## Abstract

The collection ${\mathcal{M}}_{\mathit{n}}$ of all metric spaces on *n* points whose diameter is at most 2 can naturally be viewed as a compact convex subset of ${\mathbb{R}}^{\left(\genfrac{}{}{0.0pt}{}{\mathit{n}}{2}\right)}$, 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 ${[1,2]}^{\left(\genfrac{}{}{0.0pt}{}{\mathit{n}}{2}\right)}\subseteq {\mathcal{M}}_{\mathit{n}}$ 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:

$$(\frac{1}{6}\mathbf{+}\mathit{o}(1)){\mathit{n}}^{3/2}\le log\mathrm{Vol}({\mathcal{M}}_{\mathit{n}})\le \mathit{O}\left({\mathit{n}}^{3/2}\right).$$

Second, when sampling a metric space from ${\mathcal{M}}_{\mathit{n}}$ uniformly at random, the minimum distance is at least $1-{\mathit{n}}^{-\mathit{c}}$ with high probability, for some $\mathit{c}>0$. Our proof is based on entropy techniques. We discuss alternative approaches to estimating the volume of ${\mathcal{M}}_{\mathit{n}}$ using exchangeability, Szemerédi’s regularity lemma, the hypergraph container method, and the Kővári–Sós–Turán theorem.

La collection ${\mathcal{M}}_{\mathit{n}}$ de tous les espaces métriques à *n* points de diamètre au plus 2 peut être vue naturellement comme un convexe compact de ${\mathbb{R}}^{\left(\genfrac{}{}{0.0pt}{}{\mathit{n}}{2}\right)}$, appelé le polytope métrique. Dans cet article, nous étudions le polytope métrique lorsque *n* est grand, et montrons qu’il est proche du cube ${[1,2]}^{\left(\genfrac{}{}{0.0pt}{}{\mathit{n}}{2}\right)}\subseteq {\mathcal{M}}_{\mathit{n}}$ aux deux sens suivants. Tout d’abord, le volume du polytope n’est pas beaucoup plus grand que celui du cube, avec les estimées quantitatives suivantes :

$$(\frac{1}{6}\mathbf{+}\mathit{o}(1)){\mathit{n}}^{3/2}\le log\mathrm{Vol}({\mathcal{M}}_{\mathit{n}})\le \mathit{O}\left({\mathit{n}}^{3/2}\right).$$

Ensuite, lorsqu’on échantillonne uniformément au hasard un espace métrique de ${\mathcal{M}}_{\mathit{n}}$, la distance minimale est au moins $1-{\mathit{n}}^{-\mathit{c}}$ avec grande probabilité, pour un $\mathit{c}>0$. Notre preuve utilise des techniques d’entropie. Nous discutons également d’autres approches permettant d’estimer le volume de ${\mathcal{M}}_{\mathit{n}}$, utlisant l’échangeabilité, le lemme de régularité de Szemerédi, la méthode des conteneurs d’hypergraphes, et le théorème de Kővári–Sós–Turán.

## Funding Statement

The research of G.K. was supported by the ISF, by the Jesselson foundation, and by Paul and Tina Gardner.

The research of T.M. was supported by ISF Grants 626/14 and 1052/18.

The research of R.P. was supported by ISF Grants 1048/11, 861/15 and 1971/19, by IRG Grant SPTRF, and ERC grant LocalOrder.

The research of W.S. was supported by ISF Grants 1147/14 and 1145/18.

## Dedication

To the memory of Dima Ioffe, our friend and colleague. Mathematical physicist, probabilist and a dear person who freely shared his good advice and insight. His passing is a great loss to our community.

## Acknowledgments

We thank Itai Benjamini for asking the question and Gil Kalai for informing us that this object is known as the metric polytope. We thank Omer Angel, Dor Elboim, Ronen Eldan, Ehud Friedgut, Shoni Gilboa, Rob Morris, Balász Ráth and Johan Wästlund for many interesting discussions. Special thanks are due to Rob Morris, who kindly agreed to us presenting the results of our joint work with him in Section 6.3 of this paper.

## Citation

Gady Kozma. Tom Meyerovitch. Ron Peled. Wojciech Samotij. "What does a typical metric space look like?." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 11 - 53, February 2024. https://doi.org/10.1214/22-AIHP1262

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