On the reach of isometric embeddings into Wasserstein type spaces

2023/07/03 by Casado, Javier, Cuerno, Manuel, Santos-Rodríguez, Jaime · 1 citation
#28A33 #30L15 #49Q20 #49Q22 #53C21 #55N31 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2307.01051

Abstract

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,dist) is a geodesic metric space. We prove that if a point x∈ X can be joined to another point y∈ X by two minimizing geodesics, then reach(x, X⊂ 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⊂ W_p(X)) = ∞ 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_ϑ(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 Dgm_∞, the space of persistence diagrams equipped with the bottleneck distance.

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