The Helly theorem for star-shaped sets
type: Item
instance of: scholarly article
title: The Helly theorem for star-shaped sets
zbMATH Open document ID: 1013.52006
DOI: 10.1023/A:1011367730490
published in: Journal of Mathematical Sciences (New York)
publication date: 2002-10-30T00:00:00Z
review text: The main result of this paper is the following analogue to Helly's theorem. Let \((K_{i})_{i\in I}\) be a family of star-shaped compact sets in Euclidean \(N\)-space \(E^{N}\), and let \(\operatorname {card}I\geq N+1\). If for every \(k\leq N+1\) the intersection of any \(k\) sets from this family is nonempty and star-shaped, then the intersection \(\bigcap_{i\in I}K_{i}\) is nonempty and star-shaped. There is also a dual theorem concerning the union instead of the intersection. Finally, the following theorem is proved. Let \((C_{i})_{i\in I}\) be a finite family of convex compact sets in \(E^{N}\) with \(\operatorname {card}I\geq N+1\). If for every \(k\leq N+1\) the union of any \(k\) sets from this family is star-shaped, the intersection \(\bigcap_{i\in I}C_{i}\) is nonempty.
Mathematics Subject Classification ID: 52A35
Mathematics Subject Classification ID: 52A30
zbMATH DE Number: 1821778
zbMATH Keywords: Helly-type theorems
zbMATH Keywords: star-shaped sets
author: N. A. Bobylev
reviewed by: Johann Linhart
MaRDI profile type: Publication
full work available at URL: a:1011367730490
OpenAlex ID: W55221811
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description: scientific article; zbMATH DE number 1821778