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.

翻译：我们发展了对数凹序列的离散自由度概念，并利用它证明了在固定方差条件下，几何分布是所有$\mathbb{Z}$上离散对数凹随机变量中无穷阶Rényi熵的最小化者。我们还证明，在具有固定整数均值的所有超对数凹随机变量中，泊松分布使得$\mathbb{P}(X=\mathbb{E} X)$达到最大。