  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.
