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The Fermat–Torricelli Problem in Normed Planes and Spaces

Author

Listed:
  • H. Martini

    (Technische Universität Chemnitz)

  • K.J. Swanepoel

    (University of South Africa)

  • G. Weiss

    (Technische Universität Dresden)

Abstract

We investigate the Fermat–Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat–Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat–Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach.

Suggested Citation

  • H. Martini & K.J. Swanepoel & G. Weiss, 2002. "The Fermat–Torricelli Problem in Normed Planes and Spaces," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 283-314, November.
    • Handle: RePEc:spr:joptap:v:115:y:2002:i:2:d:10.1023_a:1020884004689
    DOI: 10.1023/A:1020884004689
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    References listed on IDEAS

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    1. Durier, Roland & Michelot, Christian, 1985. "Geometrical properties of the Fermat-Weber problem," European Journal of Operational Research, Elsevier, vol. 20(3), pages 332-343, June.
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    Cited by:

    1. Thomas Jahn & Yaakov S. Kupitz & Horst Martini & Christian Richter, 2015. "Minsum Location Extended to Gauges and to Convex Sets," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 711-746, September.
    2. Simone Görner & Christian Kanzow, 2016. "On Newton’s Method for the Fermat–Weber Location Problem," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 107-118, July.
    3. Nguyen Mau Nam & Nguyen Hoang & Nguyen Thai An, 2014. "Constructions of Solutions to Generalized Sylvester and Fermat–Torricelli Problems for Euclidean Balls," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 483-509, February.
    4. Simeon Reich & Truong Minh Tuyen, 2023. "The Generalized Fermat–Torricelli Problem in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 78-97, January.
    5. Boris Mordukhovich & Nguyen Mau Nam, 2011. "Applications of Variational Analysis to a Generalized Fermat-Torricelli Problem," Journal of Optimization Theory and Applications, Springer, vol. 148(3), pages 431-454, March.
    6. H. Martini & K. J. Swanepoel & P. Oloff Wet, 2009. "Absorbing Angles, Steiner Minimal Trees, and Antipodality," Journal of Optimization Theory and Applications, Springer, vol. 143(1), pages 149-157, October.
    7. T. V. Tan, 2010. "An Extension of the Fermat-Torricelli Problem," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 735-744, September.
    8. Yaakov S. Kupitz & Horst Martini & Margarita Spirova, 2013. "The Fermat–Torricelli Problem, Part I: A Discrete Gradient-Method Approach," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 305-327, August.

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