Coefficient groups inducing nonbranched optimal transport

TL;DR

This paper characterizes normed Abelian groups that guarantee nonbranching optimal transport plans, providing a geometric classification and a duality generalization for these groups.

Contribution

It introduces an intrinsic condition on normed Abelian groups ensuring nonbranching optimal transport, with complete classifications and a duality extension.

Findings

Characterization of finitely generated normed groups inducing nonbranching transport

Classification of groups with acyclic support in optimal plans

Establishment of a global calibration and duality for nonbranching cases

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 finitely generated 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 a 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.