Four Function Theorems Through Optimal Transport and Entropic Analogs

Abstract

We study entropic versions of four-function inequalities and give transport proofs of the Euclidean Ahlswede-Daykin theorem and a four-function version of the Prekopa-Leindler inequality due to Cordero-Erausquin and Maurey. One of our main results shows that the sharp range for the four-function inequality for equally weighted power means of finite orders \(α,β\) is \(0\leqα+β\leq2\), with the Ahlswede-Daykin min-max inequality obtained as a limit. En route, we prove corresponding entropy inequalities under the quadratic optimal transport coupling, including an entropy power inequality for the minimum and maximum in Euclidean space. We also study the stability of four-function inequalities under convolution with log-concave product densities, recovering the preservation of log-supermodularity established by Karlin and Rinott, and explain its application to a conditional entropy power inequality for log-supermodular random variables.