Alexandrov-Fenchel type inequalities with convex weight in space forms

TL;DR

This paper establishes new sharp weighted Alexandrov-Fenchel and Minkowski inequalities for hypersurfaces in space forms, extending classical results by incorporating convex, non-decreasing weights across Euclidean, spherical, and hyperbolic geometries.

Contribution

It introduces a broad family of geometric inequalities with convex, non-decreasing weights, generalizing classical inequalities in space forms.

Findings

Derived sharp weighted inequalities in Euclidean, spherical, and hyperbolic spaces.

Extended classical inequalities by incorporating convex, non-decreasing weights.

Provided a flexible framework for a family of geometric inequalities.

Abstract

In this paper, we derive new sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces under various convexity assumptions in Euclidean, spherical, and hyperbolic spaces. These inequalities extend classical results by incorporating weights given by convex, non-decreasing positive functions, which are otherwise arbitrary. Our approach gives rise to a broad family of geometric inequalities, as each convex, non-decreasing function yields a corresponding inequality, providing considerable flexibility.