Sharp isoperimetric inequalities via the ABP method

TL;DR

This paper establishes new sharp isoperimetric inequalities with weights in convex cones using the ABP method, identifying minimizers as Euclidean balls and Wulff shapes, and provides new proofs for classical inequalities.

Contribution

It introduces a novel application of the ABP method to derive sharp weighted isoperimetric inequalities in convex cones, including anisotropic cases, with explicit minimizers.

Findings

Euclidean balls minimize the weighted isoperimetric quotient.

Wulff shapes minimize the anisotropic weighted perimeter.

New proofs of classical inequalities in convex cones.

Abstract

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of $\mathbb{R}^n$. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarkably, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient, even if all our weights are nonradial ---except for the constant ones. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella.