A radius sphere theorem
There is a generalization of the diameter sphere theorem due to \textit{K. Grove} and \textit{K. Shiohama} [Ann. Math. (2) 106, 201--211 (1977; Zbl 0357.53027)] to the case of Alexandrov spaces due to Perelman: An \(n\)-dimensional Alexandrov space with curvature \(\geq 1\) and diameter \(> \pi/2\) is a suspension. The main result of this paper is the following sphere theorem: An \(n\)-dimensional Alexandrov space \(M\) with curvature \(\geq 1\) and radius \(>\pi/2\) is homeomorphic to the \(n\)-sphere. Here the radius is given by \(\min_{p \in M} \max_{q \in M}\text{dist}(p,q)\).
scientific article; zbMATH DE number 549673