On the similarity of the entropy power inequality and the Brunn-Minkowski inequality

Correspondence On the Similarity of the Entropy Power Inequality The preceeding equations allow the entropy power inequality and the Brunn-Minkowski Inequality to be rewritten in the equivalent form (4) where X' and Y' are independent normal variables with corresponding entropies H(X') = H(X) and H(Y') = H(Y). Verification of this restatement follows from the use of (1) to show that Abstract-The entropy power inequality states that the effective variance (entropy power) of the sum of two independent random variables is greater than the sum of their effective variances. The Brunn-Minkowski inequality states that the effective radius of the set sum of two sets is greater than the sum of their effective radii. Both these inequalities are recast in a form that enhances their similarity. In spite of this similarity, there is as yet no common proof of the inequalities. Nevertheless, their intriguing similarity suggests that new results relating to entropies from known results in geometry and vice versa may be found. Two applications of this reasoning are presented. First, an isoperimetric inequality for entropy is proved that shows that the spherical normal distribution minimizes the trace of the Fisher information matrix given an entropy constraint just as a sphere minimizes the surface area given a volume constraint. Second, a theorem involving the effective radii of growing convex sets is proved. Let a random variable X have a probability density function f(x), x E R. Then its (differential) entropy H(X) is defined as H(X) =-sf(x)lnf(x)dx. Shannon's entropy power inequality [l] states that for X and Y independent random variables having density functions p(X+ Y) 2 pf(X) + ewY) (1) We wish to recast this inequality. First we observe that a normal random variable Z-Cp(z) = (l/ ~)e-zz~20z with variance o2 has entropy H(Z) =-J+ln+ = + ln2seu2. (4 By inverting, we see that if Z is normal with entropy H(Z), then its variance is (3) Thus, the entropy power inequality is an inequality between effective variances, where effective variance (entropy power) is simply the variance of the normal random variable with the same entropy.