A note on Gaussian correlation inequalities for nonsymmetric sets

TL;DR

This paper extends Gaussian correlation inequalities to nonsymmetric convex sets, showing that the measure of their intersection with symmetric sets like balls is at least the product of their individual measures, generalizing previous results.

Contribution

It generalizes Gaussian correlation inequalities to nonsymmetric convex sets containing the origin, broadening the scope of known correlation inequalities.

Findings

Established a lower bound for Gaussian measure of intersections with symmetric sets.

Generalized previous correlation inequalities to a wider class of convex sets.

Provided a mathematical proof extending existing propositions.

Abstract

We consider the Gaussian correlation inequality for nonsymmetric convex sets. More precisely, if $A\subset\mathbb{R}^d$ is convex and the origin $0\in A$, then for any ball $B$ centered at the origin, it holds $\gamma_d(A\cap B)\geq \gamma_d(A)\gamma_d(B)$, where $\gamma_d$ is the standard Gaussian measure on $\mathbb{R}^d$. This generalizes Proposition 1 in [Arch. Rational Mech. Anal. 161 (2002), 257--269].