What does a typical metric space look like?

TL;DR

This paper analyzes the structure of the metric polytope for large point sets, showing it closely resembles a high-dimensional cube in volume and distance properties, using entropy and combinatorial techniques.

Contribution

It provides quantitative estimates of the volume and typical distances in the metric polytope for large n, employing entropy methods and discussing alternative combinatorial approaches.

Findings

Volume of the metric polytope is close to that of the cube, with bounds involving n^{3/2}.

A random metric space from the polytope typically has minimum distance close to 1.

The paper introduces entropy-based proofs and compares them with combinatorial methods.

Abstract

The collection $\mathcal{M}_n$ of all metric spaces on $n$ points whose diameter is at most $2$ can naturally be viewed as a compact convex subset of $\mathbb{R}^{\binom{n}{2}}$, 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 $[1,2]^{\binom{n}{2}} \subseteq \mathcal{M}_n$ 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: \[ \left(\tfrac{1}{6}+o(1)\right)n^{3/2} \le \log \mathrm{Vol}(\mathcal{M}_n)\le O(n^{3/2}). \] Second, when sampling a metric space from $\mathcal{M}_n$ 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 $\mathcal{M}_n$ using exchangeability, Szemer\'edi's regularity lemma, the hypergraph container method, and the K\H{o}v\'ari--S\'os--Tur\'an theorem.