  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.
