## Abstract

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 ^{(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 ^{)} ⊆ 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: ^{(}_{6}^{1} + 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.

Original language | English |
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Pages (from-to) | 11-53 |

Number of pages | 43 |

Journal | Annales De L Institut Henri Poincare-Probabilites Et Statistiques |

Volume | 60 |

Issue number | 1 |

DOIs | |

State | Published - 1 Feb 2024 |

## Keywords

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

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty