Motivated by a generalization of Sturm-Lott-Villani theory to discrete spaces and by a conjecture stated by Shepp and Olkin about the entropy of sums of Bernoulli random variables, we prove the concavity in $t$ of the entropy of the convolution of a probability measure $a$, which has the law of a sum of independent Bernoulli variables, by the binomial measure of parameters $n\geq 1$ and $t$.
"Concavity of entropy along binomial convolutions." Electron. Commun. Probab. 17 1 - 9, 2012. https://doi.org/10.1214/ECP.v17-1707