# Further investigations of R\'enyi entropy power inequalities and an entropic characterization of s-concave densities

## TL;DR

This paper explores the role of convexity in Renyi entropy power inequalities, showing their validity for s-concave densities and providing new entropic characterizations of such densities.

## Contribution

It demonstrates the failure of Renyi entropy power inequalities for certain parameters and establishes their validity for s-concave densities, along with new convergence and characterization results.

## Key findings

- Renyi entropy power inequality fails for r in (0,1) in general
- s-concave densities satisfy Renyi entropy power inequalities
- Convergence of Renyi entropies in CLT for specific density classes

## Abstract

We investigate the role of convexity in R\'enyi entropy power inequalities. After proving that a general R\'enyi entropy power inequality in the style of Bobkov-Chistyakov (2015) fails when the R\'enyi parameter $r\in(0,1)$, we show that random vectors with $s$-concave densities do satisfy such a R\'enyi entropy power inequality. Along the way, we establish the convergence in the Central Limit Theorem for R\'enyi entropies of order $r\in(0,1)$ for log-concave densities and for compactly supported, spherically symmetric and unimodal densities, complementing a celebrated result of Barron (1986). Additionally, we give an entropic characterization of the class of $s$-concave densities, which extends a classical result of Cover and Zhang (1994).