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

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 \[ μ(λA + (1-λ)B)^{1/n} \geq λμ(A)^{1/n} + (1-λ)μ(B)^{1/n} \] holds true for an unconditional product measure $μ$ 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 $μ$ and unconditional convex bodies $A,B \subset \mathbb{R}^n$. Finally, we prove that the inequality is true for a symmetric $\log$-concave measure $μ$ 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 $t \mapsto μ(A+tB)$, Brunn's type theorem and certain analogues of Minkowski first inequality.