We study general geometric properties of cone spaces, and we apply them on
the Hellinger--Kantorovich space (M(X),HKα,β). We exploit a two-parameter scaling property of the
Hellinger-Kantorovich metric HKα,β,
and we prove the existence of a distance SHKα,β on the space of Probability measures that
turns the Hellinger--Kantorovich space
(M(X),HKα,β) into a cone
space over the space of probabilities measures
(P(X),SHKα,β). We provide a two parameter rescaling of geodesics in
(M(X),HKα,β), and for
(P(X),SHKα,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