What does a typical metric space look like?

The collection M-n of all metric spaces on n points whose diameter is at most 2 can naturally be viewed as a compact (n) convex subset of R-(n 2), known as the metric polytope. In this paper, we study the metric polytope for large n and show that it is close to the cube [1, 2]((n 2)) subset of M-n in the following two senses. First, the volume of the polytope is not much larger than that of the cube, with the following quantitative estimates: {1/6 + o(1) n(3/2) <= log Vol(M-n) <= O (n(3/2)). Second, when sampling a metric space from M-n uniformly at random, the minimum distance is at least 1-n(-c) with high probability, for some c > 0. Our proof is based on entropy techniques. We discuss alternative approaches to estimating the volume of M-n using exchangeability, Szemeredi's regularity lemma, the hypergraph container method, and the Kovari-Sos-Turan theorem.