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The entropy per coordinate in a log-concave random vector of any dimensionwith given density at the mode is shown to have a range of just 1. Uniformdistributions on convex bodies are at the lower end of this range, thedistribution with i.i.d. exponentially distributed coordinates is at the upperend, and the normal is exactly in the middle. Thus in terms of the amount ofrandomness as measured by entropy per coordinate, any log-concave random vectorof any dimension contains randomness that differs from that in the normalrandom variable with the same maximal density value by at most 1/2. Asapplications, we obtain an information-theoretic formulation of the famoushyperplane conjecture in convex geometry, entropy bounds for certain infinitelydivisible distributions, and quantitative estimates for the behavior of thedensity at the mode on convolution. More generally, one may consider so-calledconvex or hyperbolic probability measures on Euclidean spaces; we give newconstraints on entropy per coordinate for this class of measures, whichgeneralize our results under the log-concavity assumption, expose the extremalrole of multivariate Pareto-type distributions, and give some applications.

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