  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.
