Entropic analogues of Grünbaum's inequality

2026-07-25

AI Summary

This work investigates an entropy-analogue of Grünbaum’s inequality, focusing on sharp upper and lower bounds for the differential and Rényi entropies of log-concave random variables conditioned to lie on the left side of their mean. By integrating degree-of-freedom analysis with a KKT-type optimization lemma and leveraging structural properties of entropy functionals, the study establishes the first entropy counterpart to Grünbaum’s classical volume inequality. The main contributions include proving tight bounds for both conditional differential entropy and minimal Rényi entropy, fully characterizing all distributions attaining equality, and extending these results to general Rényi orders. The paper also examines the feasibility of higher-dimensional extensions and presents counterexamples demonstrating inherent limitations in such settings.

Abstract

The classical Grünbaum inequality asserts that the proportion of the volume of a convex body cut off by a halfspace containing its barycenter is at least $1/e$. From its functional counterpart, for any log-concave random variable $X$, one has $\mathbb{P}(X\ge \mathbb{E}X)\ge 1/e$, with equality if and only if $X$ is exponential. Motivated by Grünbaum's inequality for convex bodies and its functional generalizations, we prove analogous inequalities for entropy, with characterizations of the equality cases. We show that if $X$ is a log-concave random variable on $\mathbb{R}$, then $$ h(X)-\frac{e}{e-1}H_2(1/e) \leq h(X|X \leq \mathbb{E}X) \leq h(X), $$ where $h$ is the differential entropy, $H_2(\cdot)$ is the binary entropy function and $X|X\leq \mathbb{E}X$ stands for the distribution of $X$ conditional on $X\leq \mathbb{E}X$. We generalize the upper bound for all Rényi entropies and the lower bound for min-entropy. Our inequalities are sharp and we characterize all equality cases. We discuss potential generalizations in high dimensions and give counterexamples in some directions. As an intermediate step for the proof of the lower bound, we establish a new inequality that we prove using a technique known as degrees of freedom, combined with a standard KKT-type optimization lemma. Along the way, we characterize the equality case in a known comparison inequality between differential and min-entropy, which may be of independent interest.

Problem

• entropy
• log-concave
• Grünumbaum inequality
• differential entropy
• Rényi entropy

Innovation

• entropy inequalities
• log-concave distributions
• Rényi entropy
• Grünbaum inequality
• degrees of freedom