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 convex 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 close to the cube [1, 2] ⁽ⁿ2 ⁾ ⊆ 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: ⁽₆¹ + 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, Szemerédi’s regularity lemma, the hypergraph container method, and the Kővári–Sós–Turán theorem.