On the new weighted geometric inequalities near the sphere in space forms

TL;DR

This paper extends weighted geometric inequalities near spheres in space forms, providing stability estimates and generalizing previous results to broader settings and convex weights.

Contribution

It introduces new weighted Minkowski and Alexandrov-Fenchel inequalities for nearly spherical sets with convex weights in space forms.

Findings

Established weighted Minkowski inequalities for nearly spherical sets.

Proved quantitative stability estimates for Alexandrov-Fenchel inequalities.

Extended previous results to broader geometric settings and convex weights.

Abstract

In this paper, we first investigate weighted Minkowski type inequalities for nearly spherical sets in space forms, focusing on the sets that are $C^1$-close to geodesic spheres. Our results generalize the work of \cite{G22} by incorporating broader geometric settings and convex weight functions. Additionally, we establish quantitative stability estimates for weighted Alexandrov-Fenchel type inequalities in $\mathbb{R}^{n+1}$ and $\mathbb{H}^{n+1}$, extending the earlier results of \cite{VW24} and \cite{ZZ23}. These inequalities hold for nearly spherical sets that are $W^{2,\infty}$-close to geodesic spheres coupled with general convex weights.