Weak optimal transport has been recently introduced by Gozlan et al. The original motivation stems from the theory of geometric inequalities; further applications concern numerics of martingale optimal transport and stability in mathematical finance.
In this note we provide a complete geometric characterization of the weak version of the classical monotone rearrangement between measures on the real line, complementing earlier results of Alfonsi, Corbetta, and Jourdain.
"Weak monotone rearrangement on the line." Electron. Commun. Probab. 25 1 - 16, 2020. https://doi.org/10.1214/20-ECP292