A Product Inequality for Extreme Distances

Abstract

Abstract Let p 1 , … , p n be n distinct points in the plane, and assume that the minimum inter-point distance occurs s min times, while the maximum inter-point distance occurs s max times. It is shown that s min s max ≤ 9 8 n 2 + O ( n ) ; this settles a conjecture of Erdős and Pach (1990).