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

Sergey Bobkov, Mokshay Madiman

TL;DR

This work unifies entropy and convexity to show that the entropy per coordinate of any log-concave random vector is within a universal 1/2 of that of a matching Gaussian, with equality characterizing uniform and exponential cases. By extending to κ-concave and convex measures, the authors derive dimension-free bounds and extremal properties for ellipsoids, revealing the Pareto-type distributions as key extremizers. They connect these entropy bounds to the hyperplane (slicing) conjecture, present entropic reformulations, and develop applications to entropy rates, convolution behavior, and mixtures of convex densities. The results provide a versatile information-theoretic lens on convex geometry, with implications for isotropic constants, Rényi entropies, and infinite-divisible laws, and point to several open questions in extending and sharpening the framework.

Abstract

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.

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

TL;DR

This work unifies entropy and convexity to show that the entropy per coordinate of any log-concave random vector is within a universal 1/2 of that of a matching Gaussian, with equality characterizing uniform and exponential cases. By extending to κ-concave and convex measures, the authors derive dimension-free bounds and extremal properties for ellipsoids, revealing the Pareto-type distributions as key extremizers. They connect these entropy bounds to the hyperplane (slicing) conjecture, present entropic reformulations, and develop applications to entropy rates, convolution behavior, and mixtures of convex densities. The results provide a versatile information-theoretic lens on convex geometry, with implications for isotropic constants, Rényi entropies, and infinite-divisible laws, and point to several open questions in extending and sharpening the framework.

Abstract

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.

Paper Structure

This paper contains 15 sections, 28 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. One-dimensional log-concave distributions
  3. An extremal property of ellipsoids
  4. Rényi entropies of log-concave distributions
  5. An Entropic Formulation of the Slicing Problem
  6. Convexity of measures
  7. Log-concavity of norms of convex functions
  8. Entropy of $\kappa$-concave distributions
  9. Applications
  10. Entropy rates
  11. The behavior of maximum density on convolution
  12. Infinitely divisible distributions
  13. Entropy of Mixtures
  14. Discussion
  15. The multivariate Pareto distribution

Key Result

Theorem 1.1

If a random vector $X$ in ${\mathbb R}^n$ has a log-concave density $f$, let $Z$ in ${\mathbb R}^n$ be any normally distributed random vector with maximum density being the same as that of $X$. Then Equality holds in the lower bound if and only if $X$ is uniformly distributed on a convex set with non-empty interior. Equality holds in the upper bound if $X$ has coordinates that are i.i.d. exponent

Theorems & Definitions (51)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 41 more