Formulation of branched transport as geometry optimization

The branched transport problem, a popular recent variant of optimal transport, is non-convex and non-smooth variational problem on Radon measures. so-called urban planning the contrary, shape optimization that seeks geometry street or pipe network. We show with concave cost function equivalent to generalized version problem. Apart from unifying these two different models used in literature, another advantage formulation for it provides more transparent interpretation overall by separation into (Wasserstein-1-distance) network maintenance term, splits actual transportation task optimization.