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Journal of Optimization Theory and Applications

, Volume 115, Issue 2, pp 283–314 | Cite as

The Fermat–Torricelli Problem in Normed Planes and Spaces

  • H. Martini
  • K.J. Swanepoel
  • G. Weiss
Article

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.

Fermat–Torricelli problem Weber problem location science facilities location finite-dimensional normed spaces Minkowski spaces finite-dimensional Banach spaces 

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References

  1. 1.
    CHAKERIAN, G. D., and GHANDEHARI, M. A., The Fermat Problem in Minkowski Spaces, Geometriae Dedicata, Vol. 17, pp. 227–238, 1985.Google Scholar
  2. 2.
    DURIER, R., and MICHELOT, C., Geometrical Properties of the Fermat-Weber Problem, European Journal of Operational Research, Vol. 20, pp. 332–343, 1985.Google Scholar
  3. 3.
    DURIER, R., and MICHELOT, C., On the Set of Optimal Points to the Weber Problem, Transportation Science, Vol. 28, pp. 141–149, 1994.Google Scholar
  4. 4.
    KUPITZ, Y. S., and MARTINI, H., Geometric Aspects of the Generalized Fermat-Torricelli Problem, Intuitive Geometry, Bolyai Society Mathematical Studies, Vol. 6, pp. 55–127, 1997.Google Scholar
  5. 5.
    BOLTYANSKI, V., MARTINI, H., and SOLTAN, P. S., Excursions into Combinatorial Geometry, Springer Verlag, Berlin, Germany, 1997.Google Scholar
  6. 6.
    MENGER, K., Untersuchungen über allgemeine Metrik, Mathematische Annalen, Vol. 100, pp. 75–163, 1928.Google Scholar
  7. 7.
    CIESLIK, D., Steiner Minimal Trees, Nonconvex Optimization and Its Applications, Kluwer, Dordrecht, Holland, Vol. 23, 1998.Google Scholar
  8. 8.
    WENDELL, R. E., and HURTER JR., A. P., Location Theory, Dominance, and Convexity, Operations Research, Vol. 21, pp. 314–321, 1973.Google Scholar
  9. 9.
    CIESLIK, D., The Fermat-Steiner-Weber Problem in Minkowski Spaces, Optimization, Vol. 19, pp. 485–489, 1988.Google Scholar
  10. 10.
    DURIER, R., The Fermat-Weber Problem and Inner Product Spaces, Journal of Approximation Theory, Vol. 78, pp. 161–173, 1994.Google Scholar
  11. 11.
    THOMPSON, A. C., Minkowski Geometry, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, England, Vol. 63, 1996.Google Scholar
  12. 12.
    SCHNEIDER, R., Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, England, Vol. 44, 1993.Google Scholar
  13. 13.
    ROCKAFELLAR, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1997.Google Scholar
  14. 14.
    BOLTYANSKI, V., MARTINI, H., and SOLTAN, V., Geometric Methods and Optimization Problems, Kluwer Academic Publishers, Dordrecht, Holland, 1999.Google Scholar
  15. 15.
    SWANEPOEL, K. J., The Local Steiner Problem in Normed Planes, Networks, Vol. 36, pp. 104–113, 2000.Google Scholar
  16. 16.
    DREZNER, Z., Editor, Facility Location: A Survey of Applications and Methods, Springer Verlag, New York, NY, 1995.Google Scholar
  17. 17.
    LEWICKI, G., On a New Proof of Durier's Theorem, Quaestiones Mathematicae, Vol. 18, pp. 287–294, 1995.Google Scholar
  18. 18.
    BENíTEZ, C., FERNáNDEZ, M., and SORIANO, M. L., Location of the Fermat-Torricelli Medians of Three Points, Transactions of the American Mathematical Society (to appear).Google Scholar
  19. 19.
    SWANEPOEL, K. J., Balancing Unit Vectors, Journal of Combinatorial Theory, Vol. 89A, pp. 105–112, 2000.Google Scholar
  20. 20.
    TAMVAKIS, H., Problem 10526, American Mathematical Monthly, Vol. 103, p. 427, 1996.Google Scholar
  21. 21.
    BOWRON, M., and RABINOWITZ, S., Solution to Problem 10526, American Mathematical Monthly, Vol. 104, pp. 979–980, 1997.Google Scholar
  22. 22.
    DANZER, L., and GRüNBAUM, B., Ñber zwei Probleme bezüglich konvexer Körper von P. Erdos und von V. L. Klee, Mathematische Zeitschrift, Vol. 79, pp. 95–99, 1962.Google Scholar
  23. 23.
    DU, D. Z., GAO, B., GRAHAM, R. L., LIU, Z. C., and WAN, P. J., Minimum Steiner Trees in Normed Planes, Discrete and Computational Geometry, Vol. 9, pp. 351–370, 1993.Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • H. Martini
    • 1
  • K.J. Swanepoel
    • 2
  • G. Weiss
    • 3
  1. 1.Fakultät für MathematikTechnische Universität ChemnitzChemnitzGermany
  2. 2.Department of Mathematics, Applied Mathematics, and AstronomyUniversity of South AfricaPretoriaSouth Africa
  3. 3.Institut für GeometrieTechnische Universität DresdenDresdenGermany

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