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

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

.