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Mathematics, Metric Geometry, Combinatorics, Optimization and Control
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.
