Royen's proof of the Gaussian correlation inequality

TL;DR

This paper details Thomas Royen's proof of the Gaussian correlation inequality, establishing that the measure of the intersection of symmetric convex sets under a Gaussian measure is at least the product of their individual measures.

Contribution

It provides a comprehensive presentation of Royen's proof, a significant advancement in understanding Gaussian measures and convex geometry.

Findings

Proof confirms the Gaussian correlation inequality for all centered Gaussian measures and symmetric convex sets.

Clarifies the mathematical techniques used in Royen's proof.

Enhances understanding of Gaussian measures in high-dimensional spaces.

Abstract

We present in detail Thomas Royen's proof of the Gaussian correlation inequality which states that $\mu(K\cap L)\geq \mu(K)\mu(L)$ for any centered Gaussian measure $\mu$ on $R^d$ and symmetric convex sets $K,L$ in $R^d$.