λ1 λ1 all correspond to the same mixed volume V(1 , 1 , 2 ) = V(1 , 2 , 1 ) = V(2 , 1 , 2 ). The following examples are quite enlightening. Examples: In two dimensions: V (x + y) = V ()x 2 + 2V(, )x y + V ()y 2 .

12.1 Mixed Volumes

281

In three dimensions: V (x + y) = V ()x 3 + 3V(, , )x 2 y + 3V(, , )x y 2 + V ()y 3 , or V (x + y + zφ) = V ()x 3 + 3V(, , )x 2 y + 3V(, , φ)x 2 z +3V(, , )x y 2 + 6V(, , φ)x yz +3V(, φ, φ)x z 2 + 3V(, , φ)y 2 z +3V(, φ, φ)yz 2 + V ()y 3 + V (φ)z 3 . In four dimensions: V (x + y) = V ()x 4 +4V(, , , )x 3 y + 6V(, , , )x 2 y 2 +4V(, , , )x y 3 + V ()y 4 , and similarly for higher dimensions. A consequence of Theorem 12.1.1 is the uniqueness of mixed volumes. This is formulated in Lemma 12.1.2 For every dimension n, the following holds: if there is a functional V : (Kn )n → E+ that satisfies conditions i), ii) and iii), then it is unique. Proof Let V1 , V2 : (Kn )n → E+ be two mixed volumes for dimension n. Let us fix n convex bodies 1 , . . . , n (not necessarily distinct). For nonnegative reals λ1 , . . . , λn , let f (λ1 , . . . , λn ) = V (λ1 1 + · · · + λn n ). By Theorem 12.1.1, f (λ1 , . . . , λn ) can be expressed simultaneously as P1 (λ1 , . . . , λn ) and P2 (λ1 , . . . , λn ), two homogeneous polynomials of degree n in variables λ1 , . . . , λn , where the coefficients of the term λi1 , . . . , λin in P1 and P2 are, respectively, the mixed volumes V1 (i1 , . . . , in ) and V2 (i1 , . . . , in ). Since P1 (λ1 , . . . , λn ) = P2 (λ1 , . . . , λn ), for λ1 , . . . , λn ≥ 0, the polynomials P1 and P2 are equal, and so are, therefore, their respective coefficients. Since both V1 and V2 are symmetric functions, then V1 (1 , . . . , n ) = V2 (1 , . . . , n ).  Consequently, mixed volumes are, if they exist, unique. Furthermore, they are monotone and invariant under translations. Namely, we have Theorem 12.1.2 If V is the n-dimensional mixed volume, then 1) V(1 , . . . , i , . . . , n ) ≤ V(1 , . . . , i , . . . , n ), when i ⊂ i . 2) V(1 , . . . , i , . . . , n ) = V(1 , . . . , i , . . . , n ), whenever i is a translated copy of i . Whereas, in the above theorem, the proof of 2) is an interesting exercise, the proof of 1) is not obvious. It depends on the formula (12.2). The interested reader can consult the book of Schneider [1039]. Before proving the existence of mixed volumes, we shall use them in some classical theorems on volumes of constant width bodies. We begin in dimension 2 by giving a new proof of the Theorem of Barbier (see Section 5.1.2). Theorem 12.1.3 (Theorem of Barbier) The perimeter p() of , a plane figure of constant width h, is πh.

282

12 Mixed Volumes

Proof Let  be a plane figure of constant width h and let − be the reflection of  at the origin. We know that  + (−) = h B. It follows that the equality (1 − t) + t (−) = (1 − 2t) + th B holds for all t ∈ [0, 1/2]. (Note that this claim has no meaning outside the interval [0, 1/2].) Then, for all t ∈ [0, 1/2],     V (1 − t) + t (−), B = V (1 − 2t) + th B, B . By the multilinearity of mixed volumes, we have that the polynomials P1 (t) = (1 − t)V(, B) + tV(−, B) and P2 (t) = (1 − 2t)V(, B) + thV(B, B) coincide for all t ∈ [0, 1/2], and therefore for all t ∈ E. Making use of the fact that 2V(, B) = 2V(−, B) = p() and V(B, B) = π, we obtain, for t = 1, the desired equality p() = πh.  In dimension 3, precisely the same idea will be used to prove that the integral of mean curvature M() = 3V(, B, B), for a body  of constant width h, is equal to 2πh. The theorem of Blaschke [130] will then be proved in the same way, and will later be generalized to dimensions higher than 3. Theorem 12.1.4 (Theorem of Blaschke) Let  be a 3-dimensional body of constant width h. Then 3h S() − 6V () = 2πh 3 and 3V(, B, B) = M() = 2πh. Proof Let − be the reflection of  at the origin. Recall that  + (−) = h B. Then the equality (1 − t) + t (−) = (1 − 2t) + th B holds for all t ∈ [0, 1/2]. This means that for all t ∈ [0, 1/2]   V (1 − t) + t (−), (1 − t) + t (−), B   = V (1 − 2t) + th B, (1 − 2t) + th B, B ,   V (1 − t) + t (−), (1 − t) + t (−), (1 − t) + t (−)   = V (1 − 2t) + th B, (1 − 2t) + th B, (1 − 2t) + th B . From the first equation and the multilinearity of mixed volumes we have that the polynomials P1 (t) = (1 − t)2 V(, , B) + 2(1 − t)tV(, −, B) + t 2 V(−, −, B) and P2 (t) = (1 − 2t)2 V(, , B) + 2(1 − 2t)thV(, B, B) + t 2 h 2 V(B, B, B) coincide for all t ∈ [0, 1/2] and therefore for every t ∈ E. Making use of the fact that 3V(, , B) = 3V(−, −, B) = S() and 3V(B, B, B) = 4π, we obtain, for t = 1, the equality 3V(, B, B) = M() = 2πh. Similarly, using the second equation and making use of the fact that V(, , ) = V(−, −, −) = V (), 3V(, , B) = S(), 3V(, B, B) = 2πh and 3V(B, B, B) = 4π,  we obtain, for t = 1, the equation 3h S() − 6V () = 2πh 3 .

12.1 Mixed Volumes

283

In 1915, Blaschke [130] obtained the second equation of this theorem by substituting −δ for δ in Steiner’s formula for the volume of parallel bodies δ =  + δ B, but without a complete justification for this step. The justification can be found in the proof presented here, and it is based on the ideas of Debrunner [281] and Chakerian and Groemer (see [238]). Theorem 12.1.4 is easily obtained as a generalization of Theorem 12.1.3, and it was first proved by Dinghas [296]. His proof used integral geometry and a formula for the curvature of convex bodies given in Section 11.2, see also Section 11.5.2. One of the best known theorems in the theory of constant width bodies is the theorem stating that among all figures of fixed constant width, the one with least area is the Reuleaux triangle. In fact, this theorem is notorious for having a large number of essentially different proofs, see, for example, [105], [131], [308], [667], [707], and [1204]. Perhaps the simplest, but simultaneously most mysterious of these proofs is the one discovered by Chakerian [230] in 1966. Theorem 12.1.5 (Theorem of Blaschke–Lebesgue) Every planar figure of constant width one has √ area greater than or equal to (π − 3)/2, which is the area of the Reuleaux triangle of width one. Proof Let  be a planar figure with unit constant width. By the Theorem 15.2.3 of Pál we know that there exists a regular hexagon H of width one which contains . Assume that H is symmetric with respect to the origin such that if − is the reflection of  at the origin, then − is also contained in √ H . Let A() be the area of  and A(H ) = 3/2 the area of H . As  + (−) is B, the disk of radius one, we have, by the multilinearity and monotonicity of the mixed volume V, that π = A(B) = V(B, B) = V( + (−),  + (−)) = V(, ) + 2V(, −) + V(−, −) √ ≤ 2 A() + 2V(H, H ) = 2 A() + 3, which implies that 2√ A() ≥ π − then 2 A() = π − 3.

√ 3. Finally, it is quite easy to verify that if  is the Reuleaux triangle, 

Definition For k an integer between 0 and n, let Wk () = V(, . . . , , B, . . . , B), where the mixed volume has n − k ’s and k B’s. Then W0 () = V (), W1 () = S()/n, and Wn () = V (B). If n = 3, then W0 () = V (), 3W1 () = S(), 3W2 () = M() and 3W3 () = 4π. The convex functionals Wk () are known as the Minkowski functionals or quermassintegrals of the convex body . In the 3-dimensional case, these are essentially the volume V (), the surface area S(), the integral of mean curvature M(), and the integral of Gaussian curvature, which always takes the constant value C = π. It turns out that the volume of the exterior parallel body of  is a polynomial whose coefficients are the quermassintegrals of . Theorem 12.1.6 (Steiner Formula) Let  be a convex body in Kn , and let t ≥ 0. Then V (t ) = V ( + t B) =

n    n 0

i

Wi ()t i .

Proof If X and Y are convex bodies, then, using the multilinear and symmetric properties of the mixed volume, we can express the volume of X + Y in terms of the mixed volumes involving X and Y by formally expanding the product (x+y)n = n0 ni x n−i y i . We obtain the value of V(X +Y, . . . , X + Y )

284

12 Mixed Volumes

by replacing the terms of the form x n−i y i by V(X, . . . , X, Y, . . . , Y ), where in theabove  expression we have n − i X ’s and i Y ’s. If X =  and Y = B, then V ( + t B) = n0 ni Wi ()t i , as desired.  Various authors have generalized Blaschke’s Theorem 12.1.4 to dimension n by showing that the independent relations. These relations take the quermassintegrals of a set of constant width satisfy n+1 2 following form. Theorem 12.1.7 Let  be an n-dimensional body of constant width h. Then the following n + 1 equations of the quermassintegrals of  are true, k = 0, 1, . . . , n:   k  i k Wn−i ()h k−i . Wn−k () = (−1) i i=0 Proof It follows by induction that if X i = Yi + Z i , i = 1, . . . , m ≤ n, then V(Y1 , . . . , Ym , ∗) can be expressed as the sum of terms of the form (−1)σ V(X i1 , . . . , X ik , Z ik+1 , . . . , Z im , ∗). The signs (−1)σ and permutations X i1 , . . . , X ik , Z ik+1 , . . . , Z im are obtained by formally expanding the product (x1 − z 1 )(x2 − z 2 ) . . . (xm − z m ) and then replacing terms of the form (−1)σ xi1 , . . . , xik , z ik+1 , . . . , z im in the expansion by (−1)σ V(X i1 , . . . , X ik , Z ik+1 , . . . , Z im , ∗), where ∗ represents the same n − m remaining arguments in each term. For example, if X = Y + Z , we can express  V(Y, . . . , Y, ∗) in terms of mixed volumes involving Y and Z by formally expanding (x − z)k = k0 (−1)i ki x k−i z i . Thus, we have   k  i k V(X, . . . , X, Z , . . . , Z , ∗), V(Y, . . . , Y, ∗) = (−1) i 0 where in this expression we have i Z ’s and m − i X ’s. If  is an n-dimensional convex body of constant width h, then  + (−) = h B. So in the above expression we can take Y = −, Z = , X = h B, and the remaining arguments are all equal to B, to obtain our desired conclusion using the fact that Wk (−) = Wk ().  We are now ready to prove the existence of mixed volumes. The following lemma is crucial, and its proof by induction on the dimension uses the approximation theorem of convex bodies by means of convex polytopes. Lemma 12.1.3 Let 1 , . . . , r be convex bodies of dimension less than or equal to n, and λ1 , . . . , λr ≥ 0 be nonnegative real numbers. Then there exists a unique homogeneous polynomial of degree n in the variables λ1 , . . . , λr for the volume V(λ1 1 + · · · + λr r ). Proof We begin by proving the lemma for the case when the convex bodies ∇1 , . . . , ∇r are convex polytopes. Let ∇ = λ1 ∇1 + · · · + λr ∇r . As every extreme point X ∈ ∇ is a linear combination of extreme points X i ∈ ∇i of the form X = λ1 X 1 + · · · + λr X r ,

12.1 Mixed Volumes

285

it follows that ∇ has at most a finite number of extreme points and is therefore a convex polytope. Let us express the volume of ∇ in terms of λ1 , . . . , λr . Since ∇ is a convex polytope, it has a finite number of (n − 1)-dimensional faces, which we assume to be contained in the hyperplanes H1 , . . . , Hk . Let h i be the distance from the origin to Hi , where h i is positive if ∇ is on the same side of Hi as the origin, and negative otherwise. It is not difficult to see that the volume of ∇ can be expressed as k 1 V (∇) = h j Vn−1 (H j ∩ ∇), n j=1

where V(n−1) denotes the (n − 1)-volume. The convex polytope H j ∩ ∇ can in turn be expressed as the following linear combination of convex polytopes: H j ∩ ∇ = λ1 (H1, j ∩ ∇) + · · · + λr (Hr, j ∩ ∇), where Hi, j is the support plane of ∇i parallel to H j with the property that ∇i is on the same side of Hi, j as ∇ is of H j . But h j can also be expressed as a linear combination hj =

r 

λi h ij ,

i=1

where h ij denotes the distance from the origin to Hi, j , positive if ∇i is on the same side of Hi, j as the origin, and negative otherwise. Using induction to express the (n − 1)-dimensional volume Vn−1 (H j ∩ ∇) of the convex polytope H j ∩ ∇ as a homogeneous polynomial of degree n − 1 in the variables λ1 , . . . , λr , Vn−1 (H j ∩ ∇) =

r  r 

···

i 1 =1 i 2 =1

r 

κ(∇i1 ,...,∇in−1 ) λi1 · · · λin−1 .

i n−1 =1

Combining this expression with the formulas V (∇) =

k 1 h j Vn−1 (H j ∩ ∇) n j=1

we get V (∇) =

r r  

···

i 1 =1 i 2 =1

r 

and

hj =

r 

λi h ij ,

i=1

κ(∇i1 ,...,∇in ) λi1 · · · λin ,

i n =1

as desired. j For 1 ≤ i ≤ n, let {∇i }∞ j=1 be a sequence of convex polytopes which converge to i with respect to the Hausdorff metric such that if λ1 , . . . , λn are n fixed reals. Then the sequence of polytopes {∇ j = λ1 ∇1 + · · · + λn ∇nj }∞ j=1 j

286

12 Mixed Volumes

converges, also with respect to the Hausdorff metric, to . Moreover, the respective series of volumes j j j ∞ j also converge. That is, {V (∇i )}∞ j=1 converges to V (i ) and {V (∇ ) = V (λ1 ∇1 + · · · + λn ∇n )} j=1 converges to V (). Thus we know that for j = 1, . . . , ∞, V (∇ j ) =

r  r 

···

i 1 =1 i 2 =1

r  i n =1

κ(∇ j ,...,∇ j ) λi1 . . . λin . i1

in

It can now be assumed that, without loss of generality, {κ(∇ j ,...,∇ j ) }∞ 1 i1

in

converges to a real number that we shall call κ(i1 ,...,in ) , such that V () =

r r   i 1 =1 i 2 =1

···

r 

κ(i1 ,...,in ) λi1 · · · λin .

i n =1



This concludes the proof of the lemma. Theorem 12.1.8 For every dimension n, there exists a unique mixed volume V : (Kn )n → E+ .

Proof We define the mixed volume V(1 , . . . , n ) as follows. Let 1 , . . . , n be n convex bodies (not necessarily distinct) of dimension less than or equal to n and let ζ be the coefficient of the term λ1 · · · λn in the polynomial V (λ1 1 + · · · + λn n ). We define V(1 , . . . , n ) =

ζ . n!

Note first that if  is a convex body, then V (λ1  + · · · + λn ) = V ((λ1 + · · · + λn )) = (λ1 + · · · + λn )n V (), from which it follows that ζ, the coefficient of the term λ1 · · · λn , is V ()n! and therefore V(, . . . , ) = V (). Let P1 (λ1 , . . . , λn ) and P2 (λ1 , . . . , λn ) be the homogeneous polynomials of degree n that express the volumes V (λ1 1 + · · · + λi i + · · · + λn n ) and V (λ1 1 + · · · + λi (λi ) + · · · + λn n ), respectively. Then λP2 (λ1 , . . . , λi , . . . , λn ) = P1 (λ1 , . . . , λλi , . . . , λn ), from which it follows that V(1 , . . . , λi , . . . , n ) = λV(1 , . . . , i , . . . , n ). Let r > n, and P(λ1 , . . . , λn , . . . , λr ) be the polynomial that expresses the volume V (λ1 1 + · · · + λn n + · · · + λr r ). Then, if 0 < λi1 < · · · < λin , it is easy to see, by setting the remaining variables to zero, that the coefficient of the term λi1 < · · · < λin in the polynomial P is given by n!V(i1 , . . . , in ). Now let P1 (λ1 , . . . , λi , ξi , . . . , λn ) be the polynomial that expresses the volume V (λ1 1 + · · · + λi i + ξi i + · · · + λn n ). Then n!V(1 , . . . , i + i , . . . , n ) is the coefficient of the term λ1 . . . , λn in the polynomial P1 (λ1 , . . . , λi , ξi , . . . , λn ). As a consequence, from the first part of the preceding paragraph it is evident that V(1 , . . . , i + i , . . . , n ) = V(1 , . . . , i , . . . , n ) + V(1 , . . . , i , . . . , n ).

12.1 Mixed Volumes

287

Finally note that V is symmetric by definition. Together, the above serves to prove that V, the given functional of convex bodies, is a mixed volume.  It is now important to calculate the mixed volume V(1 , . . . , n ) in terms of the volume of the Minkowski sum of its arguments. In dimension 2, we have that V(1 , 2 ) =

1 (V (1 + 2 ) − V (1 ) − V (2 )) , 2

because V (1 + 2 ) = V (1 ) + 2V(1 , 2 ) + V (2 ). In dimension 3, we know that V (1 + 2 ) = V (1 ) + 3V(1 , 1 , 2 ) + 3V(1 , 2 , 2 ) + V (2 ), and V (1 + 2 + 3 ) = V (1 ) + 3V(1 , 1 , 2 ) + 3V(1 , 1 , 3 ) + 3V(1 , 2 , 2 ) + 6V(1 , 2 , 3 ) + 3V(1 , 3 , 3 ) + 3V(2 , 2 , 3 ) + 3V(2 , 3 , 3 ) + V (2 ) + V (3 ).

Hence, 6V(1 , 2 , 3 ) = (V (1 + 2 + 3 ) − (V (1 + 2 ) + V (2 + 3 ) + V (1 + 3 )) + (V (1 ) + V (2 ) + V (3 ). Following these ideas, we have the following equation whose proof is left to the reader. Theorem 12.1.9 (Polarization formula) Let 1 , . . . , r be convex bodies of dimension less than or equal to n. Then V(1 , . . . , n ) =

n  1  (−1)n+ j V (i1 + · · · + i j ). n! j=1 i 0 is submerged. If we require that any rotated copy of , with its center having the same height as the surface of the fluid, should be in equilibrium, then the center of gravity of the part below the surface level of the liquid must be at constant depth. That is,  x,y≥0

f (x)4 |x, y| d x = c

for every y ∈ S2 , where c is a constant. See Section 5.4 and Exercises 5.17 and 13.6. Since x,y≥0 |x, y| d x = π for every y ∈ S2 , there is some number k such that  x,y≥0

[ f (x)4 − k] |x, y| d x = 0.

By the third theorem of spherical integration, this implies that f (x)4 − k is odd, but since f (x)4 − k  is even, we have f (x)4 = k a.e., making  spherical. Our next application of Theorem 13.3.1 is a characterization of bodies of constant brightness.

13.3.2 Convex Bodies of Constant Brightness We know from Theorem 13.1.1 due to Minkowski that if the orthogonal projection of a convex body  ⊂ E3 onto every plane is a convex figure of fixed perimeter P, then  has constant width P/π. Following this spirit, we say that a convex body  in E3 has constant brightness A if the orthogonal projection of  onto every plane is a convex figure of area A. By the Cauchy projection formula 12.3.2,

all bodies of constant brightness A have the surface area of the unit 3-dimensional ball of radius πA . Of course, there are nonspherical 3-dimensional convex bodies of constant brightness. The first example was given by Blaschke in [130]. This convex body is a body of revolution whose principal section is reminiscent of a Reuleaux triangle, although it is not a body of constant width. In fact, since that time, it had been conjectured that a convex body of constant width and constant brightness should be a ball. In 2006, Howard [557] succeeded in proving this conjecture. In this section, we will construct a nonspherical 3-dimensional convex body of constant brightness. Furthermore, under the hypothesis that the boundary is twice continuously differentiable we shall follow the proof of Nakajima (= Matsumara) [888] that a convex body of constant width and constant brightness is a ball. Our first goal is to characterize 3-dimensional convex bodies of constant brightness in terms of the Gaussian curvature.

13.3 A Third Theorem of Spherical Integration

311

A Characterization of Bodies of Constant Brightness Theorem 13.3.3 A convex body  ⊂ E3 has constant brightness A if and only if the sum of the reciprocal Gaussian curvatures at two points having parallel support planes is constant; that is, F(2) (v) + F(2) (−v) =

2A , π

for almost every v ∈ S2 . Proof For every unit vector u ∈ S2 denote by u the orthogonal projection of  onto a plane orthogonal to u. By the generalized Cauchy projection formula of Section 12.3, we have that 1 Area u = 4

 S2

  |u, v| F(2) (v) + F(2) (−v) dv.

So  has constant brightness A if and only if A=

1 2

 u,v≥0

  |u, v| F(2) (v) + F(2) (−v) dv,

for every u ∈ S2 . Since u,v≥0 |u, v| dv = π, we have that  has constant brightness A if and only if 

 2A  |u, v| F(2) (v) + F(2) (−v) − dv = 0, π u,v≥0

for every u ∈ S2 . Therefore, if F(2) (v) + F(2) (−v) − 2πA = 0, then clearly  has constant brightness A. On the other hand, if  has constant brightness A, then by the third theorem on spherical integration 13.3.1, we conclude that the function F(2) (v) + F(2) (−v) − 2πA is an odd function, which immediately implies that F(2) (v) + F(2) (−v) = 2πA almost everywhere.  There is a mysterious result regarding convex bodies of constant brightness proved by Berwald [102]. Suppose  ⊂ E3 is a convex body and for each u ∈ S2 let λ(, u) be the mean perimeter of all sections of  by planes orthogonal to u. Theorem 13.3.4 (Berwald) Let  ⊂ E3 be a convex body. The convex body  has constant brightness if and only if λ(, u) is constant. Blaschke’s Body of Constant Brightness We shall construct a convex body of revolution  which is a body of constant brightness π. As one can see in Figure 13.1, Blaschke’s body  has two parts; the lower part belongs to a ball B of radius √ 2 centered at (0, 0, 1). The upper part belongs to the body of revolution M constructed in Section 11.2.2 for the parameters c = 2 and b = 1. Let us recall the main properties of M. 1) M is a solid of revolution around the z-axis.  π/4 cos2 t 2) M has only two singular points at (0, 0, ±a), where a = 0 √22cos dt. 2 t−1 3) The unit normal vectors of M at (0, 0, a) are all those unit vectors whose angle from (0, 0, 1) is smaller than or equal to π/4.

312

13

Bodies of Constant Width in Analysis

Figure 13.1 Blaschke’s body of constant brightness

4) At any other boundary point of M the Gaussian curvature is 2. 5) M ∩ {z = 0} is the unit disk D of the {x, y}-plane centered at the origin. Furthermore, for every point u ∈ bd M ∩ {z = 0} the normal vector of M at u is u. Note now that for every point of the boundary of B the Gaussian curvature is 2. Furthermore, B ∩ {z = 0} = D. Define  = {(x, y, z) ∈ M | z ≥ 0} ∪ {(x, y, z) ∈ B | z ≤ 0}. The singular points of  are the points of the unit circle of the {x, y}-plane together with the point (0, 0, a). At any other boundary point of , the Gaussian curvature is 2. Consider γ : S2 → bd  to be the inverse Gauss map (see Section 11.1.2). Then, as is clear from the meridian section of  shown in Figure 13.2,   √ 1) γ −1 {(x, y, z) ∈  | z = a} = {(x, y, z) ∈ S2 | z ≥ 1/ 2},   √ 2) γ −1 {(x, y, z) ∈  | 0 < z < a} = {(x, y, z) ∈ S2 | 0 < z < 1/ 2},   √ 3) γ −1 {(x, y, z) ∈  | z = 0} = {(x, y, z) ∈ S2 | −1/ 2 ≤ z ≤ 0}, and   √ 4) γ −1 {(x, y, z) ∈  | z < 0} = {(x, y, z) ∈ S2 | z < −1/ 2}. √ From this it is clear that for every point u = (x, y, z) ∈ S2 such that z = ±1/ 2, F(2) (u) + F(2) (−u) = 2, and hence, by Theorem 13.3.3, it follows that every orthogonal projection of  has area π. Constant Brightness and Constant Width Theorem 13.3.5 A convex body  ⊂ E3 with twice continuously differentiable boundary has constant brightness and constant width if and only it is a ball. Proof Suppose  has constant width h and constant brightness A. For every u ∈ S2 , let ρ1 (u) and ρ2 (u) be the principal radii of curvature of  at P, where P is the point in the boundary of 

13.3 A Third Theorem of Spherical Integration

313

Figure 13.2 Section of Blaschke body

with unit normal u. Then F(2) (u) = ρ1 (u)ρ2 (u). By Theorem 11.3.1 2), ρ1 (−u) = h − ρ2 (u) and ρ2 (−u) = h − ρ1 (u), and therefore F(2) (−u) = (h − ρ1 (u))(h − ρ2 (u)). This implies that 2A = ρ1 (u)ρ2 (u) + (h − ρ1 (u))(h − ρ2 (u)), π for every u ∈ S2 . Let u 0 be a unit normal vector with the property that the point P with unit normal u 0 is umbilical. We know that such a vector exists. In fact, suppose the contrary. Then the direction where the larger principal radius of curvature is attained, defines two opposite unit vectors tangent to the unit sphere S2 . We can locally choose one of these unit vectors, in a continuous way. This extends continuously to the whole S2 , by simple connectedness of S2 . Thus we have a continuous field of tangent unit vectors on S2 . However, this is impossible. Therefore, h2 2A = ρ1 (u 0 )2 + (h − ρ1 (u 0 ))2 ≥ . π 2 Note that for every u ∈ S2 , the area A(u) of u is A and, by Barbier’s Theorem 12.1.3, the perimeter P(u) of u is πh. Thus, by the Isoperimetric Theorem 2.7.2 for the planar case 4π A(u) − P(u)2 = 0. Consequently, every orthogonal projection u is a disk and, therefore, by Exercise 13.11 the body  is a ball. 

13.3.3 Constant Outer k-Measure Bodies Convex bodies of constant outer k-measure are the natural generalizations of constant width bodies and constant brightness bodies. If k is an integer satisfying 1 ≤ k < n, then a convex body  ⊂ En is said

314

13

Bodies of Constant Width in Analysis

to have constant outer k-measure if all its orthogonal projections into k-dimensional subspaces of En have equal k-dimensional volume. So constant width bodies are constant outer 1-measure bodies, while constant brightness bodies are constant outer (n − 1)-measure bodies. In [359], Firey gave a method for constructing a nonspherical constant outer k-measure body for every 1 ≤ k < n. His method is very similar to the one we use to construct a 3-dimensional body of constant brightness, see [401] for more details. In fact, Firey also proved the following characterization of constant outer k-measure bodies. Theorem 13.3.6 A convex body  ⊂ En with C 2 boundary (see [401, p. 26]) has constant outer k-measure if and only if F(k) (v) + F(k) (−v) = c, for every v ∈ Sn−1 , where c is a constant. Here, F(k) (v) is the k-th elementary symmetric function of the principal radii of curvature at the boundary point with unit normal vector u. To be more precise, let p ∈ bd  be such that the normal vector of φ at p is v. Let u 1 , . . . , u n be the principal curvature directions and let ρi be the principal radii of curvature of  at p. Then the k-th elementary symmetric function is defined as 

 n − 1 (k) F (v) = k 1≤i 2, which is not centrally symmetric and whose convex intersection body (see [1099] for a definition) is a Euclidean ball. Constant Girth Not directly connected with our framework of cross-section measures, but nevertheless closely related are the so-called bodies of constant girth and of constant k-girth. This topic has to do also with Minkowski’s projection theorem discussed already in our Section 13.1. A more general and broader discussion is given in § 3.3 of Gardner’s book [401]. If for 1 ≤ k ≤ n − 1 and a convex body K in En the value Vk (K |u ⊥ ) is constant for all directions u, where Vk denotes the k-th intrinsic volume of K |u ⊥ , then K is said to be of constant k-girth (this definition is due to Chakerian [231], yielding for k = n − 2 the usual notion of constant girth). It was shown in [231] that under certain smoothness assumptions every body of constant outer k-measure must have constant k-girth, and Firey [359] could remove these boundary restrictions. The converse seems to be open, although Firey [359] confirmed it for bodies of revolution. And Theorem 3.3.13 in [401] says that a convex body in En is of constant width iff it is of constant 1-girth. Extending a result of Aleksandrov from [12], Chakerian and Lutwak use in [240] also a generalization of the notion of constant girth. And in [744] bodies of constant girth occur as extremal cases in inequalities referring to quermassintegrals of mixed projection bodies. It is natural to ask the following: does a convex body in 3-space have almost constant width if the perimeter of its projections is nearly constant? Goodey and Groemer [441] treat this stability problem associated with Minkowski’s projection theorem and extend their results also to the n-dimensional situation. Combinations To combine different projection and/or section functions, we have the possibility to stay only with one type (e.g., only with projection functions), or to combine section functions with projection functions.

13.3 A Third Theorem of Spherical Integration

319

For both cases, the famous Nakajima problem (see [887]) is a good starting point: Are the balls the only convex bodies in E3 being of constant width and of constant brightness? For related observations and problems we refer also to the paper [233]. For convex bodies with twice continuously differentiable boundary, this was confirmed by Matsumura (see § 15 in [160], page 82 in [238], A10 in [272], and Theorem 3.3.20 in [401]; Chakerian [231] proved an analogue for 3-dimensional Minkowski spaces, and recently Stepanov [1098] gave an alternative proof of the original statement). These boundary conditions could be deleted by Howard [557]; he confirmed this nice ball characterization in E3 in full generality and gave also extensions of this result. Continuing this, the authors of [559] proved the following interesting result: let K and L be convex bodies in En , and let L be centrally symmetric and satisfy a weak regularity and curvature condition. Assume that K and L have proportional first and k-th projection functions, where 2 ≤ k < (n + 1)/2 or k = 2, n = 5. Then K and L are homothetic. Assuming that L is a Euclidean ball, one thus obtains characterizations of Euclidean balls as convex bodies of constant width and constant brightness. Another strong generalization in this direction was proved in [558]: in En , let L be a centrally symmetric convex body having C+2 boundary, and K be a convex body with C 2 support function. Excluding the two exceptional cases (i, j) = (1, n − 1) and (i, j) = (n − 2, n − 1), it is shown that K and L are homothetic if their i-th and j-th projection functions are proportional. The author of [500] investigated whether a convex body of class C+2 in En with constant i-brightness and constant j-brightness is a ball for i < j. Unfortunately, his approach contains gaps. Hug [564] proved, even more general, that if K is a convex body in En and i, j are two integers with 1 ≤ i < j ≤ n − 2 such that (i, j) = (1, n − 2), then the constancy of the i-th and j-th projection function implies that K is a ball. This is only a partial result from [564], more general ones on homothety classes of convex bodies are derived there. Regarding homothety classes of convex bodies and projection functions we also refer to § 3.1 of [401]. Related to these results, in [442] and [443] (the first paper being a survey) it is investigated to what extent convex bodies are determined by their projection functions, also with respect to intermediate dimensions i. If K , L are centrally symmetric convex bodies and at least one has dimension larger than i, then already Aleksandrov [12] showed that equality of both i-dimensional projection functions implies that L is a translate of K . In [442] and [443] such investigations are continued. For general convex bodies, clearly the i-dimensional projection volume cannot distinguish between K and any translate or reflection of it. First, most (in the sense of Baire category) convex bodies are determined by any single i-projection function, with i going from 2 to n − 1. Second, there is a dense family of convex bodies whose members are not determined by knowing all the i-th projection functions with i going from 1 to n − 1. And it is proved that, in the usual Baire category sense, most convex bodies are determined, up to translation or reflection, by the combination of their widths and brightnesses in all directions. Obviously, in all these considerations the subcases referring to bodies of constant i-measures are directly interesting for our purpose here. Bringing also section functions into the game, we mention once more: if Vn−1 (K |u ⊥ ) and maxVn−1 (K ∩ (u ⊥ + tu)) are both constant for all directions u, then K is a ball centered at the origin, see [755]. The 1-dimensional analogue says that if the chordal symmetral and the difference body of a convex body K are centered balls, then K is a centered ball (see again [755]). Perhaps the little field of combining suitably projection and section functions still hides many interesting questions. And we remark that also some of the problems posed at the end of Chapter 8 of Gardner’s book [401] are closely related to the philosophy of this subsection here. Methods From the viewpoint of methods used for studying projection and section functions in the way described above, especially the books [461], [1039], and [401] are fundamental. In 1998, Koldobsky [649] found a general Fourier transform formula which could be used to provide a unified approach to several geometric problems of the type discussed here. In this booklet, he recalls the basic methods of Fourier analysis and convex geometry, and some facts concerning Radon transforms and spherical harmonics,

320

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Bodies of Constant Width in Analysis

and then applies them to the relevant geometric problems, e.g., related to volumes of sections of convex bodies. We refer also to [650] with similar aims. A related work from 2014 is [992], similarly explaining analytical methods needed for the study of projection and section functions in convexity, also with many exercises. More general, we refer here also to various surveys collected in Part 4 (Analytical aspects) of the Handbook [483].

Exercises 13.1.

13.2. 13.3*.

13.4*. 13.5**. 13.6**.

13.7. 13.8.

13.9*. 13.10.

13.11. 13.12. 13.13*. 13.14*.

Prove that the following assertion is equivalent to the second  theorem of spherical integration 13.1.3. Let w : S2 → E be an even function. Then x,y=0 w(x)d x is a constant function if and only if w is a constant function. Prove that x,y≥0 |x, y| d x = π, for every y ∈ S2 . Let F be a continuous function on Sn−1 and let e be the function such that e (x) = 1, for |x| ≤ and e (x) = 0 elsewhere. Prove that   1 F(v)dv = lim e(u, v)F(v)dv. →0 2 Sn−1 u,v=0  n−3 Prove that lim →0 1 0 Pmn (t)(1 − t 2 ) 2 dt = Pmn (0). Prove the first spherical integration theorem 13.1.2 using the Funk–Hecke estimation (13.7) for a suitable function f . Let  be a convex body of uniform density one half that floats in equilibrium in every position. Assume that  is centrally symmetric with respect to the origin and let  be defined by the radial function f : S2 → E. Hence the center of gravity of the part below the surface level must be at constant depth. Prove that x,y≥0 f (x)4 |x, y| d x is constant, for every y ∈ S2 . Prove that all 3-dimensional bodies of constant brightness A have surface area 4 A. Why does Theorem 13.3.3 not imply that the body of revolution M, constructed in Section 11.2.2, with constant Gaussian curvature for almost all p ∈ bd M is a body of constant brightness? Let  ⊂ En be a convex body such that the condensed harmonic expansion of its support function P2 (u) ∼ Q 2 + 1. Prove that P (u) is the support function of an ellipsoid. A Blaschke body of constant brightness was constructed from a nonspherical body of revolution with constant curvature at regular points given in Section 11.2.2 for the parameters c = 2 and b = 1. Is it possible to use another set of parameters to construct a different body of constant brightness? Prove that if every orthogonal projection of an n-dimensional convex body  onto a plane H is a disk, then  is a ball. Is a centrally symmetric body of constant brightness always a ball? Construct a nonspherical constant outer k-measure body, for 1 < k < n. Suppose that  is an n-dimensional convex body of outer k-measure for k ∈ X ⊂ {1, . . . , n − 1}. Give conditions on X under which a convex body  with sufficiently smooth boundary conditions is a ball.

Chapter 14

Geometric Inequalities

Inspiration is needed in geometry, just as much as in poetry. A. Pushkin

14.1 Isoperimetric Inequalities Our first isoperimetric inequality is the following (see Theorem 2.7.1): For every plane convex body φ ⊂ E2 of area A and perimeter P we have P 2 − 4 Aπ ≥ 0, and equality holds only for 2-dimensional disks. The proof follows from the Brunn–Minkowski Theory in the plane, which claims that the function    f (t) = A (1 − t)φ + tψ is a concave function, where A(φ) denotes the area of a convex body φ and t ∈ [0, 1]. As an immediate consequence we have: Of all plane convex bodies of perimeter P > 0, the one that maximizes the area is the unit disk of diameter P/π. In fact, Bonnesen [158] obtained the following generalization of this inequality in terms of the circumradius R and the inradius r , see [902] for a proof: P 2 − 4 Aπ ≥ π 2 (R − r )2 .

(14.1)

Let us now consider convex bodies of constant width. Let W 2 be the set of all plane convex figures of constant width 1. Then, by Barbier’s Theorem 12.1.3, for every φ ∈ W 2 the perimeter P(φ) of φ is equal to π. Furthermore, the area A(φ) satisfies the inequality √ π− 3 2

A(φ)

π , 4

and equality holds precisely for the Reuleaux triangle and the unit disk, respectively.

© Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_14

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14 Geometric Inequalities

The first inequality is known as the Blaschke–Lebesgue Theorem 12.1.5 which claims: Of all plane convex bodies of a fixed constant width the one that minimizes the area is the Reuleaux triangle. For 3-dimensional convex bodies, we start with the following extremal theorem. Theorem 14.1.1 Of all 3-dimensional convex bodies with diameter h, the ball is the one with maximum surface area. Proof Let φ be a 3-dimensional convex body with diameter h and surface area S(φ). First we prove that the surface area of φ is smaller than or equal to the surface area of the ball with diameter h; that is, S(φ) ≤ πh 2 . By Theorem 7.2.2 of Pál, there exists a body  of constant width h that contains φ. Because the area is a monotone functional when defined on convex bodies, we have that S(φ) ≤ S(). As before, we denote by u the figure that is obtained as the orthogonal projection of  on a plane perpendicular to the unit vector u. Let P(u) and A(u) be the perimeter and the area, respectively, of u . The isoperimetric inequality for plane figures tells us that P(u)2 ≥ 4π A(u), for all u ∈ S2 . Also, by Theorem 3.3.1, u is a figure of constant width h, and by Barbier’s Theorem 12.1.3, P(u) = πh. It follows that πh 2 ≥ 4 A(u), for all u ∈ S2 . We consider the Cauchy formula (see Theorem 12.3.2) for the surface area S(φ) in terms of A(u), the area of its projections  1 A(u)du = S() . π S2  1 This can also be interpreted as 4 A = S(), where A = 4π S2 A(u)du is the mean of all the areas A(u) with u ∈ S2 . Together, this implies that πh 2 ≥ 4 A = S() ≥ S(φ). Suppose now that S(φ) = πh 2 . Then, by the above inequality, φ has constant width h, and the mean A of the areas A(u) of its projections is πh 2 . Therefore, the perimeter P(u) of all the projections is πh and, moreover, P(u)2 = 4π A(u) for every u ∈ S2 , which implies that all the projections of φ are circular disks, thus leading to the conclusion that φ is a ball (see Exercise 13.11).  Let W 3 be the set of all convex bodies of constant width 1, and for  ∈ W 3 , denote by V () and S() the volume and the surface area, respectively. Then, by Blaschke’s Theorem 12.1.4, for every  ∈ W 3, S() π − . V () = 2 3 Furthermore, the volume V() satisfies the inequality π √ π (3 6 − 7) ≤ V () ≤ , 3 6 and consequently the surface area S() satisfies the corresponding inequality √ 2π( 6 − 2) ≤ S() ≤ π.

14.1 Isoperimetric Inequalities

323

The first inequality follows from the following theorem of Chakerian [230]: Theorem 14.1.2 (Chakerian) Let  be a 3-dimensional body of constant width 1. Then √ π √ (3 6 − 7) ≤ V () and 2π( 6 − 2) ≤ S(). 3 Proof√ Let R be the radius of the circumsphere of . Then, by Theorem 3.4.2, R ≤ 3/8. By Theorem 3.4.1, we know that the insphere and the circumsphere of  are concentric and, furthermore, √ the sum of the radii is 1. Then, if r is the radius of the circumsphere of , we have that r ≥ (1 − 3/8). Let P : S2 → R be the support function of . Then, by Equation (12.2) of Section 12.2,  3V () =

S2

P(u)F(2) (u)du,

where F(2) (u) is the inverse √ of the Gaussian curvature. Since P(u) ≥ (1 − 3/8), we obtain that       F(2) (u)du = 1 − 3/8 S(), 3V () ≥ 1 − 3/8 S2

and hence, by Theorem 12.1.4 of Blaschke,     3V () ≥ 1 − 3/8 2V () + 2π/3 , √ √ which gives the inequalities π3 (3 6 − 7) ≤ V () and 2π( 6 − 2) ≤ S().

 √ We do not know a convex body  of constant width 1 for which V () = π3 (3 6 − 7). In contrast and following the spirit of the Blaschke–Lebesgue Theorem, in which the Reuleaux triangle minimizes the area among figures of given constant width, Bonnesen and Fenchel [160] conjectured that the original Meissner bodies of Section 8.3 minimize the volume among all 3-dimensional convex bodies of given constant width.

14.2 The Blaschke–Lebesgue Problem The Blaschke–Lebesgue problem asks for the bodies minimizing the volume in the class of constant width bodies of fixed width. As we know, the Reuleaux triangle is the unique solution in dimension 2 and the problem is still open in dimensions n ≥ 3. In fact, Bonnesen and Fenchel had the following conjecture. √ Conjecture The original Meissner bodies of volume 2/3 − 3/4 cos−1 (1/3) minimize the volume among all 3-dimensional convex bodies of constant width 1. It is interesting to note that for width 1, the volume of the rotated Reuleaux triangle (approximatelly 0.4494 . . . ) is greater than the volume of the two original Meissner bodies (approximatively 0.4198 . . . ) which in turn is much greater than the Chakerian bound for the volume (approximatively 0.3648 . . . ). There was the belief that the body that minimizes the volume among all 3-dimensional bodies of constant width must have the symmetry group of a regular tetrahedron. This belief was first expressed

324

14 Geometric Inequalities

by Danzer in the 1970s. Of course, this is not the case for the two original Meissner bodies M1 and M2 , but the Minkowski sum 21 M1 + 21 M2 has the symmetry group of a regular tetrahedron. This body has constant width 1, but its volume is greater than that of the original Meissner bodies, due to the Brunn–Minkowski inequality 2.7.4. It is important to identify necessary conditions that a constant width body of minimal volume and given fixed width must satisfy. The existence of such a body follows from the direct method in the calculus of variations and the Blaschke Selection Theorem 2.5.2. One of these conditions is that the boundary cannot be twice continuously differentiable. Suppose that  is a body of constant width 1 whose boundary is twice continuously differentiable and which has the property that ξ = V () minimizes the volume among all 3-dimensional convex bodies of constant width 1. We will show that this leads to a contradiction. There is  > 0 sufficiently small with the property that the ball B(o, ) slides freely inside . This means that for every point p in the boundary of  there is a translated copy B of B(o, ) such that p ∈ B ⊂ . Let  be the set of all points x such that x + B(o, ) ⊂ . Clearly,  is a convex body with the property that  + B(0, ) =  =  (see Section 2.4). Therefore, if V and S are the volume and surface area of , respectively, we have that: 1) 2) 3) 4)

 has constant width 1 − 2, and therefore V < π/6; 3S(1 − 2) − 6V = 2π(1 − 2)3 , by the Theorem 12.1.4 of Blaschke; ξ = V + S + M2 + C3 , by the Steiner formula (2.5); and 1 V ) = (1−2) V ( 1−2 3 ≥ ξ, by minimality of ξ = V (). Hence, 2) and 3) imply that ξ = V + (2V + 2π/3) + O(2 ), and therefore, by 4), V ≤ V + (4V − 2π/3) + O(2 ).

By 1), for  sufficiently small this is a contradiction. In [86], Bayen, Lachand–Robert and Oudet derive a necessary condition that a convex body must satisfy in order to minimize volume among all 3-dimensional convex bodies of constant width h. If one squeezes such a body between two parallel planes, at one of the two points of tangency its surface is not smooth or regular. Clearly, Meissner’s bodies satisfy this condition. Indeed, another necessary condition that a convex body of minimal volume and constant width h must satisfy is that the smooth parts of its boundary are spherical caps or pieces of tubes, both of them with radius equal to h, just as in the case of Meissner polyhedra described in Section 8.3. Here smooth means twice continuously differentiable. To achieve this result, we shall prove that the smooth part of the boundary of a minimizer of the Blaschke–Lebesgue problem has its smaller principal curvature constant and equal to 1/ h. We present here the proof of Anciaux and Guilfoyle [31] given in 2010. For this purpose, since the proof uses variational methods, we need first to introduce some notation and preliminaries from Section 11.5. For a function η : Sn−1 → E which admits first derivatives that are Lipschitz continuous, define  E(η) :=

Sn−1



 1 |∇η|2 − η 2 d A. n−1

The Wirtinger inequality claims that E(η) ≥ 0. This inequality can be easily proved using the theory of spherical harmonics (see [483], p. 1288, or Section 4.4 of [464]). We consider E(η) ≤ 1. J (η, h) = 1 − 4πh 2 /3

14.2 The Blaschke–Lebesgue Problem

325

If  is a body of constant width 2h and P is its support function, then η = P − h is an odd function that admits first derivatives which are Lipschitz continuous and has the property that the h 3 − hE(η). volume of  depends only on h and E(η). Indeed, by Theorem 11.5.5, V () = 4π 3 Let us introduce the radius J () =

V () E(η) V () = =1− = J (η, h). 3 V (B(0, h)) 4πh /3 4πh 2 /3

It follows from the Wirtinger inequality that the ratio J () = J (η, h) is less than or equal to 1, and equality is attained when η is a first eigenfunction of the Laplacian, as it is in the case of balls. Moreover, for a given η, J increases with respect to h. Hence it reaches its minimum at the lowest value of h such that η + h is the support function of a convex body; we define h 0 (η) to be this crucial quantity. Increasing (respectively decreasing) the value of h corresponds geometrically to flowing the boundary of  parallel to itself; i.e., along its outward (respectively inward) normal vector. Therefore the map η corresponds to a one-parameter family of parallel surfaces labeled by the parameter w ∈ [w0 (η), ∞). The inward normal flow can be continued as long as the surface is smooth. By Theorem 11.5.3, this is equivalent to the fact that the area element d A¯ is strictly positive. Hence, we deduce an explicit expression for w0 (η) h 0 (η) = inf{h ∈ E+ | h 2 + αh + β > 0 a.e. on S2 }, where α = 2η + η and β = η 2 + ηη + H (η). The convex body  of constant width whose support function is P = η + w0 (η) is always singular. One can check that w0 (η) = W (η) L ∞ (S2 ) , where W (η)(u) =

−α +

 α2 − β . 2

The directions u of S2 where the area element vanishes correspond precisely to points γ(u) of the boundary which are singular, where γ : S2 → bd  is the inverse Gauss map. The next theorem shows that in the smooth parts of a local minimizer of J such a situation actually occurs for every pair of antipodal directions (u, −u). The following lemma was also independently obtained in [86]. Lemma 14.2.1 Let (η, w0 (η)) be a local minimizer of J (η, h) and let U be an open subset of S2 where η is smooth. Then for every point u of U , the area element d A¯ vanishes at one of the points u and −u. Proof The proof is by contradiction. Assume that there is an open subset U of S2 where η is smooth and such that (w0 (η))2 + αw0 (η) + β > 0 in U ∪ (−U ). Consider a smooth map v such that v(u) + v(−u) = 0 for every u ∈ S2 , and whose support is contained in U ∪ (−U ), and define the deformation η  = η + v of η. For small , w0 (η  ) = w0 (η); hence

E(η) δE(η, v) 2 δ 2 E(η, v) E(η  ) = 2 + 2 + + o(2 ). 2  2 w02 (η) w0 (η ) w0 (η) w0 (η)

As η is a minimizer of J , and thus a maximizer of E(η)/w02 (η), we must have both δE(η, v) = 0 and δ 2 E(η, v) ≤ 0. On the other hand, the functional E is quadratic, so δ 2 E(η, v) = E(v), which is positive by the Wirtinger inequality. Finally, the support of v being contained in U ∪ (−U ), v cannot be an eigenfunction of the Laplacian. This gives us the required contradiction. 

326

14 Geometric Inequalities

Theorem 14.2.1 Let  be a convex body that minimizes volume among all 3-dimensional bodies of constant width 2h. Then the smooth part of its boundary has its smaller principal curvature constant and equal to 1/2h. Proof Assume that  is a local minimizer of J () and let η be the associated map. For the sake of brevity we set w˜ := w0 (η) in the following. Let U be an open subset of S2 such that γ(U ) is a smooth part of bd . In particular, by Theorem 11.5.3, w˜ 2 + αw˜ + β > 0 on U . Hence, by Lemma 14.2.1, w˜ 2 + α(u)w˜ + β(u) = 0, for every u ∈ −U . Since α is odd and β is even, it follows that w˜ 2 − α(u)w˜ + β(u) = 0, for every u ∈ U . Consequently, by Theorem 11.5.4, κ1,2

 √ 2w˜ + α ± α2 − 4αw˜ + 4w˜ 2 2w˜ + α ± α2 − 4β = = 2(w˜ 2 + αw˜ + β) 2(w˜ 2 + αw˜ + (αw˜ − w˜ 2 )) 2w˜ + α ± |α − 2w| ˜ = , 4αw˜

and so κ1 = α1 and κ2 = 21w˜ . This implies that the principal curvature κ2 is constant on U and equal to the inverse of the width 2w˜ of . Finally, since α(u) ≥ 0, we have 2w˜ = −α(−u) +

  α2 (−u) − 4β(−u) = α(u) + α2 (u) − 4β(u) ≥ α.

This concludes the proof of the theorem because κ1 = curvature.

1 α

≥ κ2 =

1 ; 2w˜

that is, κ2 is the smaller 

Shiohama and Takagi [1061] proved that a nonspherical smooth surface with its smaller principal curvature constant must be a piece of a tube; that is, the envelope of a one-parameter family of spheres, or, equivalently, the set of points which lie at a fixed distance from a curve. So, as a corollary of Theorem 14.2.1, we have the following statement. Corollary 14.2.1 Let  be a convex body that minimizes the volume among all 3-dimensional bodies of constant width h. Then the following two equivalent conditions hold: • The smooth parts of its boundary are spherical caps or pieces of tubes, both of them with radius equal to h. • At least one of the endpoints of every diameter of  is not a regular point of the boundary. For the original Meissner bodies, the smooth parts of their boundaries have their smaller principal curvature constant, because the smooth parts of the boundary are spherical caps or surfaces of rotation of arcs of the circle. Therefore, we cannot discard the possibility that the original Meissner bodies minimize the volume among all 3-dimensional bodies of given constant width. Another interesting constant width body is the one obtained from the Reuleaux triangle by rotation about one of its axes of symmetry. Related to the Blaschke–Lebesgue problem, Campi, Colesanti, and Gronchi [209] obtained the following result. Theorem 14.2.2 The rotated Reuleaux triangle minimizes the volume among all 3-dimensional rotational convex bodies of given constant width. The idea is that the rotated Reuleaux triangle minimizes the ratio J among constant width bodies with rotational symmetry. It is interesting to note that this body satisfies our criteria as well. Indeed, one part of its boundary is a spherical cap and the other is a tube over an arc of a circle. However, it has a larger ratio J than the Meissner bodies, which in particular proves that the solution of the Blaschke–Lebesgue problem does not have rotational symmetry.

14.2 The Blaschke–Lebesgue Problem

327

After all this discussion we can see that the most difficult issue to be addressed seems to be the regularity. We cannot a priori exclude that the boundary of the minimizer of the Blaschke–Lebesgue problem is singular everywhere. On the other hand, assuming that the minimizer is made up of a finite number of smooth parts, Corollary 14.2.1 reduces the problem to a combinatorial one: Minimize the volume among the convex bodies whose boundaries are made of spherical caps and pieces of tubes, all of them of the same radius, in particular, those obtained from a Reuleaux polyhedra by performing surgery. We refer to the discussion of Meissner polyhedra in Section 8.3.4. Instead of constant width let us consider constant brightness. Recall that in Section 13.3.2 a convex body of revolution ⊂ E3 was constructed whose orthogonal projections onto 2-dimensional subspaces all have area equal to π. It has been conjectured that this body of constant brightness due to Blaschke minimizes volume among all 3-dimensional convex bodies of constant brightness π. Indeed, Gronchi [468] proved the following partial result. Theorem 14.2.3 Blaschke’s body of constant brightness minimizes the volume among all 3-dimensional rotational convex bodies of given constant brightness. Recall that, by the Projection Formula of Cauchy (see Theorem 12.3.2), all constant brightness bodies of given brightness have the same surface area.

14.3 Measures of Asymmetry For a convex body , let R() and r () be the radius of the circumsphere and the radius of the insphere, respectively. The quotient R() J˜ () = r () is our first and simplest measure of the asymmetry of . Theorem 14.3.1 The original Meissner bodies maximize the index of asymmetry J˜ among all 3-dimensional bodies of fixed constant width. Proof In fact, every body  of constant width h that contains the regular tetrahedron of side h maximizes the index of asymmetry J˜ . Let us first prove that for every body  of constant width h the inequality √ 3 + 2 6 J˜ () ≤ 5 holds. √ The reason is that by Theorem 3.4.1, h = R() + r (), and by Jung’s Theorem 2.8.4, R() ≤ h √3/8. On the other hand, since the circumsphere of the tetrahedron of edge-length h has radius h 3/8, for every body  of constant width h that contains the regular√tetrahedron of edge-length h √ we have R() = h 3/8 and, therefore, for those bodies, J˜ () = 3+25 6 .  It is in this sense that the original Meissner bodies are more slender and should have a smaller volume than other bodies of constant width. Given a point x in the interior of a convex body φ, for a hyperplane H through x and a pair of support hyperplanes H1 and H2 , parallel to H , let γ(H, x) be the radius, not less than 1, into which H divides the distance between H1 and H2 . We write γ(φ, x) = max{γ(H, x) | x ∈ H },

328

14 Geometric Inequalities

and we define the Minkowski measure of asymmetry of φ, as introduced by Grünbaum in [487], by M(φ) = min{γ(φ, x) | x ∈ intφ}. A point x ∈ int φ satisfying γ(φ, x) = M(φ) is called a critical point. The set of all critical points of φ is denoted by φ∗ . It is known that the set of critical points φ∗ is a non-empty convex body. In fact, it is not difficult to prove that φ∗ is a singleton whenever φ is strictly convex (see Exercise 14.3). A comprehensive discussion of Klee’s critical set is given in Chapter 2 of the monograph [1135]. There is an equivalent definition of the Minkowski measure of asymmetry. Again, let x be a point in the interior of a convex body φ. For a chord l of φ through x, let γ1 (l, x) be the radius, not less than 1, into which x divides the length of l, and write γ1 (φ, x) = max{γ1 (l, x) | x ∈ int φ}. Then the Minkowski measure can also be defined as M(φ) = min{γ1 (φ, x) | x ∈ int φ}. In [628], Klee proved. Theorem 14.3.2 For any n-dimensional convex body φ, 1 ≤ M(φ) ≤ n. Equality holds on the left-hand side precisely if φ is centrally symmetric, and on the right-hand side precisely if φ is a simplex. In general, the insphere and circumsphere of a convex body are not concentric; however, as we know, if the convex body  has constant width h, then the centers of the two spheres of radius r () and R(), respectively, coincide. Moreover, we have r () + R() = h. Jin and Guo [591] proved the following. Theorem 14.3.3 For a body  of constant width h, the circumcenter is the critical point of . Furthermore, J˜ () = M(). In fact, r () =

M() 1 h and R() = h. 1 + M() 1 + M()

Corollary 14.3.1 For an n-dimensional body  of constant width h, 1 ≤ M() ≤

n+

2n(n + 1) . n+2

Equality holds on the left-hand side for the ball, and the upper bound is satisfied by those bodies that contain the regular n-dimensional simplex of side h.

14.4 Inequalities Involving the Circumradius We know from Theorem 7.2.2 of Pál that every convex body φ of diameter h can be completed to a convex body of constant width h. In this section we shall show that such a convex body  of constant width h does indeed exist, but with the additional property that  is contained inside the circumsphere of φ. This fact, first proved by Scott [1052], enables us to obtain several interesting isoperimetric inequalities involving the circumradius.

329

Theorem 14.4.1 Let φ ⊂ En be a convex body of diameter h and let C be its circumsphere. Then there is a convex body  of constant width h such that φ ⊂  ⊂ B, where B is the ball whose boundary is C. Proof Let U (φ) be the compact set of all points of the ball B with the property that the diameter of {x} ∪ φ is smaller than or equal to h, and let ρ(φ) be the maximal distance of points of U (φ) from φ. Choose a point x1 ∈ U (φ) which has distance ρ(φ) from φ and define φ1 = cc({x1 } ∪ φ). Then φ1 is a compact convex set of diameter h and φ ⊂ φ1 ⊂ B. We now repeat the process for the set φ1 and continue to obtain a nested sequence of convex bodies {φi } converging to a convex limit set. If  denotes the closure of this limit set, then  is a convex body of diameter h satisfying φ ⊂  ⊂ B. By Theorem 7.2.1, it will be enough to prove that  is complete. If  is not complete, then let y be a point such that y ∈ /  and the diameter of {y} ∪  is h. Let δ > 0 denote the distance from y to . The proof has two cases; the first case when y ∈ B, and the second case when y ∈ / B. In the first case, let xi , x j , i < j, be two of the points selected in the construction of . Then xi ∈ φi and φi ⊂ φ j−1 . Since the distance from x j to φ j−1 is ρ(φ j−1 ), then   xi − x j  ≥ ρ(φ j−1 ). We now show that ρ(φ j−1 ) ≥ δ. Since the diameter of {y} ∪  is h and φ j−1 ⊂ , it follows that the diameter of {y} ∪ φ j−1 is smaller than or equal to h and y ∈ U (φ j−1 ). Also, since φ j−1 ⊂ , the distance from y to φ j−1 is not less than δ. Then ρ(φ j−1 ) ≥ δ, because, by construction, ρ(φ j−1 ) is the maximal distance of points of U (φ j−1 ) from φ j−1 . But since all the points xi lie in B, it is impossible for any two of them to be at distance greater than or equal to δ from each other. For the second case, let us assume that y ∈ / B. Suppose, without loss of generality, that the origin is the center of B. From the above argument, we deduce that |y| = R + δ, where R is the circumradius. Choose z on the ray starting at y and passing through the origin in such a way that |z − y| = h + δ. Clearly, z ∈ /  and z ∈ B, since |z| = (h + δ) − (R + δ) = h − R ≤ R. If we take the ball B(z, h), then  ⊂ B ⊂ B(z, h), and therefore z is a point in B which can be added to  without increasing its diameter. By our first case, this is impossible. This completes the proof of the second case.  Corollary 14.4.1 Of all convex bodies having given diameter and circumradius, the body having greatest width, surface area, or volume is a body of constant width. Proof Given any body of diameter h and circumradius R, completing the body as in Theorem 14.4.1 will leave h and R unchanged, and will not decrease width, surface area, inradius, or volume.  Theorem 14.4.2 Let φ ⊂ En be a convex body of diameter h, thickness , circumradius R, and inradius r . Then  √ √  √ • (2R − h)h ≤ 2 n + 1/n 2n − n + 1 R 2 ,  √ √  √ • (2R − h) ≤ 2 n + 1/n 2n − n + 1 R 2 , and √  √  √ • (2R − h)r ≤ 2/n 3 n(n + 1) − 2(2n + 1) R 2 .

330

14 Geometric Inequalities

Proof To establish the first inequality, we observe that (2R − h)h is a decreasing function of h for fixed R. By Jung’s Theorem 2.8.4 we have √ √ h ≥ ( 2n + 2/ n)R, and hence the function takes its maximal value when h assumes this lower bound. By Corollary 14.4.1, we may assume that φ is a body of constant width h. For such bodies we have  = h, and therefore the second inequality follows immediately from the first. To establish the third inequality, we observe, by Theorem 3.4.1, that for any body of constant width√ h, r + R √ = h. Since (2R − h)(h − R) is a decreasing function of h for fixed R, substituting  h = ( 2n + 2/ n)R gives the required result. Remarks 14.4.1 When φ is a regular simplex, we obtain equality in the first inequality of the previous theorem, and equality is obtained in the second and third inequalities for any body of constant width  which contains a regular simplex of diameter . Theorem 14.4.3 Let φ ⊂ E2 be a convex figure of diameter h, area A, perimeter P, circumradius R, and inradius r . Then √ • (2R − h)P ≤ (2√ 3 − 3)π R 2 , • (2R − h)A ≤ (3 3 − 5)π R 3 , where equality in the first inequality is obtained when φ is a set of constant width. Proof As in the proof of Theorem 14.4.2, we may assume that φ is a set of constant width h. So the perimeter P = πh. Then the first inequality of this theorem follows immediately. For the second inequality, using the Bonnesen inequality (14.1), p 2 − 4π A ≥ π 2 (R − r )2 , and substituting P = πh and h = R + r , we obtain A ≤ π Rr. Now, using the third inequality of Theorem 14.4.2, we have √ (2R − h)A ≤ π R(2R − h)r ≤ (3 3 − 5)π R 3 , as we wished.



14.5 A Bonnesen-Type Isoperimetric Inequality The purpose of this section is to state a Bonnesen-type isoperimetric inequality for convex figures involving the Steiner disk. With every convex body φ ⊂ En one can associate several points that are of interest; one of them is the Steiner point of φ. This point may be regarded as the “curvature centroid” because under suitable smoothness conditions it can be defined as the center of mass of bd φ with respect to a density function that assigns to each point of bd φ the Gaussian curvature at that point. However, the Steiner point of φ can be defined without smoothness conditions as follows:  1 z(φ) = Pφ (u) · u du, κn Sn−1

14.5 A Bonnesen-Type Isoperimetric Inequality

331

integrating with respect to the Lebesgue measure on Sn−1 and the obvious extension of the Riemann integral to a vector-valued function, where κn denotes the volume of the n-dimensional unit ball. In particular, if n = 2 and if, as usual, u = (cos θ, sin θ), the Steiner point is z(φ) = (

1 π



π

Pφ (θ) cos θdθ,

−π

1 π



π −π

Pφ (θ) sin θdθ).

The ball whose center is the Steiner point and whose diameter is the mean width of φ will be called the Steiner ball of φ and denoted by Bz (φ). If n = 2, we call it the Steiner disk of , and its support function is given by 1 PBz (φ) (u) = w(φ) ˜ + z(φ), u, (14.2) 2 π where w(φ) ˜ = π1 −π Pφ (θ)dθ is the mean width of φ. Recall from Section 2.5 that given two convex bodies ψ, φ ⊂ En , we define the Hausdorff distance d(ψ, φ) as the infimum of all  > 0 such that ψ ⊂ φ and φ ⊂ ψ . It is easy to see that d(ψ, φ) can also be defined in terms of the support functions of the two bodies, namely by (14.3) d(ψ, φ) = sup{|Pψ (u) − Pφ (u)| : u ∈ Sn−1 } = Pψ − Pφ ∞ . In connection with the usage of Fourier series and spherical harmonics, often the Hausdorff metric is not the most suitable distance concept. Given two convex bodies ψ, φ ⊂ En , we define the L 2 -distance as  1 d2 (ψ, φ) =

(Pψ (u) − Pφ (u))2 du

Sn−1

2

= Pψ − Pφ .

We are ready to state our Bonnesen-type isoperimetric inequality for convex figures, which goes back to Hurwitz (1902). The poof presented here (see Section 4.3 of [464]) shows how Fourier series can be used to obtain geometric results. Theorem 14.5.1 If φ is a convex plane figure of area A(φ) and perimeter P(φ), then P(φ)2 − 4π A(φ) ≥ 6π d2 (φ, Bz (φ))2 . Proof Suppose that the Fourier series of the support function of φ is given by Pφ (θ) ∼

∞   ak cos(kθ) + bk sin(kθ) . k=0

Then, by Theorem 12.2.1, the area of φ in terms of its support function is given by 

A(φ) = 0

and since, by (13.2), Pφ (θ) ∼



 Pφ (θ)2 − Pφ (θ)2 dθ,

∞   kbk cos(kθ) − kak sin(kθ) , k=0

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14 Geometric Inequalities

Parseval’s equation and Exercise 14.13 imply A(φ) = πa02 −

∞ π 2 (k − 1)(ak2 + bk2 ). 2 k=2

Furthermore, since (see Section 5.2)  P(φ) =

Pφ (θ)dθ,

0

P(φ) = 2πa0 . Consequently, P(φ)2 − 4π A(φ) = 2π 2

∞ (k 2 − 1)(ak2 + bk2 ). k=2

On the other hand, by (14.2) and Exercise 14.12, the support function of the Steiner disk Bz (φ) is PBz (φ) = a0 + a1 cos θ + b1 sin θ, and therefore d2 (φ, Bz (φ))2 = π

(ak2 + bk2 ).

k=2

Summarizing, by all the above and Exercise 14.14, P(φ)2 − 4π A(φ) = 2π 2

∞ ∞ (k 2 − 1)(ak2 + bk2 ) ≥ 6π 2 (ak2 + bk2 ) = 6π d2 (φ, Bz (φ))2 . k=2

k=2



Notes The Blaschke–Lebesgue Problem The Blaschke–Lebesgue theorem asserts that among all plane convex sets of the same constant width only the Reuleaux triangle has the minimum area, see Theorem 12.1.5. In [160, pp. 132–133], [202], and [137] the proof of Lebesgue is reproduced, and references to proofs of Blaschke and Fujiwara are given. A variant of Lebesgue’s proof can be found in [1204]. Blaschke’s proof of the minimum property of the Reuleaux triangle (see [131] and, for older references to this topic, [598]) consists of two steps: First it is proved for Reuleaux polygons (by a geometric method developed by Steiner to prove the isoperimetric property of the circle), and then it is verified for general constant width sets via an approximation step (a variational problem with curvature restrictions). We refer to [667] for a more rigorous modification of Blaschke’s approach (all necessary lemmas are rigorously proved, thus filling gaps in Blaschke’s proof). Further on, the modifications yield a purely geometric proof of the Blaschke–Lebesgue theorem (avoiding any compactness arguments). An early analytical proof of the Blaschke–Lebesgue theorem was given by Fujiwara (see [382] and [383]), and more recently Harrell [515] presented the result also in analytic form, posing it as a variational problem for the radius of the curvature function. This admits a direct solution yielding both the uniqueness (up to rigid motions) and the structure of the minimizing set. A proof by Ghandehari [412] using Pontryagin’s maximum

14.5 A Bonnesen-Type Isoperimetric Inequality

333

principle has a few points in common with the arguments from [515], and also with the determination of the m-orbiforms of least area that can be rotated inside a given regular m-gon in [636]. Here we also mention [765], for a related formulation of the Blaschke–Lebesgue theorem in terms of optimal control theory. In [84], methods from semidefinite programming are used to attack the Blaschke–Lebesgue problem in the plane; see also the survey [85]. Ohmann [898] gave a direct generalization of Lebesgue’s theorem regarding the least area of the Reuleaux triangle among all sets of the same constant width. His approach uses affine regular hexagons inscribed to plane curves, and the result can be interpreted as an analogue of Lebesgue’s theorem for normed planes. Mayer [804] gave a broad discussion of properties of constant width sets and the Blaschke–Lebesgue problem, using notions like isoperimetric deficit (which, as area quotient of the considered set and a circle of equal perimeter, can also be interpreted as a measure of asymmetry). There are many further proofs of the Blaschke–Lebesgue theorem (see, e.g., [307], [308], [106], [230], [209], [667], and [1203]), and further elementary ones can be found in [312] and [105]. Using mixed areas, Chakerian [230] found also a short proof, allowing the extension to normed planes. See the proof of Theorem 12.1.5. In [1218], the Blaschke–Lebesgue theorem is confirmed for a special class of constant width figures. In [458] (see also the survey [460, § 6.2]), various stability results referring to the Blaschke– Lebesgue theorem are established. For example, if C is of constant width 1 and area at most A0 + ε of width 1), then there exists a Reuleaux triangle (where A0 denotes the area of the Reuleaux triangle √ whose Hausdorff distance from C is at most 10ε. In [123], the following extensions of the Blaschke–Lebesgue theorem are discussed: A disk-polygon √ is the intersection of a finite set of disks of radius 1 in the Euclidean plane. For a parameter 0 < d < 3 denote by F(d) the class of disk polygons such that the distance between any two centers is at most d. Denote by (d) the regular disk triangle whose three generating unit disks are centered at the vertices √ of a regular triangle of side length d, where 1 ≤ d < 3. The author proves that (d) is minimal in F(d) regarding area, inradius, and width. Staying with the Blaschke–Lebesgue theorem and related topics, we now turn to concrete inequalities. For a plane set C of constant width h and of area A(C) the Blaschke–Lebesgue theorem is equivalent to the inequality (see Section 14.1) √ π− 3 2 h , A(C) ≥ 2 which can also be used in normed planes if in such planes a suitably “generalized Reuleaux triangle” is analogously constructed (see Chapter 10 and [898], [661]). The latter paper contains a more general inequality for convex figures, estimating their area against minimal width and diameter and characterizing generalized Reuleaux triangles. If the diameter and the minimal width coincide, then the inequality above is obtained. Sholander [1063] adds the perimeter to these three quantities and considers triples of quantities, in each case leaving two of them fixed and extremizing the third; see also [1019] for such complete systems of inequalities and the short discussion of Blaschke–Santaló diagrams below (see also [177]). A modified Blaschke–Lebesgue problem is investigated in [912]: the authors ask for plane convex sets C of constant biwidth having minimal area (the biwidth of C in direction u is in fact the sum of two widths of C - that in direction u and that in direction u + mπ ). When m is an odd integer, the Reuleaux triangle is extremal, and for the even case certain types of Reuleaux polygons have the least area. The idea of considering the quantity “width” as sum of widths (or support values) in different directions goes back perhaps to [905], see also [1219] for properties of the related sets.

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14 Geometric Inequalities

Eggleston [314] showed that the width of a constant width set C in the plane that contains a given square S largest when C is a Reuleaux triangle, and he gives a respective bound for the edge-length of S. In [516], the concept of m-diameter of a set S is considered. This quantity is defined as the supremum of the geometric mean of all Euclidean distances among m (and not “only” two, as in the usual way) points from S. In this paper, it is shown that among all sets of constant width the Reuleaux triangle has the largest (and the circle the smallest) 3-diameter. It is proved that there are infinitely many non-circular curves of constant 3-diameter. In [570], analogous results for m = 5 and, with additional assumptions, for m = 7 are derived. Another concept of “m-width” was considered in the papers [252] and [447]. In some sense “dual” to the Blaschke–Lebesgue theorem, but restricted only to Reuleaux polygons, is the following analogue of the classical Zenodorus problem for polygons: Which Reuleaux m-gons of given width have maximal area? Firey [355] characterized the regular Reuleaux m-gons by this property, and his approach to this result and related statements was simplified by Sallee [999], therefore yielding the name “Firey–Sallee theorem” for this result. The authors of [667] used a modification of Blaschke’s original approach to the Blaschke–Lebesgue theorem for presenting also a rigorous proof of the Firey–Sallee theorem. We also mention that the Blaschke–Lebesgue theorem plays a role for studying upper bounds on the expected value of the number of normals passing through a point randomly chosen from a convex body, see [301]. The Blaschke–Lebesgue Conjecture We continue with the 3-dimensional case of the Blaschke–Lebesgue problem. Here the volumeminimizing procedure is clearly equivalent to minimizing surface area, due to Blaschke’s theorem on volume and surface area in that dimension, see Theorem 12.1.4. It is a long-standing conjecture that the 3-dimensional Meissner bodies (see, e.g., [160, pp. 135–136], [1204, § 7], [272, A. 22], [151, Chapter VIII], and [612]) are the minimizers. In [610], Kawohl mentioned that one of his students generated randomly one million of 3-dimensional bodies of constant width—none of them had smaller volume than the Meissner bodies. Assuming that this conjecture were correct, the minimum volume of a body of constant width 1 in E3 would be about 0.42. Chakerian [230] confirmed a lower bound of roughly 0.365 (see Theorem 14.1.2); see also [356]. Also, the paper [494] is devoted to the 3-dimensional situation. The comparison of the volume of a constant width body with that of the ball having the same width yields a functional which is proved to increase when going from a body to its parallel body. i.e., for improvement one has to shrink the boundary along the normals. To keep convexity, this process has to be stopped when the boundary touches the set of foci (for rational support functions explicit examples of constant width are constructed). In [31], a necessary condition in this direction is given: For a local minimizer of the Blaschke–Lebesgue problem in E3 having constant width h, the smooth parts of its boundary have their smaller principal curvature constant and equal to h1 (see Theorem 14.2.1). Hence the smooth boundary parts are spherical caps or pieces of tubes, each of radius h (see Corollary 14.2.1). In [86], 3-dimensional constant width bodies are presented analytically, using a bijection between spaces of functions and constant width bodies (see also [31] and [677]). The authors compute several quantities (like volume or surface area) of them. The paper concludes with a necessary condition for bodies of given constant width to have the minimal volume. The Blaschke–Lebesgue theorem plays also a role for determining upper bounds on the number of normals through randomly choosen points of constant width sets, see again [301]. Also in [123] the Blaschke–Lebesgue theorem is discussed for three dimensions. Campi et al. [209] showed that the restriction of the Blaschke–Lebesgue problem to 3-dimensional bodies of revolution yields the body obtained by rotating the Reuleaux triangle about one of its symmetry axes as extremum (see Theorem 14.2.2).

14.5 A Bonnesen-Type Isoperimetric Inequality

335

Going to higher dimensions, results of Firey [356] and Chakerian [230] were improved (for n > 4) by Schramm [1046], see also the discussion in [272, p. 34]. He proved that for a body C of constant width 2 in En and B as unit ball the lower bound n  3 + 2/(n + 1) − 1 · V (B) V (C) ≥ for the volume V (C) holds. These investigations on volume-minimizers led Heil [524] to define the fruitful notion of reduced bodies (see Chapter 7); see also other related problems of Heil in pp. 260– 261 of the problem collection [481] and the conjecture of Danzer formulated there, namely that the searched volume-minimizers have the symmetry group of a regular simplex. Here we also mention that Firey [356] derived inequalities for bodies of constant width also holding with respect to the minimal brightness (= minimal volume of orthogonal (n − 1)-projections). Measures of Asymmetry Before we start to discuss results on asymmetry measures (which can be defined in various ways), we refer to Grünbaum’s excellent related survey [487], nicely presenting the basic notions and the state of the art at that time, see also [272, A 15]. A recent comprehensive reference on measures of asymmetry is the book [1135]; Chapter 2 there is dedicated to bodies of constant width and related topics, such as Klee’s critical sets and affine diameters. In contrast to the general point of view presented in [487], we restrict ourselves here to bodies of constant width. To start with a suggestive example of a very natural measure of asymmetry for constant width sets, we present one studied by Besicovitch [103] for the planar case: Let C be a planar closed convex curve, and α be the ratio of the maximal area of a centrally symmetric closed curve contained in C and of the area enclosed by C itself. Then α is a measure of asymmetry of C in the sense of [457], and the smallest value 23 is known to be reached exactly by triangles. In [103], this consideration is restricted to the class of constant width curves, and it turns out that the largest asymmetry (i.e., the smallest ratio α) is attained precisely by the Reuleaux triangle. The exact value of α (roughly about 0.84) is derived, too. We note that Estermann [322] introduced the “inverse outer measure”, defined via a centrally symmetric convex body of minimal volume containing a fixed convex body; its restriction to bodies of constant width was investigated by Chakerian [230]. Another suggestive example is presented in [465]: For each direction u, a plane constant width set C has a unique diameter splitting the area of C into two subsets of areas A+ (u) and A− (u), respectively. The authors define the asymmetry function (u)/A− (u), and they show that it lies between 1, characterizing the as the maximum of√the ratio A+√ circle, and (4π − 3 3)/(2π − 3 3), characterizing the Reuleaux triangle. Continuing studies of Besicovitch’s concept, Eggleston [307] defined that for a number k with 1 ≤ k ≤ 1, Rk denotes the set of points at distance ≤ 1 − k from the Reuleaux triangle of width 2k − 1. 2 He showed that for each set C of constant width 1, such that any boundary point of it has curvature between k and 1 − k, the relation α(C) ≤ α(Rk ) holds. Chakerian and Stein [243] introduced an asymmetry measure for plane convex bodies which is defined with the help of areas created by points which occur as midpoints of a fixed number of chords passing through them. They investigated this measure also for constant width sets, getting analogously the Reuleaux triangle and the circle as extremal cases. Also in [590], [465], and [736], extremal planar constant width figures for several related measures of asymmetry are discussed, analogously yielding Reuleaux triangles as most asymmetric constant width figures. For example, in [736] the authors refer to [465] (see above), but replacing the area by perimeter. Related investigations on Reuleaux polygons are presented in [495] and [589]. Results on asymmetry measures for higher dimensions were obtained in [591] and [592]. For K a convex body in En and H a hyperplane passing through an interior point x of K , let γ(H, x) be the ratio, not less than 1, in which H divides the distance between the two supporting hyperplanes of K

336

14 Geometric Inequalities

parallel to H . Then the Minkowski measure of asymmetry of K is defined by (see Section 14.3) as (K ) := min x∈intK max x∈H γ(H, x) . In [591], the following inequalities for this measure of asymmetry are proved: 1 ≤ as (K ) ≤

n+

√ 2n(n + 1) . n+2

Equality on the left-hand side holds iff K is an Euclidean ball, and if n = 3, equality holds on the right-hand side if K is a Meissner body. Continuing this, it is shown in [592] that the upper bound is reached by bodies of constant width which are completions of a regular n-dimensional simplex, i.e., a complete convex body containing a regular n-dimensional simplex and having the same diameter as the simplex. In [587], Besicovitch’s result (that the most asymmetric constant width set in the plane is the Reuleaux triangle regarding α) is reproved, and in [588] the same author applied his methods to measures of asymmetry of rotational constant width bodies. Continuing the work [465], also in [913] asymmetry measures for constant width bodies of revolution in E3 are studied; the body obtained from a Reuleaux triangle via rotation about one of its symmetry axes is shown to be extremal. The so-called mean Minkowski measures of convex bodies (introduced by Toth, see [1135]) were investigated for the class of constant width bodies in [593]. With respect to these measures, completions of regular simplices are the most asymmetric bodies. In [294], for the class of plane convex bodies K a “measure of axial symmetry” is defined as a real-valued function f satisfying 0 ≤ f (K ) ≤ 1, with f (K ) = 1 iff K is symmetric with respect to some line, and having similarity-invariance (see also [293]). Eleven such measures of axial symmetry are studied in [294], bounds for them are established (also for the case when K ranges over all sets of constant width), and interesting questions are posed. In- and Circumradii A well known property of n-dimensional bodies of constant width h is the concentricity of their circumsphere and insphere. The respective in- and circumradius, r and R, satisfy the relations 

r + R = h and h 1 −

n 2n + 2



≤r ≤ R≤h

2 , 2n + 2

cf. Eggleston [308] and Melzak [825] (see also Section 3.4). The planar version of the left equation goes back to Minoda [840], who applied it also to some inequalities for areas. Chakerian [231] extended this relation to normed spaces, see Chapter 10. The union of all points between circum- and insphere of a constant width set C is called the minimal shell (for n = 2 minimal annulus) of C, and in [160] pp. 134–135, references are given for results to the following problem for n = 2: given a suitable shell S, where r and R satisfy the relations above, determine those constant width sets which have S as their shell and maximal/minimal area. The case of maximal area was solved (see [159] and [160], pp. 134–135), and Mayer [804] gave corresponding upper and lower bounds for areas from which also the Blaschke–Lebesgue inequality above follows. In [805], he presented a sketch of a proof that the minimal area is reached by certain Reuleaux-type polygons. We continue with the circumradius R, combining it also with other geometric quantities. For a convex body K in En , we write D and p for diameter and perimeter, respectively. Scott [1052] showed that for any convex body K there is a constant width body C containing K and having the same circumradius and diameter as K (see Section 14.4 and also Chapter 7 about completions). Let r  and

14.5 A Bonnesen-Type Isoperimetric Inequality

337

p  correspondingly denote the inradius and perimeter of C, related to K in the above sense. Scott used properties of this constant width set C and several combinations of inequalities to obtain new inequalities particularly involving the factor 2R − D, like for example   √ (2R − D) p ≤ 2 3 − 3 π R 2 , where equality holds only for Reuleaux triangles. Using  for minimum width, this inequality yields   √ (2R − D) ≤ 2 3 − 3 R 2 , obtained for all plane convex bodies already in [1051], and again with equality only for Reuleaux triangles. In [1052], the latter inequality was extended to En in the following form: (2R − D) ≤

√  √ 2 n + 1 √ 2n − n + 1 R 2 . n

In [536], it is proved that constant width bodies in En maximize the minimal width when their circumand inradius are prescribed; also the paper [535] should be mentioned here. In [436], the notions of classical radii are extended for bodies of constant width to inner and outer successive radii, and respective inequalities are given. The paper [176] contains a deep study of isoradial bodies which can be seen as a generalization of constant width bodies. Namely, the so-called strongly isoradial bodies are those of constant inner and outer constant j-radii, in the plane identical with the family of constant width sets. The paper [50] refers to m-gons inscribed to a curve C of constant width h. Given a sequence of points p0 , p1 , . . . , pn+1 = p0 in C, the author shows how closely h and the in- and circumradius of C can be approximated. Isoperimetric Inequalities and Related Topics Combining Bonnesen’s sharpening of the isoperimetric inequality for planar convex figures K (see [160, p. 83]) with relations between quantities of K and of its central symmetral K ∗ = 21 (K + (−K )), Ganapathi [391] derived the inequality p 2 − 4π A ≥ π(D − )2 + 4π(A∗ − A) , where A∗ denotes the area of K ∗ , with equality only for constant width sets, see also Theorem 14.5.1. The quantity (A∗ − A)/2 is interpreted as the area associated with the locus of all midpoints of diametral chords of K , which is maximized by the Reuleaux triangle among all sets of the same constant width (due to the Blaschke–Lebesgue theorem). This inequality was investigated further by Vincensini [1152]. Replacing perimeter by total mean curvature and area by surface area, Ganapathi [392] gave an analogue to the above inequality for the 3-dimensional situation. The value of the righthand side then becomes proportional to the signed surface area of the midpoint locus of all diametral chords of the convex body under consideration, and again equality holds only for constant width bodies. Groemer studied isoperimetric and related inequalities in Section 4.3 of his book [464], involving also constant width sets. His motivation there is to demonstrate how Fourier series can be used to get stability estimates. Taylor [1120] found the area minimizers among those plane convex bodies of given perimeter which are contained in a fixed Reuleaux triangle. On the other hand, Minoda [839] searched perimeter minimizers among plane closed curves in which a given convex figure can be rotated (see also

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14 Geometric Inequalities

Section 17.1). Besides the circle, he obtained in some cases certain non-circular constant width figures as such minimizers. In [1225], a sharpened isoperimetric inequality for closed, regular convex curves using the oriented area of their Wigner caustics (i.e., certain affine equidistants) is presented. Equality holds exactly if the original curves are of constant width. A related stability result (also for constant width curves) is given, too. In [224], an inequality between the area enclosed by a plane constant width curve (containing the origin) and the squared length of its pedal curve is obtained, characterizing the circle. In this way, also the isoperimetric deficit of the pedal curve can be estimated. Clearly, the inequality p(K ) ≤ π D(K ) for n = 2 implies that for fixed perimeter the diameter D(K ) is minimized if K is of constant width. This is a special case of results derived by Sachs [996]. He proved extremal properties of certain types of closed curves via integral power means of lengths of chords varying independently between points of these curves, and among the studied types of curves also constant width curves play a role. Further Quantities and Concepts The classical Rosenthal–Szasz theorem (see [987] and Section 44 of [160]) says that for a compact, convex figure K in the Euclidean plane with perimeter p(K ) and diameter D(K ) the inequality p(K ) ≤ π D(K ) holds, with equality iff K is of constant width D(K ). In [65] this was also confirmed for Radon planes, and a similar statement (involving the antinorm appropriately) was proved there for all normed planes. For a compact set S in En , let β(S, x) denote the maximum of distances from x to points of S. Reidemeister [964] showed that a convex body K in En is of constant width if β(K , x) is constant for x running through the boundary of K , see Theorem 7.2.3. The following nice conjecture is due to Alexander [20] (see Section 4.3): if C is a closed and rectifiable curve in En such that β(C, x) ≥ λ for all x from C, then the length of this curve is at least λπ, and it can attain this length only if it is a plane curve of constant width. This was independently proved by Sallee [1000] and Falconer [329]. Inspired by a conjecture of Herda (see [533] for a related survey), Chakerian [234] proved the following (which is more): if C is rectifiable simple closed curve in the plane, and x → f (x) with x ∈ C is any involution of C without fixed points such that the distance from x to f (x) is at least λ for all x ∈ C, then the length of C is at least λπ, and equality holds iff C is of constant width and all chords connecting x and f (x) are diametral ones. (Since β(C, x) ≥ λ for all x ∈ C, Alexander’s conjecture is more general.) Lutwak [740] extended Herda’s conjecture to integral power means of the perimeter bisectors and the area bisectors of convex curves. In [583], a theorem on lengths of chords of constant width curves which connect touching points of circumscribed squares is proved, together with a characterization of the circle. This verifies a conjecture of Green (related to is optics of given curves) at least for this class of convex curves. Kubota [660] proved the following theorem referring to four quantities: If C is a convex figure in the plane with A, p, D as its area, length, and diameter, respectively, M denoting the mixed area of C and C  (the figure obtained from C by rotation about 180◦ ), then Dp ≥ 2(A + M), with equality only for constant width curves. This brings us closer to topics related to complete systems of inequalities and the Blaschke–Santaló diagram; see, e.g., [1019], [536], [535], and [177]. E.g., in [177] a complete 3-dimensional Blaschke–Santaló diagram for plane convex bodies with respect to their four classical quantities in- and circumradius, diameter, and minimum width is presented. The large variety of extremal planar figures in this framework shows that many of them are somehow (regarding shape) close to the Reuleaux triangle, and several slight modifications of it. We continue with the same motivation (namely, that shapes of constant width figures play the key role). Again the used constant width figures are not themselves extremal figures, but can help to describe such extremal figures geometrically. This is also due to the fact that most of the questions

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339

that we discuss now refer to proper polygons. For example, area-minimal plane convex figures with prescribed diameter and perimeter were studied in [656] and [657]. This yields concave maximum problems, having a certain non-regular inpolygon of the Reuleaux triangle as area-minimal extremum. In [417], suitable circumscriptions of equiangular m-gons about constant width sets are investigated, where these m-gons should have extremal perimeters; it turns out that smallest and largest perimeters are attained when the used constant width sets are regular Reuleaux m-gons. More generally, in the surveys [45] and [47] problems of the following type are presented: let Pm be a convex polygon with m sides, and take its perimeter, diameter, area, sum of distances between the vertices, and minimum width. One is asked to minimize or maximize one of these quantities, while the others are fixed. It turns out that for solving some of these questions, polygonal shapes close to those of Reuleaux polygons (the authors also use the term “clipped Reuleaux polygons”) and diameter graphs of Reuleaux polygons play a key role. Papers from this research direction, in which geometric ideas related to Reuleaux polygons are particularly important, are [280], [109], [373], [46], and [869]. For example, Datta [280] gave a sharp upper bound for the maximum perimeter of a convex m-gon of diameter 1 in the plane. The explicit constructions of all extremal m-gons show that all of them have equal sides and are inscribed in a Reuleaux k-gon, for an odd integer k not larger than m. Gritzmann and Lassak [451] derived estimates for the thickness of polytopes inscribed to convex bodies, the latter being particularly also centrally symmetric or of constant width. The paper [899] contains a theorem about polygons with prescribed vertex angles, circumscribed about constant width curves and being extremal regarding their perimeters. Let K be a convex body in En , M(K ) its mean width, PK the parallelepiped of minimal mean width circumscribed about K , and C n a cube circumscribed about the unit ball of En . Among various other results, in [164] the inequality 1 M(PK ) ≤ M(K ) · M(C n ) 2 was established, with equality iff K is of constant width. And every parallelepiped of minimal mean width that is circumscribed about a constant width body is a cube. While all rectangular parallelepipeds circumscribed about a body of constant width have the same volume, this property does not characterize constant width sets; nonspherical bodies satisfying this property may even be centrally symmetric. Petty and McKinney [934] characterized them for n > 2, and in the plane the analogue of this characterization is still an unsettled problem. Inspired by a conjecture of Moser (on covering closed curves of certain length by rectangles, see D 18 in [272]), the authors of [1027], [241], [513], and [739] considered inequalities referring also to volumes of boxes circumscribed about convex bodies, with cases of equality only for bodies of constant width. For example, it is proved that a closed plane curve of length 2π can be covered by a rectangle of area 4. If no smaller rectangle suffices, then the covered curves have to be of constant width 2 (this is also related to [1187]). Chakerian and Logothetti [239] proved that the smallest regular m-gon (m > 3) which can cover any plane set of diameter 1 is the smallest regular m-gon that can be circumscribed about a Reuleaux triangle of width 1. Bezdek and Connelly [116] showed that any plane closed curve of length 2 can be covered by a translate of the body C if C is of constant width 1, and that, conversely, C must have constant width 1 if its perimeter is not larger than π (see Theorem 17.2.4). Somehow converse to a result in [317], where largest equilateral triangles contained in constant width sets are investigated, Eggleston proved in [309] that every plane √ set of constant width h contains a subset of constant width ≥ h/(3 − 3), this bound being attained for the equilateral triangle. In [294], the author studies inequalities referring to area and diameter of convex figures K and certain types of axially symmetric polygons inscribed to them; also the subcase that K is of constant width is taken care of. The approach to the following inequality is nicely presented in Section 7 of [238], and already Kubota [660] verified the right-hand side for n = 2. Namely, for K an n-dimensional convex body, s(K ) its surface area, and V (K − K , K , . . . , K ) symbolizing mixed volume, we have

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14 Geometric Inequalities

1 1 (K )s(K ) ≤ V (K − K , K , . . . , K ) ≤ D(K )s(K ) , n n where V (K ) is the volume of K , and equality holds iff K has constant width, see Section 12.4. Minkowski’s inequalities for mixed volumes yield V (K ) ≤

1 D(K )s(K ) , 2n

true for any convex body and obtained by Firey [356] (equality holds iff K is a ball). See also Theorem 12.4.1. Certain combinations lead to an inequality of Petermann [926], namely (K )V (K ) ≤ D(K )V (L , K , . . . , K ) , where K (being centrally symmetric) and L are convex bodies in En . Equality holds iff L is of constant width and K is a ball. If the body K above is of constant width h, we get V (K )/s(K ) ≥ h/2n. If K ∗ denotes again the central symmetral of K , we see from this that if K is a set of constant width, then V (K )/s(K ) ≤ V (K ∗ )/s(K ∗ ) , as observed by Heil in [481], p. 260. For mixed volumes of constant width sets K in En , Godbersen [422] verified

n V (K , . . . , K , −K , . . . , −K ) ≤ V (K ) , k where k = 0, 1, . . . , n, K appears k times and −K appears n − k times within the mixed volume V (K , . . . , K , −K , . . . , −K ) (see also Section 12.4). If this would hold for all convex bodies, the famous difference-body inequality (see, e.g., § 7.3 in [1039]) would directly follow from it. Also, the paper [411] deals with a variety of inequalities involving mixed volumes of a convex body K and its polar body; these results are applied to obtain inequalities for quermassintegrals of K and its polar in case that K has constant width. Leichtweiss [710] characterized affine classes of convex bodies containing a body of constant width in view of minimal diameter-thickness ratios. In [742], a special case of the Blaschke–Santaló inequality is shown to be equivalent to a power-mean inequality involving the diameters and widths of a convex body K in En . This leads to strengthened versions of known inequalities and a characterization of the affine images of bodies of constant width, see Section 11.6. In [226], a lower bound on the volume product of a constant width set and its p-centroid body (see [1039, § 7.4]) was obtained. Of course, various inequalities between geometric quantities of constant width bodies give rise to study also stability versions of them; see, e.g., [458] and [460], and we refer also to the subsection above where the Blaschke–Lebesgue theorem is discussed. Minkowski’s famous theorem about constant perimeters in E3 (see Theorem 13.1.1), yielding constant width, has the following stability variation: How much does a convex body K deviate from a convex body of constant width if the perimeter of the projections of K is approximately constant? The 3-dimensional case was studied by Campi [208], and Bourgain and Lindenstrauss [170] solved the related stability question regarding areas of projections. The paper [441] contains far-reaching generalizations of Minkowski’s theorem, involving the stability of the mean width functional in the following way. The width function of a convex body K in En is nearly constant if the mean width function of the projections of K onto hyperplanes is nearly constant. And if the latter function is assumed to be nearly constant, then there exists a set of constant width near K in the Hausdorff sense. Groemer [463] introduced and studied measures for a so-called spherical deviation of a convex body from convex

14.5 A Bonnesen-Type Isoperimetric Inequality

341

bodies of a special class, like that of constant width. Introducing also measures for normals of convex bodies, stability results are proved, i.e., inequalities which show that convex bodies cannot deviate too much from the class of constant width bodies if their corresponding deviations of the normals are small. Pommerenke [945] derived estimates on the capacity (or the transfinite diameter) of planar compact sets in terms of geometric quantities of them, such as width, area etc. within this framework, he investigated also curves of constant width. Positive centers of convex curves in the plane were introduced by Gage to sharpen Bonnesen’s isoperimetric inequality (references are given in [562]). The authors of [562] investigated positive center sets, and they also gave a characterization of them for given curves of constant width based on the notion of inner parallel bodies; this work is continued in [911], again referring to constant width sets. For a closed convex surface S in E3 , let d(K ) denote the least upper bound of the intrinsic distances between point pairs in S, i.e., the so-called geodesic diameter of S. Relating this to the usual diameter D(S) of S, Makuha (see [763] and [764]) proved the inequality d(S) ≤ π D(S)/2 , with equality iff S is the boundary of a body of revolution having constant width. The analogous statements hold in En , as confirmed in [762]. The 3-dimensional case was independently reproved by Zalgaller [1209], where also the question for the respective lower bound on d(S) for S having constant width is posed. Let there be given two parallel lines in Euclidean space which have at most distance d. In [377], it was shown that the lengths of the segments intercepted by a body C of constant width h on such lines 1 differ at most by 2(2dh) 2 , not depending on the shape of C. In [221] inequalities for so-called relative quantities (suitably defined by two planar given figures, one containing the other) are derived, and an interesting characterization of sets of constant width is obtained which is related to the notion of relative diameter.

Exercises 14.1*.

14.2.

Prove that the Minkowski sum of the two original Meissner bodies M1 + M2 has the symmetry group of a regular tetrahedron. Furthermore, prove that V ( 21 M1 + 21 M2 ) > V (M1 ) = V (M2 ). Prove that for a body of constant width h, the following two conditions are equivalent: (a) The smooth parts of its boundary are spherical caps or pieces of tubes of revolution, both of them with radius equal to h. (b) At least one of the endpoints of every diameter of  is not a regular point of the boundary.

14.3. 14.4. 14.5. 14.6.

Prove that the set of critical points φ∗ is a singleton whenever φ is strictly convex. Prove that both definitions of Minkowski measure of asymmetry given in this chapter coincide. Prove that the Minkowski measure of asymmetry of an n-dimensional simplex is n. Derive Corollary 14.3.1 from Theorem 14.3.3 and prove that for a √body  of constant 2n(n+1) . width, which contains a regular n-dimensional simplex, M() = n+ n+2

342

14.7. 14.8. 14.9. 14.10*. 14.11.

14 Geometric Inequalities

√ √ Prove, using the Theorem of Jung, that h ≥ ( 2n + 2/ n)R, where h is the diameter and R the circumradius of an n-dimensional convex body φ ⊂ En . Prove the claims of Remark√14.4.1. Show that (2R − h)P = (2 3 − 3)π R 2 for figures of constant width h. Is the Steiner point of a body of constant width its circumcenter? Prove that the Hausdorff distance between two convex bodies is given by d(ψ, φ) = sup{|Pψ (u) − Pφ (u)| : u ∈ Sn−1 } = Pψ − Pφ ∞ .

14.12.

Prove Equation (14.2). Moreover, if the Fourier series of the support function of φ is given by ∞   ak cos(kθ) + bk sin(kθ) , Pψ (θ) ∼ k=0

prove that the support function of the Steiner disk is given by PBz (φ) = a0 + a1 cos θ + b1 sin θ. 14.13.

Prove that if the Fourier series of the support function of φ is given by Pψ (θ) ∼

∞   ak cos(kθ) + bk sin(kθ) , k=0

then A(φ) = πa02 − 14.14.

Prove the inequality

∞ k=2

14.15*.

∞ π 2 (k − 1)(ak2 + bk2 ). 2 k=2

(k 2 − 1)(ak2 + bk2 ) ≥ 3

∞ (ak2 + bk2 ). k=2

Let φ be a given plane convex figure in E2 . Consider the problem of finding the subset S of φ of given perimeter L for which the area of S is maximum. Prove that the area is maximized by the convex set (φ)r for some value of r , where (φ)r consists of the union of all disks of radius r contained in φ.

Chapter 15

Bodies of Constant Width in Discrete Geometry

In mathematics the art of proposing a question must be held of higher value than solving. Georg Cantor

15.1 Helly’s Theorem and Constant Width We start with the versions of the Helly’s Theorem developed by V. Klee [628]. Let φ and ψ be two convex bodies in En , and consider the following two subsets: {x ∈ En | x + φ ⊂ ψ}, {x ∈ En | x + φ ⊃ ψ}. It is easy to see that both sets are convex bodies. From this, the following variant of Helly’s theorem is immediately obtained. Theorem 15.1.1 (Klee) Let  be a fixed convex body of En , and let F be a family of compact convex subsets of En . • If the intersection of every n + 1 or fewer members of F contains a translate of , then the intersection of all the members of F contains a translate of . • If the union of every n + 1 or fewer members of F is contained in a translate of , then the union of all members of F is contained in a translate of . Proof For each φ ∈ F, define φ∗ by φ∗ = {x ∈ En | x +  ⊂ φ} (or, respectively, φ∗ = {x ∈ En | x +  ⊃ φ}). Let us consider the following collection of compact convex subsets of En : F ∗ = {φ∗ | φ ∈ F}. By hypothesis, every n + 1 or fewer members of F ∗ have a point in common. Then, by Helly’s Theorem 2.8.1, there is a point x ∗ common to all members of F ∗ . Then x ∗ +  ⊂ φ for all φ ∈ F (or,  respectively, x ∗ +  ⊃ φ for all φ ∈ F). This completes the proof. Remarks 15.1.1 If the family F is finite, then Theorem 15.1.1 holds even when  is not necessarily compact. © Springer Nature Switzerland AG 2019 H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width, https://doi.org/10.1007/978-3-030-03868-7_15

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As a corollary of the first part of this theorem we have the following result of Chakerian [232] that claims that if X can be turned 360◦ degrees inside a Reuleaux triangle, then X can be turned 360◦ degrees inside any convex figure of the same constant width. Theorem 15.1.2 Let  ⊂ E2 be a Reuleaux triangle of width h, and let X be a compact convex subset of E2 . Suppose that every congruent copy of X can be covered by a translate of . Then, if  is any body of constant width h, every congruent copy of X can be covered by a translate of . Proof The first step of the proof is to verify the following geometric fact, whose proof is left as an exercise (namely 15.1) to the reader: the intersection of three circular disks of radius h, each containing the centers of the other two, contains a Reuleaux triangle of width h. Then let  be a body of constant width h. Consider F, the following family of compact convex subsets of En : F = {B(x, h) | x ∈ bd }.   Recall from Theorem 6.1.4 that {T ∈ F} = x∈bd  B(x, h) = . Let X  be a congruent copy of X . We shall prove that there is a translated copy of X  contained in . By Exercise 15.1, every three members of F contain a Reuleaux triangle of width h, and by hypothesis, this copy of the Reuleaux triangle contains a translated copy of X  . Then by Theorem 15.1.1 there is    a translated copy of X contained in x∈bd  B(x, h) = . This concludes the proof. As a corollary of the second variation of Helly’s Theorem 15.1.1, we have the following theorem of Buchman and Valentine [191]. Theorem 15.1.3 Let F be a family of compact convex subsets of En with the property that if the union of every subset of n + 1 or fewer members of F is contained in convex body of constant width h, then the union of all members of F is contained in a convex body of constant width h. Proof Suppose first that the family F is finite. Let u ∈ Sn−1 , and let T be a strip of width h and orthogonal to the direction u. By hypothesis, every subset of n + 1 or fewer members of F is contained in a convex body of constant width h and therefore, is contained in a translate of T . Then, by Theorem 15.1.1, all members of F are contained in a translated copy of T . Since the same is true for every direction u ∈ Sn−1 , this implies that the diameter of the union of all members of F is smaller than or equal to h. Consequently, by the Theorem of Pál (Theorem 7.2.2) the union of all members of F is contained in a convex body of constant width h. Now suppose that F is not finite. Let u ∈ Sn−1 , and let T be a strip of width h orthogonal to the direction u. For each φ ∈ F, define φ∗ by φ∗ = {x ∈ R | xu + T ⊃ φ} and consider the collection F ∗ = {φ∗ | φ ∈ F} of compact convex subsets of En . By hypothesis, every n + 1 or fewer members of F ∗ have a point in common. Then, by Helly’s theorem, there is a point x ∗ common to all members of F ∗ , and so x ∗ u + T ⊃ φ for all φ ∈ F. Since the same holds for every direction, this implies that the diameter of the union of all members of F is smaller than or equal to h. Consequently, the union of all members of F is contained in a convex body of constant width h. This completes the proof. 

15.1.1 The ( p, q)-Property and the Piercing Number We say that a family F with at least p closed convex sets in En satisfies the ( p, q)-property if among any p sets in the family there are q of them with a point in common. The piercing number, π(F), is

15.1 Helly’s Theorem and Constant Width

345

the minimum cardinality of a set S ⊂ En such that every set in the family contains at least one point of S. Hence the classical Helly theorem can be restated as follows: If F satisfies the (n + 1, n + 1)-property, then π(F) = 1. Hadwiger and Debrunner [504] conjectured that the ( p, q)-property should be enough to bound the piercing number of a finite family of convex sets, which was later confirmed by Alon and Kleitman [21]. The following theorem is now commonly known as the ( p, q)-theorem. Theorem 15.1.4 (Alon, Kleitman) Given positive integers p ≥ q ≥ n + 1, there is a constant c = c( p, q, n) such that every family F of convex bodies in En with the ( p, q)-property satisfies π(F) ≤ c. To calculate or even to give sharp bounds for the best constant c = c( p, q, n) is a difficult problem. For example, it has been conjectured that the best constant for c(4, 3, 2) is 3. Nevertheless, the best result, as we can see from the following theorem of Tancer and Kynˇcl [676], is far away from that bound. Theorem 15.1.5 Every finite family F of convex bodies in E2 with the (4, 3)-property satisfies π(F) ≤ 13. If fact, in the same paper, Tancer and Kynˇcl proved that a family F of congruent disks in E2 satisfying the (4, 3)-property can be pierced with three points. Using ideas of Chakerian and Sallee [242], we shall prove in this section that, for a family F of translates of a plane convex body of constant width satisfying the (4, 3)-property, we have π(F) ≤ 4. If p − q is sufficiently small with respect to p, then Hadwiger and Debrunner [504] proved that the piercing number of a family of convex sets in En satisfying the ( p, q)-property is smaller than or equal to p − q + 1. More precisely, we have the following Theorem 15.1.6 Every finite family F of convex bodies in En with the ( p, q)-property, p ≥ q ≥ n, n( p − q) < p − n, satisfies π(F) ≤ p − q + 1. The condition p ≥ q ≥ n + 1 in the Alon–Kleitman Theorem is important because it is possible to construct families of convex bodies in En with the (n, n)-property and piercing numbers as big as we wish. This contrasts with the following theorem of Chakerian–Sallee [242] for translates of a plane figure of constant width. Theorem 15.1.7 Every finite family F of translates of a plane figure of constant width  with the (2, 2)-property satisfies π(F) ≤ 3. Proof Suppose, without loss of generality, that  has constant width 1 and that the origin 0 ∈ . Let Q ⊂ E2 be such that F = {x +  | x ∈ Q}. The fact that F satisfies the (2, 2)-property implies that the diameter of Q is smaller than or equal √ to 1. Then, by Theorem 15.2.3 from the next section, there is a regular hexagon H of side length 1/ 3 such that Q ⊂ H . Note that H can be decomposed into three congruent irregular pentagons as shown in Figure 15.1. By inspection, we verify that each of these pentagons can be turned inside a Reuleaux triangle of diameter 1. Hence, by Theorem 15.1.2, each of these three pentagons can be turned inside . This implies that there are translated copies x1 + , x2 +  and x3 +  of , each of them covering every pentagon, and therefore with the property that Q is contained in {x1 , x2 , x3 } + . But this is equivalent to claiming that every set in the family F  contains at least one point of {x1 , x2 , x3 }. As a corollary we have that every finite family F of translates of a plane convex body of constant width,  ⊂ E2 , with the (4, 3)-property satisfies π(F) ≤ 4. This is so because it is not difficult to prove that every family F of sets satisfying the (4, 3)-property also satisfies the following: there is a set A ∈ F such that the family F \ {A} satisfies the (2, 2)-property.

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Figure 15.1

Theorem 15.1.8 Every finite family F of translates of a convex body of constant width  ⊂ E2 with the ( p, 2)-property satisfies π(F) ≤ 9( p − 1). Proof Suppose, without loss of generality, that  has constant width 1 and that 0 ∈ . Let Q ⊂ E2 be such that F = {x +  | x ∈ Q}. We shall prove that since F satisfies the ( p, 2)-property, there are p − 1 disks of radius 1 containing Q. Indeed, let Q  be a subset of Q with the property that the distance between any two members of Q  is greater than 1, and assume furthermore that Q  is maximal with at most p − 1 points; otherwise respect to this property. First note that Q  contains it would contradict the  ( p, 2)-property. Next, we prove that Q ⊂ q∈Q  B(q, 1). By definition, Q  ⊂ q∈Q  B(q, 1). Suppose now q ∈ Q \ Q  ; then q ∈ B(qi , 1), for some qi ∈ Q  , otherwise it would contradict the maximality of Q  . Our next step is to show that Q can be covered with 9( p − 1) translated copies of . To see this, let C be the incircle of the Reuleaux Triangle. It is easy to see by inspection that the unit disk can be covered by nine translated copies of the ball B whose boundary is C. But this implies that Q can be covered with 9( p − 1) translated copies of B. Note now that, by Theorem 15.1.2, there is a translated copy of B inside  and therefore Q can be covered with 9( p − 1) translated copies of . That is, 

9( p−1)

Q⊂

(xi + ).

1

It is now easy to verify that the points xi pierce F. This concludes the proof.



15.1.2 Hyperplane Systems and the Kneser Conjecture This section is devoted to prove a deep and interesting combinatorial problem. We shall generalize the notion of system of lines in the plane developed in Chapter 5 to Euclidean n-space and hyperplanes. A system of hyperplanes H in En consists of choosing continuously one hyperplane of En in every direction. That is, a system of hyperplanes H in En is completely determined by a function ϕ : Sn−1 → E such that for every x ∈ Sn−1 , ϕ(−x) = −ϕ(x), where, as usual, Sn−1 is the unit sphere in Euclidean n-space. To see this, given ϕ, it is enough to choose, for every x ∈ Sn−1 , a hyperplane perpendicular to x through ϕ(x)x, which turns out to coincide with the hyperplane perpendicular to −x through −ϕ(−x)x, and vice versa.

15.1 Helly’s Theorem and Constant Width

347

Figure 15.2 Kneser graph K G(5, 2)

In other words, for every x ∈ Sn−1 consider the hyperplane H (x) = {y ∈ En | y, x = ϕ(x)}. Then, as before, we may define a system of hyperplanes as the collection H of all hyperplanes of the form H (x) for every x ∈ Sn−1 called the system of hyperplanes with pedal function ϕ(x). Note that a system of externally simple lines is precisely a system of hyperplanes when n = 2. The following lemma follows the spirit of Dol’nikov. Lemma 15.1.1 Given n systems of hyperplanes H1 , H2 , . . . , Hn in En , there is a direction in which all of them coincide. Proof For every system of hyperplanes Hi , let ϕi : Sn−1 → E be such that ϕi (−x) = −ϕi (x) for every x ∈ Sn−1 , its correspondent function. Define  : Sn−1 → En−1 by (x) = (ϕ1 (x) − ϕn (x), . . . , ϕn−1 (x) − ϕn (x)) ∈ En−1 , for every x ∈ n−1 S . Note that (−x) = −(x). Next, by the Borsuk–Ulam Theorem (see [802]), there is x0 ∈ Sn−1 ,  such that (x0 ) = 0 ∈ En−1 . That is, ϕi (x0 ) = ϕn (x0 ), for i = 1, ..., n − 1. The following conjeture was proposed by Kneser in 1955. It was open for 22 years, until Lovász [732] found a solution of it using topological methods. Conjecture (Kneser 1955) Whenever the k-subsets of an m-set are colored with m − 2k + 1 colors, then two disjoint k-subsets end up having the same color. Lovász’s proof of the Kneser Conjecture can be considered as the starting point of a new area in mathematics: combinatorial topology. Since this proof appeared, many other proofs have been published; for example, we refer to [72], [450], [298] and [1022]. We present the easiest one here, using the idea of systems of hyperplanes. Proof of the Kneser Conjecture Suppose the conjecture is not true. Without loss of generality, we may assume that our m-set is V = {x1 , . . . , xm } ⊂ Em−2k+1 , a set of points in general position. A k-gon of V is the convex closure of a k-subset of V . We assume that the k-gons of V are colored with m − 2k + 1 colors in such a way that two k-gons with the same color always intersect. Let L be a line through the origin. Choose a color, say red. Then the projection of every red k-gon is a compact red interval contained in L. Moreover, this collection of red intervals is mutually intersecting. Then, by Helly’s Theorem for the line 2.8.1 the intersection of all these red intervals is again a compact interval. Take a hyperplane HL perpendicular to L through the midpoint of the intersection of all red

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Bodies of Constant Width in Discrete Geometry

intervals. Clearly, HL varies continuously with respect to L, forming a red system of hyperplanes H in Em−2k+1 . Similarly, we obtain a system of hyperplanes for every color. By Lemma 15.1.1, there is a direction in which all these m − 2k + 1 systems of hyperplanes in Em−2k+1 coincide. But this implies that there is a hyperplane  that transverses all k-gons of V . To the left of  in Em−2k+1 , we have fewer than k points of V ; otherwise  would not be transversal to all k-gons of V . Similarly, to the right of  in Em−2k+1 , we have fewer than k points of V , and so there are m − 2k + 2 or more points of V in the hyperplane , contradicting the fact that V ⊂ Em−2k+1 is a set of points in general position.  In graph theory, the Kneser graph K G(n, k) is the graph whose vertices correspond to the k-element subsets of a set of n elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint, see Figure 15.2. Kneser graphs are named after Martin Kneser, who first investigated them in 1955. If fact, by Exercise 15.9, Kneser’s conjecture is equivalent to the fact that if 2k < n then the chromatic number χ(K G(n, k)) = n − 2k + 2.

15.2 Universal Covers 15.2.1 Introduction A convex body U is a universal cover of En if any subset X ⊂ En of diameter 1 can be covered by a congruent copy of U. By the Theorem of Pál (Theorem 7.2.2), for any set X of diameter 1 there is a body of constant width 1 containing X , and so we only need to consider bodies of constant width 1 when verifying that U is a universal cover of En . The problem of finding the smallest universal cover of a given shape has been of interest to mathematicians for many years. For example, the solution to the problem for balls was obtained by Jung [599] as an application of the Theorem of Helly (Theorem 2.8.1), see Theorem 2.8.4.  n . Theorem 15.2.1 The smallest ball that is a universal cover of En has radius 2n+2  Proof Jung’s Theorem (Theorem 2.8.4) proves that the ball of radius

n 2n+2

is a universal cover. It is the  n smallest one, because the circumsphere of the regular simplex of diameter 1 has radius 2n+2 .  For cubes we have Theorem 15.2.2 The smallest cube that is a universal cover of En has sides of length 1. Proof It will be enough to prove that any body of constant width 1 can be covered by a cube with side length 1, but this is obvious. Furthermore, a cube of sides of length h, with h < 1, does not contain a ball of diameter 1.  In fact, in Theorem 15.2.2, we have proved a slightly stronger result than the one claimed. We have shown that the cube of side length 1 is a strong universal cover because any set of diameter 1 can be a cover by a translate of the cube of side length 1 and not merely a congruent copy. We will address this notion in a later section.

15.2 Universal Covers

349

Figure 15.3 A regular hexagon circumscribed about a Reuleaux triangle

15.2.2 Universal Covers in the Plane In 1914, the famous mathematician Lebesgue sent a letter to Pál in which he asked: what is the smallest possible area of a region S in the plane such that every planar set of diameter 1 can be rotated and translated to fit inside S? Pál set out to find universal covers, and in 1920 he wrote a paper on his results. He found a√very nice one: a regular hexagon circumscribed around a circle of diameter 1. This hexagon has area 3/2 ≈ 0.86602540, and its sides are of length √13 . (For more history we refer also to [272]). Theorem 15.2.3 The smallest regular hexagon that is a universal cover in E2 has sides of length

√1 . 3

Proof A semi-regular hexagon in the direction u(θ) is a hexagon whose opposite sides are parallel to u(θ), u(θ + π/3) and u(θ + 2π/3) respectively, and the band described by two opposite parallel sides has width one. Let  be a body of constant width one. The proof of our theorem will be complete if we circumscribe a regular hexagon to the constant width body (see Figure 15.3). For this purpose, for every θ, we will circumscribe  about a semi-regular hexagon in the direction u(θ). Denote by L(θ) the directed support line of  in the direction u(θ). That is, L(θ) is parallel to u(θ) = (cos θ, sin θ), and looking in this direction leaves  to the right. Let us denote by L + (θ) the halfplane determined by L(θ) that contains . Thus,  is circumscribed about the band L + (θ) ∩ L + (θ + π) of width 1. Let us now consider the parallel lines L(θ + π/3) and L(θ + 4π/3). The body  is now circumscribed about L + (θ) ∩ L + (θ + π) ∩ L + (θ + π/3) ∩ L + (θ + 4π/3), which is an equilateral rhombus (the side-by-side union of two equilateral triangles) of side length √23 . Let p(θ) be the intersection of the diagonals of this equilateral rhombus, see Figure 15.4. Clearly, p(θ) is a point that varies continuously with respect to θ. Finally, consider the directed lines L(θ + 2π/3) and L(θ + 5π/3). The semi-regular hexagon in the direction u(θ) limited by these six lines clearly circumscribes . Denote by a(θ) the signed distance from p(θ) to L(θ + 2π/3). Note that a(θ) ≥ 0; otherwise the width of  would be smaller than one. Of course, a(θ) varies continuously with respect to θ. Similarly, let b(θ) > 0 be the signed distance from p(θ) to L(θ + 5π/3). Here is the crucial step of the proof: if for some θ0 we have a(θ0 ) = b(θ0 ), then L + (θ0 ) ∩ L + (θ0 + π) ∩ L + (θ0 + π/3) ∩ L + (θ0 + 4π/3) ∩ L + (θ0 + 2π/3) ∩ L + (θ0 + 5π/3) is a regular hexagon that circumscribes .

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Figure 15.4 A figure  inscirbed in an equilateral rhombus of side length

√2 3

To finish the proof, let us consider the continuous function f (θ) = a(θ) − b(θ), and note that f (θ + π) = − f (θ). If f (θ) = 0, then the proof is finished; if not, we may assume, without loss of generality, that f (θ) < 0. Since f (θ + π) > 0, by continuity of the function f (θ) there is θ0 such that f (θ0 ) = 0. This completes the proof.  Since a circle of radius √13 can be inscribed in a regular hexagon of side length strengthens Jung’s Theorem (Theorem 2.8.4) considerably.

√1 , 3

this theorem

√ Theorem 15.2.4 The smallest equilateral triangle that is a universal cover in E2 has sides of length 3. √ Proof The theorem follows from the obvious fact that an equilateral triangle of side length 3 can cover a regular hexagon of side length √13 . Furthermore, that no smaller such triangle is a universal cover follows again from the disk of diameter 1.   So far, we have as universal covers of E2 the unit disk of radius 13 , the square of side length 1, the √ triangle of side 3, and the hexagon of side length √13 . Among all of these, the one with the smallest √ area is the hexagon whose area is 3/2 ≈ 0.86602540. What we know today as Lebesgue’s universal covering problem is the following: What is the greatest lower bound of the area A of the closed convex set  ⊆ E2 which is a universal cover of E2 ? In his original paper, Pál showed that two corners of√his hexagon could be cut off, leaving a smaller universal cover. The resulting hexagon has area 2 − 2/ 3 ≈ 0.84529946. He guessed that this solution

15.2 Universal Covers

351

was optimal. However, in 1936 Sprague sliced off two tiny pieces from Pál’s proposed solution and found a universal cover with area ≈ 0.84413770. But in 1975, Hansen showed that two very tiny corners could be sliced off from Sprague’s solution, one of which reduces the area by just 6 · 10−18 and the other by 4 · 10−11 . And in the very recent paper [62] the possibly newest universal cover with smaller area than those known so far is presented: it yields area at most 0.8441153. We believe this is the smallest convex universal cover known so far. All references until now can be found in [272]. In 2005, Brass and Sharifi [183] came up with a nice lower bound. They showed that any convex universal cover must have an area of least 0.832. They obtained this result by a rigorous computeraided search for the smallest possible area of a convex set containing a circle, equilateral triangle, and pentagon of diameter 1. Hence we know that A satisfies the inequality 0.832 ≤ A < 0.8441153 Another interesting problem is to determine the smallest regular k-gon that is a universal cover in E2 . We have shown the answer for k = 3, 4, 6. The general answer is the regular k-gon that circumscribes a disk of diameter 1. This is, however, not true for k > 6, as we can see from Exercise 15.15. Indeed, as k increases, the circumscribed k-gon approaches the circle, and the smallest disk which is a universal cover has diameter √23 .

15.2.3 Universal Covers in n-Space In 1953, Gale [386] proved that the regular n-simplex that circumscribes a sphere of diameter 1 will cover any n-dimensional set of diameter 1. That is, we have the following generalization of Theorem 15.2.4. (We include Gale’s original proof here as an example of geometrical and linear algebraic arguments.) Theorem 15.2.5 The smallest regular n-simplex that is a universal cover in En has sides of length  n(n+1) . 2 Proof Let S be a closed subset of n-space having diameter 1. Let e0 , . . . en be the standard orthonormal n+1 0), e1 = (0, 1, . . . , 0), . . . , en = (0, . . . , 0, 1); and let eˆ = basis n of E . That is, e0 = (1, 0, . . . ,n+1 e = (1, 1, . . . , 1). Let L = {x ∈ E | x eˆ = 0}. In this proof we identify n-space with L, and 0 i hence we may assume that S ⊂ L. Next, we have the following definitions:  • Let αi = min{xei | x ∈ S}, and let a = n0 αi ei . • Let βi = max{xei | x ∈ S}, and let b = n0 βi ei . {x | xei ≥ αi }, Hi+ = • Hi− =   {x | xei ≤ βi }. − •  = n0 Hi− ∩ L,  + = n0 Hi+ ∩ L. − The closed set S of diameter 1 is contained in both  − and  + . We shall show that   and  + are . The regular n-simplices and that at least one of them has diameter smaller than or equal to n(n+1) 2 proof will be divided into four parts.

(1) (2) (3) (4)

We first show that a eˆ < 0 and beˆ > 0. Next, we show that  − and  + are n-simplices. We show that  − and  + are regular n-simplices. √ − ˆ 2 and  + is a regular simplex Finally, we show that √  is a regular simplex of side length −a e/ of side length be/ ˆ 2.

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(1) We first show that a eˆ ≤ 0 and a eˆ = 0 only if S consists of a single point. For any x∈ S, xei ≥ αi , ˆ but x ∈ L, so x eˆ = 0; so a eˆ ≤ 0. If a eˆ = 0, then n0 αi = 0, and hence n0 xei =x eˆ ≥ n0 αi = a e, n for any x ∈ S, 0 xei = 0. But xei − αi ≥ 0, so n0 (xei − αi ) = 0 implies that xei = αi for all i, hence x = a. Similarly we show that beˆ ≥ 0, and equality holds only if S = {b}. ˆ i. (2) We will show that  − is a simplex spanned by the vectors v0 , v1 , . . . , vn , where n vi = a − (a e)e ; that is, x = λ v , where λ First, suppose that x is a convex combination of the vectors v i ≥0 0 i i  n i  (a − (a e)e ˆ  ) = a − (a e) ˆ λ e , and so xe = α − λ (a e) ˆ ≥ αi and n0 λi = 1. Then x = n0 λi i i i i 0 i i since by (1), a eˆ < 0. Also, x eˆ = n0 xei = n0 αi − a eˆ = 0, and so x ∈  − . n On the other hand, suppose that x ∈  − . Then x = n0 μi ei , where μi = i ≥ αi and 0 μi = 0. xe n = (1/a e)(a ˆ − v ), so x = (1/a e) ˆ μ (a − v ) = −(1/a e) ˆ Now, since a e ˆ  = 0, we can write e i i i 0 i n n μ v . From the definition of v we see that α v = 0, and so we may write i 0 i i 0 i i x = −(1/a e) ˆ

n 

μi vi + (1/a e) ˆ

0

n 

αi vi = (1/a e) ˆ

0

n  (αi − μi )vi .

(15.1)

0

Let us verify that  this is a convex combination. Note that (αi − μi )/a eˆ ≥ 0 since μi ≥ αi and n n a eˆ < 0. Also, (1/a e) ˆ 0 (αi − μi ) = 0 αi /a eˆ = 1. Note that S is inscribed in  − because there is an x ∈ S such that xei = μi = αi . Using (15.1), we see that x lies in the (n − 1)-face of  − opposite to the vertex vi . The proof for  + is similar. This concludes the proof of (2). (3) Working with  − for i = j, we have vi − v j = −a e(e ˆ i − e j ), and so √     2 vi − v j  = (a e) ˆ 2 ei − e j  = −a e/ ˆ 2. Similarly, each side of  + has length

√ 2be. ˆ To conclude the proof it  will be enough to prove that at

least one of the regular simplices has diameter smaller than or equal to n(n+1) . This will be done by 2 √ showing that the sum of their diameters is smaller than or equal to 2n(n + 1). Indeed, let us verify that, for all 0 ≤ i ≤ n,  βi − αi ≤ n/n + 1.

It suffices  to show this forβ0 − α0 . Choose x, y ∈ S such that xe0 = α0 , ye0 = β0 , and  let x = + n0 γi ei , y = β0 e0 + n0 δi ei . Further, let θ0 = β0 − α0 , θi = δi − γi for i > 0. Then n0 θi = 0 α0 e0 and n0 θi2 ≤ 1, using for the first time the hypothesis that the diameter of S is 1. Now, 0≤

n n n n

2     θi + θ0 /n = θi2 + (2θ0 /n) θi + θ02 /n = θi2 − θ02 /n, 1

and thus θ02 /n ≤

1

n 1

1

1

 θi2 and (n + 1)/n θ02 ≤ n0 θi2 ≤ 1, which gives β0 − α0 = θ0 ≤



n/(n + 1).

Finally, we know that the sum of the lengths of the sides of  − and  + is n  √  √ 2(beˆ − a e) ˆ = 2 (βi − αi ) ≤ 2n(n + 1) , 0

15.2 Universal Covers

353

Figure 15.5 The octahedron

 and hence the side length of at least one of them is smaller than or equal to n(n+1) . 2  This proves that the regular n-simplex of side length n(n+1) is a universal cover in En . It is the 2 smallest, because the diameter of the regular  n-simplex that circumscribes the sphere of diameter 1 can easily be computed, and it is exactly

n(n+1) . 2



In dimension 2, we can use the above theorem to conclude that every set of diameter 1 can be embedded in a regular hexagon whose opposite sides are at distance 1. In his paper, Gale claims that the above ideas can be used to show that every set of diameter 1 can be embedded in a regular octahedron (Figure 15.5) whose opposite faces are distance 1 apart. With an approach following the spirit of the proof of Theorem 15.2.3, we next present a proof of Gale’s claim. As a consequence, we will obtain a new proof of Theorem 15.2.5. Assume that the ball of diameter 1 centered at the origin is circumscribed about a regular n-simplex  T whose sides have length

n(n+1) . 2

Let us consider the following polyhedra: n = T ∩ (−T ).

For n = 2, 2 is a regular hexagon with sides of length √ sides of length 3/2.

√1 . 3

For n = 3, 3 is an octahedron with

Theorem 15.2.6 The polyhedron n is a universal cover in En . Proof It will be enough to prove that every convex body  ⊂ En of constant width 1 can be inscribed in n . For each isometry g ∈ S O(n), let T (g) be the regular n-simplex that circumscribes  and is positively homothetic to gT . Of course, the side length s(g) of T (g) varies continuously with g ∈ S O(n). Now let T ∗ (g) be the regular n-simplex that circumscribes  and is positively homothetic to −gT . Denote by s ∗ (g) the side length of T ∗ (g). Let f : S O(n) → E be the continuous function defined by f (g) = s(g) − s ∗ (g). Let g˜ ∈ S O(n) be such that gT ˜ = −T (as a set). Since f (id) = − f (g), ˜ by continuity and the fact that S O(n) is arc-connected, there has to be g0 ∈ S O(n) such that f (g0 ) = 0 and, therefore, such that T (g0 ) and T ∗ (g0 ) are congruent. Additionally, the corresponding pair of facets of T (g0 ) and T ∗ (g0 ) are parallel and distance 1 apart. So it is easy to see that the centroids of T (g0 ) and T ∗ (g0 ) coincide, and

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Figure 15.6 A minimal universal cover

that T ∗ (g0 ) is obtained as a central reflection from T (g0 ) through its centroid. It follows then that T is congruent to T (g0 ), −T is congruent to T ∗ (g0 ), and n is congruent to T (g0 ) ∩ T ∗ (g0 ). To conclude  the proof, we recall that  is circumscribed about T (g0 ) and also about T ∗ (g0 ). Corollary 15.2.1 The smallest octahedron that is a universal cover in E3 has sides of length

3/2.

In Section 16.3 we will characterize when a centrally symmetric polyhedron that circumscribes the sphere of diameter 1 is a universal cover in En , by using the theory of fiber bundles and also the following characterization of universal covers in En . A finite subset X of Sn−1 is called an affinely Knaster set if for every continuous odd function f : Sn−1 → E there is ρ ∈ S O(n) and a linear function L : En → E such that the function f coincides with L in ρX . The importance of affinely Knaster sets for the theory of universal covers is the following result: If a polytope P circumscribing the (n − 1)-dimensional sphere of diameter 1 has the property that its tangent points are an affinely Knaster set, then P is a universal cover, see Theorem 16.3.1. This kind of characterization is important because it will allow us in Section 16.3 to use the theory of fiber bundles to prove that the rhombic dodecahedron that circumscribes the sphere of diameter 1 is a universal cover in E3 .

15.2.4 Minimal Universal Covers A minimal universal cover is a universal cover no proper subset of which is also a universal cover. An explicit example of a plane minimal universal cover is the following. Let ϒ be the union of a disk and a Reuleaux triangle, both of diameter 1, placed in such a way that two of the vertices of the Reuleaux triangle are on diametrically opposite points on the disk; see Figure 15.6. Theorem 15.2.7 The compact set ϒ described above is a minimal universal cover of E2 . Proof Let us prove first that ϒ is a universal cover. For this purpose it will be enough to prove that any plane figure  of constant width 1 can be covered by a congruent copy of ϒ. By Theorem 5.3.3, there is a semicircle of diameter 1 contained in . Thus, there is a diameter pq of  with the property that at least one of the two semicircles of radius 1/2 that passes through pq does not meet the interior of . If this is so, since every point of  is at most at distance 1 from both p and q, it follows that  is contained in a convex figure bounded by a semicircle and, on the opposite side of pq, two arcs of unit radius with centers at p and q. Therefore  is contained in a congruent copy of ϒ.

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355

If ϒ is a universal cover, it must be a minimal one. Indeed no proper subset of ϒ can contain both a disk and a Reuleaux triangle of diameter 1.  Minimal universal covers of En also exist. Moreover, for n > 2 Eggleston [315] was able to construct, for any positive number N , a compact minimal universal cover of En whose diameter is greater than N . In contrast, for the plane the diameter of any minimal universal cover is less than 3.

15.2.5 Strong Universal Covers A convex body U is a strong universal cover of En if any subset X ⊂ En of diameter 1 can be covered by a translated copy of U. Roughly speaking, it is required that any set X of diameter 1 can be placed inside U in any orientation, in contrast to the case of ordinary universal covers, where X must  be placed inside U in n some orientation. For example, a cube of length side 1 and a ball of radius 2n+2 are strong universal covers of En . Of course every strong universal cover is a particular universal cover, but the regular hexagon of side length √13 is an example of a universal cover of E2 which is not a strong universal cover. Note that U ⊂ E2 is a strong universal cover if and only if every plane set X of diameter 1 can be “turned” through 360◦ inside U . Theorem 15.1.2 enables us to detect strong universal covers, because every compact set X that can be turned 360◦ degrees inside the Reuleaux triangle can be turned 360◦ degrees inside any constant width figure of the same width. Theorem 15.2.8 Any plane convex body of constant width λ0 is a strong universal cover, where λ0 be the smallest value of h such that the Reuleaux triangle of width h is a strong universal cover, Using Theorem 15.1.2, we will prove in Section 17.2 the following interesting covering result for curves in the plane, originally proved by Bezdek and Connelly in [116]. Let  be a figure of constant width 1 in E2 . Then every closed curve C of perimeter 2 can be covered by a translate of . Furthermore, if φ is any compact convex set with the property that every closed curve C of perimeter 2 can be covered by a translate of φ, then the perimeter of φ is equal to or greater than π, with equality if and only if φ has constant width 1.

15.3 Packing, Covering, Lattice Points 15.3.1 Packing and Covering Let , 1 , 2 , . . . , be convex bodies in En . The family {i } is a packing in  if i ⊂  and int i ∩ int  j = ∅, for i = j. The packing {i } is said to be translative if there is a convex body  such that i is a translated copy of ; i.e., if for every i > 0, there is xi ∈ En such that i =  + xi . The family {i } is a covering in  if  ⊂ i>0 i . The covering {i } is said to be translative if there is a convex body  such that i is a translated copy of , for every i > 0.

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If U is an open set of En and U ∗ =

1 2

Bodies of Constant Width in Discrete Geometry

U + (−U ) , then

U ∩ (U + x) = ∅ if and only if U ∗ ∩ (U ∗ + x) = ∅.

(15.2)

This is so because the condition U ∩ (U + x) = ∅ is equivalent to the existence of two points p, q ∈ U such that p = q + x. This can also be stated as 21 x ∈ U ∗ . Correspondingly, the condition U ∗ ∩ (U ∗ + x) = ∅ is equivalent to 21 x ∈ U ∗∗ = U ∗ . This relation enables one to reduce many transitive packing problems for bodies of constant width to corresponding problems for balls. In particular, the problem of finding densest translative packings in En of bodies of constant width is equivalent to the problem of densest packings of balls. For this reason, the density of a densest translative packing of plane convex bodies of constant width is always √ π− 3 between √3 = 0.81379 . . . and √π12 = 0.90689 . . . For more about packings of balls see the survey [337] by G. Fejes Tóth.

15.3.2 The Borsuk Conjecture It is easy to see that it is possible to divide a square into two pieces in such a way that both pieces have diameter less than that of the original square. The same is also true, for example, for a non-equilateral triangle or, in fact, any rectangle, parallelogram, or ellipse (see Figure 15.7). It is, however, impossible to divide an equilateral triangle into two pieces that both have diameter less than that of the original triangle. No matter how the triangle is cut, one piece will contain two of its vertices; that piece must therefore have the same diameter as the triangle. It is easy, however, to verify that an equilateral triangle can be divided into three pieces that all have diameter strictly less than that of the original equilateral triangle. We consider the problem of dividing any set φ into pieces of smaller diameter. A partition of φ into pieces of smaller diameter is a collection of subsets of φ, each of which has diameter less than that of φ, and whose union is all of φ. We call the minimum number of necessary pieces β(φ). If φ is, for example, an equilateral triangle, then β(φ) is three; if φ is an ellipse or a parallelogram, β(φ) is two. Is there a plane figure for which β(φ) is four or even five? The eminent Polish mathematician Karol Borsuk [168] proposed the following problem in 1933. Is it true that for a set φ in n-dimensional Euclidean space β(φ) ≤ n + 1. holds? Although this problem is usually called “Borsuk’s conjecture”, we underline that Borsuk was careful enough to pose it only as a problem.

Figure 15.7

15.3 Packing, Covering, Lattice Points

357

Since every set of diameter h is contained in a body of constant width h, we may restrict our study of Borsuk’s conjecture to constant width sets, that is, the conjecture is equivalent to the following claim: if  is an n-dimensional constant width body, then β() ≤ n + 1. The reader should also consult Section 7.5 regarding this partition conjecture, and in Sections 10.6 and 10.7 it is also discussed for normed spaces. Let us prove the Borsuk question for smooth convex bodies. Theorem 15.3.1 (Hadwiger) If  is an n-dimensional smooth convex body, then β() ≤ n + 1. Proof The proof of the case of a ball will be left as an exercise to the reader. Let us consider the Gauss map f : bd  → Sn−1 between the points in the boundary of  and the unit vectors as follows. For each point x of bd , let f (x) be the unique outward normal unit vector to  at x; that is, a unique support half-space of  touching x, f (x) then being the unit vector perpendicular to this half-space that points away from it. Let A0 , A1 , . . . , An be a partition of Sn−1 into pieces of diameter less than two. Note that each of the subsets f −1 (A0 ), f −1 (A1 ), . . . , f −1 (An ) of bd  have diameter smaller than the diameter of . Otherwise, if p and q are in f −1 (Ai ) and the segment pq is a diameter of , we have, since pq is a binormal of , that there is v ∈ Sn−1 such that f ( p) = v and f (q) = −v, which is a contradiction, since the diameter of Ai is less than 2. Finally, to partition  into n + 1 pieces of  smaller diameter, it suffices to take an interior point a of  and then each cone f −1 (Ai ) from a. Note that the boundary of a body  of constant width h cannot be the union of two sets R and T of diameter smaller than h. Assume that bd  = R ∪ T and, without loss of generality, that R and T are both closed subset of . Since the boundary of  is connected, there must be a point p ∈ R ∩ T . Let pq be a diameter of . Then, as q is either in R or in T , one of these two pieces will have diameter h. At this point it is interesting to note that every partition of the sphere Sn−1 into n pieces has the property that some of the pieces contain two antipodal points, which implies that β(Sn−1 ) = n + 1. The same is true for the boundary of every n-dimensional body of constant width , implying that β() ≥ n + 1. The proof of this assertion uses sophisticated topological techniques; in particular, it uses the fact that the Lyusternik–Schnirelmann category [738] of the n-dimensional projective plane is n + 1 (see also [843]). For n = 2, the conjecture claims that every planar set can be partitioned into three pieces whose diameter is less than that of the set. In fact, for this case Borsuk himself gave a proof based on the elegant Theorem of Pál (Theorem 15.2.3), which says that every plane figure of unit diameter is contained in a regular hexagon of unit width, which in turn may easily be partitioned into three congruent pieces √ of diameter 3/2 (see Exercise 15.10 e). Next we present another proof based on our knowledge of constant width figures. Theorem 15.3.2 For every set S in the plane, β(S) ≤ 3. Proof By the Theorem 7.2.2 (of Pál), we can also assume that S is a figure of constant width one; furthermore, since the conjecture is true for smooth convex bodies (see Hadwiger’s Theorem 15.3.1), we can also assume that S has corner points. It suffices to partition the boundary of S into three subarcs of diameter less than one. Let p be a corner point of bd S. Then the boundary of S contains a circular arc with center at p and unit radius. Let q be an interior point of this arc and note that q is not a corner point of bd S. We can then choose points s and t in bd S sufficiently close to p that the diameter of the subarc spt of bd S is less than one and that neither sq nor tq is a diameter of S. We claim that the subarcs sq and tq of bd S, which do not contain p, have diameter less than one. For otherwise there would be a diameter mn of S with endpoints m and n, say in the subarc sq. By

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Theorem 3.1.6, the diameter mn must intersect the diameter pq, but if this is so, then, without loss of generality, m = q. Since q is a regular point of the boundary of S, this implies that n = p, which is a contradiction. This partitions the boundary of S into three subarcs of diameter less than one, which results in a partition of S into three pieces of diameter less than one.  We now use spindle h-convexity to characterize, in terms of constant width extensions, for which sets S the value of β(S) is two, and for which it is three? There are cases in which the completion of a figure of diameter h can only lead to a unique figure of constant width h. This is, however, not true for a square of diameter h, since it is contained in a circle of diameter h and also in the noncircular shape of constant width h shown in Figure 1.9. In contrast, as has already been shown, an equilateral triangle (and, more general, any regular polygon with an odd number of sides) is contained in a unique constant width set of the same diameter. As seen by the following theorem, Boltyanski and Gohberg [150] related this concept to the conjecture of Borsuk in the planar case. Theorem 15.3.3 A planar set S of diameter h may be uniquely completed to form a figure of constant width h if and only if β(S) = 3. Proof Suppose that S may be completed in two different ways to form two different constant width figures. Our first purpose is to prove that there are two disks of radius h that contain S and whose centers x1 and x2 are more than h apart. If this is so, it is left to the reader to show that the line that passes through the points x1 and x2 partitions the intersection of the disks, and therefore S, into two pieces of diameter less than h. So, let us assume that S ⊂ 1 ∩ 2 , where 1 and 2 are two different figures of constant width. Take x ∈ 1 \ 2 . Then 1 ⊂ B(x, h), and therefore S ⊂ B(x, h). If 2 is not contained in B(x, h), take x2 ∈ 2 \ B(x, h). By the previous arguments we have S ⊂ B(x2 , h) and, furthermore, |x − x2 | > h. If 2 ⊂ B(x, h), for every x ∈ 1 \ 2 , then by Theorem 6.1.4, 2 ⊂ 1 , and therefore 2 = 1 . Suppose now that S is uniquely completed to form a figure of constant width h, but that S can be partitioned into two pieces R and T of diameter less than h. We will arrive at a contradiction. By Lemma 6.1.4, cch (S) is a figure of constant width h. Let us start proving that bd cch (S) ⊂ cch (R) ∪ cch (T ). Take m to be a point in cch (S). If m is a point of S, then m must be either in cch (R) or in cch (T ). If m is not a point of S, then, by Lemma 6.1.5, m is in a subarc of bd cch (S) with endpoints a and b in S, which is a circular arc with radius h and center in n. As cch (S) has constant width h, n is a vertex point of bd cch (S) which, again by Lemma 6.1.5, must be in S and consequently either in cch (R) or in cch (T ). Suppose that n is in cch (R). Since the distance from n to a and to b is h and the diameter of cch (R) is less than h, it follows that the points a and b are in cch (T ) and, by spindle h-convexity, that the point m is in cch (T ). The fact that bd cch (S) ⊂ cch (R) ∪ cch (T ) is a contradiction, because by Exercise 6.12, the h-convex hull of a set of diameter smaller than h has also diameter smaller than h, and hence the  boundary of cch (S) can be partitioned into two sets of diameter smaller than h. The higher dimensional analogue of this theorem does not appear to be true. The spindle h-convex hull of an equilateral tetrahedron with unit sides is the intersection of the four balls with unit radius whose centers are at its vertices. We know that this body is not of constant width, and therefore the equilateral tetrahedron cannot be uniquely completed to a body of constant width. Nevertheless, the equilateral tetrahedron necessarily needs to be partitioned into four pieces whose diameter is less than one. In three dimensions, the Borsuk conjecture was independently verified by Perkal and Eggleston [310] using some rather complex arguments. Some time later, and also independently, Grünbaum [486] and

15.3 Packing, Covering, Lattice Points

359

Heppes [530] showed that every 3-dimensional set of unit diameter is contained in a regular octahedron of unit width, three of whose vertices are cut by planes perpendicular to the main diagonals, 1/2 unit from the center, giving a partition into four pieces with diameter less than 0.98867. In higher dimensions, the conjecture has been proved in certain special cases: when φ is a smooth convex body (see Hadwiger [501]), when φ is centrally symmetric (see Riesling [972]), when φ is a body of revolution (cf. [651] and [290]), and when φ is invariant under the group of symmetries of the regular n-dimensional simplex (Rogers [982]). The Borsuk problem was negatively solved in 1993 by Kahn and Kalai [600], who showed that n + 1 pieces do not suffice for n = 1325 and for all n > 2014. After that, Bondarenko [157] showed that there are counterexamples for the conjecture for n ≥ 65. The current best bound, due to Jenrich [580], is 64. Apart from finding the minimum number n of dimensions such that the number of pieces α(n) > n + 1, we are also interested in finding the general behavior of the function α(n). Kahn and Kalai √ showed that for n large enough, the number of pieces needed is α(n) ≥ (1.2) n . The correct order of magnitude of α(n) is still unknown; however, it is conjectured that there is a constant c such that α(n) > cn for all n ≥ 1.

Notes Helly’s Theorem and Discrete Geometry A prominent role in combinatorial geometry is played by Helly’s theorem, it has stimulated numerous generalizations and variants. There are many interesting connections between Helly’s theorem and its relatives, the theorems of Radon, of Carathéodory and of Tverberg as we partially mentioned already in Section 2.8. These theorems have been the object of active research, and they inspired many problems in the field of discrete geometry. We mention here the excellent surveys [278] and [306] to see a sample of numerous problems associated to Helly’s theorems. One of the most beautiful theorems in combinatorial convexity is due to Tverberg, that is, the r -partite version of Radon’s Lemma. To be more precise, Tverberg’s theorem states that every (n + 1)(r − 1) + 1 points in Euclidean n-space En can be partitioned into r parts such that the convex hulls of these parts have nonempty intersection, see [1138]. This theorem still remains central and is one of the most intriguing results of discrete geometry. It has been shown that there are many close relations between Tverberg’s theorem and several important results in mathematics, such as Rado’s Central Theorem on general measures, the Ham-Sandwich Theorem, and the Four-Color Theorem, just to mention some examples. Tverberg’s theorem is closely connected with the multiplied or colorful versions of the Theorems of Helly, Hadwiger, and Carathéodory, first studied by Bárány and Lovasz, see [40]. In fact, there is a topological version of Tverberg’s theorem that has received much attention in the last decades. During the 90s, techniques and ideas of algebraic topology were used in a relevant and deep manner to study this version, and nowadays, due to the influence of Gromov’s topological ideas, the late developments of this problem became an important area of research. See [74] and the beautiful book of Matoušek [802]. Helly-type theorems explicitly referring to constant width sets can be found in [29] and [30]. For example, in [30] it is proved that if every n + 1 sets of a family of compact convex sets in nspace simultaneously contain a set of width h > 0 (respectively, a set of constant √ width h), then all , where h = h/ n if n is odd, and members of this collection contain a set of constant width h 1 1 √ h 1 = h n + 2/(n + 1) if n is even. The author calls such a set a common set (of constant width h 1 ) of the whole family.

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Universal Covers The notion of universal covers clearly plays an essential role regarding Borsuk’s partition problem (namely, by dissecting universal covers of diameter-one sets into pieces each having diameter less than 1); see [488], [272, D 15], [151, Chapters V and VII], and [182, Section 11.4]. A nice survey on constant width sets mainly devoted to universal covers (but discussing also other problems from discrete geometry) is [1192], also updating [238]. To construct universal covers having constant width sets in mind, Hansen [514] used coverings of Reuleaux polygons by polygons. Chakerian and Logothetti [239] proved that the smallest convex m-gon (m > 3) being a universal cover coincides with the smallest regular m-gon that covers the Reuleaux triangle of width 1. This is based on Pál’s classical result that a plane constant width set has a circumscribed regular hexagon [909] (see Theorem 15.2.3). For solving a problem of Klee, Eggleston [315] showed that the known union of a circular disk and of a Reuleaux triangle, both of unit diameter and placed so that two of the vertices of the Reulaux triangle are contained in the disk, is a universal cover, see Section 15.2.4. Makeev [758] proved the following: let M be the intersection of finitely many half-spaces in En , and X (M) be the union of all bodies of constant width 1 contained in M. Then the polyhedral set X (M) is a semi-algebraic set, and by this construction a possible universal cover M can be made smaller by substituting X (M) for M. Also the paper [757] refers to universal covers and constant width bodies. Namely, a value m(ε) is established such that, for m > m(ε), each constant width body in En has an ε-aspherical orthogonal projection of dimension m (meaning the deviation from the spherical shape measured via the ratio of diameters of in- and circumspheres). The proof is based on universal covers considered in Em which are m-polytopes with 2m facets. Related is also the exposition [760], presenting results on 3-dimensional polytopes having homothets that are inscribed in or circumscribed about a 3-dimensional compact set. The cases with 3, 4, 5, 6, and 7 facets are discussed in detail, and also the problem of polyhedra circumscribed about constant width bodies is taken care of. Related results are also presented in [762]; e.g., for n = 3 two one-parameter families of polytopes with 12 facets circumscribed about any constant width body are presented. Packing and Covering We start with packings. As we already know, for an open convex subset G of En and p ∈ En , G ∩ (G + p) = ∅ ⇔ G ∗ ∩ (G ∗ + p) = ∅ holds (see [238, p. 77]), where G ∗ denotes the central symmetral 1 (G + (−G)) of G. This equivalence allows the reduction of many translative packing problems for 2 constant width bodies to corresponding problems for balls (taking G as their interior), and thus literature about ball packings is also relevant here (see, for example, the excellent surveys [341], [342], [338], and [339], as well as the book [163]). To give an example: finding the densest translative packings of constant width bodies in En is equivalent to finding the densest corresponding ball packings. (Unfortunately, there is no corresponding reduction for analogous covering problems; see [453].) For a convex body K in E2 , let K 1 , K 2 be two nonoverlapping smaller homothets of K contained in K . It is easy to see that if K is of constant width, then for the perimeters the inequality p(K 1 ) + p(K 2 ) ≤ p(K ) holds. Somehow conversely, Beck and Bleicher [94] showed that the following holds: if for K , K 1 , K 2 being such convex bodies in the plane the same inequality holds, then K is either of constant width or a regular polygon. They also considered the concept of constant minimal width, see Section 3.2. Finite packings of Reuleaux triangles are sometimes nicely visible in Gothic church windows, see Figures 1.4 and 18.2. The authors of [138] give an estimate for the packing density of the Reuleaux triangle, understanding it as a good example for investigating packings of nonsymmetric figures. To find the densest translative packings of Reuleaux triangles is not hard, but if congruent copies are allowed, the problem is difficult and (to our best knowledge) unsolved. Continuing [138], in [139] the case is studied when the Reuleaux triangles are not necessarily translates of each other, but the packings still satisfy a certain regularity condition. If I is one of the 2-dimensional discrete groups of

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isometries and no two copies of a Reuleaux triangle R following I overlap, then we have an I -packing of Reuleaux triangles. Besides other results, it is also shown in [139] that the densest I -packing of a Reuleaux triangle occurs when I is the group generated by two rotations of period 3. We refer also to [970], where results on densest packings of Reuleaux triangles are independently re-obtained. Investigating densities of packings in E3 , the authors of [110] used also certain affine images of bodies of constant width, see Section 11.6. Another interesting problem in which only the case of balls has to be considered is that of finding the largest number of nonoverlapping translates of a given body of constant width  that can touch  without intersecting int . We now discuss some results regarding a translative packing problem that cannot be reduced to a corresponding packing of balls. It concerns the permeability of a layer of plane convex figures in E2 . For a given figure of constant width , let {i } be a translative packing in a strip of width t. Assume that t is minimal for the given collection {i } and let λ be the infimum of the lengths of all continuous rectifiable curves connecting the two boundary lines without meeting interior points of any i . The permeability of {i } is defined by t α({i }) = . λ In [552], Hortobágyi proved that √ α({i }) ≥

27

√ , 2π(t − h) + h 27

√ is an integer. There are some improvements of where h is the width of . Equality is possible if 2(t−h) 3h this estimate due to Florian [364, 365]; he also studied the possibility of relating permeability estimates of {i } to those for {i∗ }. Packing problems of constant width bodies which are not necessarily translative were also studied. A typical problem of this kind concerns the Newton number. Given a figure of constant width , the Newton number N () of  is the largest number of congruent nonoverlapping copies of  that touch . It has been shown by L. Fejes Tóth [344] that N (T ) ≤ 7 for the Reuleaux triangle T . Moreover, it has been proved by Schopp [1044] that N () ≤ 7 for any figure of constant width  ⊂ E2 . On the other hand, Hortobágyi [551] has shown that any eight figures of the same constant width , 1 , · · · , 7 can always be arranged so that each i touches  and any two i do not overlap, thus proving that for any plane convex body of constant width , the largest number of congruent nonoverlapping copies of  that touch  is 7. That is, its Newton number N () = 7. We come now to coverings. A convex set K is called a covering set of a class M of sets in E2 if any member of M can be covered by a congruent copy of K ; and K is said to be a minimal covering set if it cannot be replaced in this framework by a proper subset. Answering Problem 28 from Chapter VIII of [151], Weissbach [1195] showed that there are sets which are minimal covering sets for the class of all planar sets of constant width, except for closed Reuleaux triangles, thus also suggesting a sharpened reformulation of this problem. Bezdek and Connelly [116] defined a set K to be a translation cover for a class W of sets if any member of W can be covered by a translate of K . They showed that every planar set of constant width 1 is a translation cover for the class of planar closed curves of length 1, and that in fact these are the 2 convex translation covers of minimum perimeter for this class, see Theorem 17.2.4. Their investigations were continued by [766]. In [513], it was proved that a closed plane curve of length 2π can be covered by a rectangle of area 4; if no smaller rectangle will do this, the curve under consideration has to be of constant width 2. Buchman and Valentine [191] proved that if each n + 1 members of a family of convex bodies in En can be covered by some set of constant width 1, say, then all members of the family can be covered this

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way, see Theorem 15.1.3. Further Helly-type conjectures posed in [191] (see also [272, pp. 131–132]) are investigated in [30]: if, for a given positive number h, every n + 1 members of a collection of of this compact convex sets in En simultaneously contain a set of constant width h, then all members √ collection contain a set of constant width h ∗ , where h ∗ = √hn for odd n, and h ∗ = 2h − h 2n/(n + 1) for even n. Moreover, an oracle-based algorithm is presented which determines a set of constant width that the members of the given collection have in common. Analogous results for a generalization of the notion of convex sets were derived in [29]. Eggleston [309] proved √ that any planar convex set of width h contains a convex subset of constant width at least h/(3 − 3); this bound is sharp for an equilateral triangle. On the other hand, it was proved in [317] that among the largest equilateral triangles which may be inscribed in different types of curves of constant width that one inscribed in a Reuleaux pentagon is the least. In [1094], it is shown that the sum of the diameters of three sets, each of diameter smaller than h, which can cover a set of constant width h, is larger than 2h. The author uses the fact that the perimeter of a triangle whose vertices are from the distinct arcs of Reuleaux triangles of width d has to be larger than 2h. Besicovitch [106] confirmed partially the following question of Klee: is it possible to inscribe any semicircle of diameter h > 0 in every set of constant width h. With restrictive boundary conditions he proved that three semicircles of diameter h can be inscribed in a set of constant width h. Eggleston [315] continued this by proving that every set of constant width h (without any boundary condition) contains a semicircle of diameter h. Cooke [264] proved that, again without boundary conditions, there are always at least three distinct such semicircles, and that this is best possible. In a simple geometric way this was reproved by [641], and in [330] this was finally extended to the configurations of plane convex containers of minimal width h, covering three semicircles of diameter h, see Theorem 5.3.3. Lutwak [741] showed that any closed curve of length 1 can be covered by a semicircular disk of radius π1 . Moreover, he showed that this radius cannot be decreased if and only if the curve has the same constant width. This implies that any two closed curves of length 1 can be positioned inside a circle of radius 1 such that their interiors are disjoint. Taking two sets of constant width π1 , one sees that the radius of π the circle cannot be decreased, and there are no other cases of this kind. As we already mentioned, the article [1192] refers to several interesting results on sets of constant width which are not covered by the survey [238]; these refer to universal covers, to the least number of translates of the interior of a set of constant width required to cover the original set, to cylindrical pipes of minimal radius through which any set of fixed constant width passes, and to equiangular polygons circumscriptible about constant width sets. For a given family W of convex bodies in En , let am (W ) be the largest number α such that every set A ∈ W contains a polytope P with at most m > n vertices such that the minimal width of P, denoted by w(P), has the lower bound αw(A). Among many other results, the authors of [451] take W also as class of all bodies of constant width, and they obtain lower bounds for am (W ), getting, e.g., the Reuleaux pentagon as an extremal figure. It is well known that the minimal number of directions necessary to illuminate the whole boundary of a convex body K equals the minimal number h(K ) of smaller homothets of K sufficient to cover K (see, e.g., [202, Chapter VI]). For this covering problem (often called Hadwiger’s covering problem) also results referring to constant width bodies were derived. For example, it was shown by Lassak [692] that for any 3-dimensional constant width body K the inequality h(K ) ≤ 6 holds. The same estimate can be obtained from more general results of Weissbach [1192]: if K is an n-dimensional body of constant width and N (π/4) denotes the smallest positive integer m such that the unit sphere can be covered by m congruent caps of spherical radius π/4, then N (π/4) is an upper bound for h(K ). Going √ further with this method, one gets h(K ) ≤ 2(1 + o(1)), n → ∞, for K ⊂ En having constant √width. n(4 + With a more complicated method, Schramm [1047] obtained the upper bound h(K ) ≤ 5n √ log n)(3/2)n/2 for constant width bodies K in En , yielding also h(K ) ≤ 3/2(1 + o(1)), n → ∞. For small dimensions, the estimate of Weissbach is better, in higher dimensions Schramm’s estimate is much stronger. Weissbach [1193] also studied a modification of the number h(K ) (in terms of

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363

illumination), namely suitably referring to all congruent copies of a convex body. He obtained respective upper bounds for constant width bodies in En . The Borsuk Problem We first refer to some basic surveys on Borsuk’s partition problem taking also the role of constant width sets into consideration, namely to [488], [272, D 14], [151, Chapter V, and Problems 13–17 in Chapter VIII], [960], [602], and many references therein; see also [238, pp. 79–80]. A part of the information about the history of Borsuk’s problem are already given at the end of Section 15.3. Boltyanski [146] proved that a plane convex figure has the (maximal) Borsuk number 3 if and only if it can be completed to a figure of constant width in a unique way, see Theorem 15.3.3. In [652] it was proved that a planar convex body has a unique completion iff for every of its non-diametral chords (say ab) there exists a diametral chord of it whose relative interior is disjoint from ab. Lenz [716] proved that the Borsuk number of smooth convex bodies K in En is smaller than n + 1 if and only if K is not of constant width (see also [824] and [501]). Dol’nikov [299] used bodies of constant width to study properties of (geometric) diameter graphs in E3 and thus to answer the Borsuk problem for finite sets in E3 . To solve the 3-dimensional case, in [962] universal covering systems are used which are based on some kind of Reuleaux polyhedra. For more about translating Borsuk’s problem into the language of graph theory and similar methods, we refer to the survey [961]. Also the paper [300] refers to Borsuk’s problem; one of the numerous related corollaries of that paper concerns the partition of sets of constant width and yields a generalization of Borsuk’s problem. In [532], a special type of Borsuk partition (called cylindrical partition) is studied, where each partition piece can be represented as the intersection of the starting set K and some cylinder; it turns out that K is of constant width if and only if it does not admit a cylindrical partition. In [652], the following was proved: The Borsuk number β(En ) equals n + 1 if and only if for all constant width bodies K and M in En the equality β(K + M) = min{β(K ), β(M)} holds. For K a convex body of diameter 2 in En , Schulte [1048] showed that a body C of constant width 2 containing K exists such that every symmetry of K is one of C and every singular boundary point of C is a boundary point of K for which the set of antipodes in C is the convex hull of the antipodes in K . Based on this property he proved Borsuk’s conjecture for convex bodies having no point as endpoint of more than one diameter, see Section 7.5. In [272, A 27], midpoints of diameters of sets C of constant width 1 are discussed, based on the papers [825] and [823]: how large can the diameter of the set M of all midpoints of diametrical chords of such sets C be? If one could always find a simplex containing M and being contained in C, then the Borsuk problem would have an affirmative answer. Ivanov [573] described geometric properties of constant width bodies such that the Borsuk number n + 1 is guaranteed; see also [574], [575], and [576] for related results and extensions to strictly convex normed spaces. The main result of [1047] on Hadwiger’s covering problem for constant width bodies (see also [1192] and above) yields also an interesting new bound for Borsuk’s partition problem of constant width bodies. For further results and observations regarding Borsuk partitions and constant width bodies we refer also to [651]. The k-fold Borsuk number of a set S of diameter d > 0 is the smallest integer m such that there is a k-fold cover of S with m sets of smaller diameters. Among other results, the authors of [565] characterize the k-fold Borsuk numbers of bodies of constant width in En . An interesting covering problem related to the Borsuk conjecture is one in which the symmetry group plays an important role. In this vein Rogers [982] constructed an 8-dimensional body of constant width 1 that is invariant under the symmetry group determined by a regular simplex in E8 but cannot be covered by nine convex bodies of diameter less than 1, each having this same symmetry group. This contrasts with the fact, also proved by Rogers, that any n-dimensional body of constant width 1 with the symmetry group of the regular simplex can be covered by n + 1 sets of diameter less than 1 if there are no symmetry requirements on the covering sets. Some results related to the Borsuk conjecture in the plane are the following: Let  ⊂ E2 be a figure of constant width 1.

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In [824], Melzak proved that  can be covered by three subsets of diameter at most min{e+ (),

3 − e+ ()},

where e+ () is the perimeter of the largest equilateral triangle inscribed in . In [822] an upper bound for the diameters of three subsets, into which any planar convex body can be partitioned, is derived. It is given in terms of the edge-length of the largest equilateral triangle whose vertices sit on the boundary of that body, and it improves a result of Gale [386]. Schopp [1045] proved that  can be covered with three circular disks of diameter e− ()/6, where − e () is the perimeter of the smallest equilateral triangle containing . Chakerian and Sallee [242] proved that  can be covered by three translates of any figure of constant width not less than 0.9101. There are some results that go in the opposite direction by showing that the covering sets cannot be too few or too small. In the case of a figure  of√constant width 1, Lenz√ [717] proved that at least one of the covering sets must have diameter at least 3 − 1 and greater than 3 − 1 if  is not a Reuleaux triangle. Lattice Points Let us consider an n-dimensional lattice n in En , and let us denote by n0 the lattice of all points of En with integral coordinates. Sallee [997] proved that there exists an essentially unique body of maximal constant width h 0 that contains no point of 20 in its interior. This is a Reuleaux triangle whose width h 0 is a root of a certain polynomial of fourth degree. Numerically, h 0 is between 1.545 and 1.546. Also of interest is the existence of figures of maximal constant width whose interior avoids a locally finite collection of convex sets in E2 . The resulting maximal sets, if they exist, are certainly Reuleaux polygons. The proof of these results uses an interesting procedure that enables one to construct new Reuleaux polygons from a given one by a replacement procedure for boundary arcs. The following result, proved by Elkington and Hammer [319] using Sallee’s theorem, is related. Let g(c) denote the minimum of the number of points of 20 in a figure of constant width greater than c. Then √ c2 c ( )2 ≤ g(c) ≤ (π − 3). h0 2 Hence, if one is interested in densest lattice packings of  (that is, packings of the form { + p | p ∈ n }) it follows from (15.2) that the direction of one of the basic vectors of n may be selected arbitrarily. L. Fejes Tóth [343] noticed this property of sets of constant width (when n = 2) and proposed to characterize all convex bodies that permit such direction-invariant densest packings. A lattice in the plane is called a holding-lattice of a planar set S if any set congruent to S contains at least one lattice point; if its fundamental triangle has the greatest possible area, the lattice is called “thinnest”. In [111] it is proved that, also for certain types of constant width sets, the thinnest holdinglattice is based on an equilateral triangle. Let P ⊂ E2 be a convex polygon, and K be a planar convex body. Peri [925] calculated explicitly the measure of all sets congruent to K which can be contained in P in the cases when P is a parallelogram and K is either a constant width set or a regular convex polygon. Identifying P as the fundamental region of a lattice, these results are applied to corresponding problems in geometric probability. Similarly, in [10] the measure of families of congruent constant width bodies in E3 is computed which, under certain conditions, are entirely contained in a rectangular parallelepiped or in a tetrahedron. Based on this one can compute the probability that such a body, placed uniformly at random in the space of a lattice based on parallelepipeds or regular tetrahedra, meets the planes of the lattices. The paper [211] refers to an extension of Buffon’s needle problem, namely taking (instead of segments) convex test bodies of given perimeter and diameter in the Euclidean plane

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in view of a lattice of lines. The cases of ellipses and constant width sets are also studied. Similarly, in [17] two events are taken into consideration, namely that a random congruent copy of a convex body in the plane meets each of two given families of equidistant lines, with angle θ between them. It is shown that there is always a value of θ such that the two events are independent. Moreover, they are independent for any θ iff the test body is of constant width. Here we mention also the paper [547]. Also (but differently) connected with the concept of distinct families of parallel lines and planar convex bodies is the paper [303]. Given such a body K and a sweep line as a tool, one might ask for the best procedure to reduce K by a sequence of sweeps to a point. The authors consider a variant of this problem, for which also constant width sets play a role. Also in [547] results connecting Buffon’s needle problem and sets of constant width are announced. In- and Circumscribed Sets Clearly, some of the covering results discussed above might be also interpreted as inscriptioncircumscription results, such as those with inscribed semicircles. It is a question of personal judgement where they should be located in our framework. Regarding regular simplices which are in- and circumscribed about bodies of constant width we refer the reader to a nice and comprehensive discussion on page 74 of [238]. Specializing the results presented there, it was first observed in [309] √ that an equilateral triangle of minimal width  contains no set of constant width larger than (3 − 3). See also [311, pp. 140–149], where it is furthermore shown that √ any plane convex body of minimal width  contains a set whose constant width equals (3 − 3). Using suitable intersections of regular simplices S and −S as polytopes circumscribed about bodies of constant width, one gets for n = 2 the result of Pál [909] that each plane set of constant width admits a regular circumscribed hexagon (see also [160, p. 131]), and for n = 3 the observation of Gale [386] that any 3-dimensional body of constant width has a circumscribed regular octahedron, see also the proof of Theorem 15.2.5. Pál’s theorem was generalized in [250]: If the width function of a strictly convex body K ⊂ E2 has period π3 , then there is a regular hexagon circumscribing K . Clearly, if some polytope P circumscribes every convex body of constant width h, it contains every set of diameter h. Motivated by the facet number n(n + 1), which is the largest possible for the circumscribing property, Makeev [758] conjectured that every constant width body in En is inscribed in a polytope similar to the dual of the difference body of a regular n-simplex. In [664], Makeev’s conjecture was confirmed for n = 3, and also further related results were obtained there, see Section 16.3. A convex body in En , n > 2, is of constant width if and only if all its circumscribed boxes are mutually congruent cubes. This is not true for n = 2 (already the square itself is such a set). Such sets are investigated in [1204, p. 97], and polygonal versions are studied in [607] and [608]. While all circumscribed boxes of an n-dimensional constant width body have the same volume, the converse is not true. Petty and McKinney [934] investigated this family of bodies, and in [1204, p. 93] a detailed representation of plane convex sets with constant perimeter of their circumscribed rectangles is given. Characterizing affine images of n-dimensional constant width bodies, n ≥ 3, Chakerian studied in [229] convex sets with the property that all their circumscribed boxes have constant diagonal lengths, see Theorem 11.6.3. Toranzos [1133] characterized sets of constant width in terms of circumscribed rhombuses by the following property: if three sides of a rhombus span supporting lines of a plane curve, then also the fourth. (If “rhombus” is replaced by the union of two isosceles triangles, this condition characterizes circles.) For convex figures in the plane, De Valcourt (see [293], [294], and [295]) defined measures of axial symmetry by means of different types of inscribed figures; he paid special attention to the situation when the given figures are of constant width, and he posed several interesting questions.

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Further Topics Another interesting topic from the combinatorial geometry of convex bodies, which can be described in terms of illumination systems, is that of primitive fixing systems of a convex body K in En (see [151, § 44]). Such a point system F from the boundary of K will stabilize K against translations, and no proper subset of F does the same. If in this procedure frictions are not allowed, then we call this variation of F a primitive hindering system of K . The papers [148] and [149] contain also results on the cardinalities of maximal primitive fixing and hindering systems of plane sets constant width. For example, in [149] it is shown that for such sets the maximal cardinality of a primitive hindering system equals 3 for Reuleaux triangles, 5 for Reuleaux pentagons, and 4 otherwise. A further variation of illuminating convex bodies K in En is that of X -raying them. The X -ray number X (K ) is defined as the smallest number of lines {L i } such that each point p ∈ K is contained in some line parallel to one element of {L i } and intersecting the interior of K . The authors of [119] proved that for K a constant width body, X (K ) ≤ 6 holds for n = 4, and X (K ) ≤ 2n−1 holds for n = 5 as well as n = 6.

Exercises 15.1*. 15.2. 15.3. 15.4*. 15.5**.

Prove that the intersection of three unit disks of radius h, each containing the centers of the other two, contains a Reuleaux triangle of width h. Let F be a collection of sets satisfying the (4, 3)-property. Prove that there is S ∈ F with the property that the family F − {S} satisfies the (2, 2)-property. Show that for any integer N > 0 sufficiently large, there is a family F of convex sets in the plane satisfying the (2, 2)-property and π(F) > N . Suppose F is a finite family of translates of a convex body  ⊂ Ed of constant width with the (2, 2)-property. Is π(F) bounded ? Let X ⊂ E2 be a finite set of cardinality 4k + 1. Furthermore, let F = {cc(Y ) : Y ⊂ X, |Y | = 2k + 1}.

15.6*. 15.7. 15.8. 15.9. 15.10.

Prove that F has the (4, 3)-property and that π(F) ≤ 5. Can you improve the bound of Theorem 15.1.8 when  is a disk? Prove that the circumradius of a figure of constant width is at most the circumradius of the Reuleaux triangle of the same width. Is a system of externally simple lines a system of hyperplanes, when n = 2? Prove that if 2k < n then the chromatic number χ(K G(n, k) ≤ n − 2k + 2. Let P be the regular hexagon of side length √13 . Prove that √

(a) (b) (c) (d) (e)

15.11. 15.12.

the area of P is 23 , P can be inscribed in a circle of radius √13 , P circumscribes a disk of diameter 1, √ P is contained in an equilateral triangle of side length 3, and √ P can be partitioned into three congruent pieces of diameter 3/2.

 n . Prove that the circumsphere of the regular n-simplex of diameter 1 has radius 2n+2 Prove that the regular n-simplex that circumscribes a sphere of diameter 1 has length side  n(n+1) . 2

Exercises

15.13. 15.14. 15.15*. 15.16. 15.17. 15.18. 15.19**. 15.20**. 15.21. 15.22. 15.23. 15.24. 15.25**.

367

In the proof of Theorem 15.2.3, prove that if a(θ0 ) = b(θ0 ), then the semi-regular hexagon in the direction u(θ0 ), circumscribed about , is actually a regular hexagon. Let X ⊂ E2 be a compact set of √ diameter 1. Prove that X is the union of three sets each of diameter less than or equal to 3/2. Let P be a regular k-gon in the plane, k > 6, circumscribed about a circle of diameter 1. Find a set X which cannot be covered by a congruent copy of P. Let T ⊂ E3 be a tetrahedron circumscribing the sphere of diameter 1 centered at the origin. Prove that T ∩ −T is a octahedron. Prove that the octahedron that circumscribes the sphere of diameter 1 has sides of length √ 3/2. Find the lengths of the edges of the rhombic dodecahedron that circumscribes the sphere of diameter 1. Prove that the diameter of a minimal universal cover of E2 is less than three. Prove that the Newton number of the disk and the Reuleaux triangle is smaller than or equal to 7. Prove that an n-dimensional ball can be partitioned into n + 1 pieces of smaller diameter. Prove that an equilateral triangle, or, in fact, any regular polygon with an odd number of sides, is contained in a unique constant width figure of the same diameter. If |x1 − x2 | > h, prove that the line that passes through the points x1 and x2 partitions B(x1 , h) ∩ B(x2 , h) into two pieces of diameter less than h. Let X ⊂ E2 be a compact set of √ diameter 1. Prove that X is the union of three sets each of diameter less than or equal to 3/2. n-cube of By using the fact that each subset of En of diameter 1 is contained in an √

nedgen length 1, show that every bounded subset φ of E can be covered by  n + 1 sets, each of which is of smaller diameter than φ.

Chapter 16

Bodies of Constant Width in Topology

Mathematics is the most beautiful and most powerful creation of the human spirit. Stefan Banach

Four sections integrate this chapter. In the first section we shall study the hyperspace of all convex sets in Euclidean space En and, within this, the hyperspace of all bodies of constant width. In the second section, differential and algebraic topology are needed to study an amazing generalization of bodies of constant width known as transnormal manifolds. Section 16.3 is devoted to polyhedra that circumscribe the sphere of diameter 1; within this family, we will characterize those polyhedra which are universal covers. In particular, we will use the theory of fiber bundles to prove that the rhombic dodecahedron circumscribing the sphere of diameter 1 is a universal cover in E3 . Finally, in Section 16.4 the topology and the geometry of Grassmannian spaces are used to see how big or complicated a collection of constant width sections should be such that the origina