A note on Gaussian correlation inequalities for nonsymmetric sets

We consider the Gaussian correlation inequality for nonsymmetric convex sets. More precisely, if A⊂RdA⊂Rd is convex and the origin 0∈A0∈A, then for any ball BB centered at the origin, it holds γd(A∩B)≥γd(A)γd(B)γd(A∩B)≥γd(A)γd(B), where γdγd is the standard Gaussian measure on RdRd. This generalizes Proposition 1 in [Cordero-Erausquin, Dario, 2002. Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 161, 257–269].