Brunn-Minkowski inequalities for sprays on surfaces

We propose a generalization of the Minkowski average of two subsets of a Riemannian manifold, in which geodesics are replaced by an arbitrary family of parametrized curves. Under certain assumptions, we characterize families of curves on a Riemannian surface for which a Brunn-Minkowski inequality holds with respect to a given volume form. In particular, we prove that under these assumptions, a family of constant-speed curves on a Riemannian surface satisfies the Brunn-Minkowski inequality with respect to the Riemannian area form if and only if the geodesic curvature of its members is determined by a function $\kappa$ on the surface, and $\kappa$ satisfies the inequality $$K+{\kappa }^2-\mathrm{\mid }\mathrm{\nabla }\kappa \mathrm{\mid }\ge 0$$ where $K$ is the Gauss curvature.