On regularity of maximal distance minimizers

We study the properties of sets $\mathrm{\Sigma }$ which are the solutions of the maximal distance minimizer problem, id est of sets having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $\mathrm{\Sigma }\subset {\mathbb{R}}^2$ satisfying the inequality $\max_{y \in M} dist(y,\Sigma) \leq r$foragivencompactset$M \subset $\mathbb{R}$^2$ and some given $r>0$. Such sets can be considered as the shortest networks of radiating cables arriving to each customer (from the set $M$ of customers) at a distance at most $r$. In this work it is proved that each maximal distance minimizer is a union of finite number of simple curves, having one-sided tangents at each point. Moreover the angle between these rays at each point of a maximal distance minimizer is greater or equal to $2\pi \mathrm{/}3$. It shows that a maximal distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact $M$, which differs it from a solution of the Steiner problem (there exists an example of a Steiner tree with an infinite number of branching points). Also we classify the behavior of a minimizer in a neighbourhood of an arbitrary point of $\mathrm{\Sigma }$. In fact, all the results are proved for more general class of local minimizer, id est sets which are optimal in a neighbourhood of its arbitrary point.