For every finite measure μ on R^n we define a decomposability bundle V(μ, ·) related to the decompositions of μ in terms of rectifiable one-dimensional measures. We then show that every Lipschitz function on R^n is differentiable at μ-a.e. x with respect to the subspace V(μ, x), and prove that this differentiability result is optimal, in the sense that, following [4], we can construct Lipschitz functions which are not differentiable at μ-a.e. x in any direction which is not in V(μ,x). As a consequence we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) k-dimensional normal currents, which we use to extend certain basic formulas involving normal currents and maps of class C^1 to Lipschitz maps

Abstract. The Steiner tree problem can be stated in terms of finding a connected set of minimal length containing a given set of finitely many points. We show how to formulate it as a mass-minimization problem for 1-dimensional currents with coefficients in a suitable normed group. The representation used for these currents allows to state a calibration principle for this problem. We also exhibit calibrations in some examples.

In this article we prove that the singular set of Dirichlet-minimizing Q-valued functions is countably (m − 2)-rectifiable and we give upper bounds for the (m − 2)-dimensional Minkowski content of the set of singular points with multiplicity Q.2010 Mathematics Subject Classification. 49Q20, 53A10, 49N60.

The Gilbert-Steiner problem is a mass transportation problem, where the cost of the transportation depends on the network used to move the mass and it is proportional to a certain power of the "flow". In this paper, we introduce a new formulation of the problem, which turns it into the minimization of a convex functional in a class of currents with coefficients in a group. This framework allows us to define calibrations, which can be used to prove the optimality of concrete configurations. We apply this technique to prove the optimality of a certain irrigation network, having the topological property mentioned in the title.

In this note we prove an abstract version of a recent quantitative stratification principle introduced by Cheeger and Naber (2013) [6,7]. Using this general regularity result paired with an ε-regularity theorem we provide a new estimate of the Minkowski dimension of the set of higher multiplicity points of a Dir-minimizing Q-valued function. The abstract principle is applicable to several other problems: we recover recent results in the literature and we obtain also some improvements in more classical contexts.