Lectures on Sphere Arrangements – the Discrete Geometric Side

verfasst von: Károly Bezdek

Verlag: Springer New York

Buchreihe : Fields Institute Monographs

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Über dieses Buch

This monograph gives a short introduction to the relevant modern parts of discrete geometry, in addition to leading the reader to the frontiers of geometric research on sphere arrangements. The readership is aimed at advanced undergraduate and early graduate students, as well as interested researchers. It contains more than 40 open research problems ideal for graduate students and researchers in mathematics and computer science. Additionally, this book may be considered ideal for a one-semester advanced undergraduate or graduate level course.

The core part of this book is based on three lectures given by the author at the Fields Institute during the thematic program on “Discrete Geometry and Applications” and contains four core topics. The first two topics surround active areas that have been outstanding from the birth of discrete geometry, namely dense sphere packings and tilings. Sphere packings and tilings have a very strong connection to number theory, coding, groups, and mathematical programming. Extending the tradition of studying packings of spheres, is the investigation of the monotonicity of volume under contractions of arbitrary arrangements of spheres. The third major topic of this book can be found under the sections on ball-polyhedra that study the possibility of extending the theory of convex polytopes to the family of intersections of congruent balls. This section of the text is connected in many ways to the above-mentioned major topics and it is also connected to some other important research areas as the one on coverings by planks (with close ties to geometric analysis). This fourth core topic is discussed under covering balls by cylinders.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Unit Sphere Packings

Abstract

Unit sphere packings are the classical core of (discrete) geometry. We survey old as well new results giving an overview of the art of the matters. The emphases are on the following five topics: the contact number problem (generalizing the problem of kissing numbers), lower bounds for Voronoi cells (studying Voronoi cells from volumetric point of view), dense sphere packings in Euclidean 3-space (studying a strong version of the Kepler conjecture), sphere packings in Euclidean dimensions higher than 3, and uniformly stable sphere packings.

Károly Bezdek

Chapter 2. Proofs on Unit Sphere Packings

Abstract

The proofs presented in this chapter can be grouped as follows. We prove lower and upper estimates for the contact numbers of packings of n unit balls in Euclidean 3-space. One can regard this problem as a combinatorial relative of the Kepler problem on the densest unit sphere packings. Next, we give lower estimates for the surface volume of Voronoi cells in packings of unit balls in Euclidean d-space for all d ≥ 2 and then we improve those estimates in dimensions d ≥ 8. All these results imply upper bounds for the usual density of unit ball packings. Returning to the 3-dimensional Euclidean space we give lower bounds for the average surface area (resp., average edge curvature) of the cells in an arbitrary normal tiling with each cell holding a unit ball. On the one hand, it leads to a new version of the Kepler problem on unit sphere packings on the other hand, it generates a new relative of Kelvin’s foam problem. Finally, we find sufficient conditions for sphere packings being uniformly stable, a property that holds for all densest lattice sphere packings up to dimension 8.

Károly Bezdek

Chapter 3. Contractions of Sphere Arrangements

Abstract

Extending the tradition of studying packings of spheres we investigate the monotonicity of volume under contractions of arbitrary arrangements of spheres. The research on this fundamental topic started with the conjecture of E. T. Poulsen and M. Kneser in the late 1950s. In this chapter we survey the status of the long-standing Kneser–Poulsen conjecture in Euclidean as well as in non-Euclidean spaces.

Károly Bezdek

Chapter 4. Proofs on Contractions of Sphere Arrangements

Abstract

First, we give a proof of the long-standing Kneser–Poulsen conjecture in the Euclidean plane. Although that result is 2-dimensional its proof is higher dimensional. Second, we prove an analogue of the Kneser–Poulsen conjecture for hemispheres in spherical d-space. Third, we give a proof of a Kneser–Poulsen-type theorem for convex polyhedra in hyperbolic 3-space.

Károly Bezdek

Chapter 5. Ball-Polyhedra and Spindle Convex Bodies

Abstract

In this chapter, we introduce an extension of the theory of convex polyhedral sets (resp., convex bodies) to the family of intersections of finitely many congruent balls, called ball-polyhedra (resp., to the family of intersections of not necessarily finitely many congruent balls, called spindle convex bodies). The basic properties of ball-polyhedra (resp., spindle convex bodies) that are discussed here include separation and support properties for spindle convex bodies, a Carathéodory-type theorem for spindle convex hulls, and an Euler–Poincaré-type formula for ball-polyhedra in d-dimensional Euclidean space. In addition, we discuss (generalized) billiard trajectories in disk-polygons and an analogue of Blaschke–Lebesgue theorem for disk-polygons. Furthermore, we investigate the problem of characterizing the edge-graphs of ball-polyhedra in Euclidean 3-space. Another topic, discussed in more details, is on global and local rigidity of ball-polyhedra in Euclidean 3-space. Finally, we investigate the status of the long-standing illumination conjecture of V. G. Boltyanski and H. Hadwiger from 1960 for ball-polyhedra (resp., spindle convex bodies) in Euclidean d-space.

Károly Bezdek

Chapter 6. Proofs on Ball-Polyhedra and Spindle Convex Bodies

Abstract

First, we prove that any of the shortest generalized billiard trajectories in an arbitrary convex body of Euclidean d-space is of period of at most d + 1. Second, we prove an analogue of Stoker’s rigidity theorem for standard ball-polyhedra. Third, we give a proof of the global rigidity analogue of Alexandrov’s theorem for normal ball-polyhedra. Next, we show that every simple and standard ball-polyhedron of Euclidean 3-space is locally rigid with respect to its inner dihedral angles (resp., face angles). Then we prove some basic separation and support properties for spindle convex bodies as well as give a proof of a Charathéodory-type theorem for spindle convex hulls. Furthermore, we prove an Euler-Poincaré-type formula for standard ball-polyhedra in Euclidean d-space. Finally, we give a proof of the long-standing Boltyanski-Hadwiger illumination conjecture for fat spindle convex bodies in Euclidean dimensions greater than or equal to 15.

Károly Bezdek

Chapter 7. Coverings by Cylinders

Abstract

In the 1930s, A.Tarski introduced his plank problem at a time when the field discrete geometry was about to born. It is quite remarkable that Tarski’s question and its variants continue to generate interest in the geometric and analytic aspects of coverings by cylinders in the present time as well. This chapter surveys plank theorems, covering convex bodies by cylinders, Kadets–Ohmann-type theorems and investigates partial coverings of balls by planks.

Károly Bezdek

Chapter 8. Proofs on Coverings by Cylinders

Abstract

First, we give a proof of the long-standing affine plank conjecture of Bang for successive hyperplane cuts and then for inductive partitions. Second, we prove a lower estimate for the sum of the cross-sectional volumes of cylinders covering a convex body in Euclidean d-space. Then we prove a Kadets–Ohmann-type theorem in spherical d-space for coverings of balls by convex bodies via volume maximizing lunes. Finally, we give estimates for partial coverings of balls by planks in Euclidean d-space.

Károly Bezdek

Chapter 9. Research Problems: An Overview

Abstract

In this chapter we give a short summary of the 30 open research problems that have been raised in this book. Some of them are well known however, the majority of them seems to be new. For more details we refer the interested reader to the relevant sections of this book.

Károly Bezdek

Backmatter

Titel: Lectures on Sphere Arrangements – the Discrete Geometric Side
verfasst von: Károly Bezdek
Verlag: Springer New York
Electronic ISBN: 978-1-4614-8118-8
Print ISBN: 978-1-4614-8117-1
DOI: https://doi.org/10.1007/978-1-4614-8118-8