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 \((X,{{\,\textrm{dist}\,}})\) 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 \({{\,\textrm{reach}\,}}(x, X\subset W_p(X)) = 0\) . This includes the cases where X is a compact manifold or a non-simply connected one. On the other hand, we show that \({{\,\textrm{reach}\,}}(X\subset W_p(X)) = \infty \) 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 \(X\hookrightarrow {{\,\textrm{Dgm}\,}}_\infty \) , the space of persistence diagrams equipped with the bottleneck distance.
