Strange phenomena in convex and discrete geometry. Ed. by James J. Dudziak
type: Item
instance of: scholarly article
title: Strange phenomena in convex and discrete geometry. Ed. by James J. Dudziak
zbMATH Open document ID: 0865.52001
published in: Universitext
publication date: 1996-08-18T00:00:00Z
review text: The field of Convex and Discrete Geometry is rich in intuitive problems which are easy to state but hard to solve, and sometimes the answers to plausible, but longstanding conjectures have been unexpected and surprising. In this exciting book, the author describes in seven chapters such ``strange phenomena'', where the solutions have been counterintuitive, or spectacular in some other way. In general, complete proofs are provided, without presupposing special knowledge, except that occasionally older results from number theory, analysis, or the theory of convex bodies have to be applied without proof. The first chapter is devoted to the famous Borsuk problem. It asks whether every bounded set in Euclidean space \(R^n\) is a union of \(n+1\) sets of smaller diameter (this question can be considered as one in convex geometry). The surprising answer (after 60 years) was that there exist finite sets in \(R^n\) for which any decomposition into sets of smaller diameter requires more than \(1.07^{\sqrt n}\) such sets. Chapter 2 treats finite packing problems. L. Fejes Tóth's sausage conjecture says that the volume of the convex hull of \(m\) nonoverlapping unit balls in \(R^n\), \(n\geq 5\), is minimal only when the centres of the balls are collinear. The chapter reproduces the proof (after almost 20 years) of the sausage conjecture in high dimensions. In contrast to this (but not so surprising, perhaps), it is shown that a convex body of minimal surface area into which \(m\) unit balls can be packed is roughly spherical when \(m\) is large. In Chapter 3, it is first proved that every convex body that tiles \(R^n\) by translations also admits a lattice tiling of \(R^n\). Then a star body is constructed that tiles \(R^{10}\), but not as a lattice. The first topic of Chapter 4, on local packing phenomena, is the existence of a convex body in \(R^n\), for large \(n\), which has smaller kissing number than a second one, but larger blocking number. Then two packing lattices for…
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 52-02
zbMATH DE Number: 917410
zbMATH Keywords: Borsuk problem
zbMATH Keywords: finite packing problems
zbMATH Keywords: sausage conjecture
zbMATH Keywords: lattice tiling
zbMATH Keywords: kissing number
zbMATH Keywords: Baire category
zbMATH Keywords: Busemann-Petty problem
zbMATH Keywords: Dvoretzky's theorem
author: Chuan Ming Zong
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 917410