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Constructions of Solutions to Generalized Sylvester and Fermat–Torricelli Problems for Euclidean Balls

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  • Nguyen Mau Nam

    (Portland State University)

  • Nguyen Hoang

    (Hue University)

  • Nguyen Thai An

    (Thua Thien Hue College of Education)

Abstract

The classical problem of Apollonius is to construct circles that are tangent to three given circles in the plane. This problem was posed by Apollonius of Perga in his work “Tangencies.” The Sylvester problem, which was introduced by the English mathematician J.J. Sylvester, asks for the smallest circle that encloses a finite collection of points in the plane. In this paper, we study the following generalized version of the Sylvester problem and its connection to the problem of Apollonius: given two finite collections of Euclidean balls, find the smallest Euclidean ball that encloses all the balls in the first collection and intersects all the balls in the second collection. We also study a generalized version of the Fermat–Torricelli problem stated as follows: given two finite collections of Euclidean balls, find a point that minimizes the sum of the farthest distances to the balls in the first collection and shortest distances to the balls in the second collection.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:160:y:2014:i:2:d:10.1007_s10957-013-0366-9
    DOI: 10.1007/s10957-013-0366-9
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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. E. Weiszfeld & Frank Plastria, 2009. "On the point for which the sum of the distances to n given points is minimum," Annals of Operations Research, Springer, vol. 167(1), pages 7-41, March.
    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.
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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. Sorin-Mihai Grad & Oleg Wilfer, 2019. "A proximal method for solving nonlinear minmax location problems with perturbed minimal time functions via conjugate duality," Journal of Global Optimization, Springer, vol. 74(1), pages 121-160, May.
    3. Marta Cavaleiro & Farid Alizadeh, 2021. "A dual simplex-type algorithm for the smallest enclosing ball of balls," Computational Optimization and Applications, Springer, vol. 79(3), pages 767-787, July.

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