  We develop the notion of discrete degrees of freedom of a log-concave
sequence and use it to prove that geometric distribution minimises R\'enyi
entropy of order infinity under fixed variance, among all discrete log-concave
random variables in $\mathbb{Z}$. We also show that the quantity
$\mathbb{P}(X=\mathbb{E} X)$ is maximised, among all ultra-log-concave random
variables with fixed integral mean, for a Poisson distribution.
