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We study the reach (in the sense of Federer) of the natural isometricembedding $X\hookrightarrow W_p(X)$ of $X$ inside its $p$-Wasserstein space,where $(X,\operatorname{dist})$ is a geodesic metric space. We prove that if apoint $x\in X$ can be joined to another point $y\in X$ by two minimizinggeodesics, then $\operatorname{reach}(x, X\subset W_p(X)) = 0$. This includesthe cases where $X$ is a compact manifold or a non-simply connected one. On theother 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 regularityof the projection map. Furthermore, we replicate these findings by consideringthe isometric embedding $X\hookrightarrow W_\vartheta(X)$ into anOrlicz--Wasserstein space, a generalization by Sturm of the classicalWasserstein space. Lastly, we establish the nullity of the reach for theisometric embedding of $X$ into $\operatorname{Dgm}_\infty$, the space ofpersistence diagrams equipped with the bottleneck distance.

On the reach of isometric embeddings into Wasserstein type spaces