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For a compact [[Riemannian manifold|Riemannian manifold]] $M = M ^ { n }$, let

\begin{equation*} \operatorname {inj} M = \operatorname { inf } _ { p \in M } \operatorname { sup } \{ r : \operatorname { exp } _ { p } \text { injective on } B _ { r } ( 0 ) \subset T _ { p } M \}, \end{equation*}

where $B _ { r } ( 0 )$ is the ball around $0$ with radius $r$, be the injectivity radius, and set $\alpha ( n ) = \text { Vol } ( S ^ { n } )$. Then the inequality

\begin{equation*} \operatorname {Vol} ( M ) \geq \alpha ( n ) \left( \frac { \operatorname { inj } M } { \pi } \right) ^ { n } \end{equation*}

holds, with equality if and only if $M$ is isometric to the standard sphere with diameter $\operatorname { inj} M$.

This inequality relies on the [[Kazdan inequality|Kazdan inequality]] applied to the Jacobi equation $X ^ { \prime \prime } ( t ) + {\cal {R}} ( t ) \circ X ( t ) = 0$ for operators $X ( t )$ on $v ^ { \perp }$ for a unit vector $v \in T _ { p } M$. Here, $R ( t ) = R ( \gamma ^ { \prime } ( t ) , . ) \gamma ^ { \prime } ( t )$ is the [[Curvature|curvature]] operator, $\tau _ { t , v } : T _ { p } M \rightarrow T _ { \gamma ( t ) } M$ is the parallel transport along the geodesic ray $\gamma ( t ) = \operatorname { exp } _ { p } ( t v )$, and $\mathcal{R} ( t ) = \tau ^ { - 1 _ { t , v } } \circ R ( t ) \circ \tau _ { t , v }$ is the parallel translated curvature operator on $v ^ { \perp } \subset T _ { p } M$.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  M. Berger,   "Une borne inférieure pour le volume d'une variété riemannienes en fonction du rayon d'injectivité"  ''Ann. Inst. Fourier (Grenoble)'' , '''30'''  (1980)  pp. 259–265</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  I. Chavel,   "Riemannian geometry: A modern introduction" , Cambridge Univ. Press  (1995)</td></tr></table>