Brunn-Minkowski inequalities for sprays on surfaces

TL;DR

This paper generalizes the Brunn-Minkowski inequality to families of curves on Riemannian surfaces, characterizing when the inequality holds based on curvature conditions involving geodesic curvature and Gauss curvature.

Contribution

It introduces a new framework for Brunn-Minkowski inequalities using arbitrary families of curves on surfaces and characterizes the curvature conditions needed for the inequality to hold.

Findings

The inequality holds if the geodesic curvature satisfies a specific differential inequality.

Constant-speed curves on surfaces satisfy the inequality under the curvature condition.

The curvature condition involves the Gauss curvature and the gradient of the geodesic curvature.

Abstract

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 $$K + \kappa^2 - |\nabla\kappa| \ge 0$$ where $K$ is the Gauss curvature.