A note on the Ehrhard inequality

Abstract

We prove that for λ ∈ [0, 1] and A, B two Borel sets in ℝn with A convex, Φ-1(γn(λA + (1 - λ)B)) ≥ λΦ-1(γn(A)) + (1 - λ)Φ-1(γn(B)), where γn is the canonical gaussian measure in ℝn and Φ-1 is the inverse of the gaussian distribution function.