Nearly equal distances in metric spaces

Abstract

Let (X,d) be any finite metric space with n elements. We show that there are two pairs of distinct elements in X that determine two nearly equal distances in the sense that their ratio differs from 1 by at most 9lognn2. This bound (apart for the multiplicative constant) is best possible and we construct a metric space that attains this bound.We discuss related questions and consider in particular the Euclidean metric space.