A Gaussian correlation inequality for convex sets

Michael R. Tehranchi

Published 2013-09-06, updated 2015-12-31Version 2

A Gaussian correlation inequality is proven which generalises results of Schechtman, Schlumprecht \& Zinn , Li and Shao. One implication of this inequality is that, for the standard Gaussian measure $\gamma$ on $R^n$, the inequality $$ |\sin(\alpha + \beta)|^n \gamma( \sin \alpha \ A) \gamma( \sin \beta \ B) \le \gamma( \sin(\alpha+\beta) A \cap B) \gamma\big( \sin \alpha \sin \beta \ (A+B) \big) $$ holds for all symmetric convex sets $A, B \subseteq R^n $ and real $\alpha, \beta$. Furthermore, connections to the Gaussian correlation conjecture are explored.