11. (20 points) The volume of a regular $n$-sided pyramid is $V$. Find a necessary and sufficient condition independent of $n$ for the surface area of the regular $n$-sided pyramid to be minimized.
11. Let the radius of the base of a regular $n$-sided pyramid be $r$, and the angle between the side and the base be $\theta\left(0<\theta<\frac{\pi}{2}\right)$.
Then the height of the pyramid is
$$
h=r \cos \frac{\pi}{n} \cdot \tan \theta,
$$

The area of the base, a regular $n$-sided polygon, is
$$
\begin{array}{l}
n r^{2} \sin \frac{\pi}{n} \cdot \cos \frac{\pi}{n} . \\
\text { Also, } V=\frac{n r^{2}}{3} \sin \frac{\pi}{n} \cdot \cos \frac{\pi}{n} \cdot r \cos \frac{\pi}{n} \cdot \tan \theta \\
\Rightarrow r=\left(\frac{3 V}{n \sin \frac{\pi}{n} \cdot \cos ^{2} \frac{\pi}{n} \cdot \tan \theta}\right)^{\frac{1}{3}},
\end{array}
$$

Therefore, the surface area of the regular $n$-sided pyramid is
$$
\begin{array}{l} 
S=n r^{2} \sin \frac{\pi}{n} \cdot \cos \frac{\pi}{n}\left(1+\frac{1}{\cos \theta}\right) \\
=n \sqrt[3]{\frac{9 V^{2} \tan \frac{\pi}{n}}{n^{2}}} \cdot \sqrt[3]{\frac{(1+\cos \theta)^{3}}{\cos \theta \cdot \sin ^{2} \theta}} \\
=\sqrt[3]{9 n V^{2} \tan \frac{\pi}{n}} \cdot \sqrt[3]{\frac{(1+\cos \theta)^{2}}{\cos \theta(1-\cos \theta)}} . \\
\text { and } \\
\geqslant \sqrt[3]{\frac{(1+\cos \theta)^{2}}{\cos \theta(1-\cos \theta)}}=\sqrt[3]{\frac{2(1+\cos \theta)^{2}}{2 \cos \theta(1-\cos \theta)}} \\
\sqrt[3]{\left(\frac{1+\cos \theta)^{2}}{2}\right.}=2,
\end{array}
$$

where the equality holds if and only if $2 \cos \theta=1-\cos \theta$, i.e., $\cos \theta=\frac{1}{3}$.