Brunn–Minkowski Inequalities for Sprays on Surfaces

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 $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> on the surface, and $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> satisfies the inequality $$\begin{aligned} K + \kappa ^2 - |\nabla \kappa | \ge 0 \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>κ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>-</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>∇</mml:mi> <mml:mi>κ</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where K is the Gauss curvature.