The Radius Rigidity Theorem for Manifolds of Positive Curvature

Recall that the radius of a compact metric space $(X, dist)$ is given by $rad\ X = \min_{x\in X} \max_{y\in X} dist(x,y)$. In this paper we generalize Berger's $\frac{1}{4}$-pinched rigidity theorem and show that a closed, simply connected, Riemannian manifold with sectional curvature $\geq 1$ and radius $\geq \frac{\pi}{2}$ is either homeomorphic to the sphere or isometric to a compact rank one symmetric space.