Anisotropic Isoperimetric Inequalities involving Boundary Momentum, Perimeter and Volume

TL;DR

This paper proves that the Wulff shape uniquely minimizes a scale-invariant anisotropic functional involving boundary momentum, perimeter, and volume, extending isoperimetric inequalities in anisotropic settings.

Contribution

It establishes the Wulff shape as the unique minimizer for a new anisotropic functional combining boundary momentum, perimeter, and volume.

Findings

Wulff shape is the unique minimizer of the functional.

The functional is scale-invariant and involves anisotropic measures.

The result extends classical isoperimetric inequalities to anisotropic contexts.

Abstract

We consider a scale invariant functional involving the anisotropic $p-$momentum, the anisotropic perimeter and the volume. We show that the Wulff shape, associated with the Finsler norm $F$ considered and centered at the origin, is the unique minimizer of the anisotropic functional taken into consideration among all bounded convex sets.