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

2018/07/18 by Kolesnikov, Alexander V., Livshyts, Galyna V. · 2 citations
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.1807.06952

Abstract

In this paper, we study the conjecture of Gardner and Zvavitch from \citeGZ, which suggests that the standard Gaussian measure γ enjoys (1)/(n)-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, γ(λK+(1-λ)L)^(1)/(2n)≥ λγ(K)^(1)/(2n)+(1-λ)γ(L)^(1)/(2n). 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.

Cited by

• On The Minkowski Problem With Respect To A Mixed Euclidean-Gaussian Density
• Grünbaum's inequality for Gaussian and convex probability measures

Related

• The dimensional Brunn-Minkowski inequality in Gauss space
• On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities
• A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures
• On the stability of Brunn-Minkowski type inequalities
• On L_p Brunn-Minkowski type inequalities for a general class of functionals