What question did this study set out to answer?

The aim is to propose a corrected extension of the classical Gaussian correlation inequality (GCI) for non-centered Gaussian measures on compact convex sets.

March 27, 2026Open Access

The Bounded Shifted Gaussian Correlation Inequality Conjecture

Key Points

The aim is to propose a corrected extension of the classical Gaussian correlation inequality (GCI) for non-centered Gaussian measures on compact convex sets.
Propose the Bounded Shifted Gaussian Correlation Inequality (BSGCI) as an extension of Royen's GCI.
Analyze the effects of compactness in closing gaps found in the original inequality.
Conjecture how the inequality holds for non-centered Gaussian measures and compact convex bodies.
Conjectured inequality μ(A∩B) ≥ μ(A)μ(B) holds for given conditions regarding shifts and compact bodies.
Demonstrated that unbounded half-spaces led to disproving the original shifted generalization in previous studies.

Abstract

We propose the Bounded Shifted Gaussian Correlation Inequality (BSGCI) — a corrected extension of Royen's classical GCI (2014) to non-centred Gaussian measures on compact convex sets. The original shifted generalisation was disproved via unbounded half-spaces; we show compactness closes the gap. For any N(h,Σ) and compact origin-symmetric convex bodies A, B ⊂ ℝⁿ, we conjecture μ(A∩B) ≥ μ(A)μ(B) for all shifts h.

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