An optimal irrigation network with infinitely many branching points (Q2808058)

In his well-known paper [``Minimum cost communication networks'', Bell Syst. Tech. J. 46, 2209--2227 (1967)], \textit{E. N. Gilbert} considered the problem of minimizing the cost of networks, taking into account the paths between terminals, eventually leading to the nowadays called Gilbert-Steiner Mass Transportation Problem (GSMTP, for short). For details, see e.\,g. [\textit{M. Bernot} et al., Optimal transportation networks. Models and theory. Lecture Notes in Mathematics 1955. Berlin: Springer (2009; Zbl 1163.90001)] and [\textit{Q. Xia}, Commun. Contemp. Math. 5, No. 2, 251--279 (2003; Zbl 1032.90003)]. In present paper, the authors give a new general approach to the GSMTP by describing it as the minimization of a convex functional on currents with coefficients in a group. This procedure has already been used by the authors in a special case, see [Adv. Calc. Var. 9, No. 1, 19--39 (2016; Zbl 1334.49143)]. Here, their new extended framework allows the introduction of calibrations, which, beyond interesting technical new tools, yield the optimality of certain irrigation networks with a countable number of branching points in separable Hilbert spaces.