Coefficient Groups Inducing Nonbranched Optimal Transport

Abstract

In this work we consider an optimal transport problem with coefficients in a normed Abelian group $G$, and extract a purely intrinsic condition on $G$ that guarantees that the optimal transport (or the corresponding minimum filling) is not branching. The condition turns out to be equivalent to the nonbranching of minimum fillings in geodesic metric spaces. We completely characterize discrete normed groups and finite-dimensional normed vector spaces of coefficients that induce nonbranching optimal transport plans. We also provide a complete classification of normed groups for which the optimal transport plans, besides being nonbranching, have acyclic support. This seems to initiate new geometric classifications of certain normed groups. In the nonbranching case we also provide a global version of calibration, i.e. a generalization of Monge–Kantorovich duality.