Duality in branched transport and urban planning

In recent work arXiv:2109.07820 we have shown the equivalence of the widely used nonconvex (generalized) branched transport problem with a shape optimization problem of a street or railroad network, known as (generalized) urban planning problem. The argument was solely based on an explicit construction and characterization of competitors. In the current article we instead analyse the dual perspective associated with both problems. In more detail, the shape optimization problem involves the Wasserstein distance between two measures with respect to a metric depending on the street network. We show a Kantorovich$\unicode{x2013}$Rubinstein formula for Wasserstein distances on such street networks under mild assumptions. Further, we provide a Beckmann formulation for such Wasserstein distances under assumptions which generalize our previous result in arXiv:2109.07820. As an application we then give an alternative, duality-based proof of the equivalence of both problems under a growth condition on the transportation cost, which reveals that urban planning and branched transport can both be viewed as two bilinearly coupled convex optimization problems.