The Brascamp-Lieb inequality and the reverse form generalize the Holder and Prekopa-Leindler inequality. The equality case in the Brascamp-Lieb inequality has been characterized by Valdimarsson.  Partially building on the work of Bennett, Carbery, Christ and Tao we characterize the equality case in the Reverse Brascamp-Lieb inequality. The proof builds on the structure theory of ‘’Brascamp-Lieb data’’ and uses a variant of Caffarelli's contraction principle. We will also discuss some geometric applications, concerning volume estimates from orthogonal projections and sections. This is based on joint work with Karoly Boroczky and Dongmeng Xi.