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

@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://www.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://www.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://www.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, Tome 24 (2018) no. 3, pp. 1015-1041. doi : 10.1051/cocv/2017025. http://www.numdam.org/articles/10.1051/cocv/2017025/

Bibliographie
Cité par

[1] G. Bouchitté, C. Jimenez and R. Mahadevan, Asymptotic analysis of a class of optimal location problems. J. Math. Pures Appl. 95 (2011) 382–419 | DOI | MR | Zbl

[2] A. Brancolini, G. Buttazzo, F. Santambrogio and E. Stepanov, Long-term planning versus short-term planning in the asymptotical location problem. ESAIM Control Optim. Calc. Var. 15 (2009) 509–524 | DOI | Numdam | MR | Zbl

[3] G. Buttazzo, E. Oudet and E. Stepanov, Optimal transportation problems with free Dirichlet regions. In Variational methods for discontinuous structures, Vol. 51 of Progr. Nonlinear Differential Equations Appl. Birkhäuser, Basel (2002) 41–65 | MR | Zbl

[4] G. Buttazzo, A. Pratelli, S. Solimini and E. Stepanov, Optimal urban networks via mass transportation, Vol. 1961 of Lecture Notes in Mathematics. Springer-Verlag, Berlin 2009 | MR | Zbl

[5] G. Buttazzo and E. Stepanov, Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem. Ann. Sc. Norm. Super. Pisa Cl. Sci. 2 (2003) 631–678 | Numdam | MR | Zbl

[6] G. Buttazzo and E. Stepanov, Minimization problems for average distance functionals. Calculus of Variations: Topics from the Mathematical Heritage of Ennio De Giorgi, edited by D. Pallara, Quaderni di Matematica, Seconda Università di Napoli, Caserta 14 (2004) 47–83 | MR | Zbl

[7] S. Eilenberg and O. G. Harrold, Continua of finite linear measure I. Am. J. Math. 65 (1943) 137–146 | DOI | MR | Zbl

[8] S. Graf and H. Luschgy, Foundations of quantization for probability distributions, Vol. 1730 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2000) | MR | Zbl

[9] A.O. Ivanov and A.A. Tuzhilin, Minimal Networks: The Steiner Problem and Its Generalizations. CRC Press (1994) | MR | Zbl

[10] A. Lemenant, A presentation of the average distance minimizing problem. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 390 (Teoriya Predstavlenii, Dinamicheskie Sistemy, Kombinatornye Metody. XX) 308 (2011) 117–146 | MR | Zbl

[11] X.Y. Lu and D. Slepčev, Properties of minimizers of average-distance problem via discrete approximation of measures. SIAM J. Math. Anal. 45 (2013) 820–836 | MR | Zbl

[12] M. Miranda Jr. E. Paolini and E. Stepanov, On one-dimensional continua uniformly approximating planar sets. Calc. Var. Partial Diff. Eq. 27 (2006) 287–309 | DOI | MR | Zbl

[13] E. Paolini and E. Stepanov, Qualitative properties of maximum distance minimizers and average distance minimizers in ℝ. J. Math. Sci. (NY) 122 (2004) 3290–3309. Problems in mathematical analysis. | DOI | MR | Zbl

[14] E. Paolini and E. Stepanov, Existence and regularity results for the Steiner problem. Calc. Var. Partial Diff. Eq. 46 (2013) 837–860 | DOI | MR | Zbl

[15] A. Suzuki and Z. Drezner, The p-center location problem in an area. Location Science 4 (1996) 69–82 | DOI | Zbl

[16] A. Suzuki and A. Okabe, Using Voronoi diagrams, edited by Z. Drezner, Facility location: A survey of applications and methods, Springer series in operations research (1995), pp. 103–118 | DOI | MR

Cité par Sources :