  We study the fine regularity properties of optimal potentials for the dual
formulation of the Hellinger--Kantorovich problem (HK), providing sufficient
conditions for the solvability of the primal Monge formulation. We also
establish new regularity properties for the solution of the Hamilton--Jacobi
equation arising in the dual dynamic formulation of HK, which are sufficiently
strong to construct a characteristic transport-dilation flow driving the
geodesic interpolation between two arbitrary positive measures. These results
are applied to study relevant geometric properties of HK geodesics and to
derive the convex behaviour of their Lebesgue density along the transport flow.
Finally, exact conditions for functionals defined on the space of measures are
derived that guarantee the geodesic $\lambda$-convexity with respect to the
Hellinger--Kantorovich distance. Examples of geodesically convex functionals
are provided.
