On the horseshoe conjecture for maximal distance minimizers

¹ Chebyshev Laboratory, St. Petersburg State University, 14th Line V. O., 29B, 199178 Saint Petersburg, Russia
janejashka@gmail.com
² Moscow Institute of Physics and Technology, Lab. of advanced combinatorics and network applications, Institutsky lane 9, Dolgoprudny, 141700 Moscow region, Russia
³ St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences, Russia

^a Corresponding author: matelk@mail.ru

Received: 18 March 2016
Revised: 31 December 2016
Accepted: 17 March 2017

Abstract

We study the properties of sets Σ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Σ ⊂ ℝ² satisfying the inequality max_y∈M dist (y,Σ) ≤ r for a given compact set M ⊂ ℝ² and some given r > 0. Such sets play the role of 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. We describe the set of minimizers for M a circumference of radius R > 0 for the case when r < R ∕ 4 .98, thus proving the conjecture of Miranda, Paolini and Stepanov for this particular case. Moreover we show that when M is the boundary of a smooth convex set with minimal radius of curvature R, then every minimizer Σ has similar structure for r < R ∕ 5. Additionaly, we prove a similar statement for local minimizers.