Gromov-compactifiability of Wasserstein spaces

Abstract

We characterize Gromov-compactifiability of Wasserstein spaces over Polish metric spaces for $1\leq p\leq\infty$. The space $\mathcal{P}_1(X)$ is Gromov-compactifiable if and only if $X$ is bounded and Gromov-compactifiable, whereas $\mathcal{P}_p(X)$ is Gromov-compactifiable for every $1<p<\infty$. At the other endpoint, $\mathcal{P}_\infty(X)$ is Gromov-compactifiable if and only if $X$ is. For $p=1$, the sufficiency argument combines an averaged triangle defect with tightness. For $1<p<\infty$, the main tool is the uniform convexity of $L^p$. The $p=\infty$ proof uses the finite-set criterion and a mass redistribution construction.