New bounds on the Katchalski-Lewis transversal problem
type: Item
instance of: scholarly article
title: New bounds on the Katchalski-Lewis transversal problem
zbMATH Open document ID: 1030.52009
published in: Discrete \& Computational Geometry
publication date: 2003-08-21T00:00:00Z
review text: Let \({\mathcal F}\) be a family of disjoint translates of a compact convex set in the plane with the property that each three members of \({\mathcal F}\) are met by a line. Katchalski and Lewis showed in 1982 that there is a number \(k\), independent of \({\mathcal F}\), so that some line meets all but at most \(k\) members of \({\mathcal F}\). They also conjectured that one can choose \(k=2\). The present author sets a new record by showing that \(k\leq 22\), and he disproves the Katchalski-Lewis conjecture by constructing a family of disjoint translates of a parallelogram, each three met by a line, but where any line misses at least four members.
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 1969092
zbMATH Keywords: transversal
zbMATH Keywords: Katchalski-Lewis conjecture
author: Andreas Holmsen
reviewed by: Rolf Schneider
MaRDI profile type: Publication
full work available at URL: s00454-002-0755-6
OpenAlex ID: W2038112877
DOI: 10.1007/S00454-002-0755-6
description: scientific article; zbMATH DE number 1969092