Sharp Poincaré-type inequality for the Gaussian measure on the boundary of convex sets

Abstract

A sharp Poincaré-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.

14 pages, to appear in GAFA Seminar Notes (Springer's Lecture Notes in Math. 2169)