On different notions of calibrations for minimal partitions and minimal networks in \(\mathbb{R}^2\) (Q2039531)

Given a collection \(S\) of \(n\) points in the Euclidean plane, the Steiner problem is finding a connected set containing \(S\) that has minimal length. The first part of this paper compares different notions of calibrations determining when different notions of calibrations are equivalent. The authors show that when the points of \(S\) are on the boundary of a convex set then calibrations on coverings are paired calibrations. It is also shown that the notion for currents is stronger than the one on coverings. In the second part, they prove that the existence of a calibration for a constrained set \(E\) in a covering \(Y\) implies the minimality of \(E\) among the class of finite linear combinations of characteristic functions of finite perimeter sets.