Gromov-compactifiability of Wasserstein spaces

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