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

TL;DR

This paper introduces new sharp weighted Alexandrov-Fenchel and Minkowski inequalities for convex hypersurfaces in hyperbolic space and spheres, utilizing inverse curvature flows to establish these geometric inequalities.

Contribution

It develops a broad class of new inequalities involving convex weights, extending classical results to space forms with novel methods.

Findings

Established sharp weighted Alexandrov-Fenchel inequalities in hyperbolic space.

Derived new weighted Minkowski inequalities in hyperbolic space and spheres.

Utilized locally constrained inverse curvature flows as the main tool.

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.