# Weak monotone rearrangement on the line

## TL;DR

This paper provides a complete geometric characterization of the weak monotone rearrangement between measures on the real line, expanding understanding in optimal transport and its applications.

## Contribution

It offers a comprehensive geometric description of the weak monotone rearrangement, complementing previous results and enhancing theoretical understanding.

## Key findings

- Complete geometric characterization of weak monotone rearrangement
- Extension of classical monotone rearrangement theory
- Implications for optimal transport and related fields

## Abstract

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.