Minimal time functions and the smallest intersecting ball problem generated by unbounded dynamics

The smallest enclosing circle problem introduced in the 19th century by J. J. Sylvester [20] aks for the circle of smallest radius enclosing a given set of finite points in the plane. An extension of the smallest enclosing circle problem called the smallest intersecting ball problem was considered in [17,18]: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that intersects all of the sets. In this paper we initiate the study of minimal time functions generated by unbounded dynamics and discuss their applications to extensions of the smallest intersecting ball problem. This approach continues our effort in applying convex and nonsmooth analysis to the well-established field of facility location.