On the horseshoe conjecture for maximal distance minimizers

Cherkashin, Danila; Teplitskaya, Yana

doi:10.1051/cocv/2017025

Cherkashin, Danila ¹ ; Teplitskaya, Yana ¹

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1015-1041.

Résumé

We study the properties of sets $Σ$ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $Σ \subset ℝ^{2}$ satisfying the inequality satisfying the inequality $\max_{y \in M} dist (y, Σ) \leq r$ for a given compact set $M \subset ℝ^{2}$ 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.

MR Zbl

DOI : 10.1051/cocv/2017025

Classification : 49Q10, 49Q20, 49K30, 90B10, 90C27
Mots clés : Steiner tree, locally minimal network, maximal distance minimizer

Affiliations des auteurs :

Cherkashin, Danila ¹ ; Teplitskaya, Yana ¹

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     title = {On the horseshoe conjecture for maximal distance minimizers},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1015--1041},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {3},
     year = {2018},
     doi = {10.1051/cocv/2017025},
     mrnumber = {3877191},
     zbl = {1405.49031},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2017025/}
}

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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://archive.numdam.org/articles/10.1051/cocv/2017025/

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