# A note on the c-monotonicity in optimal transport with capacity constraints

## TL;DR

This paper explores the geometric properties of optimal transport solutions under capacity constraints, introducing c-capacity monotonicity as a generalization of c-cyclical monotonicity, and proves that optimal solutions satisfy this property.

## Contribution

It introduces the concept of c-capacity monotonicity and proves that optimal transport plans with capacity constraints exhibit this property.

## Key findings

- Optimal transport solutions with capacity constraints are c-capacity monotone.
- c-capacity monotonicity generalizes c-cyclical monotonicity in optimal transport.
- The paper provides a geometric characterization of optimal plans under capacity constraints.

## Abstract

This paper studies the geometry of the optimizer for the optimal transport problem with capacity constraints. We introduce the concept of c-capacity monotonicity, which is a generalization of c-cyclical monotonicity in optimal transport. We show that the optimizer of the optimal transport problem with capacity constraints is c-capacity monotone.

---
Source: https://tomesphere.com/paper/2508.20428