Karoly Bezdek - Volumetric Discrete Geometry (Discrete Mathematics and Its Applications)

Series Editors

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Donald L. Kreher

Patrice Ossona de Mendez

Douglas West

Handbook of Discrete and Computational Geometry, Third Edition

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Handbook of Discrete and Combinatorial Mathematics, Second Edition

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Crossing Numbers of Graphs

Marcus Schaefer

Graph Searching Games and Probabilistic Methods

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Handbook of Geometric Constraint Systems Principles

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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

Kroly Bezdek and Zsolt Lngi

https://www.crcpress.com/Discrete-Mathematics-and-Its-Applications/book-series/CHDISMTHAPP?page=1&order=dtitle&size=12&view=list&status=published,forthcoming

Kroly Bezdek

University of Calgary

Zsolt Lngi

Budapest University of Technology and Economics

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Library of Congress Cataloging-in-Publication Data

Names: Bezdek, Kroly, author. | Lngi, Zsolt, author.

Title: Volumetric discrete geometry / Kroly Bezdek and Zsolt Lngi.

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

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To Krolys wife, va, and Zsolts wife, Kornlia, for their exceptional and continuous support.

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 d

3.5 The Kneser-Poulsen conjecture for contractions of unions and intersections of disks in S2 and 2

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

5.1 Solution of the contact number problem for smooth strictly convex domains in E2

5.2 The separable Olers inequality and its applications in E2

5.3 Higher dimensional results: minimizing the mean projections of finite -separable packings in Ed

The volume of geometric objects has been studied by the ancient Greek mathematicians and it is quite remarkable that even today volume continues to play an important role in applied as well as pure mathematics. So, it did not come as a surprise to us that also in discrete geometry, which is a relatively new branch of geometry, volume played a significant role in generating interesting new topics for research. When writing this book, our goal was to demonstrate the more recent aspects of volume within discrete geometry. We found it convenient to split the book into two parts, with is of graduate level reading that intends to lead the interested reader to the frontiers of research in discrete geometry. In what follows, we give a brief description of the topics discussed in our book.

In .

In .

The monotonicity of volume under contractions of arbitrary arrangements of spheres is a well-known fundamental problem in discrete geometry. The research on this topic started with the conjecture of Poulsen and Kneser in the late 1950s. In .

The well-known kissing number problem asks for the largest number of non-overlapping unit balls touching a given unit ball in the Euclidean d-space Ed. Generalizing the kissing number, the Hadwiger number or the translative kissing number H(K) of a convex body K in Ed is the maximum number of non-overlapping translates of K that all touch K. In we study and survey the following natural extension of these problems. A finite translative packing of the convex body K in Ed is a finite family P of non-overlapping translates of K in Ed. Furthermore, the contact graph G(P) of P is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if and only if the corresponding two packing elements touch each other. The number of edges of G(P) is called the contact number of P. Finally, the contact number problem asks for the largest contact number, that is, for the maximum number c(K, n, d) of edges that a contact graph of n