Applications of weak transport theory

Motivated by applications to geometric inequalities, Gozlan, Roberto, Samson, and Tetali (J. Funct. Anal. 273 (2017) 3327–3405) introduced a transport problem for ‘weak’ cost functionals. Basic results of optimal transport theory can be extended to this setup in remarkable generality. In this article, we collect several problems from different areas that can be recast in the framework of weak transport theory, namely: the Schrödinger problem, the Brenier–Strassen theorem, optimal mechanism design, linear transfers, semimartingale transport. Our viewpoint yields a unified approach and often allows to strengthen the original results.