Helly-type theorems on common supporting lines for nonoverlapping families of convex bodies in the plane
type: Item
instance of: scholarly article
title: Helly-type theorems on common supporting lines for nonoverlapping families of convex bodies in the plane
zbMATH Open document ID: 0958.52010
DOI: 10.1023/A:1005063503371
author: Sorin Revenko
author: Valeriu Soltan
published in: Geometriae Dedicata
publication date: 2001-04-16T00:00:00Z
review text: Let us say that a family \({\mathbf F}\) of convex bodies has the property \(S\) if some line supports all the bodies, and the property \(D\) if in addition all the bodies are on the same side of the line. If every \(n\)-element subfamily of \({\mathbf F}\) has the property \(S\) (resp. \(D)\) we say that \({\mathbf F}\) has the property \(S(n)\) (resp. \(D(n))\). A Helly-type theorem on common supporting lines typically asserts that, for some class of family, \(S(n) \Rightarrow S\) or \(D(n)\Rightarrow D\). Typically, the configurations of bodies which prevent the Helly number from being lowered have rather few members; and restricting \({\mathbf F}\) to contain more than a certain number of bodies may allow \(n\) to be reduced. The first theorems of this type were obtained for families of disjoint bodies. In this paper, the authors consider families of bodies that are not necessarily disjoint, but are nonoverlapping -- that is, with pairwise disjoint interiors. They show that for nonoverlapping families of convex bodies, as for disjoint families, \(D(4)\Rightarrow D\), and give an example to show that (in contrast with the disjoint case) there is no threshold above which \(D(3)\Rightarrow D\). It is shown that a nonoverlapping family may have the property \(S(n)\) but not \(S\) for any \(n\). However, unless there is a point common to all the bodies, \(S(6)\Rightarrow S\). Moreover, if no \(k\) of the bodies have a common point, and there are at least \(12k-13\) bodies, \(S(5)\Rightarrow S\); and if there are at least \(72k-75\) bodies, \(S(4)\Rightarrow S\).
reviewed by: Robert Dawson
Mathematics Subject Classification ID: 52A35
Mathematics Subject Classification ID: 52A10
zbMATH DE Number: 1374910
zbMATH Keywords: Helly-type theorem
zbMATH Keywords: common supports
zbMATH Keywords: nonoverlapping family
MaRDI profile type: Publication
full work available at URL: a:1005063503371
OpenAlex ID: W151326233
description: scientific article; zbMATH DE number 1374910