Rényi entropy power inequality and a reverse

Abstract

This paper is twofold. In the first part, we present a refinement of the Rényi Entropy Power Inequality (EPI) recently obtained in BM16. The proof largely follows the approach in DCT91 of employing Young's convolution inequalities with sharp constants. In the second part, we study the reversibility of the Rényi EPI, and confirm a conjecture in BNT15, MMX16 in two cases. Connections with various $p$-th mean bodies in convex geometry are also explored.