A case when the union of polytopes is convex
type: Item
instance of: scholarly article
title: A case when the union of polytopes is convex
zbMATH Open document ID: 1072.52007
author: Imre Bárány
author: Komei Fukuda
published in: Linear Algebra and its Applications
publication date: 2005-05-04T00:00:00Z
review text: The authors present the following necessary and sufficient condition for the union of a finite number of convex polytopes in \(\mathbb{R}^d\) to be convex. Let \(P_1,\ldots,P_n\) (\(n\geq 2\)) be convex polytopes with vertex-sets \(X_1,\ldots,X_n\), respectively, and let \(P:=\bigcup_{i=1}^{n}P_i\), \(X:=\bigcup_{i=1}^{n}X_i\) and \(Q:=\text{conv}(X)\) (convex hull of \(X\)). Then \(P=Q\) if and only if \(\text{conv}(S) \subset P\) for each \(S\subset X\) with \(| S| \leq d+1\) and \(| S\cap X_i| \leq 1\) for \(i=1,\dots,n\).
reviewed by: Egon Schulte
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A35
Mathematics Subject Classification ID: 52B55
zbMATH DE Number: 2165369
zbMATH Keywords: convex polytopes
zbMATH Keywords: union
zbMATH Keywords: convexity characterization
zbMATH Keywords: Caratheodory's Theorem
MaRDI profile type: Publication
full work available at URL: j.laa.2004.11.018
OpenAlex ID: W2066684552
cites work: A generalization of Caratheodory's theorem
cites work: Convexity recognition of the union of polyhedra
cites work: Q5341481
DOI: 10.1016/J.LAA.2004.11.018
description: scientific article; zbMATH DE number 2165369