Gaussian Correlation Inequality: Optimal Extensions

§ Problem Statement

Setup

Let $n\in\mathbb{N}$ . A Borel set $K\subseteq\mathbb{R}^n$ is called symmetric if $x\in K\Rightarrow -x\in K$ , and convex if $(1-t)x+ty\in K$ for all $x,y\in K$ and $t\in[0,1]$ .

A Gaussian probability measure on $\mathbb{R}^n$ is a law $N(m,\Sigma)$ with mean $m\in\mathbb{R}^n$ and covariance matrix $\Sigma\in\mathbb{R}^{n\times n}$ symmetric positive semidefinite. It is centered when $m=0$ . For a probability measure $\mu$ and Borel set $E$ , write $\mu(E)$ for its probability.

The Gaussian-correlation line predates Pitt’s 1977 paper (see, e.g., Das Gupta et al. (1972)); Pitt (1977) proved the $n=2$ case and stated the all-dimensions symmetric-convex conjecture. Royen (2014) proved the centered case: for every centered Gaussian measure $\gamma=N(0,\Sigma)$ on $\mathbb{R}^n$ and all symmetric convex Borel sets $A,B\subseteq\mathbb{R}^n$ ,

\gamma(A\cap B)\ge \gamma(A)\,\gamma(B).

2025 progress: Nakamura & Tsuji (2025) proved a strict extension for centered Gaussian measure: if convex sets $K_1,K_2\subseteq\mathbb{R}^n$ have the same Gaussian barycenter (in particular, if both are symmetric), then

\gamma(K_1\cap K_2)\ge \gamma(K_1)\,\gamma(K_2).

So the equal-barycenter centered-convex case is now resolved.

Unsolved Problem

Non-centered Gaussian extension for symmetric convex sets in full generality: does

N(m,\Sigma)(A\cap B)\ge N(m,\Sigma)(A)\,N(m,\Sigma)(B)

hold for all $m\in\mathbb{R}^n$ , all symmetric positive semidefinite $\Sigma$ , and all symmetric convex Borel $A,B\subseteq\mathbb{R}^n$ ?

Log-concave extension constant problem: for each dimension $n$ , determine the optimal $c(n)\in[0,1]$ such that for every log-concave probability measure $\mu$ on $\mathbb{R}^n$ and all symmetric convex Borel sets $A,B$ ,

\mu(A\cap B)\ge c(n)\,\mu(A)\,\mu(B).

§ Discussion

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§ Significance & Implications

The GCI is a central positive-correlation theorem in Gaussian probability and convex geometry, with a long development culminating in Royen (2014). It is closely connected to classical multivariate-normal rectangle/orthant probability inequalities used in simultaneous inference (e.g., Šidák (1967), Khatri (1967)). The 2025 equal-barycenter extension shows meaningful progress, but full non-centered and log-concave sharp extensions remain open.

§ References

[1]

A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions

Thomas Royen (2014)

📍 Section 1 (Introduction), Eq. (1.1): centered Gaussian and symmetric convex sets.

arXiv ↗

[2]

Inequalities on the probability content of convex regions for elliptically contoured distributions

Somesh Das Gupta, Morris L. Eaton, Ingram Olkin, Michael Perlman, Leonard J. Savage, Milton Sobel (1972)

Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II

📍 Historical early formulation of correlation-type conjectures for convex regions in the Gaussian/elliptical setting (pp. 241-265).

[3]

A Gaussian correlation inequality for symmetric convex sets

Loren D. Pitt (1977)

Annals of Probability

📍 Theorem 1 (dimension 2) and all-dimensions symmetric-convex statement.

DOI ↗

[4]

Royen's proof of the Gaussian correlation inequality

Rafał Latała, Dariusz Matlak (2017)

Geometric Aspects of Functional Analysis (Lecture Notes in Mathematics, vol. 2169), Springer

📍 arXiv:1512.08776, Abstract and opening setup restate Royen's centered Gaussian/symmetric-convex theorem; this source does not itself state the non-centered or log-concave open extensions.

DOI ↗arXiv ↗

[5]

The Gaussian correlation inequality for centered convex sets and the case of equality

Shohei Nakamura, Hiroshi Tsuji (2025)

📍 Abstract (v3, 12 Nov 2025): proves GCI for convex sets with the same barycenter and gives an affirmative answer to the Szarek-Werner problem.

arXiv ↗

[6]

Rectangular Confidence Regions for the Means of Multivariate Normal Distributions

Zbyněk Šidák (1967)

Journal of the American Statistical Association

📍 Classical Gaussian rectangle-probability inequality used in simultaneous inference.

DOI ↗