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

$$\begin{aligned} K + \kappa ^2 - |\nabla \kappa | \ge 0 \end{aligned}$$

where K is the Gauss curvature.