10. If a convex $n$-sided polygon has exactly 4 obtuse interior angles, then the maximum number of sides $n$ of this polygon is $\qquad$ .
10.7.

Since a convex $n$-sided polygon has exactly 4 obtuse interior angles, the sum of these 4 angles is greater than $360^{\circ}$ and less than $720^{\circ}$. The other $n-4$ angles are right angles or acute angles, so the sum of these $n-4$ angles is no more than $(n-4) \times 90^{\circ}$ and greater than $0^{\circ}$. Therefore, we have the inequality
$$
360^{\circ}+0^{\circ}<(n-2) \times 180^{\circ}<720^{\circ}+(n-4) \times 90^{\circ} \text {. }
$$

Solving this, we get $4<n<8$. Therefore, the number of sides $n$ is at most 7.