On regularity of maximal distance minimizers

Yana Teplitskaya

Published 2019-10-16Version 1

We study the properties of sets $\Sigma$ 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 $\Sigma \subset \mathbb{R}^2$ satisfying the inequality $max_{y \in M} dist(y,\Sigma) \leq r$ 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 work it is proved that each maximal distance minimizer is a union of finite number of curves, having one-sided tangent lines at each point. Moreover the angle between these lines at each point of a maximal distance minimizer is greater or equal to $2\pi/3$. 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. In fact, all the results are proved for more general class of local minimizer, i.e. sets which are optimal in any neighbourhood of its arbitrary point.