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arXiv · 2410.14648

Isometric Rigidity of Metric Constructions with respect to Wasserstein Spaces

Mauricio Che·Fernando Galaz-García·Martin Kerin·Jaime Santos-Rodríguez
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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 $p\geq 1$, 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 $p>1$.

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Mauricio Che, Fernando Galaz-García, Martin Kerin, Jaime Santos-Rodríguez. 2024-10-18. Isometric Rigidity of Metric Constructions with respect to Wasserstein Spaces. https://arxiv.org/abs/2410.14648

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