Characterization of Gromov-type geodesics

It is well known that, when endowed with the Gromov-Hausdorff distance dGH, collection M of all isometry classes compact metric spaces a complete and separable space. also that (M,dGH) geodesic space, but there no structural characterization geodesics in M. In this paper we provide two characterizations We call γ:[0,1]→M Hausdorff-realizable if exists space Z containing isometric copies γ(t) for each t∈[0,1] such Hausdorff satisfies dHZ(γ(s),γ(t))=dGH(γ(s),γ(t)) s,t∈[0,1]. way, γ actually hyperspace Z, it geodesic. prove fact every Hausdorff-realizable. Inspired by characterization, further elucidate connection between Wasserstein geodesics: show equivalent to so-called displacement interpolation. This equivalence allows us establish dynamic, notion which develop analogy dynamic optimal couplings theory transport. Besides M, study on Mw isomorphism measure spaces. Sturm constructed family Gromov-type distances Mw, denote dGW,pS (for p∈[1,∞)), as an analogue proved (Mw,dGW,pS) define Wasserstein-realizable sense similar set dense geodesics. identify rich class are Wasserstein-realizable.