Formulas for the h-mass on $1$-currents with coefficients in Rᵐ

We consider the minimization of the h-mass over normal $1$-currents in Rⁿ with coefficients in Rᵐ 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 Mₕ,|·|H, and Mₕ, whose equality is the main result of this article: Mₕ is a functional on $1$-currents in the spirit of Federer and Fleming, norm |·|H denotes the total variation of a Radon measure with respect to H induced by h, and Mₕ 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.