Favourite distances in 3-space

2019/07/19 by Konrad J. Swanepoel, Swanepoel, Konrad J.
Computer Science · Mathematics · #52C10 #Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Metric Geometry (math.MG) #cs.CG #math.CO #math.MG #msc:52C10

paper · pdf · doi:10.48550/arxiv.1907.08402

10 pages, 4 figures

arxiv created 2019/07/19 · arxiv updated 2019/07/22

Abstract

Let S be a set of n points in Euclidean 3-space. Assign to each x∈ S a distance r(x)>0, and let e_r(x,S) denote the number of points in S at distance r(x) from x. Avis, Erdős and Pach (1988) introduced the extremal quantity f₃(n)=max∑_{x∈ S}e_r(x,S), where the maximum is taken over all n-point subsets S of 3-space and all assignments r\colon S→(0,∞) of distances. We show that if the pair (S,r) maximises f₃(n) and n is sufficiently large, then, except for at most 2 points, S is contained in a circle C and the axis of symmetry L of C, and r(x) equals the distance from x to C for each x∈ S\capL. This, together with a new construction, implies that f₃(n)=n²/4 + 5n/2 + O(1).

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