Covariance inequalities for convex and log-concave functions

TL;DR

This paper extends covariance inequalities to convex and log-concave functions under general product measures, broadening the scope of previous Gaussian-specific results.

Contribution

It generalizes covariance inequalities from Gaussian measures to a wider class of product measures for convex and log-concave functions.

Findings

Established new covariance inequalities for convex functions

Extended inequalities to log-concave functions with symmetries

Broadened applicability beyond Gaussian measures

Abstract

Extending results of Harg{\'e} and Hu for the Gaussian measure, we prove inequalities for the covariance Cov$_\mu(f, g)$ where $\mu$ is a general product probability measure on $\mathbb{R}^d$ and $f,g: \mathbb{R}^d \to \mathbb{R}$ satisfy some convexity or log-concavity assumptions, with possibly some symmetries.