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A sharp Poincare-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 equivalent form of Ehrhard’s inequality for the Gaussian measure. While Ehrhard’s inequality does not extend to general CD(1, ∞) measures, we formulate a sufficient condition for the validity of Ehrhard-type inequalities for general measures on \(\mathbb{R}^{n}\) via a certain property of an associated Neumann-to-Dirichlet operator.

Sharp Poincaré-Type Inequality for the Gaussian Measure on the Boundary of Convex Sets