2021/05/11 by Facundo Mémoli, Mémoli, Facundo, Zhengchao Wan +1
Mathematics · #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2105.05369
The collection M of all isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance dGH is known to be a geodesic space. However, there is no known structural characterization of geodesics in M. In this paper we provide two such characterizations. We first prove that every Gromov-Hausdorff geodesic is in fact a geodesic in the Hausdorff hyperspace of some compact metric space, which we call a Hausdorff geodesic. Inspired by this characterization, we further elucidate a structural connection between Hausdorff geodesics and Wasserstein geodesics: every Hausdorff geodesic is equivalent to a so-called Hausdorff displacement interpolation. This equivalence allows us to establish that every Gromov-Hausdorff geodesic is dynamic, a notion which we develop in analogy with dynamic optimal couplings in the theory of optimal transport. Besides geodesics in M, we also study geodesics on the collection Mw of isomorphism classes of compact metric measure spaces. Sturm constructed a family of Gromov-type distances on Mw, which we denote d_GW,pS (for p∈[1,∞)), and proved that (Mw,d_GW,pS) is also a geodesic space. We are interested in d_GW,pS geodesics which are (essentially) Wasserstein geodesics. We prove the set of such geodesics is dense in the set of all d_GW,pS geodesics and identify a rich class of such geodesics.