A fractional Helly theorem for convex lattice sets
type: Item
instance of: scholarly article
title: A fractional Helly theorem for convex lattice sets
zbMATH Open document ID: 1028.52003
DOI: 10.1016/S0001-8708(02)00037-3
published in: Advances in Mathematics
publication date: 2003-05-04T00:00:00Z
review text: Let \({\mathbb Z}^d\) denote the integer lattice in Euclidean \(d\)-space \({\mathbb R}^d\). The intersection of any convex set of \({\mathbb R}^d\) with \({\mathbb Z}^d\) is called a convex lattice set. The authors prove that if for every \(\alpha \in (0,1]\) there is a positive number \(\beta\) such that for arbitrary convex lattice sets \(F_1, \dots ,F_n\) in \({\mathbb Z} ^d\) with \(\bigcap_{i\in I} F_i \not = \emptyset\) for at least \(\alpha {n \choose d+1}\) index sets \(I \subset \{1, \dots,n\}\) of size \(d+1\), then at least \(\beta n\) of the sets \(F_1,\dots,F_n\) have a common point.
reviewed by: Marek Lassak
Mathematics Subject Classification ID: 52A35
Mathematics Subject Classification ID: 52C07
zbMATH DE Number: 1903515
zbMATH Keywords: convex body
zbMATH Keywords: lattice
zbMATH Keywords: Helly's theorem
zbMATH Keywords: fractional Helly number
author: Imre Bárány
author: Jiří Matoušek
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1903515