Approximation of rectifiable 1-currents and weak-⁎ relaxation of the h-mass

Based on Smirnov's decomposition theorem we prove that every rectifiable 1-current T with finite mass M(T) and finite mass M(∂T) of its boundary ∂T can be approximated in mass by a sequence of rectifiable 1-currents Tn with polyhedral boundary ∂Tn and M(∂Tn) no larger than M(∂T). Using this result we can compute the relaxation of the h-mass for polyhedral 1-currents with respect to the joint weak-⁎ convergence of currents and their boundaries. We obtain that this relaxation coincides with the usual h-mass for normal currents. This shows that the concepts of so-called generalized branched transport and the h-mass are equivalent. © 2019 Elsevier Inc. All rights reserved.