We show that for any$\alpha >0$the Rényi entropy of order$\alpha $is minimized, among all symmetric log-concave random variables with fixed variance, either for a uniform distribution or for a two sided exponential distribution. The first case occurs for$\alpha \in (0,\alpha ^{\ast}]$and the second case for$\alpha \in [\alpha ^{\ast},\infty$), where$\alpha ^{\ast}$satisfies the equation$2 \log \alpha ^{\ast}= (\alpha ^{\ast}-1) \log 6$, that is$\alpha ^{\ast} \approx 1.241$. We deduce that the one-sided exponential distribution minimizes Rényi entropy of order$\alpha \geq 2$among all log-concave random variables with fixed variance.

In this article we study weighted sums of$n$i.i.d. Gamma($\alpha $) random variables with nonnegative weights. We show that for$n \geq 1/\alpha $the sum with equal coefficients maximizes differential entropy when variance is fixed. As a consequence, we prove that among nonnegative quadratic forms in$n$independent standard Gaussian random variables, a diagonal form with equal coefficients maximizes differential entropy, under a fixed variance. This provides a sharp lower bound for the relative entropy between a nonnegative quadratic form and a Gaussian random variable. Bounds on capacities of transmission channels subjects to$n$independent additive gamma noises are also derived.

We show that the uniform distribution minimizes entropy among all one-dimensional symmetric log-concave distributions with fixed variance, as well as various generalizations of this fact to Rényi entropies of orders less than 1 and with moment constraints involving p-th absolute moments with p ≤ 2. As consequences, we give new capacity bounds for additive noise channels with symmetric log-concave noises, as well as for timing channels involving positive signal and noise where the noise has a decreasing log-concave density. In particular, we show that the capacity of an additive noise channel with symmetric, log-concave noise under an average power constraint is at most 0.254 bits per channel use greater than the capacity of an additive Gaussian noise channel with the same noise power. Consequences for reverse entropy power inequalities and connections to the slicing problem in convex geometry are also discussed.

In 1948, Shannon showed that the worst additive noise channel for a given noise power is the additive white Gaussian noise channel. We pose the question of the best additive noise within a natural class of noise distributions- namely, symmetric and log-concave distributions on the real line. While we are unable to answer the question, we do completely solve two related optimization problems. In particular, we identify the distribution in this class that minimizes differential entropy when the variance is fixed, and thereby give refined capacity bounds for channels with symmetric log-concave noises. A full version of this paper which also contains more general results and some additional theorems and corollaries is accessible at: https://arxiv.org/abs/1811.00345.