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On the horseshoe conjecture for maximal distance minimizers. / Cherkashin, Danila; Teplitskaya, Yana.

In: ESAIM - Control, Optimisation and Calculus of Variations, Vol. 24, No. 3, 01.01.2018, p. 1015-1041.

Research output: Contribution to journal › Article › peer-review

Harvard

Cherkashin, D & Teplitskaya, Y 2018, 'On the horseshoe conjecture for maximal distance minimizers', ESAIM - Control, Optimisation and Calculus of Variations, vol. 24, no. 3, pp. 1015-1041. https://doi.org/10.1051/cocv/2017025

APA

Cherkashin, D., & Teplitskaya, Y. (2018). On the horseshoe conjecture for maximal distance minimizers. ESAIM - Control, Optimisation and Calculus of Variations, 24(3), 1015-1041. https://doi.org/10.1051/cocv/2017025

Vancouver

Cherkashin D, Teplitskaya Y. On the horseshoe conjecture for maximal distance minimizers. ESAIM - Control, Optimisation and Calculus of Variations. 2018 Jan 1;24(3):1015-1041. https://doi.org/10.1051/cocv/2017025

Author

Cherkashin, Danila ; Teplitskaya, Yana. / On the horseshoe conjecture for maximal distance minimizers. In: ESAIM - Control, Optimisation and Calculus of Variations. 2018 ; Vol. 24, No. 3. pp. 1015-1041.

BibTeX

@article{593720e7135c4a80a8ac88617a284314,
title = "On the horseshoe conjecture for maximal distance minimizers",
abstract = "We study the properties of sets S having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets σ R2 satisfying the inequality maxy2M dist (y; S) = 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 S has similar structure for r < R=5. Additionaly, we prove a similar statement for local minimizers.",
keywords = "Locally minimal network, Maximal distance minimizer, Steiner tree, CONTINUA, DIRICHLET REGIONS, locally minimal network, maximal distance minimizer",
author = "Danila Cherkashin and Yana Teplitskaya",
year = "2018",
month = jan,
day = "1",
doi = "10.1051/cocv/2017025",
language = "English",
volume = "24",
pages = "1015--1041",
journal = "ESAIM - Control, Optimisation and Calculus of Variations",
issn = "1292-8119",
publisher = "EDP Sciences",
number = "3",

}

RIS

TY - JOUR

T1 - On the horseshoe conjecture for maximal distance minimizers

AU - Cherkashin, Danila

AU - Teplitskaya, Yana

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We study the properties of sets S having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets σ R2 satisfying the inequality maxy2M dist (y; S) = 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 S has similar structure for r < R=5. Additionaly, we prove a similar statement for local minimizers.

AB - We study the properties of sets S having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets σ R2 satisfying the inequality maxy2M dist (y; S) = 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 S has similar structure for r < R=5. Additionaly, we prove a similar statement for local minimizers.

KW - Locally minimal network

KW - Maximal distance minimizer

KW - Steiner tree

KW - CONTINUA

KW - DIRICHLET REGIONS

KW - locally minimal network

KW - maximal distance minimizer

UR - http://www.scopus.com/inward/record.url?scp=85056345193&partnerID=8YFLogxK

U2 - 10.1051/cocv/2017025

DO - 10.1051/cocv/2017025

M3 - Article

VL - 24

SP - 1015

EP - 1041

JO - ESAIM - Control, Optimisation and Calculus of Variations

JF - ESAIM - Control, Optimisation and Calculus of Variations

SN - 1292-8119

IS - 3

ER -

ID: 7631684