Fine Properties of Geodesics and Geodesic [Formula: see text] -Convexity for the Hellinger–Kantorovich Distance

We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger–Kantorovich problem ([Formula: see text] ), 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 [Formula: see text] , which are sufficiently strong to construct a characteristic transport-growth flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of [Formula: see text] 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 [Formula: see text] -convexity with respect to the Hellinger–Kantorovich distance. Examples of geodesically convex functionals are provided.