The collection Mn 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 ) ⊆ Mn 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) ) 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. We discuss alternative approaches to estimating the volume of Mn using exchangeability, Szemerédi’s regularity lemma, the hypergraph container method, and the Kővári–Sós–Turán theorem.

Original languageEnglish
Pages (from-to)11-53
Number of pages43
JournalAnnales De L Institut Henri Poincare-Probabilites Et Statistiques
Issue number1
StatePublished - 1 Feb 2024


  • Entropy method
  • Finite metric space
  • Metric polytope
  • Random metric space

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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