A reverse entropy power inequality for i.i.d. log-concave random variables

Let $X$ and $Y$ be independent identically distributed log-concave random variables. We show that $h_\infty(X+Y)-h_\infty(X)$ is maximized when $X$ and $Y$ have exponential distributions. Here, $h_\infty(\cdot)$ is the R\'enyi entropy of order $\infty$. Analogs for integer-valued log-concave random variables are also obtained.