The Gilbert-Steiner Problem

Abstract

In this chapter, we consider the irrigation problem for two finite atomic measures μ⁺ and μ⁻. This problem was first set and studied in a finite graph setting by Gilbert in [44]. Following his steps, we first consider the irrigation problem from a source to two Dirac masses. If the optimal structure is made of three edges, the first order condition for a local optimum yields constraints on the angles between the edges at the bifurcation point (see Lemma 12.2). Conversely, thanks to the central angle property, Lemma 12.6 describes how these angle constraints permit to construct the bifurcation point directly from the source and the two masses. Proposition 12.10 then completely describes and proves the structure of an optimum for given locations and weights of the source and the two Dirac masses. Let us mention that the classification of the different possible optimal structures is given in Gilbert’s article, yet without proof. In the second section, we generalize the construction already made for two masses to any number of masses. More precisely, if we prescribe a particular topology of a tree irrigating n masses from a Dirac mass, we describe a recursive procedure that permits to construct a local optimum that has this topology. Finally, in the last section, we raise the question of the number of edges meeting at a bifurcation point in an optimal traffic plan with graph structure. In particular, Proposition 12.17 proves that for $\alpha \leqslant \tfrac{1} {2}$ and for an optimal traffic plan in ℝ², there cannot be more than three edges meeting at a bifurcation point. It is an open question and a numerically plausible conjecture that the result holds for general α.