论文标题
Brunn-Minkowski在表面上喷雾剂的不平等现象
Brunn-Minkowski inequalities for sprays on surfaces
论文作者
论文摘要
我们提出了对riemannian歧管的两个子集的平均水平的概括，其中大地测量被任意的参数化曲线家族取代。在某些假设下，我们表征了在riemannian表面上的曲线家族，而布鲁恩 - 金斯基（Brunn-Minkowski）不平等相对于给定的体积形式存在。特别是，我们证明在这些假设下，在riemannian表面上的恒定曲线家族满足了布鲁恩 - 米科夫斯基相对于riemannian区域形式的不平等，并且仅当其成员的地理曲率弯曲时，当它的地面$κ$ $κ$κ; \ ge 0 $$其中$ k $是高斯曲率。
We propose a generalization of the Minkowski average of two subsets of a Riemannian manifold, in which geodesics are replaced by an arbitrary family of parametrized curves. Under certain assumptions, we characterize families of curves on a Riemannian surface for which a Brunn-Minkowski inequality holds with respect to a given volume form. In particular, we prove that under these assumptions, a family of constant-speed curves on a Riemannian surface satisfies the Brunn-Minkowski inequality with respect to the Riemannian area form if and only if the geodesic curvature of its members is determined by a function $κ$ on the surface, and $κ$ satisfies the inequality $$K + κ^2 - |\nablaκ| \ge 0$$ where $K$ is the Gauss curvature.
