A Generalized Sylvester Problem and a Generalized Fermat-Torricelli Problem

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.