  We prove a nonlinear variant of the general Brascamp-Lieb inequality.
Instances of this inequality are quite prevalent in analysis, and we illustrate
this with substantial applications in harmonic analysis and partial
differential equations. Our proof consists of running an efficient, or "tight",
induction on scales argument, which uses the existence of gaussian
near-extremisers to the underlying linear Brascamp-Lieb inequality (Lieb's
theorem) in a fundamental way. A key ingredient is an effective version of
Lieb's theorem, which we establish via a careful analysis of near-minimisers of
weighted sums of exponential functions.
