The Fermat–Torricelli Problem, Part I: A Discrete Gradient-Method Approach (original) (raw)

Access this article

Log in via an institution

Subscribe and save

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Kupitz, Y.S., Martini, H.: Geometric aspects of the generalized Fermat–Torricelli problem. In: Intuitive Geometry. Bolyai Society of Mathematical Studies, vol. 6, pp. 55–127 (1997)
    Google Scholar
  2. Boltyanski, V., Martini, H., Soltan, V.: Geometric Methods and Optimization Problems. Kluwer, Dordrecht (1999)
    MATH Google Scholar
  3. Courant, R., Robbins, H.: What Is Mathematics? Oxford University Press, Oxford (1941)
    Google Scholar
  4. Martini, H., Swanepoel, K.J., Weiss, G.: The Fermat–Torricelli problem in normed planes and spaces. J. Optim. Theory Appl. 115, 283–314 (2002)
    Article MathSciNet MATH Google Scholar
  5. Tan, T.V.: An extension of the Fermat–Torricelli problem. J. Optim. Theory Appl. 146, 735–744 (2010)
    Article MathSciNet MATH Google Scholar
  6. Mordukhovich, B., Nam N, N.: Applications of variational analysis to a generalized Fermat–Torricelli problem. J. Optim. Theory Appl. 148, 431–454 (2011)
    Article MathSciNet MATH Google Scholar
  7. Cockayne, E.J., Melzak, Z.A.: Euclidean constructibility in graph-minimization problems. Math. Mag. 42, 206–208 (1969)
    Article MathSciNet MATH Google Scholar
  8. Bajaj, C.: The algebraic degree of geometric optimization problems. Discrete Comput. Geom. 3, 177–191 (1988)
    Article MathSciNet MATH Google Scholar
  9. Mehlhos, S.: Simple counter-examples for the unsolvability of the Fermat- and Steiner–Weber problem by compass and ruler. Beitr. Algebra Geom. 41, 151–158 (2000)
    MathSciNet MATH Google Scholar
  10. Chandrasekaran, R., Tamir, A.: Algebraic optimization: the Fermat–Weber location problem. Math. Program. 46, 219–224 (1990)
    Article MathSciNet MATH Google Scholar
  11. Martini, H., Weissbach, B.: Napoleon’s theorem with weights in _n_-space. Geom. Dedic. 74, 213–223 (1999)
    Article MathSciNet MATH Google Scholar
  12. Hajja, M., Martini, H., Spirova, M.: New extensions of Napoleon’s theorem to higher dimensions. Beitr. Algebra Geom. 49, 253–264 (2008)
    MathSciNet MATH Google Scholar
  13. Erdős, P., Vincze, I.: On the approximation of convex, closed plane curves by multifocal ellipses. J. Appl. Probab. 19A, 89–96 (1982). Special Volume: Essays in Statist. Science; Papers in Honour of P.A.P. Moran
    Article Google Scholar
  14. Gross, C., Strempel, T.-K.: On generalizations of conics and on a generalization of the Fermat–Torricelli problem. Am. Math. Mon. 105, 732–743 (1998)
    Article MathSciNet MATH Google Scholar
  15. Nie, J., Parillo, P., Sturmfels, B.: Semidefinite representation of the _k_-ellipse. In: Algorithms in Algebraic Geometry. IMA Vol. Math. Appl., vol. 146, pp. 117–132. Springer, New York (2008)
    Chapter Google Scholar
  16. Kuhn, H.W.: Steiner’s problem revisited. In: Dantzig, G.B., Eaves, B.C. (eds.) Studies in Optimization. MAA Studies in Mathematics, vol. 10, pp. 52–70 (1974). Math. Association of America
    Google Scholar
  17. Gilbert, E.N., Pollack, H.O.: Steiner minimal trees. SIAM J. Appl. Math. 16, 1–29 (1968)
    Article MathSciNet MATH Google Scholar
  18. Cieslik, D.: Steiner Minimal Trees. Kluwer, Dordrecht (1998)
    Book MATH Google Scholar
  19. Cieslik, D.: Shortest Connectivity. Springer, New York (2005)
    MATH Google Scholar
  20. Drezner, Y. (ed.): Facility Location: A Survey of Applications and Methods. Springer, New York (1995)
    Google Scholar
  21. Eckhoff, J.: Helly, Radon, and Carathéodory type theorems. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, pp. 389–448. North-Holland, Amsterdam (1993)
    Google Scholar
  22. Lindelöf, L.L.: Sur les maxima et minima, d’une fonction des reyons vecteurs menés d’un point mobile à plusieurs centres fixes. Acta Soc. Sci. Finnic. 8, 191–207 (1867)
    Google Scholar
  23. Kupitz, Y.S., Martini, H.: The Fermat–Torricelli point and isosceles tetrahedra. J. Geom. 49, 150–162 (1994)
    Article MathSciNet MATH Google Scholar
  24. Heath, Th.: A History of Greek Mathematics, vol. II. Clarendon, Oxford (1921). Reprinted by Dover Publications, 1981
    MATH Google Scholar
  25. Durier, R., Michelot, C.: Geometrical properties of the Fermat–Weber problem. Eur. J. Oper. Res. 20, 332–343 (1985)
    Article MathSciNet MATH Google Scholar
  26. Baronti, M., Casini, E., Papini, P.L.: Centroids, centers, medians: what is the difference? Geom. Dedic. 68, 157–168 (1997)
    Article MathSciNet MATH Google Scholar
  27. Martini, H., Schöbel, A.: Median and center hyperplanes in Minkowski spaces—a unified approach. Discrete Math. 241, 407–426 (2001)
    Article MathSciNet MATH Google Scholar
  28. Nievergelt, Y.: Median spheres: theory, algorithms, applications. Numer. Math. 114, 573–606 (2010)
    Article MathSciNet MATH Google Scholar
  29. Körner, M.-C., Martini, H., Schöbel, A.: Minsum hyperspheres in normed spaces. Discrete Appl. Math. 160, 2221–2233 (2012)
    Article MathSciNet MATH Google Scholar

Download references