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.