Isometric Rigidity of Metric Constructions with respect to Wasserstein Spaces

Abstract

In this paper we study the isometric rigidity of certain classes of metric spaces with respect to the $p$-Wasserstein space. We prove that spaces that split a separable Hilbert space are not isometrically rigid with respect to ${\mathbb{P}}_2$. We then prove that infinite rays are isometrically rigid with respect to ${\mathbb{P}}_p$ for any $p\ge 1$, whereas taking infinite half-cylinders (i.e.\ product spaces of the form $X\times [0,\infty )$) over compact non-branching geodesic spaces preserves isometric rigidity with respect to ${\mathbb{P}}_p$, for $p>1$. Finally, we prove that spherical suspensions over compact spaces with diameter less than $\pi /2$ are isometrically rigid with respect to ${\mathbb{P}}_p$, for $p>1$.