Rényi entropy and variance comparison for symmetric log-concave random variables

Abstract

We show that for any $\alpha >0$ the Rényi 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 $\alpha \in (0,{\alpha }^{\ast }]$ and the second case for $\alpha \in [{\alpha }^{\ast },\infty )$, where ${\alpha }^{\ast }$ satisfies the equation $\frac{1}{{\alpha }^{\ast }-1}\log {\alpha }^{\ast }=\frac{1}{2}\log 6$, that is ${\alpha }^{\ast }\approx 1.241$. Using those results, we prove that one-sided exponential distribution minimizes Rényi entropy of order $\alpha \ge 2$ among all log-concave random variables with fixed variance.