The collection Mnβ of all metric spaces on n points whose
diameter is at most 2 can naturally be viewed as a compact convex subset of
R(2nβ), 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](2nβ)βMnβ 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: (61β+o(1))n3/2β€logVol(Mnβ)β€O(n3/2). Second, when sampling a metric space from Mnβ
uniformly at random, the minimum distance is at least 1βnβc with high
probability, for some c>0. Our proof is based on entropy techniques. We
discuss alternative approaches to estimating the volume of Mnβ
using exchangeability, Szemer\'edi's regularity lemma, the hypergraph container
method, and the K\H{o}v\'ari--S\'os--Tur\'an theorem.Comment: 64 pages, 2 figures. v2: Swapped Sections 5 and 6 and added a
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