On the distribution of distances in finite sets in the plane

AbstractLet nk denote the number of times the kth largest distance occurs among a set S of n points. We show that if S is the set of vertices of a convex polygone in the euclidean plane, then n1+2n2⩽3n and n2⩽n+n1. Together with the well-known inequality n1⩽n and the trivial inequalities n1⩾0 and n2⩾0, all linear inequalities which are valid for n, n1 and n2 are consequences of these. Similar results are obtained for the hyperbolic plane

The shortest distance among points in general position

AbstractWe prove that among n points in the plane in general position, the shortest distance can occur at most (2+37)n times. We also give a construction where the shortest distance occurs more than (2+516)N−10⌊n⌊ times

The two largest distances in finite planar sets

AbstractWe determine all homogenous linear inequalities satisfied by the numbers of occurrences of the two largest distances among n points in the plane