  In this paper we study the following problems: given a finite number of
nonempty closed subsets of a normed space, find a ball with the smallest radius
that encloses all of the sets, and find a ball with the smallest radius that
intersects all of the sets. These problems can be viewed as generalized
versions of the 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 will focus on the sufficient
conditions for the existence and uniqueness of an optimal solution for each
problem, while the study of optimality conditions and numerical implementation
will be addressed in our next projects.
