On almost-equidistant sets - II

2017/08/07 by Polyanskii, Alexandr
#05A99 #05C50 #51F99 #51K99 #52C99 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1708.02039

Abstract

A set in \mathbb R^d is called almost-equidistant if for any three distinct points in the set, some two are at unit distance apart. First, we give a short proof of the result of Bezdek and Lángi claiming that an almost-equidistant set lying on a (d-1)-dimensional sphere of radius r, where r<1/√(2), has at most 2d+2 points. Second, we prove that an almost-equidistant set V in \mathbb R^d has O(d) points in two cases: if the diameter of V is at most 1 or if V is a subset of a d-dimensional ball of radius at most 1/√(2)+cd^-2/3, where c<1/2. Also, we present a new proof of the result of Kupavskii, Mustafa and Swanepoel arXiv:1708.01590 that an almost-equidistant set in \mathbb R^d has O(d^4/3) elements.

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• Mutually Equidistant Spheres that Intersect
• Banach spaces in which large subsets of spheres concentrate
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