A Weighted Prékopa-Leindler inequality and sumsets with quasicubes

We give a short, self-contained proof of two key results from a paper of four of the authors. The first is a kind of weighted discrete Prékopa-Leindler inequality. This is then applied to show that if \(A, B \subseteq \mathbb{Z}^d\) are finite sets and \(U\) is a subset of a "quasicube" then \(|A + B + U| \geq |A|^{1/2} |B|^{1/2} |U|\). This result is a key ingredient in forthcoming work of the fifth author and P\"alv\"olgyi on the sum-product phenomenon.