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Difference between revisions of "Jung theorem"

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Every set of diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054430/j0544301.png" /> in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054430/j0544302.png" /> is contained in a ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054430/j0544303.png" /> of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054430/j0544304.png" />. There are analogues and generalizations of Jung's theorem (e.g. replacing the Euclidean distance by other metrics) (cf. [[#References|[2]]]).
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Every set of diameter $d$ in a Euclidean space $E_n$ is contained in a ball in $E_n$ of radius $r=d(n/2(n+1))^{1/2}$. There are analogues and generalizations of Jung's theorem (e.g. replacing the Euclidean distance by other metrics) (cf. [[#References|[2]]]).
  
 
The theorem was proved by H.W.E. Jung [[#References|[1]]].
 
The theorem was proved by H.W.E. Jung [[#References|[1]]].

Latest revision as of 19:34, 29 April 2014

Every set of diameter $d$ in a Euclidean space $E_n$ is contained in a ball in $E_n$ of radius $r=d(n/2(n+1))^{1/2}$. There are analogues and generalizations of Jung's theorem (e.g. replacing the Euclidean distance by other metrics) (cf. [2]).

The theorem was proved by H.W.E. Jung [1].

References

[1] H.W.E. Jung, "Ueber den kleinsten Kreis, der eine ebene Figur einschliesst" J. Reine Angew. Math. , 130 (1901) pp. 310–313
[2] L. Danzer, B. Grünbaum, V.L. Klee, "Helly's theorem and its relatives" V.L. Klee (ed.) , Convexity , Proc. Symp. Pure Math. , 7 , Amer. Math. Soc. (1963) pp. 101–180
[3] H. Hadwiger, H. Debrunner, "Combinatorial geometry in the plane" , Holt, Rinehart & Winston (1964) (Translated from German)


Comments

The counterpart for inscribed balls is Steinhagen's theorem (cf. [a1]).

References

[a1] T. Bonnesen, W. Fenchel, "Theorie der konvexen Körper" , Springer (1974)
How to Cite This Entry:
Jung theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jung_theorem&oldid=18343
This article was adapted from an original article by P.S. Soltan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article