2022/07/27 by Gordeev, Alexey, Teplitskaya, Yana · 1 citation
#FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2207.13745
We study the properties of sets Σ 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 Σ⊂ ℝn satisfying the inequality maxy ∈ M dist(y,Σ) ≤ r for a given compact set M ⊂ ℝn 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 Σ⊂ ℝn has at most 3 tangent rays at each point and the angle between any two tangent rays at the same point is at least 2π/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.