Jung's theorem

In geometry, Jung's theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set. It is named after Heinrich Jung, who first studied this inequality in 1901.
Statement
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.
Jung's theorem in the plane
Most common is the case of Jung's theorem in the plane, that is n = 2. In this case the theorem states that there exists a circle enclosing all points whose radius satisfies
No tighter bound on r can be shown: when S is an equilateral triangle (or its...