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.