We study the properties of sets \n \Sigma \n having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets \n \Sigma \subset {\mathbb{R}}^{2}\n satisfying the inequality satisfying the inequality \n {\mathrm{max}}_{y\in M}\phantom{\rule{0.166667em}{0ex}}\mathrm{dist}(y,\Sigma )\le r\n for a given compact set \n M\subset {\mathbb{R}}^{2}\n^{} and some given \n r\hspace{0.17em}>\hspace{0.17em}0\n. Such sets play the role of shortest possible pipelines arriving at a distance at most \n r\n to every point of \n M\n, where \n M\n is the set of customers of the pipeline. We describe the set of minimizers for \n M\n a circumference of radius \n R>0\n for the case when \n r\hspace{0.17em}<R\hspace{0.17em}\u2215\hspace{0.17em}4.98\n, thus proving the conjecture of Miranda, Paolini and Stepanov for this particular case. Moreover we show that when \n M\n is the boundary of a smooth convex set with minimal radius of curvature \n R\n, then every minimizer \n \Sigma \n has similar structure for \n r<R/5\n. Additionaly, we prove a similar statement for local minimizers.

Keywords: Steiner tree, locally minimal network, maximal distance minimizer

^{1}; Teplitskaya, Yana

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@article{COCV_2018__24_3_1015_0, author = {Cherkashin, Danila and Teplitskaya, Yana}, 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/} }

TY - JOUR AU - Cherkashin, Danila AU - Teplitskaya, Yana TI - On the horseshoe conjecture for maximal distance minimizers JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1015 EP - 1041 VL - 24 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017025/ DO - 10.1051/cocv/2017025 LA - en ID - COCV_2018__24_3_1015_0 ER -

%0 Journal Article %A Cherkashin, Danila %A Teplitskaya, Yana %T On the horseshoe conjecture for maximal distance minimizers %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1015-1041 %V 24 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017025/ %R 10.1051/cocv/2017025 %G en %F COCV_2018__24_3_1015_0

Cherkashin, Danila; Teplitskaya, Yana. On the horseshoe conjecture for maximal distance minimizers. ESAIM: Control, Optimisation and Calculus of Variations, Volume 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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