Favourite distances in 3-space
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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 f_3(n)=max∑_x∈ Se_r(x,S), where the maximum is taken over all n-point subsets S of 3-space and all assignments r S→(0,∞) of distances. We show that if the pair (S,r) maximises f_3(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∩L. This, together with a new construction, implies that f_3(n)=n^2/4 + 5n/2 + O(1).
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