Table Of Content

1 2 Volumetric Discrete Geometry 3 Discrete Mathematics and Its Applications Series Editors Miklos Bona Donald L. Kreher Patrice Ossona de Mendez Douglas West Handbook of Discrete and Computational Geometry, Third Edition C. Toth, Jacob E. Goodman and Joseph O’Rourke Handbook of Discrete and Combinatorial Mathematics, Second Edition Kenneth H. Rosen Crossing Numbers of Graphs Marcus Schaefer Graph Searching Games and Probabilistic Methods Anthony Bonato and Pawel Pralat Handbook of Geometric Constraint Systems Principles Meera Sitharam, Audrey St. John, and Jessica Sidman, Additive Combinatorics Béla Bajnok Algorithmics of Nonuniformity: Tools and Paradigms Micha Hofri and Hosam Mahmoud Extremal Finite Set Theory Daniel Gerbner and Balazs Patkos Cryptology: Classical and Modern Richard E. Klima and Neil P. Sigmon Volumetric Discrete Geometry Károly Bezdek and Zsolt Lángi https://www.crcpress.com/Discrete-Mathematics-and-Its-Applications/book-series/CHDISMTHAPP? page=1&order=dtitle&size=12&view=list&status=published,forthcoming 4 Volumetric Discrete Geometry Károly Bezdek University of Calgary Zsolt Lángi Budapest University of Technology and Economics 5 CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works International Standard Book Number-13: 978-0-367-22375-5 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Bezdek, Károly, author. | Lángi, Zsolt, author. Title: Volumetric discrete geometry / Károly Bezdek and Zsolt Lángi. Description: Boca Raton : CRC Press, Taylor & Francis Group, 2019. | Includes bibliographical references. Identifiers: LCCN 2018061556 | ISBN 9780367223755 Subjects: LCSH: Volume (Cubic content) | Geometry, Solid. | Discrete geometry. Classification: LCC QC105 .B49 2019 | DDC 516/.11--dc23 LC record available at https://lccn.loc.gov/2018061556 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 6 To Károly’s wife, Éva, and Zsolt’s wife, Kornélia, for their exceptional and continuous support. 7 Contents Preface Authors Symbols I Selected Topics 1 Volumetric Properties of (m, d)-scribed Polytopes 1.1 The isoperimetric inequality 1.2 Discrete isoperimetric inequalities: volume of polytopes circumscribed about a sphere 1.3 Volume of polytopes inscribed in a sphere 1.4 Polyhedra midscribed to a sphere 1.5 Research Exercises 2 Volume of the Convex Hull of a Pair of Convex Bodies 2.1 Volume of the convex hull of a pair of convex bodies in Euclidean space 2.2 Volume of the convex hull of a pair of convex bodies in normed spaces 2.3 Research Exercises 3 The Kneser-Poulsen Conjecture Revisited 3.1 The Kneser-Poulsen conjecture 3.2 The Kneser-Poulsen conjecture for continuous contractions of unions and intersections of balls 3.3 The Kneser-Poulsen conjecture for contractions of unions and intersections of disks in E2 3.4 The Kneser-Poulsen conjecture for uniform contractions of r-ball polyhedra in Ed,Sd and Hd 3.5 The Kneser-Poulsen conjecture for contractions of unions and intersections of disks in S2 and H2 3.6 Research Exercises 4 Volumetric Bounds for Contact Numbers 4.1 Description of the basic geometric questions 4.2 Motivation from materials science 4.3 Largest contact numbers for congruent circle packings 4.3.1 The Euclidean plane 4.3.2 Spherical and hyperbolic planes 4.4 Largest contact numbers for unit ball packings in E3 4.5 Upper bounding the contact numbers for packings by translates of a convex body in Ed 4.6 Contact numbers for digital and totally separable packings of unit balls in Ed 4.7 Bounds for contact numbers of totally separable packings by translates of a convex body in Ed with d = 1, 2, 3, 4 4.7.1 Separable Hadwiger numbers 4.7.2 One-sided separable Hadwiger numbers 4.7.3 Maximum separable contact numbers 4.8 Appendix: Hadwiger numbers of topological disks 4.9 Research Exercises 5 More on Volumetric Properties of Separable Packings 5.1 Solution of the contact number problem for smooth strictly convex domains in E2 5.2 The separable Oler’s inequality and its applications in E2 5.2.1 Oler’s inequality 8 5.2.2 An analogue of Oler’s inequality for totally separable translative packings 5.2.3 On the densest totally separable translative packings 5.2.4 On the smallest area convex hull of totally separable translative finite packings 5.3 Higher dimensional results: minimizing the mean projections of finite ρ-separable packings in Ed 5.4 Research Exercises II Selected Proofs 6 Proofs on Volumetric Properties of (m, d)-scribed Polytopes 6.1 Proof of Theorem 3 6.2 Proofs of Theorems 10 and 11 6.3 Proof of Theorem 14 6.3.1 Preliminaries 6.3.2 Proof of Theorem 14 for n ≤ 6 6.3.3 Proof of Theorem 14 for n = 7 6.3.4 Proof of Theorem 14 for n = 8 6.4 Proofs of Theorems 16, 17 and 18 6.4.1 Proof of Theorem 16 and some lemmas for Theorems 17 and 18 6.4.2 Proofs of Theorems 17 and 18 6.5 Proof of Theorem 21 6.6 Proof of Theorem 22 6.7 Proof of Theorem 27 6.8 Proofs of Theorems 28 and 29 6.8.1 Preliminaries and the main idea of the proofs 6.8.2 The main lemma of the proofs 6.8.3 Proof of Theorem 28 6.8.4 Proof of Theorem 29 7 Proofs on the Volume of the Convex Hull of a Pair of Convex Bodies 7.1 Proofs of Theorems 32 and 33 7.1.1 Proof of Theorem 32 7.1.2 Proof of Theorem 33 7.2 Proofs of Theorems 34, 36, 37 and 40 7.2.1 Preliminaries 7.2.2 Proofs of the Theorems 7.3 Proofs of Theorems 41 and 46 7.3.1 Proof of Theorem 41 7.3.2 Proof of Theorem 46 7.4 Proof of Theorem 53 7.5 Proof of Theorem 54 7.6 Proofs of Theorems 57 and 58 7.6.1 Proof of Theorem 57 7.6.2 Proof of Theorem 58 7.7 Proofs of Theorems 59 and 60 7.7.1 The proof of the left-hand side inequality in (ii) 7.7.2 The proof of the right-hand side inequality in (ii) 7.7.3 The proofs of (i), (iii) and (iv) 8 Proofs on the Kneser-Poulsen Conjecture 8.1 Proof of Theorem 67 8.2 Proof of Theorem 68 8.3 Proof of Theorem 69 8.4 Proof of Theorem 72 8.5 Proof of Theorem 73 8.5.1 Proof of (i) in Theorem 73 8.5.2 Proof of (ii) in Theorem 73 9