The multiplicity of the two smallest distances among points
type: Item
instance of: scholarly article
title: The multiplicity of the two smallest distances among points
zbMATH Open document ID: 0951.52014
DOI: 10.1016/S0012-365X(98)00116-2
author: György Csizmadia
published in: Discrete Mathematics
publication date: 2000-12-18T00:00:00Z
review text: The author studies distances occuring in configurations of \(n\) points in the plane, where the minimal distance between two points is assumed to be \(1=:d_1\). The multiplicity of the smallest occuring distances and \(d_1\) and \(d_2\) is smaller than \(6n\) and maximized by the triangular lattice with \(d_2= \sqrt 3\). Partially answering a question of P. Erdős and K. Vesztergombi, the author proves that \(d_2\neq\sqrt 3\) implies that the multiplicity \(d_1\) and \(d_2\) is at most \(4n\) unless \(d_2\) is \((\sqrt 5+1)/2\) or \(1/(2\sin 15)\). In the first case, the multiplicity is at most \(4,5 n\). In addition, the author presents same interesting extremal point configurations for different values of \(d_2\).
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52C10
zbMATH DE Number: 1321838
zbMATH Keywords: triangular lattice
zbMATH Keywords: Erdős type problem
zbMATH Keywords: point configurations
MaRDI profile type: Publication
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full work available at URL: s0012-365x(98)00116-2
OpenAlex ID: W1996122332
description: scientific article; zbMATH DE number 1321838