We utilize a discrete version of the notion of degree of freedom to prove a sharp min-entropy-variance inequality for integer valued log-concave random variables. More specifically, we show that the geometric distribution minimizes the min-entropy within the class of log-concave probability sequences with fixed variance. As an application, we obtain a discrete R\'enyi entropy power inequality in the log-concave case, which improves a result of Bobkov, Marsiglietti and Melbourne (2022).

翻译：我们利用自由度的离散版本概念，证明了整数值对数凹随机变量在给定方差下的尖锐最小熵-方差不等式。具体而言，我们表明几何分布是具有固定方差的对数凹概率序列中最小熵的最小化分布。作为应用，我们得到了对数凹情形下的离散Rényi熵幂不等式，该结果改进了Bobkov、Marsiglietti和Melbourne（2022）的结论。