Jung's theorem

Algorithms also exist to solve the smallest-circle problem explicitly.

Consider a compact set and let be the diameter of K, that is, the largest Euclidean distance between any two of its points.

Jung's theorem states that there exists a closed ball with radius that contains K. The boundary case of equality is attained by the regular n-simplex.

In this case the theorem states that there exists a circle enclosing all points whose radius satisfies and this bound is as tight as possible since when K is an equilateral triangle (or its three vertices) one has

Both these inequalities are tight: Versions of Jung's theorem for various non-Euclidean geometries are also known (see e.g. Dekster 1995, 1997).