  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.
