  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.
