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

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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.

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Acknowledgements

The research of Nguyen Mau Nam was partially supported by the Simons Foundation under Grant #208785. The research of Nguyen Hoang was partially supported by the NAFOSTED, Vietnam, under Grant # 101.01-2011.26. The authors would like to thank the referees for reading the paper carefully and giving valuable comments/suggestions that help improve the paper significantly.

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Authors and Affiliations

  1. Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, PO Box 751, Portland, OR, 97207, USA

    Nguyen Mau Nam

  2. Department of Mathematics, College of Education, Hue University, Hue City, Vietnam

    Nguyen Hoang

  3. Thua Thien Hue College of Education, 123 Nguyen Hue, Hue City, Vietnam

    Nguyen Thai An

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  1. Nguyen Mau Nam
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  2. Nguyen Hoang
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Correspondence to Nguyen Mau Nam.

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Communicated by Horst Martini.

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Nam, N.M., Hoang, N. & An, N.T. Constructions of Solutions to Generalized Sylvester and Fermat–Torricelli Problems for Euclidean Balls. J Optim Theory Appl 160, 483–509 (2014). https://doi.org/10.1007/s10957-013-0366-9

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