A sphere theorem for reverse volume pinching on even-dimensional manifolds

Let $M$ be a compact simply connected riemannian manifold of even dimension $d$. It is well known that if the sectional curvature of $M$ lies in the range $\left ( {0,\lambda } \right ]$, then $M$ has volume greater than or equal to that of the $d$-dimensional euclidean sphere $S_\lambda ^d$ of constant curvature $\lambda$. We prove that if the volume of $M$ is no greater than 3/2 times that of $S_\lambda ^d$, then $M$ is homeomorphic with the sphere.