The nonlinear Brascamp-Lieb inequality for simple data

We establish a nonlinear generalisation of the classical Brascamp-Lieb inequality in the case where the Lebesgue exponents lie in the interior of the finiteness polytope. As a corollary we show that the best constant in Young's convolution inequality in a small neighbourhood of the identity of a general Lie group, approaches the euclidean constant as the size of the neighbourhood approaches zero, answering a question of Cowling, Martini, M\"uller and Parcet. Our proof consists of running an efficient, or "tight", induction on scales argument which uses the existence of gaussian extremisers to the underlying linear Brascamp-Lieb inequality in a fundamental way.