Berger-Kazdan Comparison Theorem

Let $M$ be a compact $n$-D Manifold with Injectivity radius $\mathop{\rm inj}(M)$. Then

$$\mathop{\rm Vol}(M)\geq {c_n\mathop{\rm inj}(M)\over\pi},$$

with equality Iff $M$ is Isometric to the standard round Sphere $S^n$ with Radius $\mathop{\rm inj}(M)$, where $c_n(r)$ is the Volume of the standard $n$-Hypersphere of Radius $r$.

See also Blaschke Conjecture, Hypersphere, Injective, Isometry

References

Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994.