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

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.