Journal of Optimization Theory and Applications

, Volume 146, Issue 3, pp 735–744 | Cite as

An Extension of the Fermat-Torricelli Problem

  • T. V. Tan


The Fermat-Torricelli problem is an optimization problem associated with a finite subset \(\{a_{j}\}_{j=1}^{q}\) of ℝ N and a family \(\{c_{j}\}_{j=1}^{q}\) of positive weights. The function F to be minimized is defined by \(F(x)=\sum _{j=1}^{q}c_{j}\Vert x-a_{j}\Vert\). In this paper, we extend this problem to the case of volumes.


Fermat-Torricelli problem Weber problem Finite-dimensional space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boltyanski, V., Martini, H., Stoltan, V.: Geometric Methods and Optimization Problems. Kluwer Academic, Dordrecht (1999) MATHGoogle Scholar
  2. 2.
    Cieslik, D.: The Fermat-Steiner-Weber problem in Minkowski spaces. Optimization 19, 485–489 (1988) CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Dalla, L.: A note on the Fermat-Torricelli point of a d-simplex. J. Geom. 70, 38–43 (2001) CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Durier, R.: The Fermat-Weber problem and inner product spaces. J. Approx. Theory 78, 161–173 (1994) CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Kupitz, Y.S., Martini, H.: The Fermat-Torricelli point and the isosceles tetrahedra. J. Geom. 49, 150–162 (1994) CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Kupitz, Y.S., Martini, H.: Geometric aspects of the generalized Fermat-Torricelli problem. In: Intuitive Geometry. Bolyai Society Math. Studies, vol. 6, pp. 55–127 (1997) Google Scholar
  7. 7.
    Martini, H., Swanepoel, K.J., Weiss, G.: The Fermat-Torricelli in normed planes and spaces. J. Optim. Theory Appl. 115, 283–314 (2002) CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Sturm, R.: Über die Punkte kleinster Entfernungssumme von gegebenen Punkten. J. Reine Angew. Math. 97, 49–61 (1884) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsHanoi National University of EducationHanoiVietnam

Personalised recommendations