Rényi entropy power inequality and a reverse

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

[1]  Sergey G. Bobkov,et al.  Entropy Power Inequality for the Rényi Entropy , 2015, IEEE Transactions on Information Theory.

[2]  Igal Sason,et al.  On Rényi Entropy Power Inequalities , 2016, IEEE Transactions on Information Theory.

[3]  Assaf Naor,et al.  On the rate of convergence in the entropic central limit theorem , 2004 .

[4]  Jose A. Costa,et al.  On Solutions to Multivariate Maximum α-Entropy Problems , 2003 .

[5]  C. Borell.  Convex set functions ind-space , 1975 .

[6]  P-CROSS-SECTION BODIES , 1999 .

[7]  Olivier Rioul,et al.  Yet Another Proof of the Entropy Power Inequality , 2016, IEEE Transactions on Information Theory.

[8]  Assaf Naor,et al.  Entropy jumps in the presence of a spectral gap , 2003 .

[9]  Max H. M. Costa,et al.  A new entropy power inequality , 1985, IEEE Trans. Inf. Theory.

[10]  INVERSE FORMULA FOR THE BLASCHKE-LEVY REPRESENTATION WITH APPLICATIONS TO ZONOIDS AND SECTIONS OF STAR BODIES. , 1996, math/9605212.

[11]  H. Busemann,et al.  A Theorem on Convex Bodies of the Brunn-Minkowski Type. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[12]  Sergey G. Bobkov,et al.  Convex bodies and norms associated to convex measures , 2010 .

[13]  Gaoyong Zhang,et al.  Convolutions, Transforms, and Convex Bodies , 1999 .

[14]  R. J. Gardner,et al.  Affine inequalities and radial mean bodies , 1998 .

[15]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[16]  E. Lieb.  Proof of an entropy conjecture of Wehrl , 1978 .

[17]  Olivier Rioul,et al.  Information Theoretic Proofs of Entropy Power Inequalities , 2007, IEEE Transactions on Information Theory.

[18]  A. J. Stam.  Some Inequalities Satisfied by the Quantities of Information of Fisher and Shannon , 1959, Inf. Control..

[19]  Sergey G. Bobkov,et al.  The Entropy Per Coordinate of a Random Vector is Highly Constrained Under Convexity Conditions , 2010, IEEE Transactions on Information Theory.

[20]  Stanislaw J. Szarek,et al.  Shannon’s entropy power inequality via restricted minkowski sums , 2000 .

[21]  Zhen Zhang,et al.  On the maximum entropy of the sum of two dependent random variables , 1994, IEEE Trans. Inf. Theory.

[22]  Jean Bourgain,et al.  ON HIGH DIMENSIONAL MAXIMAL FUNCTIONS ASSOCIATED TO CONVEX BODIES , 1986 .

[23]  S. Bobkov,et al.  Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures , 2011, 1109.5287.

[24]  Sergio Verdú,et al.  A simple proof of the entropy-power inequality , 2006, IEEE Transactions on Information Theory.

[25]  Sergey G. Bobkov,et al.  Variants of the Entropy Power Inequality , 2017, IEEE Transactions on Information Theory.

[26]  Piotr Nayar,et al.  A reverse entropy power inequality for log-concave random vectors , 2015, 1509.05926.

[27]  Giuseppe Toscani,et al.  The Concavity of Rényi Entropy Power , 2014, IEEE Transactions on Information Theory.

[28]  Amir Dembo,et al.  Information theoretic inequalities , 1991, IEEE Trans. Inf. Theory.

[29]  E. Carlen.  Superadditivity of Fisher's information and logarithmic Sobolev inequalities , 1991 .

[30]  Van Hoang Nguyen,et al.  Entropy jumps for isotropic log-concave random vectors and spectral gap , 2012, 1206.5098.

[31]  K. Ball.  Logarithmically concave functions and sections of convex sets in $R^{n}$ , 1988 .

[32]  Peng Xu,et al.  Forward and Reverse Entropy Power Inequalities in Convex Geometry , 2016, ArXiv.

[33]  Nelson M. Blachman,et al.  The convolution inequality for entropy powers , 1965, IEEE Trans. Inf. Theory.