The oriented mailing problem and its convex relaxation

In this note we introduce a new model for the mailing problem in branched transportation that takes into account the orientation of the moving particles. This gives an effective answer to Bernot et al. (2009, Problem 15.9). Moreover we define a convex relaxation in terms of rectifiable currents with group coefficients. We provide the problem with a notion of calibration. Using similar techniques we define a convex relaxation and a corresponding notion of calibration for a variant of the Steiner tree problem in which a connectedness constraint is assigned only among a certain partition of a given set of finitely many points.