Characterization of Gromov-type geodesics

Abstract

It is well known that, when endowed with the Gromov-Hausdorff distance dGH, the collection M of all isometry classes of compact metric spaces is a complete and separable space. It is also known that (M,dGH) is a geodesic metric space, but there is no known structural characterization of geodesics in M.In this paper we provide two characterizations of geodesics in M. We call a Gromov-Hausdorff geodesic γ:[0,1]→MHausdorff-realizable if there exists a compact metric space Z containing isometric copies of γ(t) for each t∈[0,1] such that the Hausdorff distance satisfies dHZ(γ(s),γ(t))=dGH(γ(s),γ(t)) for all s,t∈[0,1]. In this way, γ is actually a geodesic in the Hausdorff hyperspace of Z, and we call it a Hausdorff geodesic. We prove that in fact every Gromov-Hausdorff geodesic is Hausdorff-realizable. Inspired by this characterization, we further elucidate a structural connection between Hausdorff geodesics and Wasserstein geodesics: we show that 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 dGW,pS (for p∈[1,∞)), as an analogue of dGH, and proved that (Mw,dGW,pS) is also a geodesic space. We define a notion of Wasserstein-realizabledGW,pS geodesics in a sense similar to Hausdorff-realizable geodesics and show that the set of all Wasserstein-realizabledGW,pS geodesics is dense in the set of all dGW,pS geodesics. We further identify a rich class of dGW,pS geodesics which are Wasserstein-realizable.