We consider the minimization of the h-mass over normal 1-currents in \(\mathbb {R}^n\) with coefficients in \(\mathbb {R}^m\) and prescribed boundary. This optimization is known as multi-material transport problem and used in the context of logistics of multiple commodities, but also as a relaxation of nonconvex optimal transport tasks such as so-called branched transport problems. The h-mass with norm h can be defined in different ways, resulting in three functionals \(\mathscr {M}_h,|\cdot |_H\) , and \(\mathbb {M}_h\) , whose equality is the main result of this article: \(\mathscr {M}_h\) is a functional on 1-currents in the spirit of Federer and Fleming, norm \(|\cdot |_H\) denotes the total variation of a Radon measure with respect to H induced by h, and \(\mathbb {M}_h\) is a mass on flat 1-chains in the sense of Whitney. On top we introduce a new and improved notion of calibrations for the multi-material transport problem: we identify calibrations with (weak) Jacobians of optimizers of the associated convex dual problem, which yields their existence and natural regularity.
