Nearly equal distances in metric spaces

Let (X,d)(X,d) be any finite metric space with nn elements. We show that there are two pairs of distinct elements in XX 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.