Geometric properties of cones with applications on the Hellinger-Kantorovich space, and a new distance on the space of probability measures

TL;DR

This paper explores the geometric structure of cone spaces, introduces a new distance on probability measures, and characterizes geodesics in the Hellinger-Kantorovich space, with implications for gradient flow analysis.

Contribution

It establishes a cone space structure over probability measures with a new distance, providing a full characterization of geodesics and geometric properties of the Hellinger-Kantorovich space.

Findings

Existence of a new distance on probability measures that forms a cone space.

Complete characterization of geodesics in the probability measure space.

Proven geometric properties like local-angle condition and partial K-semiconcavity.

Abstract

We study general geometric properties of cone spaces, and we apply them on the Hellinger--Kantorovich space $(\mathcal{M}(X),\mathsf{H\hspace{-0.25em} K}_{\alpha,\beta}).$ We exploit a two-parameter scaling property of the Hellinger-Kantorovich metric $ \mathsf{H\hspace{-0.25em} K}_{\alpha,\beta},$ and we prove the existence of a distance $\mathsf{S\hspace{-0.18em} H\hspace{-0.25em} K}_{\alpha,\beta}$ on the space of Probability measures that turns the Hellinger--Kantorovich space $(\mathcal{M}(X),\mathsf{H\hspace{-0.25em} K}_{\alpha,\beta})$ into a cone space over the space of probabilities measures $(\mathcal{P}(X),\mathsf{S\hspace{-0.18em} H\hspace{-0.25em} K}_{\alpha,\beta}).$ We provide a two parameter rescaling of geodesics in $(\mathcal{M}(X),\mathsf{H\hspace{-0.25em} K}_{\alpha,\beta}),$ and for $(\mathcal{P}(X),\mathsf{S\hspace{-0.18em} H\hspace{-0.25em} K}_{\alpha,\beta})$ we obtain a full characterization of the geodesics. We finally prove finer geometric properties, including local-angle condition and partial $K$-semiconcavity of the squared distances, that will be used in a future paper to prove existence of gradient flows on both spaces.