Characterization of Gromov-type geodesics

Abstract

The collection $\mathcal{M}$ of all isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance $d_\mathcal{GH}$ is known to be a geodesic space. However, there is no known structural characterization of geodesics in $\mathcal{M}$. 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 $\mathcal{M}$, we also study geodesics on the collection $\mathcal{M}^w$ of isomorphism classes of compact metric measure spaces. Sturm constructed a family of Gromov-type distances on $\mathcal{M}^w$, which we denote $d_{\mathcal{GW},{p}}^\mathrm{S}$ (for $p\in[1,\infty)$), and proved that $(\mathcal{M}^w,d_{\mathcal{GW},{p}}^\mathrm{S})$ is also a geodesic space. We are interested in $d_{\mathcal{GW},{p}}^\mathrm{S}$ geodesics which are (essentially) Wasserstein geodesics. We prove the set of such geodesics is dense in the set of all $d_{\mathcal{GW},{p}}^\mathrm{S}$ geodesics and identify a rich class of such geodesics.