The Brunn--Minkowski inequality for the Gaussian measure

Abstract

Let $γ_n$ be the standard Gaussian measure on $\mathbb{R}^n$, $n\ge2$, and let $α_γ(n)$ be the largest number for which \[ γ_n(λK+(1-λ)L)^{α_γ(n)} \ge λγ_n(K)^{α_γ(n)} +(1-λ)γ_n(L)^{α_γ(n)} \] holds for all convex bodies $K,L\subset\mathbb{R}^n$ containing the origin and all $λ\in[0,1]$. In this paper, we prove that \[ α_γ(n) =1-\frac{2}{n-1} \frac{Γ(\frac n2)^2}{Γ(\frac{n-1}{2})^2}. \] The core of the proof is a raywise radial--tangential localization of the Hessian energy of a solution of a Neumann problem, which reduces source selection of the Neumann problem to a one-dimensional optimization. Monotonicity in the segment length and Laguerre spectral analysis determine the sharp one-dimensional value, whereas the planar endpoint is treated separately.