On branching points in the Gilbert-Steiner problem

TL;DR

This paper characterizes when the Gilbert--Steiner problem in Euclidean space admits only branching points of degree 3, depending on the dimension and the parameter p, revealing a precise condition for such structure.

Contribution

It provides a complete characterization of the pairs (p, d) where the Gilbert--Steiner problem has exclusively degree 3 branching points.

Findings

Branching points are degree 3 if and only if d=2 or p<1/2.

The paper identifies the exact conditions on p and d for the structure of optimal transport networks.

Results clarify the geometric structure of solutions in the Gilbert--Steiner problem.

Abstract

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