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

Kolesnikov A., Milman E.
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.