An entropic analogue of the MMS conjecture

Abstract

Let $P=\{x_1,\ldots,x_n\}$ be a multiset consisting of $n\ge 2$ real numbers such that $\sum_{i=1}^{n}x_i=0$ and $\sum_{i=1}^{n}|x_i|>0$, and let $k <n$ be a positive integer. We sample $k$ elements from $P$ without replacement and set $X_P$ be the sum of the elements in our sample. It is shown that the Shannon entropy of $X_P$ satisfies \[ \mathbf{H}(X_P) \ge \mathbf{H}(\text{Ber}(k/n)) \, , \] where $\text{Ber}(k/n)$ is a Bernoulli random variable of mean $k/n$. The result is sharp, and may be seen as an entropic analogue of the Manickam-Miklós-Singhi (MMS) conjecture.