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Gaussian correlation inequality

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Let n≥1n\ge 1 and let μ\mu be a centered Gaussian probability measure on Rn\mathbb{R}^n, i.e. the distribution of a random vector X=(X1,…,Xn)X=(X_1,\dots,X_n) having a multivariate normal law with mean vector 00 and some covariance matrix Σ∈Rn×n\Sigma\in\mathbb{R}^{n\times n} (positive semidefinite, possibly singular). Call a set A⊂RnA\subset\mathbb{R}^n symmetric about the origin if A=−AA=-A, that is, x∈Ax\in A implies −x∈A-x\in A.

Then for all convex sets E,F⊂RnE,F\subset\mathbb{R}^n that are symmetric about the origin,

μ(E∩F)  ≥  μ(E)⋅μ(F).\mu(E\cap F)\;\ge\;\mu(E)\cdot\mu(F).
References

Primary source

Wikipedia

Additional references

  1. Wikipedia, Gaussian correlation inequality, the article this problem comes from.
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Progress summary

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Claimed solved

The decades-old conjecture was proved by Thomas Royen in 2014, with later papers providing simpler proofs and extensions.

The inequality asserts that two centrally symmetric convex regions are at least as likely to overlap as the product of their individual probabilities under a centered Gaussian distribution. It arose in the 1950s and took its modern form in the 1970s; Royen proved it in 2014.

Known results

  • Khatri and Šidák established the one-block case; Pitt proved the formulation for n=2n=2.
  • Borell and Schechtman, Schlumprecht, and Zinn obtained important partial results before Royen.
  • Royen, 2014: complete proof in every dimension, for symmetric convex sets; his method also covers a broader gamma-distribution setting.
  • Latała and Matlak, 2015: simplified and made Royen’s Gaussian proof self-contained.

Alternative proofs and extensions, January–April 2025

A 2025 preprint gives a different proof via a generalized symmetric inverse Brascamp–Lieb inequality. Another proves the inequality for convex sets with the same Gaussian barycenter, including centered sets, and characterizes equality; these developments corroborate and extend the settled theorem.

Current status (as of August 2026): The Gaussian correlation inequality is settled; Royen’s proof is supported by subsequent expositions and alternative proofs, while stronger variants continue to be studied.

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