On the Smallest Enclosing Balls

Abstract

In the paper a theoretical analysis is given for the smallest ball that covers a finite number of points $p_1, p_2, \cdots, p_N \in \Bbb R^n$. 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.