On the horseshoe conjecture for maximal distance minimizers

• Published in: ESAIM. Control, optimisation and calculus of variations Vol. 24; no. 3; pp. 1015 - 1041
• Main Authors: Cherkashin, Danila, Teplitskaya, Yana
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 2018
• Subjects: 49K30; 49Q10; 49Q20; 90B10; 90C27; locally minimal network; Mathematical analysis; Mathematics; maximal distance minimizer; Radius of curvature; Set theory; Steiner tree; 90B10; Mathematics Subject Classification. 49Q10; 90C27; 49K30; Mathematics Subject Classification. 49Q10 49Q20 49K30 90B10 90C27; 49Q20
• ISSN: 1292-8119; 1262-3377; 1262-3377
• DOI: 10.1051/cocv/2017025

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.