New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms

Abstract

In this paper, we establish a broad class of new sharp Alexandrov-Fenchel inequalities involving general convex weight functions for static convex hypersurfaces in hyperbolic space. Additionally, we derive new weighted Minkowski-type inequalities for static convex hypersurfaces in hyperbolic space $\mathbb{H}^n$ and for convex hypersurfaces in the sphere $\mathbb{S}^n$. The tools we shall use are the locally constrained inverse curvature flows in hyperbolic space and in the sphere.