典型的度量空间是什么样的？
What does a typical metric space look like?
摘要：
在$n$点上的所有度量空间的集合$\mathcal{M}_n$，其直径至多为$2$，可以自然地看作是$\mathbb{R}^{\binom{n}{2}}$的紧凸子集，称为度量多面体。本文研究了大$n$的度量多面体，并证明了它在以下两个意义上接近立方体$[1,2]^{\binom{n}{2}} \subseteq \mathcal{M}_n$。首先，多面体的体积并不比立方体大多少，有以下的定量估计：\[ \left(\tfrac{1}{6}+o(1)\right)n^{3/2} \le \log \mathrm{Vol}(\mathcal{M}_n)\le O(n^{3/2}). \]其次，当从$\mathcal{M}_n$均匀随机地采样一个度量空间时，对于某些$c > 0$，最小距离至少是$1 - n^{-c}$，概率很高。我们的证明是基于熵技术。我们讨论了替代方法来估计$\mathcal{M}_n$的体积使用交换，Szemerédi的正则引理，超图容器方法，和Komvári-Sós-Turán定理。
英文摘要：
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, Szemerédi's regularity lemma, the hypergraph container method, and the Kővári--Sós--Turán theorem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
