The stability of log-supermodularity under convolution

We study the behavior of log-supermodular functions under convolution. In particular we show that log-concave product densities preserve log-supermodularity, confirming in the special case of the standard Gaussian density, a conjecture of Zartash and Robeva. Additionally, this stability gives a ``conditional'' entropy power inequality for log-supermodular random variables. We also compare the Ahlswede-Daykin four function theorem and a recent four function version of the Prekopa-Leindler inequality due to Cordero-Erausquin and Maurey and giving transport proofs for the two theorems. In the Prekopa-Leindler case, the proof gives a generalization that seems to be new, which interpolates the classical three and the recent four function versions.