  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 R\'enyi entropies of orders
less than 1 and with moment constraints involving $p$-th absolute moments with
$p\leq 2$. 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.
