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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Jung Theorem in metric spaces
of curvature bounded above


Author: B. V. Dekster
Journal: Proc. Amer. Math. Soc. 125 (1997), 2425-2433
MSC (1991): Primary 52A40, 53C20
MathSciNet review: 1389515
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Abstract | References | Similar Articles | Additional Information

Abstract: The classical Jung Theorem states in essence that the diameter $D$ of a compact set $X$ in $E^n$ satisfies $D \geq R[(2n+2)/n]^{1/2}$ where $R$ is the circumradius of $X$. The theorem was extended recently to the hyperbolic and the spherical $n$-spaces. Here, the estimate above is extended to a class of metric spaces of curvature $\leq K$ introduced by A. D. Alexandrov. The class includes the Riemannian spaces. The extended estimate is of the form $D \geq f(R,K,n)$ where $n$ is a positive integer suitably defined for the set $X$ and its circumcenter. It can be that $n$ is not unique or does not exist. In the latter case, no estimate is derived. In case of a Riemannian $d$-dimensional space, an integer $n$ always exists and satisfies $n \leq d$. Then $D \geq f(R,K,n) \geq f(R,K,d)$. In case of $E^d$, one has $D \geq R[(2n+2)/n]^{1/2} \geq R[(2d+2)/d]^{1/2}$.


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Additional Information

B. V. Dekster
Affiliation: Department of Mathematics and Computer Science, Mount Allison University, Sack- ville, New Brunswick, Canada E0A 3C0
Email: bdekster@mta.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03842-2
PII: S 0002-9939(97)03842-2
Keywords: Jung Theorem, metric spaces of curvature $\leq K$
Additional Notes: Supported by a Canadian NSERC grant
Communicated by: Christopher Croke
Article copyright: © Copyright 1997 American Mathematical Society