Jung's theorem

Statement

Consider a compact set

:K \subset \mathbb{R}^n

and let

:d = \max_{p,q,\in, K} | p - q |_2

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

:r \leq d \sqrt{\frac{n}{2(n+1)}}

that contains K. The boundary case of equality is attained by the regular n-simplex.

Jung's theorem in the plane

The most common case of Jung's theorem is in the plane, that is, when n = 2. In this case the theorem states that there exists a circle enclosing all points whose radius satisfies

:r \leq \frac{d}{\sqrt{3}},

and this bound is as tight as possible since when K is an equilateral triangle (or its three vertices) one has r = \frac{d}{\sqrt{3}}.

General metric spaces

For any bounded set S in any metric space, d/2\le r\le d. The first inequality is implied by the triangle inequality for the center of the ball and the two diametral points, and the second inequality follows since a ball of radius d centered at any point of S will contain all of S. Both these inequalities are tight:

• In a uniform metric space, that is, a space in which all distances are equal, r=d.
• At the other end of the spectrum, in an injective metric space such as the Manhattan distance in the plane, r=d/2: any two closed balls of radius d/2 centered at points of S have a non-empty intersection, therefore all such balls have a common intersection, and a radius d/2 ball centered at a point of this intersection contains all of S.

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

References

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