By Theorem 6.1, we have

$$Vol_{h_{1}}(B_{h_{1}}(r))\leq Vol_{g}(B_{g}(p,r)\cap L)\leq Vol_{g}(L)=\int_{L}\Theta,$$

for any $r\leq\min\{i_{g}(p),\frac{\pi}{2}\}$, where $h_{1}$ denotes the standard metric on $S^{n}$ with constant curvature 1, and $B_{h_{1}}(r)$ denotes a metric $r$-ball in $S^{n}$. Since $h_{1}=dr^{2}+\sin^{2}rh_{S^{n-1}}$ where $h_{S^{n-1}}$ is the standard metric on $S^{n-1}$ with constant curvature 1, we obtain $\sin r\geq\frac{2}{\pi}r$, and

$$\frac{2^{n-1}}{n\pi^{n-1}}r^{n}\varpi_{n-1}\leq\int_{0}^{r}\sin^{n-1}rdr\varpi_{n-1}=Vol_{h_{1}}(B_{h_{1}}(r))\leq\int_{L}\Theta.$$

If $i_{g}(p)\geq\frac{\pi}{2}$, by letting $r=\frac{\pi}{2}$, we obtain

$$\frac{\pi}{2n}\varpi_{n-1}\leq\int_{L}\Theta<\frac{\pi}{2n}\varpi_{n-1},$$

which is a contradiction. Thus $i_{g}(p)<\frac{\pi}{2}$. By letting $r=i_{g}(p)$, we obtain

$$i_{g}(p)^{n}\leq\frac{n\pi^{n-1}}{2^{n-1}\varpi_{n-1}}\int_{L}\Theta.$$

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