Anisotropic fractional area measures

TL;DR

This work introduces the anisotropic $s$-fractional area measures $\mathcal{A}_s(K,L,\cdot)$ as the first variation of the anisotropic $s$-perimeter $P_s(K,L)$ with respect to perturbations of $K$, linking fractional perimeters to classical convex-geometry invariants. It proves a sharp variational formula, derives limiting relations $s\mathcal{A}_s(K,L,\cdot)\to \frac{|L|}{2}S_{n-1}(K,\cdot)$ as $s\to 0^+$ and $(1-s)\mathcal{A}_s(K,L,\cdot)\to S_{n-2}(K,ZL,\cdot)$ as $s\to 1^-$ for smooth $K$, and solves the Minkowski problem for these measures in full generality. The paper also provides a variational route to a Monge–Ampère reformulation in the smooth setting and gives a necessary condition for the convexity of optimizers in the anisotropic fractional isoperimetric inequality, tying together dual mixed volumes, chord measures, and moment bodies. Overall, it extends classical convex-geometry concepts to a fractional, anisotropic setting, enabling new isoperimetric-type results and measure-based characterizations.

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.

Theorems & Definitions (26)

• Theorem 1.1
• Definition 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Lemma 3.5
• Lemma 3.6
• proof