On the Gardner-Zvavitch conjecture: Symmetry in inequalities of Brunn-Minkowski type

Abstract

In this paper, we study the conjecture of Gardner and Zvavitch from [22], which suggests that the standard Gaussian measure γ enjoys 1n-concavity with respect to the Minkowski addition of symmetric convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex K and L, and λ∈[0,1],γ(λK+(1−λ)L)12n≥λγ(K)12n+(1−λ)γ(L)12n. More generally, this inequality holds for convex sets containing the origin. Further, we show that under suitable dimension-free uniform bounds on the Hessian of the potential, the log-concavity of even measures can be strengthened to p-concavity, with p>0, with respect to the addition of symmetric convex sets.