Brunn-Minkowski inequalities for sprays on surfaces (Q6611193)

Let \((M,g)\) be a two-dimensional Riemannian manifold. The author assumes that on \(M\) a family \(\Gamma\) of smooth constant-speed curves, called \(\Gamma\)-geodesics, is given, which is determined by a vector field on \(TM\), called a spray, satisfying certain explicit assumptions. (Examples are the flat spray on \({\mathbb R}^2\), the geodesic spray on a Riemannian manifold, the horocyclic spray on the hyperbolic plane.) For nonempty subsets \(A,B\subseteq M\) and \(\lambda\in (0,1)\) one defines \(M_\Gamma(A,B;\lambda)\) as the set of all points of the form \(\gamma(\lambda)\), where \(\gamma\) is a \(\Gamma\)-geodesic satisfying \(\gamma(0)\in A\) and \(\gamma(1)\in B\). \N\NThe main result is a necessary and sufficient condition on \(\Gamma\), involving the Gauss curvature of \(g\), in order that for every pair of nonempty Borel sets \(A,B\subseteq M\) and every \(0<\lambda<1\) one has \[ \mathrm{Area}(M_\Gamma(A,B;\lambda))^{1/2} \ge (1-\lambda)\cdot\mathrm{Area}(A)^{1/2}+\lambda\cdot\mathrm{Area}(B)^{1/2},\] where \(\mathrm{Area}\) denotes the Riemannnian area (this is also extended to more general volume forms on \(M\)). This theorem generalizes some earlier results (for example, the horocyclic Brunn-Minkowski inequality due to the author and \textit{B. Klartag} [Adv. Math. 436, Article ID 109381, 39 p. (2024; Zbl 1543.52003)]). Techniques of the proof comprise a needle decomposition theorem and projective Finsler metrizability. The paper concludes with a number of examples.