Sharp Poincar\'e-type inequality for the Gaussian measure on the   boundary of convex sets
  A sharp Poincar\'e-type inequality is derived for the restriction of the Gaussian measure on the boundary of a convex set. In particular, it implies a Gaussian mean-curvature inequality and a Gaussian iso second-variation inequality. The new inequality is nothing but an infinitesimal form of Ehrhard's inequality for the Gaussian measure. 
Alexander V. Kolesnikov and Emanuel Milman
math.FA math.PR