TY - JOUR

T1 - The entropy per coordinate of a random vector is highly constrained under convexity conditions

AU - Bobkov, Sergey G

AU - Madiman, Mokshay

PY - 2011/8/1

Y1 - 2011/8/1

N2 - 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.

AB - 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.

KW - Convex measures

KW - inequalities

KW - log-concave

KW - maximum entropy

KW - slicing problem

UR - http://www.scopus.com/inward/record.url?scp=79960976529&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960976529&partnerID=8YFLogxK

U2 - 10.1109/TIT.2011.2158475

DO - 10.1109/TIT.2011.2158475

M3 - Article

AN - SCOPUS:79960976529

VL - 57

SP - 4940

EP - 4954

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 8

M1 - 5961831

ER -