In his survey paper in the Bulletin of the AMS from 2002, R. J. Gardner discussed the Brunn-Minkowski inequality, stating that it deserves to be better known and painted a beautiful picture of its relationship with other inequalities in analysis and geometry. At the top of a hierarchical tree of inequalities in this survey paper came the Brascamp-Lieb inequality, which was originally motivated as a natural generalisation of the sharp Young convolution inequality. In this talk I will explain recent developments on the Brascamp-Lieb inequality, most of which took place after 2002, and applications to fundamental problems in analysis, geometry and beyond.
