Optimal transport paths with capacity induced cost function

This article generalizes the study of ramified optimal transport with capacity constraint in transport multi-paths by generalizing the ${\mathbf{M}}_{\alpha }$ cost to ${\mathbf{M}}_{\alpha ,c}$, which incorporates capacity constraints into the cost function. Equipped with ${\mathbf{M}}_{\alpha ,c}$ cost, we prove the existence of optimal transport path, ${\mathbf{M}}_{\alpha ,c}$ related inequalities, decomposition of any general transport paths, and occurrence of direct line segments in an optimal transport path.