On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities

TL;DR

This paper extends the Brunn-Minkowski inequality to various classes of measures and sets, providing new isoperimetric inequalities and settling a Gaussian measure conjecture in two dimensions.

Contribution

It establishes the Brunn-Minkowski inequality for unconditional product and log-concave measures with convex bodies, and confirms the two-dimensional Gaussian case of a conjecture.

Findings

Inequality holds for unconditional product measures with decreasing density.

Inequality valid for unconditional log-concave measures and convex bodies.

Confirmed the 2D Gaussian measure case of the conjecture.

Abstract

In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality \[ \mu(\lambda A + (1-\lambda)B)^{1/n} \geq \lambda \mu(A)^{1/n} + (1-\lambda)\mu(B)^{1/n} \] holds true for an unconditional product measure $\mu$ with decreasing density and a pair of unconditional convex bodies $A,B \subset \mathbb{R}^n$. We also show that the above inequality is true for any unconditional $\log$-concave measure $\mu$ and unconditional convex bodies $A,B \subset \mathbb{R}^n$. Finally, we prove that the inequality is true for a symmetric $\log$-concave measure $\mu$ and a pair of symmetric convex sets $A,B \subset \mathbb{R}^2$, which, in particular, settles two-dimensional case of the conjecture for Gaussian measure proposed by R. Gardner and the fourth named author. In addition, we deduce the $1/n$-concavity of the parallel volume $t \mapsto \mu(A+tB)$, Brunn's type theorem and certain analogues of Minkowski first inequality.