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arXiv · 2508.04067

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

Weimin Sheng·Yinhang Wang
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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.

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Weimin Sheng, Yinhang Wang. 2025-08-06. On the new weighted geometric inequalities near the sphere in space forms. https://doi.org/10.4153/s0008414x26102223

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