On the Smallest Enclosing Balls

In the paper a theoretical analysis is given for the smallest ball that covers a finite number of points p1, p2, · · · , pN ∈ R . Several fundamental properties of the smallest enclosing ball are described and proved. Particularly, it is proved that the k-circumscribing enclosing ball with smallest k is the smallest enclosing ball, which dramatically reduces a possible large number of computations in the higher dimensional case. General formulas are deduced for calculating circumscribing balls. The difficulty of the closed-form description is discussed. Finally, as an application, the problem of finding a common quadratic Lyapunov function for a set of stable matrices is considered.