Anisotropic fractional area measures

Xiaxing Cai

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arXiv · 2510.05279

Anisotropic fractional area measures

Xiaxing Cai

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Abstract

The anisotropic $s$-fractional area measures are introduced as the first variation of the anisotropic fractional $s$-perimeter $P_s(K,L)$, with $L$ an origin symmetric convex body and $s\in(0,1)$. As $s\rightarrow 1^-$, the anisotropic $s$-fractional area measure converges to the mixed area measure of $K$ and the moment body of $L$. The Minkowski problem of these measures are solved. Finally, a necessary condition for the convexity of optimizers in the anisotropic fractional isoperimetric inequality is derived.

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Xiaxing Cai. 2025-10-06. Anisotropic fractional area measures. https://arxiv.org/abs/2510.05279

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