The Gaussian Correlation Conjecture

From papers

Let $\gamma_n$ be the Gaussian probability measure on $\mathbb{R}^n$ , and let $K_1$ and $K_2$ be symmetric convex bodies in $\mathbb{R}^n$ . The Gaussian Correlation Conjecture. The following inequality holds:

\gamma_n(\mathbb{R}^n)\gamma_n(K_2\cap K_2)\geq \gamma_n(K_1)\gamma_n(K_2).

The displayed statement appears to contain typographical errors: the standard Gaussian correlation inequality concerns $\gamma_n(K_1\cap K_2)$ and the product $\gamma_n(K_1)\gamma_n(K_2)$ . The conjecture was resolved affirmatively, but the exact intended formulation should be checked against the source.

Progress summary

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The Gaussian correlation conjecture
Let $n\geq 1$ , let $\mu_n$ be a mean-zero Gaussian measure on $\mathbb{R}^n$ , and let $K,M\subseteq\mathbb{R}^n$ be convex closed subsets symmetric around the origin. Gaussian correlation conjecture. One has
$\mu_n(K\cap M)\geq \mu_n(K)\mu_n(M).$
This is a central problem in convex geometry and probability theory. The source presents it as a major open question and claims to prove it in its most general form, but the supplied parser status is unknown; its resolution should therefore be checked against the paper and subsequent literature.
source: Yashar Memarian, “A Geometric Approach to Radial Correlation Type Problems”, arXiv:1310.8099 (2017).

Sources & referencesView supporting material

Primary source

Yashar Memarian, “Spherical Localisation in Convex and Metric Geometry”, arXiv:1507.00915 (2015).

1 week ago

Progress summary

Equivalent formulations 1

The Gaussian correlation conjecture

Primary source