On the reach of isometric embeddings into Wasserstein type spaces

We study the reach (in the sense of Federer) of the natural isometric embedding $X\hookrightarrow W_p(X)$ of $X$ inside its $p$-Wasserstein space, where $(X,\operatorname{dist})$ is a geodesic metric space. We prove that if a point $x\in X$ can be joined to another point $y\in X$ by two minimizing geodesics, then $\operatorname{reach}(x, X\subset W_p(X)) = 0$. This includes the cases where $X$ is a compact manifold or a non-simply connected one. On the other hand, we show that $\operatorname{reach}(X\subset W_p(X)) = \infty$ when $X$ is a CAT(0) space. The infinite reach enables us to examine the regularity of the projection map. Furthermore, we replicate these findings by considering the isometric embedding $X\hookrightarrow W_\vartheta(X)$ into an Orlicz--Wasserstein space, a generalization by Sturm of the classical Wasserstein space. Lastly, we establish the nullity of the reach for the isometric embedding of $X$ into $\operatorname{Dgm}_\infty$, the space of persistence diagrams equipped with the bottleneck distance.