On the distribution of distances in finite sets in the plane
type: Item
instance of: scholarly article
title: On the distribution of distances in finite sets in the plane
zbMATH Open document ID: 0591.52014
DOI: 10.1016/0012-365X(85)90162-1
author: B. George
published in: Discrete Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let \(n_ k\) denote the number of times the kth largest distance occurs among a set S of n points. The author shows that if S is the set of vertices of a convex polygon in the Euclidean plane, then \(n_ 1+2n_ 2\leq 3n\) and \(n_ 2\leq n+n_ 1.\) (In the paper two another inequalities are proved as well, which are valid in any metric space, but they are weaker than the mentioned ones.) Together with the well-known inequality \(n_ 1\leq n\) and the trivial inequalities \(n_ 1\geq 0\), \(n_ 2\geq 0\), all linear inequalities which are valid for n, \(n_ 1\) and \(n_ 2\) are consequences of these.
reviewed by: Károly Bezdek
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52Bxx
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 51M25
zbMATH DE Number: 3949181
zbMATH Keywords: finite point set
zbMATH Keywords: convex n-gon
zbMATH Keywords: hyperbolic plane
zbMATH Keywords: kth largest distance
zbMATH Keywords: convex polygon
zbMATH Keywords: Euclidean plane
MaRDI profile type: Publication
cites work: The Number of Furthest Neighbour Pairs of a Finite Planar Set
full work available at URL: 0012-365x(85)90162-1
OpenAlex ID: W1984221423
description: scientific article; zbMATH DE number 3949181