Volume 24, Number 3, July–September 2018
|Page(s)||1015 - 1041|
|Published online||07 June 2018|
On the horseshoe conjecture for maximal distance minimizers
Chebyshev Laboratory, St. Petersburg State University,
14th Line V. O., 29B,
Saint Petersburg, Russia
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: firstname.lastname@example.org
Revised: 31 December 2016
Accepted: 17 March 2017
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 maxy∈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.
Mathematics Subject Classification: 49Q10 / 49Q20 / 49K30 / 90B10 / 90C27
Key words: Steiner tree / locally minimal network / maximal distance minimizer
© EDP Sciences, SMAI 2018
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