On Branching Points in the Gilbert-Steiner Problem

The Gilbert-Steiner problem is a generalization of the Steiner tree problem and specific optimal mass transportation, which allows the use of additional (branching) points in a transport plan. A specific feature of the problem is that the cost of transporting a mass m m along a segment of length l l is equal to l \cdot m^p l ⋅ m p for a fixed 0 < p < 1 0 < p < 1 and segments may end at points not belonging to the supports of given measures (branching points). The main result of this paper determines all pairs of (p,d) ( p , d ) for which the Gilbert-Steiner problem in \mathbb{R}^d R d admits only branching points of degree 3. Namely, it happens if and only if d = 2 d = 2 or p < 1/2 p < 1 / 2 .