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The Jung Theorem in metric spaces of curvature bounded above

Authors:	B V Dekster

Institution:	Department of Mathematics and Computer Science, Mount Allison University, Sack- ville, New Brunswick, Canada E0A 3C0

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

Keywords:	Jung Theorem metric spaces of curvature $\leq K$

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