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

Shohei Nakamura, Hiroshi Tsuji

TL;DR

The paper proves a non-symmetric Gaussian correlation inequality for centered convex sets: for any convex $K_1,K_2\subset\mathbb{R}^n$ with the same Gaussian barycenter, ${\gamma}(K_1\cap K_2) \ge {\gamma}(K_1){\gamma}(K_2)$, thus affirmatively answering Szarek–Werner’s question in the centered case and fully characterizing equality. The authors develop a centered Gaussian saturation principle for the inverse Brascamp--Lieb inequality, showing that the optimal constant under centering is attained by Gaussian inputs, extending Milman’s approach beyond even inputs. They derive a multilinear centered GCI, prove a non-symmetric GCI via translations that preserve centering, and provide a detailed equality-case analysis linking independence of Gaussian events and independence of the Minkowski norms. The work also connects to rigidity phenomena in spectral-gap problems and CD(κ,∞) geometry, offering a broad framework that unifies convex-geometry, probability, and Brascamp--Lieb-type inequalities with potential implications for independent Gaussian structures and curvature-dimension analyses.

Abstract

Inspired by Milman's recent observation, we prove that the Gaussian correlation inequality holds for convex sets having the same barycenter, and especially for centered ones. This gives an affirmative answer to the problem proposed by Szarek and Werner. We also characterize the equality case. The study of the equality case in the non-symmetric Gaussian correlation inequality relates to the following question: Let $X$ be a standard Gaussian random vector in $\mathbb{R}^n$. For which convex sets $K_1,K_2 \subset \mathbb{R}^n$, are the two events $\{X\in K_1\}$ and $\{X\in K_2\}$ independent? By imposing an additional normalization that $K_1$ and $K_2$ have the same barycenter, we give the necessary and sufficient conditions for this independence. The conditions also identify when $\|X\|_{K_1}$ and $\|X\|_{K_2}$ are independent as random variables.

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

TL;DR

The paper proves a non-symmetric Gaussian correlation inequality for centered convex sets: for any convex with the same Gaussian barycenter, , thus affirmatively answering Szarek–Werner’s question in the centered case and fully characterizing equality. The authors develop a centered Gaussian saturation principle for the inverse Brascamp--Lieb inequality, showing that the optimal constant under centering is attained by Gaussian inputs, extending Milman’s approach beyond even inputs. They derive a multilinear centered GCI, prove a non-symmetric GCI via translations that preserve centering, and provide a detailed equality-case analysis linking independence of Gaussian events and independence of the Minkowski norms. The work also connects to rigidity phenomena in spectral-gap problems and CD(κ,∞) geometry, offering a broad framework that unifies convex-geometry, probability, and Brascamp--Lieb-type inequalities with potential implications for independent Gaussian structures and curvature-dimension analyses.

Abstract

Inspired by Milman's recent observation, we prove that the Gaussian correlation inequality holds for convex sets having the same barycenter, and especially for centered ones. This gives an affirmative answer to the problem proposed by Szarek and Werner. We also characterize the equality case. The study of the equality case in the non-symmetric Gaussian correlation inequality relates to the following question: Let be a standard Gaussian random vector in . For which convex sets , are the two events and independent? By imposing an additional normalization that and have the same barycenter, we give the necessary and sufficient conditions for this independence. The conditions also identify when and are independent as random variables.

Paper Structure

This paper contains 12 sections, 28 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. The inverse Brascamp--Lieb inequality
  3. Proof of Theorem \ref{['t:CenteredIBL']}
  4. Preliminaries
  5. The Gaussian saturation in the case of $0< G_i \le H_i <\infty$
  6. The case of $G_i >0$ and $H_i = \infty$: a limiting argument
  7. Proof of Theorems \ref{['t:NonSymGCI']} and \ref{['t:GenCor']}
  8. Proof of Theorem \ref{['t:GenCor']}
  9. Proof of Theorem \ref{['t:NonSymGCI']}
  10. Proof of Theorem \ref{['t:EqualityCase']}
  11. Proof of Corollary \ref{['cor:Translate']}
  12. Formal proof of Theorem \ref{['t:GKKO']} for linear $\phi_i$ and regular $\mu$

Key Result

Theorem 1.1

For any convex sets $K_1,K_2 \subset {\mathbb R}^n$ with ${\rm bar}_\gamma(K_1)={\rm bar}_\gamma(K_2)$, we have e:OriginGCI:

Theorems & Definitions (54)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 2.1: Lieb Lieb
  • Theorem 2.2: NT3
  • Definition 2.3
  • Theorem 2.4
  • Example 2.5
  • ...and 44 more