Gaussian Maximum of Entropy and Reversed Log-sobolev Inequality

The aim of this note is to connect a reversed form of the Gross logarithmic Sobolev inequality with the Gaussian maximum of Shannon’s entropy power. There is thus a complete parallel with the well-known link between logarithmic Sobolev inequalities and their information theoretic counterparts. We moreover provide an elementary proof of the reversed Gross inequality via a two-point inequality and the Central Limit Theorem. 1. Shannon’s entropy power and Gross’s inequality In the sequel, we denote by Entμ(f) the entropy of a non-negative integrable function f with respect to a positive measure μ, defined by Entμ(f) := ∫ f log fdμ − ∫ fdμ log ∫ fdμ. The Shannon entropy [15] of an n-variate random vector X with probability density function (pdf) f is given by H(X) := −Entλn(f) = − ∫ f log f dx, where dx denotes the n-dimensional Lebesgue measure on Rn. The Shannon entropy power [15] of X is then given by N(X) := 1 2πe exp ( 2 n H(X) ) . It is well-known (cf. [15, 8]) that Gaussians saturates this entropy at fixed covariance. Namely, for any n-variate random vector X with covariance matrix K(X), one have N(X) ≤ |K(X)|, (1) and |K| is the entropy power of the n-dimensional Gaussian with covariance K. The logarithmic Sobolev inequality of Gross [11] expresses that for any nonnegative smooth function f : Rn → R+ 2Entγn(f) ≤ Eγn ( |∇f | f )