On the distribution of distances in finite sets in the plane

Abstract Let n k denote the number of times the k th 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 n 1 +2 n 2 ⩽3 n and n 2 ⩽ n + n 1 . Together with the well-known inequality n 1 ⩽ n and the trivial inequalities n 1 ⩾0 and n 2 ⩾0, all linear inequalities which are valid for n , n 1 and n 2 are consequences of these. Similar results are obtained for the hyperbolic plane.