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

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 non-increasing 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 the two-dimensional case of the conjecture for Gaussian measure proposed by Gardner and Zvavitch in 2010.

In addition, we note that in the cases when the above inequality is true, one can deduce from it the $1/n$-concavity of the parallel volume $t \mapsto \mu (A+tB)$, Brunn’s type theorem and certain analogues of Minkowski’s first inequality.