10. Let $S=x^{2}+y^{2}-2(x+y)$, where 1, $y$ satisfy $\log _{2} x+\log _{2} y=1$. Then the minimum value of $S$ is $\qquad$ .
$$
10.4-4 \sqrt{2}
$$

From $\log _{2} x+\log _{2} y=1$, we get $x y=2$.
$$
\begin{array}{l}
\text { Also } S=x^{2}+y^{2}-2(x+y) \\
=(x+y)^{2}-2(x+y)-2 x y \\
=(x+y)^{2}-2(x+y)-4 \\
=[(x+y)-1]^{2}-5 \geqslant[2 \sqrt{x y}-1]^{2}-5 \\
=(2 \sqrt{2}-1)^{2}-5=4-4 \sqrt{2} .
\end{array}
$$