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arXiv · 2303.03719

Anisotropic weighted isoperimetric inequalities for star-shaped and $F$-mean convex hypersurface

Rong Zhou·Tailong Zhou
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Abstract

We prove two anisotropic type weighted geometric inequalities that hold for star-shaped and $F$-mean convex hypersurfaces in $\mathbb{R}^{n+1}$. These inequalities involve the anisotropic $p$-momentum, the anisotropic perimeter and the volume of the region enclosed by the hypersurface. We show that the Wulff shape of $F$ is the unique minimizer of the corresponding functionals among all star-shaped and $F$-mean convex sets. We also consider their quantitative versions characterized by the Hausdorff distance between the hypersurface and a rescaled Wulff shape. As a corollary, we obtain the stability of the Weinstock inequality for star-shaped and strictly mean convex domains, which requires weaker convexity compared to \cite{Gavitone}.

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Rong Zhou, Tailong Zhou. 2023-03-07. Anisotropic weighted isoperimetric inequalities for star-shaped and $F$-mean convex hypersurface. https://arxiv.org/abs/2303.03719

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