The weighted Fermat-Torricelli problem for tetrahedra and an ``inverse'' problem
type: Item
instance of: scholarly article
title: The weighted Fermat-Torricelli problem for tetrahedra and an ``inverse'' problem
zbMATH Open document ID: 1163.52005
author: Gerasimos Zouzoulas
author: Anastasios N. Zachos
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-04-29T00:00:00Z
review text: Let \(A_1\), \(A_2\), \(A_3\), \(A_4\) be four affinely independent points in \(\mathbb{R}^3\) with corresponding positive weights in each case. The authors ask for the weighted Fermat-Torricelli point \(A_1\) of that point system, i.e., for the unique and existing point \(A_0\) minimizing the sum of weighted distances to the four given points. They propose a new method and solve an ``inverse'' Fermat-Torricelli problem by differentiating some geometric relations for tetrahedra with respect to specific dihedral angles. This reads as follows: Given \(A_0\) with the four points from above on prescribed rays, find the ratios between every pair of nonnegative weights of two corresponding rays such that the sum of the four nonnegative weights is a constant number. The authors point out that this inverse problem cannot be solved via the usual numerical approach, also known as ``Weiszfeld algorithm''. An application of their results is the introduction of some specific classes of tetrahedra.
Mathematics Subject Classification ID: 52B10
Mathematics Subject Classification ID: 51M04
Mathematics Subject Classification ID: 51M16
zbMATH DE Number: 5549409
zbMATH Keywords: (weighted) Fermat-Torricelli problem
zbMATH Keywords: tetrahedra
zbMATH Keywords: dihedral angles
zbMATH Keywords: location science
reviewed by: Horst Martini
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.11.057
OpenAlex ID: W2028402141
cites work: Geometric methods and optimization problems
cites work: Q4352313
cites work: The Fermat-Torricelli point and isosceles tetrahedra
cites work: Q5447471
DOI: 10.1016/J.JMAA.2008.11.057
description: scientific article; zbMATH DE number 5549409