On the horseshoe conjecture for maximal distance minimizers
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1015-1041.

We study the properties of sets  Σ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Σ ⊂ ℝ 2 satisfying the inequality satisfying the inequality max y ∈ M dist ( y , Σ ) ≤ r for a given compact set M ⊂ ℝ 2 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.

DOI : 10.1051/cocv/2017025
Classification : 49Q10, 49Q20, 49K30, 90B10, 90C27
Mots-clés : Steiner tree, locally minimal network, maximal distance minimizer
Cherkashin, Danila 1 ; Teplitskaya, Yana 1

1 
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Cherkashin, Danila; Teplitskaya, Yana. On the horseshoe conjecture for maximal distance minimizers. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1015-1041. doi : 10.1051/cocv/2017025. http://www.numdam.org/articles/10.1051/cocv/2017025/

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