We exploit a recent approach to Brascamp–Lieb inequalities, due to Caffarelli [5], and reconsider earlier approaches to establish stochastic domination inequalities between Gaussian variables and random variables with density of the form $g\cdot h, g$ a Gaussian density and $h$ a log-concave or log-convex function. These extend to inequalities on random vectors via a classical result by Prékopa and Leindler and they complement the Brascamp–Lieb moment inequalities. Some applications to a class of Gibbs measures, the anharmonic crystals, are developed.