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A Generalized Sylvester Problem and a Generalized Fermat-Torricelli Problem for Euclidean Balls

Nguyen Mau Nam (nmnam2007***at***gmail.com)
Nguyen Hoang (nguyenhoanghue***at***gmail.com)
Nguyen Thai An (thaian2784***at***gmail.com)

Abstract: The classical Apollonius' problem is to construct circles that are tangent to three given circles in a 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 in $\Bbb R^n$, find the smallest Euclidean ball that encloses all of the balls in the first collection and intersects all of the balls in the second collection. We also study a generalized version of the Fermat-Torricelli problem stated as follows: given two finite collections composed of three Euclidean balls in $\Bbb R^n$, 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.

Category 1: Convex and Nonsmooth Optimization

Category 2: Nonlinear Optimization

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Entry Submitted: 07/09/2012
Entry Accepted: 07/09/2012
Entry Last Modified: 10/03/2012

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