10. Given the set $\Omega=\left\{(x, y) \mid x^{2}+y^{2} \leqslant 2008\right\}$. If points $P(x, y)$ and $P^{\prime}\left(x^{\prime}, y^{\prime}\right)$ satisfy $x \leqslant x^{\prime}$ and $y \geqslant y^{\prime}$, then point $P$ is said to dominate $P^{\prime}$. If a point $Q$ in the set $\Omega$ satisfies: there does not exist any other point in $\Omega$ that dominates $Q$, then the set of all such points $Q$ is
10. $\left\{(x, y) \mid x^{2}+y^{2}=2008, x \leqslant 0\right.$ and $\left.y \geqslant 0\right\}$.
$P$ is better than $P^{\prime}$, meaning $P$ is located to the left and above $P^{\prime}$. “There does not exist any other point in $\Omega$ that is better than $Q$”, means “there is no point in $\Omega$ to the left and above $Q$”. Therefore, the set of points that satisfy the condition is $\left\{(x, y) \mid x^{2}+y^{2}=2008, x \leqslant 0\right.$ and $\left.y \geqslant 0\right\}$.