Variations on the theme of repeated distances
type: Item
instance of: scholarly article
title: Variations on the theme of repeated distances
zbMATH Open document ID: 0722.52009
DOI: 10.1007/BF02122780
author: János Pach
author: P. Erdős
published in: Combinatorica
publication date: 1990-01-01T00:00:00Z
review text: Given a set \(X=\{x_1,...x_n\}\) of n points in \({\mathbb{R}}^d\), let \(f(X)\) denote the number of pairs \(\{x_i,x_j\}\) whose Euclidean distances \(||x_i-x_j|| = 1\). Let \[ f_d(n) = \max_{X\subset {\mathbb{R}}^d,\quad |X| \leq n}f(X). \] An asymptotically sharp estimate for the error term in this maximum is given for \(d\geq 4\). Tight upper bounds are also determined for the total number of occurrences of what are called ``favourite'' distances from \(n\) points in \({\mathbb{R}}^d\), \(d\geq 4\). Some related results are also proved for distances determined by \(n\) disjoint compact convex sets in \({\mathbb{R}}^2\).
Mathematics Subject Classification ID: 52C10
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 4190577
zbMATH Keywords: repeated distances
zbMATH Keywords: points in \({\mathbb{R}}^d\)
zbMATH Keywords: convex sets
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02122780
OpenAlex ID: W2033259203
description: scientific article; zbMATH DE number 4190577