Calculus of Variations and Geometric Measure Theory
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A. Gordeev - Y. Teplitskaya

On regularity of maximal distance minimizers in Euclidean Space

created by teplitskaya1 on 29 Jul 2022

[BibTeX]

preprint

Inserted: 29 jul 2022

Year: 2022

ArXiv: 2207.13745 PDF

Abstract:

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, at the plane (for $n=2$) we show that the number of points with three tangent rays is finite and maximal distance minimizer is a finite union of simple curves with continuous one-sided tangents. 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.

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