Geometric properties of cones with applications on the Hellinger-Kantorovich space, and a new distance on the space of probability measures
Abstract
We study general geometric properties of cone spaces, and we apply them on the Hellinger--Kantorovich space (M(X),H-0.25em Kα,β). We exploit a two-parameter scaling property of the Hellinger-Kantorovich metric H-0.25em Kα,β, and we prove the existence of a distance S-0.18em H-0.25em Kα,β on the space of Probability measures that turns the Hellinger--Kantorovich space (M(X),H-0.25em Kα,β) into a cone space over the space of probabilities measures (P(X),S-0.18em H-0.25em Kα,β). We provide a two parameter rescaling of geodesics in (M(X),H-0.25em Kα,β), and for (P(X),S-0.18em H-0.25em Kα,β) 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.
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