Piercing convex sets and the Hadwiger-Debrunner \((p,q)\)-problem
type: Item
instance of: scholarly article
title: Piercing convex sets and the Hadwiger-Debrunner \((p,q)\)-problem
zbMATH Open document ID: 0768.52001
DOI: 10.1016/0001-8708(92)90052-M
published in: Advances in Mathematics
publication date: 1993-04-01T00:00:00Z
review text: Let \(p\), \(q\), \(d\) be natural numbers with \(p\geq q\geq d+1\). Then, a \(c=c(p,q,d)<\infty\) exists with the property: for each family \(\mathcal F\) of compact convex sets in \(\mathbb{R}^ d\) with the \((p,q)\)-property (i.e., among any \(p\) members of \(\mathcal F\) some \(q\) have a nonempty intersection) there is a set of at most \(c\) points in \(\mathbb{R}^ n\) that intersects each member of \(\mathcal F\). This result settles an old problem of \textit{H. Hadwiger} and \textit{H. Debrunner} [Arch. Math. 8, 309-313 (1957; Zbl 0080.154)].
reviewed by: Mihai Turinici
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 149037
zbMATH Keywords: compact convex set
zbMATH Keywords: \((p,q)\)-property
zbMATH Keywords: piercing number
zbMATH Keywords: Helly type theorem
author: Noga Alon
author: Daniel J. Kleitman
MaRDI profile type: Publication
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full work available at URL: 0001-8708(92)90052-m
OpenAlex ID: W2009545408
description: scientific article; zbMATH DE number 149037