Renyi entropy power inequality and a reverse

The aim of this paper is twofold. In the first part, we present a refinement of the Renyi Entropy Power Inequality (EPI) recently obtained by Bobkov and Marsiglietti (2016). The proof largely follows the approach of Dembo et al. (1991) of employing Young's convolution inequalities with sharp constants. In the second part, we study the reversibility of the Renyi EPI, and confirm a conjecture of Ball et al. (2016) and Madiman et al. (2016) in two cases. Connections with various pth mean bodies in convex geometry are also explored.