A Radius Sphere Theorem

The purpose of this paper is to present an optimal sphere theorem for metric spaces analogous to the celebrated Rauch-Berger-Klingenberg Sphere Theorem and the Diameter Sphere Theorem in riemannian geometry. There has lately been considerable interest in studying spaces which are more singular than riemannian manifolds. A natural reason for doing this is because Gromov-Hausdorff limits of riemannian manifolds are almost never riemannian manifolds, but usually only inner metric spaces with various nice properties. The kind of spaces we wish to study here are the so-called Alexandrov spaces. Alexandrov spaces are finite dimensional inner metric spaces with a lower curvature bound in the distance comparison sense. This definition might seem a little ambiguous since there are many ways in which one can define finite dimensionality and lower curvature bounds. The foundational work by Plaut in [PI], however, shows that these different possibilities for definitions are equivalent. The structure of Alexandrov spaces was studied in [BGP], [PI] and [P]. In particular if X is an Alexandrov space and p e X then the space of directions Ep at p is an Alexandrov space of one less dimension and with curvature > 1. Furthermore a neighborhood of p in X is homeomorphic to the linear cone over £„ . One of the important implications of this is that the local structure of n-dimensional Alexandrov spaces is determined by the structure of (n-l)-dimensional Alexandrov spaces with curvature > 1. Sphere theorems in this context seem to be particularly interesting. For if one can give geometric characterizations of