Gaussian Correlation via Inverse Brascamp-Lieb
arXiv:2501.11018
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.
14 pages, simplified computation of Gaussian constant following referee's suggestion, and included analysis of equality conditions. Added section on comparison with Royen's approach. To appear in Probability Theory and Related Fields (PTRF)