On a conjecture of A. Bezdek
type: Item
instance of: scholarly article
title: On a conjecture of A. Bezdek
zbMATH Open document ID: 0999.52009
published in: Discrete \& Computational Geometry
publication date: 2002-05-30T00:00:00Z
review text: In 1985, V. I. Proizvolov posed the question whether any \(n\) points \(A_1\dots,A_n\) inside a convex \(n\)-gon with sides \(s_1,\dots,s_n\) can be renumbered such that the triangles with bases \(s_i\) and a vertex \(A_i\) cover the given \(n\)-gon. A. Bezdek proved a generalization of this, and the author of the present paper verifies a related conjecture of A. Bezdek by showing the following: Let \(k\) points \(a_1,\dots,a_k\in R^n\) and a convex polytope \(P\subset R^n\) be given. Then one can choose \(k\) facets \(F_1,\dots,F_k\) of \(P\) such that the convex hulls \(\text{conv}(\{a_i\}\cup F_i)\) either cover the polytope \(P\) or do not pairwise intersect in interior points. Some further statements (referring to polyhedral sets, i.e., to possibly unbounded intersections of halfspaces) are proved, too. At the end of the article some interesting related conjectures and questions are given.
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 1747951
zbMATH Keywords: convex polytope
zbMATH Keywords: polyhedral set
zbMATH Keywords: simple polytope
zbMATH Keywords: covering theorems
author: Roman Karasev
MaRDI profile type: Publication
full work available at URL: s00454-001-0072-5
OpenAlex ID: W2045638219
Wikidata QID: Q123341713
DOI: 10.1007/S00454-001-0072-5
description: scientific article; zbMATH DE number 1747951