Wasserstein Rigidity over $\mathbb{R}^n$ with smooth norms

TL;DR

This paper investigates the geometric rigidity of Wasserstein spaces over Euclidean spaces with smooth norms, establishing conditions under which these spaces are isometrically rigid, especially for p ≠ 2 and certain q-norms.

Contribution

It proves isometric rigidity of Wasserstein spaces over bR^n with smooth norms for p  2, extending to specific cases where p=2 and N is an l_q-norm with q>2.

Findings

Wasserstein spaces are isometrically rigid for smooth norms when p  2.

Rigidity also holds for p=2 when the norm is an l_q-norm with q>2.

Rigidity results depend on the smoothness of the norm and the value of p.

Abstract

We study $p-$Wasserstein spaces $ \mathcal{W}_p(\mathbb{R}^n, d_N)$ over $\mathbb{R}^n$ equipped with a norm metric $d_N$. We show that, if the norm is smooth enough, then the Wasserstein space is isometrically rigid whenever $p \neq 2$. We also show that, even when $p=2$, we can recover the isometric rigidity of the Wasserstein space $\mathcal{W}_2(\mathbb{R}^n, d_N)$ when $N$ is an $l_q-$norm and $q>2$.