  In this paper, we introduce and study the following problem and its further
generalizations: given two finite collections of sets in a normed space, find a
ball whose center lies in a given constraint set with the smallest radius that
encloses all the sets in the first collection and intersects all the sets in
the second one. This problem can be considered as a generalized version of the
Sylvester smallest enclosing circle problem introduced in the 19th century by
Sylvester which asks for the circle of smallest radius enclosing a given set of
finite points in the plane. We also consider a generalized version of the
Fermat-Torricelli problem: given two finite collections of sets in a normed
space, find a point in a given constraint set that minimizes the sum of the
farthest distances to the sets in the first collection and shortest distances
(distances) to the sets in the second collection.
