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

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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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Preface

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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13 13 14 18 19 22 24 27 29 34 37 37 41 41 44 50 53 55

3

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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4

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

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Contents

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

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5

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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95 95 95 98 101 101 102 105 105 109 111 112 114 114 116 120 121 125

6

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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127 127 132 134 136 139 141

7

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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143 143 144 145 147 148 149 154 159 164

8

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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Contents

ix

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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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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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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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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

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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

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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

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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

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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 co