We show that for any $\alpha>0$ the R\'enyi entropy of order $\alpha$ is minimized, among all symmetric log-concave random variables with fixed variance, either for a uniform distribution or for a two sided exponential distribution. The first case occurs for $\alpha \in (0,\alpha^*]$ and the second case for $\alpha \in [\alpha^*,\infty)$, where $\alpha^*$ satisfies the equation $\frac{1}{\alpha^*-1}\log \alpha^*= \frac12 \log 6$, that is $\alpha^* \approx 1.241$. Using those results, we prove that one-sided exponential distribution minimizes R\'enyi entropy of order $\alpha \geq 2$ among all log-concave random variables with fixed variance.

翻译：我们显示,对于任何$alpha>0美元,R\'enyi entropy, $\ alpha$, 在所有具有固定差异的对称对数对数对数随机变数中, 无论是统一分布还是两个侧面指数分布, 都尽量减少R\'enyi entropy $\ alpha>0美元( 0,\ alpha ⁇,\ inty) 和第二个情况[\ alpha,\ fty]$, 其中, $\alpha\\ $满足公式$\frac{1, halpha\\\\\\\\\ lapha\\\\\\\ frac12\log 6美元, 即$\ alpha\\ approx 1. 241美元。使用这些结果, 我们证明, 单面指数分布将所有有固定差异的log- cocave 随机变数的R\'enpyenpy uny pordy $ $\ e 2$ ge 2$。