A Helly-type theorem for countable intersections of starshaped sets
type: Item
instance of: scholarly article
title: A Helly-type theorem for countable intersections of starshaped sets
zbMATH Open document ID: 1080.52006
author: Marilyn Breen
published in: Archiv der Mathematik
publication date: 2005-04-28T00:00:00Z
review text: Let \(S\) be a set in \(\mathbb{R}^{d}\). For every pair of points \(x\) and \(y\) in \(S\), \(x\) is visible from \(y\) via \(S\) if and only if the line segment \([x,y]\) lies in \(S\). A set \(S\) is called starshaped if and only if for some point \(p\) in \(S\), \(p\) sees each point of \(S\) via \(S\) and the set of all such points \(p\) is the (convex) kernel of \(S\). In the first theorem taking \(k\) and \(d\) as fixed integers \(0\leq k\leq d\) and \(\mathcal{K} =\{K_{\alpha }:\alpha \) in some index set\(\}\) as a collection of sets in \( \mathbb{R}^{d}\) the author proves that if every countable subfamily of \(\mathcal{K}\) has a starshaped intersection whose kernel is at least \(k\)-dimensional then \(\bigcap \{K_{\alpha }:K_{\alpha }\) in \(\mathcal{K}\}\) also is a starshaped set whose kernel is at least \(k\)-dimensional. In the second theorem, again taking \(k\) and \(d\) as fixed integers \(0\leq k\leq d\) and \(\mathcal{K} =\{K_{\alpha }:\alpha \) in some index set\(\}\) as a collection of sets in \( \mathbb{R}^{d}\) it is proved that if every countable subfamily of \(\mathcal{K}\) has a starshaped union then \(\bigcup \{K_{\alpha }:K_{\alpha }\) in \(\mathcal{K}\}\) also is a starshaped set whose kernel is at least \(k\)-dimensional.
reviewed by: Semin Akdoǧan
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 2162320
zbMATH Keywords: starshaped set
zbMATH Keywords: convex kernel
zbMATH Keywords: Helly type theorem
MaRDI profile type: Publication
full work available at URL: s00013-004-1120-1
OpenAlex ID: W2088725538
DOI: 10.1007/S00013-004-1120-1
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description: scientific article; zbMATH DE number 2162320