Gaussian Correlation via Inverse Brascamp–Lieb

Abstract We give a simple alternative proof of Royen’s Gaussian Correlation inequality by using (a slightly generalized version of) Nakamura–Tsuji’s symmetric inverse Brascamp–Lieb inequality for even log-concave functions. We explain that this inverse inequality is in a certain sense a dual counterpart to the forward inequality of Bennett–Carbery–Christ–Tao and Valdimarsson, and that the log-concavity assumption therein cannot be omitted in general.