This is an announcement for the paper “On isometric embeddings of Wasserstein spaces -- the discrete case” by György Pál Gehérhttps://arxiv.org/search/math?searchtype=author&query=Geh%C3%A9r%2C+G+P, Tamás Titkoshttps://arxiv.org/search/math?searchtype=author&query=Titkos%2C+T, Dániel Virosztekhttps://arxiv.org/search/math?searchtype=author&query=Virosztek%2C+D. Abstract: The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space $\mathcal{W}_p(\mathcal{X})$, where $\mathcal{X}$ 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 $\mathcal{X}\times(0,1]$-indexed family of nonnegative finite measures. Our result implies that a typical non-surjective isometric embedding of $\mathcal{W}_p(\mathcal{X})$ 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 $\mathcal{W}_p(\mathcal{X})$ is isometrically rigid for all $0<p<\infty$.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1809.01101