Volumetric Discrete Geometry

237 Pages·2019·6.729 MB·English

by  Karoly Bezdek, Zsolt Langi

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Volumetric Discrete Geometry
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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
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Volumetric Discrete Geometry
Károly Bezdek
University of Calgary
Zsolt Lángi
Budapest University of Technology and Economics
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CRC Press
Taylor & Francis Group
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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
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To Károly’s wife, Éva, and Zsolt’s wife, Kornélia, for their exceptional and continuous support.
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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
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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
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