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The Fermat–Torricelli Problem, Part I: A Discrete Gradient-Method Approach

Author

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  • Yaakov S. Kupitz

    (The Hebrew University of Jerusalem)

  • Horst Martini

    (University of Technology)

  • Margarita Spirova

    (University of Technology)

Abstract

We give a discrete geometric (differential-free) proof of the theorem underlying the solution of the well known Fermat–Torricelli problem, referring to the unique point having minimal distance sum to a given finite set of non-collinear points in d-dimensional space. Further on, we extend this problem to the case that one of the given points is replaced by an affine flat, and we give also a partial result for the case where all given points are replaced by affine flats (of various dimensions), with illustrative applications of these theorems.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:158:y:2013:i:2:d:10.1007_s10957-013-0266-z
    DOI: 10.1007/s10957-013-0266-z
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. 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.
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    Cited by:

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    2. 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.
    3. 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.
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