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

Nguyen Mau Nam (), Nguyen Hoang () and Nguyen Thai An ()
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Nguyen Mau Nam: Portland State University
Nguyen Hoang: Hue University
Nguyen Thai An: Thua Thien Hue College of Education

Journal of Optimization Theory and Applications, 2014, vol. 160, issue 2, No 7, 483-509

Abstract: 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.

Keywords: Convex analysis and optimization; Generalized differentiation; Smallest enclosing circle problem; Fermat–Torricelli problem (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10957-013-0366-9

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