Nine convex sets determine a pentagon with convex sets as vertices
type: Item
instance of: scholarly article
title: Nine convex sets determine a pentagon with convex sets as vertices
zbMATH Open document ID: 0677.52003
DOI: 10.1007/BF00184161
author: S. H. Smith
published in: Geometriae Dedicata
publication date: 1989-01-01T00:00:00Z
review text: The paper deals with a generalization of a famous Erdős-Szekeres problem about convexly independent points. Let \({\mathcal F}\) be a family of mutually disjoint ovals (i.e. compact convex subsets) in the plane. Any three ovals from \({\mathcal F}\) are called collinear if the convex hull of two of them contains the third. Some ovals \(A_ 1,...,A_ n\) from \({\mathcal F}\) form an n-gon provided \(A_ i\not\subset conv\{UA_ j:\) \(j\neq i\}\), \(i=1,...,n.\) Theorem. If \(| {\mathcal F}| >8\), then some five ovals from \({\mathcal F}\) determine a 5-gon.
Mathematics Subject Classification ID: 52A10
zbMATH DE Number: 4109500
zbMATH Keywords: convex sets in the plane
zbMATH Keywords: Erdős-Szekeres problem
reviewed by: Valeriu Soltan
MaRDI profile type: Publication
cites work: A generalization of the Erdös-Szekeres convex n-gon theorem.
cites work: On Convex Polygons Determined by a Finite Planar Set
cites work: Q5759552
cites work: Q3286826
cites work: Q5628820
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description: scientific article; zbMATH DE number 4109500