The dimensional Brunn-Minkowski inequality in Gauss space

2020/04/15 by Eskenazis, Alexandros, Moschidis, Georgios · 3 citations
#FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2004.07146

Abstract

Let γ_n be the standard Gaussian measure on ℝⁿ. We prove that for every symmetric convex sets K,L in ℝⁿ and every λ∈(0,1), γ_n(λK+(1-λ)L)^(1)/(n) ≥ λγ_n(K)^(1)/(n)+(1-λ)γ_n(L)^(1)/(n), thus settling a problem raised by Gardner and Zvavitch (2010). This is the Gaussian analogue of the classical Brunn-Minkowski inequality for the Lebesgue measure. We also show that, for a fixed λ∈(0,1), equality is attained if and only if K=L.

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• From Brunn-Minkowski to Prékopa-Leindler and Borell-Brascamp-Lieb: discrete inequalities
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