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

2015/04/19 by Livshyts, Galyna, Marsiglietti, Arnaud, Nayar, Piotr +1 · 4 citations
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1504.04878

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 μ(λA + (1-λ)B)^1/n ≥ 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 ⊂ ℝⁿ. We also show that the above inequality is true for any unconditional log-concave measure μ and unconditional convex bodies A,B ⊂ ℝⁿ. Finally, we prove that the inequality is true for a symmetric log-concave measure μ and a pair of symmetric convex sets A,B ⊂ ℝ², 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 ↦ μ(A+tB), Brunn's type theorem and certain analogues of Minkowski first inequality.

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