Uniformly distributed distances: A geometric application of Jansen's inequality

Abstract

Let $d_1\le d_2\le \dots \le d_{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}$ denote the distances determined by $n$ points in the plane. It is shown that $\min {\sum }_i(d_{i+1}-d_i{)}^2=O(n^{-6/7})$, where the minimum is taken over all point sets with minimal distance $d_1\ge 1$. This bound is asymptotically tight.