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 $(\mathcal{M}(X),{\mathsf{H}\mathsf{K}}_{\alpha ,\beta }).$ We exploit a two-parameter scaling property of the Hellinger-Kantorovich metric ${\mathsf{H}\mathsf{K}}_{\alpha ,\beta },$ and we prove the existence of a distance ${\mathsf{S}\mathsf{H}\mathsf{K}}_{\alpha ,\beta }$ on the space of Probability measures that turns the Hellinger--Kantorovich space $(\mathcal{M}(X),{\mathsf{H}\mathsf{K}}_{\alpha ,\beta })$ into a cone space over the space of probabilities measures $(\mathcal{P}(X),{\mathsf{S}\mathsf{H}\mathsf{K}}_{\alpha ,\beta }).$ We provide a two parameter rescaling of geodesics in $(\mathcal{M}(X),{\mathsf{H}\mathsf{K}}_{\alpha ,\beta }),$ and for $(\mathcal{P}(X),{\mathsf{S}\mathsf{H}\mathsf{K}}_{\alpha ,\beta })$ 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.