Geometric properties of cones with applications on the Hellinger--Kantorovich space, and a new distance on the space of probability measures
- Laschos, Vaios
- Mielke, Alexander
2010 Mathematics Subject Classification
- Geometry on cones, local angle condition, K-semiconcavity, Hellinger-Kantorovich, Spherical Hellinger-Kantorovich
By studying general geometric properties of cone spaces, we prove the existence of a distance on the space of Probability measures that turns the Hellinger--Kantorovich space into a cone space over the space of probabilities measures. Here we exploit a natural two-parameter scaling property of the Hellinger-Kantorovich distance. For the new space, we obtain a full characterization of the geodesics. We also provide new geometric properties for the original space, including a two-parameter rescaling and reparametrization of the geodesics, local-angle condition and some partial K-semiconcavity of the squared distance, that it will be used in a future paper to prove existence of gradient flows.