Reversals of R\'enyi Entropy Inequalities under Log-Concavity

TL;DR

This paper explores how R'enyi entropy inequalities behave under log-concavity, providing discrete analogs and sharp versions of known inequalities, with implications for entropy comparison and bounds.

Contribution

It introduces a discrete analog of R'enyi entropy comparison for log-concave variables and establishes sharp R'enyi versions of the Rogers-Shephard inequality.

Findings

Min entropy is within log e of Shannon entropy for log-concave variables on integers

Established a sharp R'enyi version of the Rogers-Shephard inequality

Provided both continuous and discrete case results

Abstract

We establish a discrete analog of the R\'enyi entropy comparison due to Bobkov and Madiman. For log-concave variables on the integers, the min entropy is within log e of the usual Shannon entropy. Additionally we investigate the entropic Rogers-Shephard inequality studied by Madiman and Kontoyannis, and establish a sharp R\'enyi version for certain parameters in both the continuous and discrete cases