Complemented Brunn-Minkowski Inequalities and Isoperimetry for Homogeneous and Non-Homogeneous Measures

TL;DR

This paper introduces new isoperimetric and Brunn-Minkowski inequalities for measures with weights that are homogeneous or satisfy specific concavity conditions, extending classical results and providing elementary proofs.

Contribution

It develops a novel complemented Brunn-Minkowski inequality for certain measures, broadening the scope of isoperimetric inequalities beyond previous homogeneous cases.

Findings

Elementary proofs of sharp isoperimetric inequalities in normed spaces.

Introduction of complemented Brunn-Minkowski inequality for non-homogeneous measures.

Extension of inequalities to functional, Sobolev, and Nash-type forms.

Abstract

Elementary proofs of sharp isoperimetric inequalities on a normed space $(\mathbb{R}^n,||\cdot||)$ equipped with a measure $\mu = w(x) dx$ so that $w^p$ is homogeneous are provided, along with a characterization of the corresponding equality cases. When $p \in (0,\infty]$ and in addition $w^p$ is assumed concave, the result is an immediate corollary of the Borell-Brascamp-Lieb extension of the classical Brunn-Minkowski inequality, providing an elementary proof of a recent result of Cabr\'e-Ros Oton-Serra. When $p \in (-1/n,0)$, the relevant property turns out to be a novel "complemented Brunn-Minkowski" inequality, which we show is always satisfied by $\mu$ when $w^p$ is homogeneous. This gives rise to a new class of measures, which are "complemented" analogues of the class of convex measures introduced by Borell, but which have vastly different properties. The resulting isoperimetric inequality and characterization of isoperimetric minimizers extends beyond the recent results of Ca\~{n}ete--Rosales and Howe. The isoperimetric and Brunn-Minkowski type inequalities extend to the non-homogeneous setting, under a certain log-convexity assumption on the density. Finally, we obtain functional, Sobolev and Nash-type versions of the studied inequalities.