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

:$Ksubset\; mathbb\{R\}^n$

and let

:$d\; =\; max\; olimits\_\{p,qin\; K\}\; |\; p\; -\; q\; |\_2$

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

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

:$r\; leq\; frac\{d\}\{sqrt\{3.$

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

:$r\; =\; frac\{d\}\{sqrt\{3.$

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).

References

*Katz, M.: Jung's theorem in complex projective geometry, Quart. J. Math. Oxford (2) 36 (1985) 451-466.

*cite journal
author = Dekster, B. V.
title = The Jung theorem for the spherical and hyperbolic spaces
journal = Acta Math. Sci. Hungar.
volume = 67
issue = 4
year = 1995
pages = 315–331
doi = 10.1007/BF01874495

*cite journal
author = Dekster, B. V.
title = The Jung theorem in metric spaces of curvature bounded above
journal = Proceedings of the American Mathematical Society
volume = 125
issue = 8
year = 1997
pages = 2425–2433
doi = 10.1090/S0002-9939-97-03842-2

*cite journal
author = Jung, Heinrich
title = Über die kleinste Kugel, die eine räumliche Figur einschließt
journal = J. Reine Angew. Math.
volume = 123
year = 1901
pages = 241–257
format = in German

*cite journal
author = Jung, Heinrich
title = Über den kleinsten Kreis, der eine ebene Figur einschließt
journal = J. Reine Angew. Math.
volume = 137
year = 1910
pages = 310–313
format = in German

*cite book
author = Rademacher, Hans; Toeplitz, Otto
title = The Enjoyment of Mathematics
year = 1990
publisher = Dover
id = ISBN 978-0-486-26242-0
pages = chapter 16

External links

*mathworld | title = Jung's Theorem | urlname = JungsTheorem