Jung's theorem

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In geometry, Jung's theorem is an inequality between the diameter and the radius of the minimum enclosing ball of a set of points in any Euclidean space. It is named after Heinrich Jung, who first studied this inequality in 1901.

1 Statement
2 Jung's theorem in the plane
3 General metric spaces
4 References
5 External links

[edit] Statement

Consider a compact set

<math>K\subset \mathbb{R}^n</math>

and let

<math>d = \max\nolimits_{p,q\in K} \| p - q \|_2</math>

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.

[edit] Jung's theorem in the plane

Most common is the case of Jung's theorem in the plane, that is <math>n = 2</math>. In this case the theorem states that the radius of a ball containing all points satisfies

<math>r \leq d/\sqrt{3}</math>.

No tighter bound on r can be shown: when S is an equilateral triangle (or its three vertices),

holds.

[edit] General metric spaces

For any bounded set S in any metric space, d/2 ≤ r ≤ 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. 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 nonempty 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).

[edit] References

Dekster, B. V. (1995). "The Jung theorem for the spherical and hyperbolic spaces". Acta Math. Sci. Hungar. 67 (4): 315–331.

Dekster, B. V. (1997). "The Jung theorem in metric spaces of curvature bounded above". Proceedings of the American Mathematical Society 125 (8): 2425–2433.