The dimensional Brunn–Minkowski inequality in Gauss space

Abstract

Let γn be the standard Gaussian measure on Rn. We prove that for every symmetric convex sets K,L in Rn and every λ∈(0,1),γn(λK+(1−λ)L)1n⩾λγn(K)1n+(1−λ)γn(L)1n, 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.