Jung's theorem

Consider a compact set

${\displaystyle K\subset \mathbb {R} ^{n}}$

and let

${\displaystyle d=\max _{p,q\,\in \,K}\|p-q\|_{2}}$

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

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

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

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

${\displaystyle r\leq {\frac {d}{\sqrt {3}}},}$

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

For any bounded set ${\displaystyle S}$ in any metric space, ${\displaystyle d/2\leq r\leq d}$. 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 ${\displaystyle d}$ centered at any point of ${\displaystyle S}$ will contain all of ${\displaystyle S}$. Both these inequalities are tight:

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

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

• Katz, M. (1985). "Jung's theorem in complex projective geometry". Quart. J. Math. Oxford. 36 (4): 451–466. doi:10.1093/qmath/36.4.451.
• Dekster, B. V. (1995). "The Jung theorem for the spherical and hyperbolic spaces". Acta Mathematica Hungarica. 67 (4): 315–331. doi:10.1007/BF01874495.
• Dekster, B. V. (1997). "The Jung theorem in metric spaces of curvature bounded above". Proceedings of the American Mathematical Society. 125 (8): 2425–2433. doi:10.1090/S0002-9939-97-03842-2.
• Jung, Heinrich (1901). "Über die kleinste Kugel, die eine räumliche Figur einschließt". J. Reine Angew. Math. (in German). 123: 241–257.
• Jung, Heinrich (1910). "Über den kleinsten Kreis, der eine ebene Figur einschließt". J. Reine Angew. Math. (in German). 137: 310–313.
• Rademacher, Hans; Toeplitz, Otto (1990). The Enjoyment of Mathematics. Dover. chapter 16. ISBN 978-0-486-26242-0.

• Weisstein, Eric W. "Jung's Theorem". MathWorld.