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The classical Apollonius' problem is to construct circles that are tangent tothree given circles in a plane. This problem was posed by Apollonius of Pergain his work "Tangencies". The Sylvester problem, which was introduced by theEnglish mathematician J.J. Sylvester, asks for the smallest circle thatencloses a finite collection of points in the plane. In this paper, we studythe following generalized version of the Sylvester problem and its connectionto the problem of Apollonius: given two finite collections of Euclidean ballsin $\Bbb R^n$, find the smallest Euclidean ball that encloses all of the ballsin the first collection and intersects all of the balls in the secondcollection. We also study a generalized version of the Fermat-Torricelliproblem stated as follows: given two finite collections composed of threeEuclidean balls in $\Bbb R^n$, find a point that minimizes the sum of thefarthest distances to the balls in the first collection and shortest distancesto the balls in the second collection.

Solutions Constructions of a Generalized Sylvester Problem and a Generalized Fermat-Torricelli Problem for Euclidean Balls