Formulas for the $h$-mass on $1$-currents with coefficients in ${\mathbb{R}}^m$

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 ${\mathcal{M}}_h,\mathrm{\mid }\cdot {\mathrm{\mid }}_H$, and ${\mathbb{M}}_h$, whose equality is the main result of this article: ${\mathcal{M}}_h$ is a functional on $1$-currents in the spirit of Federer and Fleming, norm $\mathrm{\mid }\cdot {\mathrm{\mid }}_H$ 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.