On a theorem of Helly
type: Item
instance of: scholarly article
title: On a theorem of Helly
zbMATH Open document ID: 0860.52001
DOI: 10.1007/BF02304884
published in: Mathematical Notes
publication date: 1997-04-21T00:00:00Z
review text: Let \(K\) be a compact convex set in Euclidean \(d\)-space. The symbol \({\mathcal H}_K\) denotes the set of all mappings \(f:K\to K\), and \({\mathcal F}_K\) denotes the subfamily of continuous mappings. For every \(f\in {\mathcal H}_K\), the convex hull of the set of all points of the form \({d\over {d+1}}x+ {1\over {d+1}} f(x)\), where \(x\in K\), is denoted by \(K(f)\). The author proves that \(\bigcap\{K(f)\); \(f\in{\mathcal F}_K\}\neq \emptyset\). Moreover, he shows that if \(K\) is a convex polytope, then \(\bigcap\{K(f)\); \(f\in{\mathcal H}_K\}\neq \emptyset\). The author also presents a proof of Helly's theorem based on the theorem of Brouwer.
reviewed by: Marek Lassak
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A35
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 921635
zbMATH Keywords: Helly theorem
zbMATH Keywords: Brouwer theorem
zbMATH Keywords: continuous mapping
zbMATH Keywords: convex set
zbMATH Keywords: Euclidean \(d\)-space
author: N. A. Bobylev
MaRDI profile type: Publication
cites work: Q5341481
cites work: Convex Analysis
full work available at URL: https://doi.org/10.1007/bf02304884
OpenAlex ID: W2094095243
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description: scientific article; zbMATH DE number 921635