The Gaussian correlation inequality for centered convex sets and the case of equality

Inspired by Milman's recent observation, we prove that the Gaussian correlation inequality holds for convex sets having the same barycenter, and especially for centered ones. This gives an affirmative answer to the problem proposed by Szarek and Werner. We also characterize the equality case. The study of the equality case in the non-symmetric Gaussian correlation inequality relates to the following question: Let $X$ be a standard Gaussian random vector in $\mathbb{R}^n$. For which convex sets $K_1,K_2 \subset \mathbb{R}^n$, are the two events $\{X\in K_1\}$ and $\{X\in K_2\}$ independent? By imposing an additional normalization that $K_1$ and $K_2$ have the same barycenter, we give the necessary and sufficient conditions for this independence. The conditions also identify when $\|X\|_{K_1}$ and $\|X\|_{K_2}$ are independent as random variables.