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Article

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

Kolesnikov A., Livshyts G.

In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure γ enjoys 1n-concavity with respect to the Minkowski addition of \textbf{symmetric} convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex K and L,

γ(λK+(1−λ)L)12n≥λγ(K)12n+(1−λ)γ(L)12n.

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.