Combinatorial geometry of belt bodies
type: Item
instance of: scholarly article
title: Combinatorial geometry of belt bodies
zbMATH Open document ID: 0853.52004
DOI: 10.1007/BF03322255
author: Horst Martini
author: V. G. Boltyanski
published in: Results in Mathematics
publication date: 1996-02-01T00:00:00Z
review text: Belt polytopes are natural generalizations of zonotopes. A polytope \(P \subset \mathbb{R}^d\) is a belt polytope, if in every 2-face of \(P\) the sides occur in parallel pairs. If, in addition, the parallel sides have the same length, then \(P\) is a zonotope. The authors introduce a new class of convex bodies, the belt bodies. For that purpose, a unit vector \(u\) is called a belt vector of a convex body \(K\), if every supporting line \(L\) parallel to \(u\) meets \(K\) at a unique point \(a\) and if one of the two rays with direction \(\pm u\) starting in \(a\) is a tangential ray at \(K\). Let \(B(K)\) be the set of all belt vectors of \(K\) and \(p(B(K))\) the unit vectors orthogonal to linearly independent \((d-1)\)-tuples from \(B(K)\). \(K\) is called a belt body if \(p(B(K))\) is dense in the set of outer unit normals of \(K\) at regular boundary points. The authors show that zonoids are belt bodies and that a polytope is a belt body if and only if it is a belt polytope. Consequently, belt bodies are dense in the class of all convex bodies. The main purpose of the work is to solve three well-known (and still unsolved) combinatorial problems for the class of belt bodies. The first problem is to cover a convex body \(K\) with a minimal number \(b(K)\) of translated smaller homothets of \(K\) (a problem due to Hadwiger, Gohberg and Markus). The covering problem has an equivalent formulation in terms of illumination of \(K\). For belt bodies \(K\) (which are not parallelotopes), the inequality \[ b(M) \leq 3 \cdot 2^{d-2} \] is shown. The second problem is to find the Helly dimension of the family of all translates of a given convex body \(K\) (Szökefalvi-Nagy problem). Here, the authors give a complete description of the Helly dimension of belt bodies. Finally, the latter result is applied to minimal fixing systems of convex bodies (a notion which goes back to Fejes Tóth). Again, for belt bodies \(K\), a complete description of the minimal fixing number is gi…
Mathematics Subject Classification ID: 52A37
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A35
Mathematics Subject Classification ID: 52A21
zbMATH DE Number: 828274
zbMATH Keywords: zonoid
zbMATH Keywords: equatorial polytope
zbMATH Keywords: covering problem
zbMATH Keywords: Helly dimension
zbMATH Keywords: fixing system
zbMATH Keywords: belt polytope
zbMATH Keywords: belt body
zbMATH Keywords: illumination
reviewed by: Wolfgang Weil
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03322255
OpenAlex ID: W1993604621
description: scientific article; zbMATH DE number 828274