On the question of the best additive noise among symmetric log-concave noises

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.