On the Reach of Isometric Embeddings into Wasserstein Type Spaces

Resumen

We study the reach (in the sense of Federer) of the natural isometric embedding X hooked right arrow Wp(X) of X inside its p-Wasserstein space, where (X,dist) is a geodesic metric space. We prove that if a point x is an element of X can be joined to another point y is an element of X by two minimizing geodesics, then reach (x, X subset of 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 reach (X subset of Wp(X))=infinity 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 hooked right arrow W-v(X) into an Orlicz-Wasserstein space, a generalization by St urm of the classical Wasserstein space. Lastly, we establish the nullity of the reach for the isometric embedding of X hooked right arrow Dgm(infinity), the space of persistence diagrams equipped with the bottleneck distance.