$\Phi-$entropy inequalities and asymmetric covariance estimates for convex measures

TL;DR

This paper extends $\

Contribution

It introduces $\\Phi$-entropy inequalities and asymmetric covariance estimates for convex measures, generalizing known results for log-concave measures using semi-group and $L^2$ methods.

Findings

$\\Phi$-entropy inequalities are sharp for Cauchy measures.

The results generalize inequalities from log-concave to convex measures.

Similar inequalities for log-concave measures are derivable as limits.

Abstract

In this paper, we use the semi-group method and an adaptation of the $L^2-$method of H\"ormander to establish some $\Phi-$entropy inequalities and asymmetric covariance estimates for the strictly convex measures in $\mathbb R^n$. These inequalities extends the ones for the strictly log-concave measures to more general setting of convex measures. The $\Phi-$entropy inequalities are turned out to be sharp in the special case of Cauchy measures. Finally, we show that the similar inequalities for log-concave measures can be obtained from our results in the limiting case.