An isoembolic pinching theorem (Q1105205)

For \(\alpha\) (n) the volume of the n-sphere, Berger's isoembolic inequality states \(V/i^ n\geq \alpha (n)/\pi^ n\), where V denotes the volume, i the injectivity radius of a compact Riemannian n-manifold M. Moreover, equality holds if M is a round sphere. The author shows for an explicit constant c(n) that \[ \alpha (n)/\pi^ n+c(n)\geq V/i^ n \] implies that M is homeomorphic to \(S^ n\).