1. Let the function $y=\left|\log _{2} \frac{x}{2}\right|$ have the domain $[m, n]$, and the range $[0,2]$. Then the minimum length of the interval $[m, n]$ is $\qquad$
$-1 . \frac{3}{2}$.
Notice, $y=\left|\log _{2} \frac{x}{2}\right|=\left|\log _{2} x-1\right|$.
When $y=0$, $\log _{2} x=1 \Rightarrow x=2$;
When $y=2$, $\log _{2} x=3$ or $1 \Rightarrow x=8$ or $\frac{1}{2}$.
Since the function is monotonically decreasing in the interval $(0,2]$ and monotonically increasing in the interval $[2,+\infty)$, we have
$$
\left[\frac{1}{2}, 2\right] \subseteq[m, n] \text { or }[2,8] \subseteq[m, n] \text {. }
$$

In summary, the answer is $\frac{3}{2}$.