Minimum entropy of a log-concave variable for fixed variance

TL;DR

This paper proves that among log-concave variables with fixed variance, the exponential distribution minimizes Shannon entropy, and applies this to bound channel capacities and refine entropy power inequalities.

Contribution

It establishes the minimal entropy property of exponential variables among log-concave distributions with fixed variance and improves related entropy inequalities.

Findings

Exponential distribution minimizes entropy for fixed variance among log-concave variables.

Derived upper bounds on additive noise channel capacities with log-concave noise.

Improved constants in reverse entropy power inequalities for log-concave variables.

Abstract

We show that for log-concave real random variables with fixed variance the Shannon differential entropy is minimized for an exponential random variable. We apply this result to derive upper bounds on capacities of additive noise channels with log-concave noise. We also improve constants in the reverse entropy power inequalities for log-concave random variables.