On regularity of maximal distance minimizers in Euclidean Space

TL;DR

This paper investigates the geometric regularity of minimal-length sets that cover a compact set within a fixed radius, establishing bounds on tangent rays and their angles, with specific results in the plane.

Contribution

It proves that maximal distance minimizers have at most three tangent rays at each point and a minimum angle of 120 degrees, with finiteness of triple-ray points in the plane.

Findings

Maximal distance minimizers have at most 3 tangent rays at each point.

Angles between tangent rays are at least 120 degrees.

In the plane, the set of points with three tangent rays is finite.

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, 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.