The weighted Fermat triangle problem
type: Item
instance of: scholarly article
title: The weighted Fermat triangle problem
zbMATH Open document ID: 1151.90034
DOI: 10.1155/2008/283846
author: Yujin Shen
author: Juan Tolosa
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2008-08-19T00:00:00Z
full work available at URL: https://eudml.org/doc/54836
review text: Let there be given a triangle \(p_1p_2p_3\) and three positive numbers \(\lambda_1, \lambda_2, \lambda_3\), called the weights of \(p_1, p_2,\) and \(p_3\), respectively. The weighted Fermat-Torricelli problem (in location science also called the Steiner-Weber problem) asks for a point \(p\) minimizing the sum \[ \sum_{i=1}^3\lambda_i\| p_i-p\| . \] For \(m\) non-collinear points \(p_1, \ldots, p_m\in\mathbb{R}^n\), \(n\geq 2\), with corresponding positive weights \(\lambda_1, \ldots\lambda_m\) there exists an unique point \(p\in \mathbb{R}^n\) that minimizes the sum \[ \sum_{i=1}^m\lambda_i\| p_i-p\| ; \] see, e.g., Theorem 18.37 in [\textit{V. Boltyanski, H. Martini} and \textit{V. Soltan}, Geometric methods and optimization problems, Combinatorial Optimization. 4, Dordrecht: Kluwer Academic Publishers (1999; Zbl 0933.90002)]. More about the location of the point \(p\) as well as about different aspects and history of this famous problem can be found in Chapter 2 of [loc. cit.]. In case \(\lambda_1=\dots=\lambda_m=1\) the Fermat-Torricelli problem is even generalized for an arbitrary normed linear space [\textit{H. Martini, K. J. Swanepoel} and \textit{G. Weiss}, J. Optimization Theory Appl. 115, No. 2, 283--314 (2002; Zbl 1047.90032)]. In the paper under review a solution is given for the usual Euclidean norm in the case \(n=2\) and \( m=3\).
reviewed by: Margarita Spirova
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 51M04
Mathematics Subject Classification ID: 90B85
zbMATH DE Number: 5313278
zbMATH Keywords: Fermat-Torricelli (weighted) problem
zbMATH Keywords: location science
zbMATH Keywords: Steiner-Weber problem
MaRDI profile type: Publication
OpenAlex ID: W2060387711
cites work: Q3273607
cites work: Was ist Mathematik?
cites work: The Fermat-Torricelli problem once more
cites work: Q4767256
cites work: The Three Factory Problem
cites work: A Geometrical Solution of the Three Factory Problem
cites work: The Generalized Fermat's Point
cites work: Q5708363
Wikidata QID: Q58645443
DBLP publication ID: journals/ijmmsc/ShenT08
Recommended article: Q4670516
Recommended article: The Fermat-Steiner Problem
Recommended article: Two new analytical solutions and two new geometrical solutions for the weighted Fermat-Torricelli problem in the Euclidean plane
Recommended article: Q3840330
Recommended article: Q5285729
Recommended article: A complete analytical treatment of the weighted Fermat-Torricelli point for a triangle
Recommended article: Geometrical solution to the Fermat problem with arbitrary weights
Recommended article: An analytical solution of the weighted Fermat-Torricelli problem on a unit sphere
Recommended article: The Fermat-Torricelli problem on surfaces
description: scientific article; zbMATH DE number 5313278