A Helly type conjecture
type: Item
instance of: scholarly article
title: A Helly type conjecture
zbMATH Open document ID: 0927.52011
DOI: 10.1007/PL00009408
author: Meir Katchalski
author: D. Nashtir
published in: Discrete \& Computational Geometry
publication date: 1999-12-14T00:00:00Z
review text: We say that a family of sets is \(\Pi^n\) provided there exists a set of \(n\) points such that every set from the family contains at least one of those points. A family of sets is said to be \(\Pi^n_k\) if each its subfamily of at most \(k\) sets is \(\Pi^n\). The authors conjecture that there exists a positive integer \(k_0\) such that if a family of convex sets in the plane is \(\Pi^2_{k_0}\), then it is also \(\Pi^3\). They show that if such an integer \(k_0\) exists, then it is over 5. The paper confirms the above conjecture for two very special cases. One of them is when \(k_0= 10\) and when the family of convex sets contains only intersections of translates of any four fixed closed half-planes.
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 1271007
zbMATH Keywords: Helly's theorem
zbMATH Keywords: \(n\)-pierceable family
zbMATH Keywords: convex \(n\)-gon
zbMATH Keywords: convex quadrilateral
Wikidata QID: Q123121323
reviewed by: Marek Lassak
MaRDI profile type: Publication
full work available at URL: pl00009408
OpenAlex ID: W2018313938
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description: scientific article; zbMATH DE number 1271007