Gaussian Correlation via Inverse Brascamp-Lieb

TL;DR

This paper presents a simplified proof of Royen's Gaussian Correlation inequality using a generalized inverse Brascamp-Lieb inequality, highlighting its duality with the forward inequality and emphasizing the importance of log-concavity.

Contribution

It offers a new, simpler proof of a key Gaussian correlation inequality by leveraging a generalized inverse Brascamp-Lieb inequality, clarifying its duality with existing inequalities.

Findings

Simplified proof of Royen's Gaussian Correlation inequality

Identification of the inverse inequality as a dual to the forward inequality

Highlighting the necessity of log-concavity in the inequalities

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.