# On isometric embeddings of Wasserstein spaces - the discrete case

Geher, G. P., Titkos, T. and Virosztek, D. (2019) On isometric embeddings of Wasserstein spaces - the discrete case. Journal of Mathematical Analysis and Applications, 480 (2). 123435. ISSN 0022-247X

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To link to this item DOI: 10.1016/j.jmaa.2019.123435

## Abstract/Summary

The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space $\mathcal{W}_p(\X)$, where $\X$ is a countable discrete metric space and $0<p<\infty$ is any parameter value. Roughly speaking, we will prove that any isometric embedding can be described by a special kind of $\X\times(0,1]$-indexed family of nonnegative finite measures. Our result implies that a typical non-surjective isometric embedding of $\ws$ splits mass and does not preserve the shape of measures. In order to stress that the lack of surjectivity is what makes things challenging, we will prove alternatively that $\ws$ is isometrically rigid for all $0<p<\infty$.

Item Type: Article Yes Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics 85683 Elsevier