On the distribution of distances in finite sets in the plane

Let n k 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 +n 1. Together with the well-known inequality n~<~n and the trivial inequalities n~>~O and n2>~O, all linear inequalities which are valid for n, n 1 and n2 are consequences of these. Similar results are obtained for the hyperbolic plane.