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Gaussian Correlation via Inverse Brascamp-Lieb

Emanuel Milman

TL;DR

The paper provides a simple proof of Royen's Gaussian Correlation Inequality by leveraging a generalized symmetric inverse Brascamp–Lieb inequality for even, log-concave test functions. It clarifies a natural duality between forward and inverse Brascamp–Lieb inequalities and demonstrates that the log-concavity hypothesis is essential; the argument reduces to Gaussian extremizers via a central limit-type convergence. The approach yields the Gaussian correlation bound and identifies equality precisely when the cross-covariance block vanishes, offering a complementary perspective to Royen's monotonicity-based method. The work also situates these results within broader extensions to centered Gaussian barycenters and convex sets, highlighting the structural role of log-concavity in correlation inequalities.

Abstract

We give a simple alternative proof of Royen's Gaussian Correlation inequality by using (a slightly generalized version of) Nakamura-Tsuji's symmetric inverse Brascamp-Lieb inequality for even log-concave functions. We explain that this inverse inequality is in a certain sense a dual counterpart to the forward inequality of Bennett-Carbery-Christ-Tao and Valdimarsson, and that the log-concavity assumption therein cannot be omitted in general.

Gaussian Correlation via Inverse Brascamp-Lieb

TL;DR

The paper provides a simple proof of Royen's Gaussian Correlation Inequality by leveraging a generalized symmetric inverse Brascamp–Lieb inequality for even, log-concave test functions. It clarifies a natural duality between forward and inverse Brascamp–Lieb inequalities and demonstrates that the log-concavity hypothesis is essential; the argument reduces to Gaussian extremizers via a central limit-type convergence. The approach yields the Gaussian correlation bound and identifies equality precisely when the cross-covariance block vanishes, offering a complementary perspective to Royen's monotonicity-based method. The work also situates these results within broader extensions to centered Gaussian barycenters and convex sets, highlighting the structural role of log-concavity in correlation inequalities.

Abstract

We give a simple alternative proof of Royen's Gaussian Correlation inequality by using (a slightly generalized version of) Nakamura-Tsuji's symmetric inverse Brascamp-Lieb inequality for even log-concave functions. We explain that this inverse inequality is in a certain sense a dual counterpart to the forward inequality of Bennett-Carbery-Christ-Tao and Valdimarsson, and that the log-concavity assumption therein cannot be omitted in general.
Paper Structure (9 sections, 8 theorems, 34 equations)

This paper contains 9 sections, 8 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Log-concavity assumption
  3. Gaussian Correlation Inequality
  4. Proof of Theorem \ref{['thm:intro-IBL2']}
  5. Comparison with Royen's approach
  6. The equality case in Theorem \ref{['thm:intro-GCI']}
  7. Original argument
  8. Monotonicity
  9. Subsequent developments

Key Result

Theorem 1.1

Let $\bar{Q}$ be an $N \times N$ symmetric matrix (of arbitrary signature), and let $Q_i \geq 0$ denote positive semi-definite $n_i \times n_i$ matrices, for $i=1,\ldots,m$. Let $c_1,\ldots,c_m>0$. Then for all (non-zero) even log-concave$h_i \in L^1(\mathbb{R}^{n_i},g_{Q_i}(x_i) dx_i)$, $i=1,\ldots where the infimum is over $n_i \times n_i$ symmetric matrices $A_i$, $i=1,\ldots,m$.

Theorems & Definitions (11)

  • Theorem 1.1: Symmetric Inverse Brascamp--Lieb Inequality, after Nakamura--Tsuji NakamuraTsuji-InverseBrascampLieb
  • Theorem 1.2: Theorem \ref{['thm:intro-IBL']} again
  • Theorem 1.3: After BCCT-BrascampLiebValdimarsson-GenCaffarelliKolesnikovContractionSurvey
  • Theorem 1.4: Gaussian Correlation Inequality Royen-GaussianCorrelation
  • Lemma 1.5: Fischer's inequality
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • ...and 1 more