A note on the Ehrhard inequality

We prove that for λ ∈ [0,1] and A, B two Borel sets in ${\displaystyle {ℝ}^n}$ with A convex, ${\displaystyle {\Phi }^{-1}\left({\gamma }_n(\lambda A+(1-\lambda )B)\right)\ge \lambda {\Phi }^{-1}\left({\gamma }_n\left(A\right)\right)+(1-\lambda ){\Phi }^{-1}\left({\gamma }_n\left(B\right)\right)}$, where ${\displaystyle {\gamma }_n}$ is the canonical gaussian measure in ${\displaystyle {ℝ}^n}$ and ${\displaystyle {\Phi }^{-1}}$ is the inverse of the gaussian distribution function.