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 \(\kappa \) $\kappa$ on the surface, and \(\kappa \) $\kappa$ satisfies the inequality \(\begin{aligned} K + \kappa ^2 - |\nabla \kappa | \ge 0 \end{aligned}\) $$\begin{array}{c}\hfill K+{\kappa }^2-|\mathrm{\nabla }\kappa |\ge 0\end{array}$$ where K is the Gauss curvature.