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.

Contents

Statement

Jung's theorem in the plane

General metric spaces

References

External links

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
:
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 the radius of a ball containing all points satisfies
:$r\; leq\; d/sqrt\{3\}.$
No tighter bound on ''r'' can be shown: when ''S'' is an equilateral triangle (or its three vertices),
:$r\; =\; d/sqrt\{3\}$
holds.

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

★ The Jung theorem for the spherical and hyperbolic spaces, Dekster, B. V., , , Acta Math. Sci. Hungar., 1995

★ The Jung theorem in metric spaces of curvature bounded above, Dekster, B. V., , , Proceedings of the American Mathematical Society, 1997

★ Über die kleinste Kugel, die eine räumliche Figur einschließt, Jung, Heinrich, , , J. Reine Angew. Math., 1901

★ Über den kleinsten Kreis, der eine ebene Figur einschließt, Jung, Heinrich, , , J. Reine Angew. Math., 1910

★ The Enjoyment of Mathematics, Rademacher, Hans; Toeplitz, Otto, , , Dover, 1990, ISBN 978-0-486-26242-0

External links

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