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In this paper we study the following problems: given a finite number ofnonempty closed subsets of a normed space, find a ball with the smallest radiusthat encloses all of the sets, and find a ball with the smallest radius thatintersects all of the sets. These problems can be viewed as generalizedversions of the smallest enclosing circle problem introduced in the 19thcentury by Sylvester which asks for the circle of smallest radius enclosing agiven set of finite points in the plane. We will focus on the sufficientconditions for the existence and uniqueness of an optimal solution for eachproblem, while the study of optimality conditions and numerical implementationwill be addressed in our next projects.

The Smallest Enclosing Ball Problem and the Smallest Intersecting Ball Problem: Existence and Uniqueness of Solutions