Rényi entropy power inequality and a reverse. (English) Zbl 1421.94021
Summary: The aim of this paper is twofold. In the first part, we present a refinement of the Rényi Entropy Power Inequality (EPI) recently obtained by S. G. Bobkov and A. Marsiglietti [IEEE Trans. Inf. Theory 63, No. 12, 7747–7752 (2017; Zbl 1390.94613)]. The proof largely follows the approach of A. Dembo et al. [IEEE Trans. Inf. Theory 37, No. 6, 1501–1518 (1991; Zbl 0741.94001)] 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 of K. Ball et al. [Stud. Math. 235, No. 1, 17–30 (2016; Zbl 1407.94055)] and M. Madiman et al. [The IMA Volumes in Mathematics and its Applications 161, 427–485 (2017; Zbl 1381.52013)] in two cases. Connections with various \(p\)th mean bodies in convex geometry are also explored.
MSC:
| 94A17 | Measures of information, entropy |
| 62B10 | Statistical aspects of information-theoretic topics |
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
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