Sets in \(\mathbb{R}^ d\) with no large empty convex subsets
type: Item
instance of: scholarly article
title: Sets in \(\mathbb{R}^ d\) with no large empty convex subsets
zbMATH Open document ID: 0766.52003
DOI: 10.1016/0012-365X(92)90665-3
author: Pavel Valtr
published in: Discrete Mathematics
publication date: 1993-01-17T00:00:00Z
review text: Let \(A\) be a finite set of points in general position in \(\mathbb{R}^ d\), \(d\geq 2\). A subset \(B\subseteq A\) of \(n\) points is called an \(n\)-hole in \(A\) if the points in \(B\) are the vertices of a convex polytope which contains no other point of \(A\). Let \(h(d)\) be the maximum number \(n\) such that any sufficiently large set of points in general position in \(\mathbb{R}^ d\) contains an \(n\)-hole. The author shows that \[ 2d+1\leq h(d)\leq 2^{d-1}(P(d-1)+1), \] where \(P(d-1)\) denotes the product of the smallest \(d-1\) prime numbers. He also gives an improved upper bound in \(\mathbb{R}^ 3\), namely \(h(3)\leq 22\).
reviewed by: Wolfgang Weil
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 05B99
zbMATH DE Number: 97492
zbMATH Keywords: convex hull
zbMATH Keywords: convex hole
zbMATH Keywords: finite sets
MaRDI profile type: Publication
cites work: Konvexe Fünfecke in ebenen Punktmengen
cites work: Sets with No Empty Convex 7-Gons
cites work: Convex independent sets and 7-holes in restricted planar point sets
full work available at URL: 0012-365x(92)90665-3
OpenAlex ID: W2012104218
description: scientific article; zbMATH DE number 97492