What does a typical metric space look like?

The collection Mn of all metric spaces on n pointswhose diameter is at most 2 can naturally be viewed as a compactconvex subset of R(n2), known as the metric polytope. In this paper,we study the metric polytope for large n and show that it is closeto the cube [1, 2](n2) ⊆ Mn in the following two senses. First, thevolume of the polytope is not much larger than that of the cube,with the following quantitative estimates:(1/6 + o(1))n3/2 ≤ log Vol(Mn) ≤ O(n3/2).Second, when sampling a metric space from Mn 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. Wediscuss alternative approaches to estimating the volume of Mn using exchangeability, Szemer´edi’s regularity lemma, the hypergraphcontainer method, and the K˝ov´ari–S´os–Tur´an theorem.