By Topic

The Entropy Per Coordinate of a Random Vector is Highly Constrained Under Convexity Conditions

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$31 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

2 Author(s)
Bobkov, S. ; Sch. of Math., Univ. of Minnesota, Minneapolis, MN, USA ; Madiman, M.

The entropy per coordinate in a log-concave random vector of any dimension with given density at the mode is shown to have a range of just 1. Uniform distributions on convex bodies are at the lower end of this range, the distribution with i.i.d. exponentially distributed coordinates is at the upper end, and the normal is exactly in the middle. Thus, in terms of the amount of randomness as measured by entropy per coordinate, any log-concave random vector of any dimension contains randomness that differs from that in the normal random variable with the same maximal density value by at most 1/2. As applications, we obtain an information-theoretic formulation of the famous hyperplane conjecture in convex geometry, entropy bounds for certain infinitely divisible distributions, and quantitative estimates for the behavior of the density at the mode on convolution. More generally, one may consider so-called convex or hyperbolic probability measures on Euclidean spaces; we give new constraints on entropy per coordinate for this class of measures, which generalize our results under the log-concavity assumption, expose the extremal role of multivariate Pareto-type distributions, and give some applications.

Published in:

Information Theory, IEEE Transactions on  (Volume:57 ,  Issue: 8 )