A combinatorial result on points and circles on the plane
type: Item
instance of: scholarly article
title: A combinatorial result on points and circles on the plane
zbMATH Open document ID: 0645.05024
DOI: 10.1016/0012-365X(88)90015-5
published in: Discrete Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let \(P_ n\) be a collection of n points on the plane. For a pair of points u and \(v\in P_ n\) let C(u,v) be the minimum number of points of \(P_ n\) contained in any circle containing u and v. In this paper we prove the result that there exist two points \(u_ 0\) and \(v_ 0\in P_ n\) such that any circle containing \(u_ 0\) and \(v_ 0\) contains at least \(\lceil (n-2)/60\rceil\) elements of \(P_ n\) (other than \(u_ 0\) and \(v_ 0)\). We also prove that the average value of C(u,v) over all pairs \(\{u,v\}\subset P_ nis\geq \lceil (n-2)/60\rceil.\) For the case when \(P_ n\) are the vertices of a convex polygon, we prove that there exist two vertices \(u,v\in P_ n\) such that any circle containing them contains at least \(\lceil (n-2)/4\rceil\) elements of \(P_ n\).
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 05C38
zbMATH DE Number: 4051646
zbMATH Keywords: collection of n points on the plane
zbMATH Keywords: circle
zbMATH Keywords: convex polygon
author: V. Neumann-Lara
author: J. Urrutia
MaRDI profile type: Publication
cites work: Q5543311
cites work: Q3931992
cites work: Q5547252
cites work: Q5587734
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description: scientific article; zbMATH DE number 4051646