On regularity of maximal distance minimizers in Euclidean Space

We study the properties of sets $\Sigma$ which are the solutions of the maximal distance minimizer problem, i.e. of sets having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $\Sigma \subset \mathbb{R}^n$ satisfying the inequality $$max_{y \in M} dist(y,\Sigma) \leq r$$ for a given compact set $M \subset \mathbb{R}^n$ and some given $r > 0$. Such sets can be considered as the shortest networks of radiating Wi-Fi cables arriving to each customer (for the set $M$ of customers) at a distance at most $r$. In this paper we prove that any maximal distance minimizer $\Sigma \subset \mathbb{R}^n$ has at most $3$ tangent rays at each point and the angle between any two tangent rays at the same point is at least $2\pi/3$. Moreover, in the plane (for $n=2$) we show that the number of points with three tangent rays is finite and every maximal distance minimizer is a finite union of simple curves with one-sided tangents continuous from the corresponding side. All the results are proved for the more general class of local minimizers, i.e. sets which are optimal under a perturbation of a neighbourhood of their arbitrary point.