Further investigations of Renyi entropy power inequalities and an entropic characterization of s-concave densities

We investigate the role of convexity in Renyi entropy power inequalities. After proving that a general Renyi entropy power inequality in the style of Bobkov-Chistyakov (2015) fails when the Renyi parameter $rin(0,1)$, we show that random vectors with $s$-concave densities do satisfy such a Renyi entropy power inequality. Along the way, we establish the convergence in the Central Limit Theorem for Renyi entropies of order $rin(0,1)$ 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).