The Jung Theorem in metric spaces of curvature bounded above

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 $D \geq R[(2n+2)/n]^{1/2} \geq R[(2d+2)/d]^{1/2}$.