Éléments extrémaux pour les inégalités de Brunn-Minkowski gaussiennes (Extreme elements for the Gaussian Brunn-Minkowski inequalities) (Q1077800)

A general approach for investigating extremal sets w.r.t. the following C. Borell's inequality is realized: \[ \Phi^{-1}\circ \gamma_ n(A_ r)\geq \Phi^{-1}\circ \gamma_ n(A)+r, \] where A is a Borel set of \({\mathbb{R}}^ n\), \(r\geq 0\), \(A_ r=A+B_ n(0,r),\) \(B_ n(0,r)\) is the Euclidean ball with center 0 and radius r, \(\gamma_ n\) is the standard normal density in \({\mathbb{R}}^ n\) and \(\Phi\) is the standard normal distribution function. As a consequence it is proved that \[ \int f^ 2\gamma_ n\leq \int \| \nabla f\|^ 2\gamma_ n \] for each Lipschitz function on \({\mathbb{R}}^ n\) such that \(\gamma_ n(\{f=0\})\geq 1/2.\)