Snowflake universality of Wasserstein spaces

TL;DR

This paper demonstrates the universality of Wasserstein spaces over -dimensional Euclidean space for embedding finite metric spaces with low distortion, revealing their rich geometric structure and implications for metric geometry open problems.

Contribution

It establishes sharp embedding results for Wasserstein spaces, showing their universality for finite metric spaces and implications for Alexandrov spaces and expander graphs.

Findings

Wasserstein spaces -dimensional are universal for finite metric spaces with low distortion

Sharpness of embedding exponents for Wasserstein spaces when p

Existence of metric spaces that do not embed into Wasserstein spaces with low distortion

Abstract

For $p\in (1,\infty)$ let $\mathscr{P}_p(\mathbb{R}^3)$ denote the metric space of all $p$-integrable Borel probability measures on $\mathbb{R}^3$, equipped with the Wasserstein $p$ metric $\mathsf{W}_p$. We prove that for every $\varepsilon>0$, every $\theta\in (0,1/p]$ and every finite metric space $(X,d_X)$, the metric space $(X,d_{X}^{\theta})$ embeds into $\mathscr{P}_p(\mathbb{R}^3)$ with distortion at most $1+\varepsilon$. We show that this is sharp when $p\in (1,2]$ in the sense that the exponent $1/p$ cannot be replaced by any larger number. In fact, for arbitrarily large $n\in \mathbb{N}$ there exists an $n$-point metric space $(X_n,d_n)$ such that for every $\alpha\in (1/p,1]$ any embedding of the metric space $(X_n,d_n^\alpha)$ into $\mathscr{P}_p(\mathbb{R}^3)$ incurs distortion that is at least a constant multiple of $(\log n)^{\alpha-1/p}$. These statements establish that there exists an Alexandrov space of nonnegative curvature, namely $\mathscr{P}_{\! 2}(\mathbb{R}^3)$, with respect to which there does not exist a sequence of bounded degree expander graphs. It also follows that $\mathscr{P}_{\! 2}(\mathbb{R}^3)$ does not admit a uniform, coarse, or quasisymmetric embedding into any Banach space of nontrivial type. Links to several longstanding open questions in metric geometry are discussed, including the characterization of subsets of Alexandrov spaces, existence of expanders, the universality problem for $\mathscr{P}_{\! 2}(\mathbb{R}^k)$, and the metric cotype dichotomy problem.