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

Inspired from Milman's observation, we prove that the Gaussian correlation inequality holds for centered convex sets and characterize the case of equality. Furthermore, we confirm a nonsymmetric version of the inequality proposed by Szarek--Werner. The existing arguments due to Royen and Milman are seemingly difficult to be applied for these purposes since both of them involve certain approximation arguments that reduce the inequality to some weaker form. To overcome this difficulty, we generalize our previous work on the inverse Brascamp--Lieb inequality.