The Brascamp-Lieb Inequalities

The Brascamp-Lieb inequalities are a family of inequalities giving a general form for a host of important inequalities such as Hölder, Young, and Loomis-Whitney. A particularly nice subfamily is the geometric Brascamp-Lieb inequalities. Following part of a paper of Bennett, Carbery, Christ, and Tao (https://arxiv.org/abs/math/0505065), I will describe a heat flow method for proving an optimal constant for geometric Brascamp-Lieb inequalities, and show that such inequalities are extremised by Gaussians.