Regularity of maximum distance minimizers

We study properties of sets having the minimum length (one-dimensional Hausdorff measure) over the class of closed connected sets $\Sigma\subset\mathbb R^2$ satisfying the inequality $\max_{y\in M}\operatorname{dist}(y,\Sigma)\leq r$ for a given compact set $M\subset\mathbb R^2$ and some given $r>0$. Such sets play the role of the shortest possible pipelines arriving at a distance at most $r$ to every point of $M$, where $M$ is the set of customers of the pipeline. In this paper, it is proved that each maximum distance minimizer is a union of a finite number of curves having one-sided tangent lines at each point. This shows that a maximum distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact set $M$, which distinguishes it from a solution of the Steiner problem (an example of a Steiner tree with an infinite number of branching points can be found in a paper by Paolini, Stepanov, and Teplitskaya). Moreover, the angle between these lines at each point of a maximum distance minimizer is greater than or equal to $2\pi/3$. Also, we classify the behavior of a minimizer in a neighborhood of any point of $\Sigma$. In fact, all the results are proved for a more general class of local minimizers.