11. Let $x_{1}, x_{2}, x_{3}, x_{4}$ be positive real numbers, and $x_{1}+x_{2}+x_{3}+x_{4}=\pi$. Find the minimum value of the expression $\left(2 \sin ^{2} x_{1}+\frac{1}{\sin ^{2} x_{1}}\right)\left(2 \sin ^{2} x_{2}+\frac{1}{\sin ^{2} x_{2}}\right)\left(2 \sin ^{2} x_{3}+\frac{1}{\sin ^{2} x_{3}}\right)\left(2 \sin ^{2} x_{4}+\frac{1}{\sin ^{2} x_{4}}\right)$. (1991 Chinese National Training Team Exam)
I1. Let $x_{1}+x_{2}$ be a constant. Since
$$\sin x_{1} \sin x_{2}=\frac{1}{2}\left[\cos \left(x_{1}-x_{2}\right)-\cos \left(x_{1}+x_{2}\right)\right]$$

we know that the value of $\sin x_{1} \sin x_{2}$ increases as $\left|x_{1}-x_{2}\right|$ decreases. Denote the given expression as $f\left(-x_{1}, x_{2}, x_{3}, x_{4}\right)$. If they are not all equal, assume without loss of generality that $x_{1}>\frac{\pi}{4}>x_{2}$.

Let $x_{1}^{\prime}=\frac{\pi}{4}, x_{2}^{\prime}=x_{1}+x_{2}-\frac{\pi}{4}, x_{3}^{\prime}=x_{3}, x_{4}^{\prime}=x_{4}$, then we have $x_{1}^{\prime}+x^{\prime}{ }_{2}=x_{1}+x_{2}$, $x_{1}^{\prime}-x_{2}^{\prime} 1 \\
2\left(\sin ^{2} x_{1}^{\prime} \sin ^{2} x_{2}^{\prime}+\frac{1}{2 \sin ^{2} x_{1}^{\prime} \sin ^{2} x_{2}^{\prime}}\right)+2\left(\frac{\sin ^{2} x_{2}^{\prime}}{\sin ^{2} x_{1}^{\prime}}+\frac{\sin ^{2} x_{1}^{\prime}}{\sin ^{2} x_{2}^{\prime}}\right)= \\
\left(2 \sin ^{2} x_{1}^{\prime}+\frac{1}{\sin ^{2} x_{1}^{\prime}}\right)\left(2 \sin ^{2} x_{2}^{\prime}+\frac{1}{\sin ^{2} x_{2}^{\prime}}\right)
\end{array}$$

Thus, we have
$$f\left(x_{1}, x_{2}, x_{3}, x_{4}\right)>f\left(x_{1}^{\prime}, x_{2}^{\prime}, x_{3}^{\prime}, x^{\prime}{ }_{4}\right)$$

If $x_{2}^{\prime}, x_{3}^{\prime}, x_{4}$ are not all equal, then we can perform a similar smoothing transformation to prove that: when $x_{1}, x_{2}, x_{3}, x_{4}$ are not all equal, we always have
$$f\left(x_{1}, x_{2}, x_{3}, x_{4}\right)>f\left(\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{4}\right)$$

Therefore, the minimum value sought is $f\left(\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{4}\right)=81$, which is achieved if and only if $x_{1}=x_{2}=x_{3}=x_{4}=\frac{\pi}{4}$.