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

Abstract

We show that $h_{\infty }(X+Y)\le h_{\infty }(Z+W)$, where $X,Y$ are independent log-concave random variables, and $Z,W$ are exponential random variables having the same respective $\infty$-Rényi entropies. Analogs for integer-valued monotone log-concave random variables are also obtained. Our main tools are decreasing rearrangement, majorization, and the change of measure.