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In this paper we present new versions of the classical Brunn-Minkowskiinequality for different classes of measures and sets. We show that theinequality \[  \mu(\lambda A + (1-\lambda)B)^{1/n} \geq \lambda \mu(A)^{1/n} +(1-\lambda)\mu(B)^{1/n} \] 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 anyunconditional $\log$-concave measure $\mu$ and unconditional convex bodies $A,B\subset \mathbb{R}^n$. Finally, we prove that the inequality is true for asymmetric $\log$-concave measure $\mu$ and a pair of symmetric convex sets $A,B\subset \mathbb{R}^2$, which, in particular, settles two-dimensional case ofthe conjecture for Gaussian measure proposed by R. Gardner and the fourth namedauthor.  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 firstinequality.

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