  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 Pr\'ekopa-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.
