The Gaussian Correlation Conjecture Proof

Yashar Memarian

Published 2013-10-30, updated 2015-01-08Version 2

A radial probability measure in the $n$-dimensional Euclidean space is a probability measure which has a density function with respect to the Lebesgue measure which depends only on the distances to the origin of this space. A radial correlation problem concerns showing whether the radial measure of the intersection of two symmetric convex bodies is greater than the product of the radial measures of the two convex bodies. A major open question in this field concerns when the radial measure is the Gaussian measure. The main result in this paper concerns the proof of the Gaussian Correlation Conjecture in its most general form for symmetric convex sets. The proof is presented through the use of localisation on the canonical sphere. Furthermore, I present a general theorem concerning (at least four dimensional) radial correlation problems. This theorem simplifies their study to the study of an anisotropic 2-dimensional correlation problem with regards to symmetric strips in this space.