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) ≥ λμ(A) + (1− λ)μ(B) holds true for an unconditional product measure μ with decreasing density and a pair of unconditional convex bodies A,B ⊂ R. We also show that the above inequality is true for any unconditional logconcave measure μ and unconditional convex bodies A,B ⊂ R. Finally, we prove that the inequality is true for a symmetric log-concave measure μ and a pair of symmetric convex sets A,B ⊂ R, which, in particular, settles two-dimensional case of the conjecture for Gaussian measure proposed in [13]. In addition, we deduce the 1/n-concavity of the parallel volume t 7→ μ(A + tB), Brunn’s type theorem and certain analogues of Minkowski first inequality. 2010 Mathematics Subject Classification. Primary 52A40; Secondary 60G15.