On the distribution of distances in finite sets in the plane (Q1074869)

Let \(n_ k\) denote the number of times the kth largest distance occurs among a set S of n points. The author shows that if S is the set of vertices of a convex polygon in the Euclidean plane, then \(n_ 1+2n_ 2\leq 3n\) and \(n_ 2\leq n+n_ 1.\) (In the paper two another inequalities are proved as well, which are valid in any metric space, but they are weaker than the mentioned ones.) Together with the well-known inequality \(n_ 1\leq n\) and the trivial inequalities \(n_ 1\geq 0\), \(n_ 2\geq 0\), all linear inequalities which are valid for n, \(n_ 1\) and \(n_ 2\) are consequences of these.