10. In the unit cube $A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, place an inscribed sphere $O$. Then place a small sphere at each of the eight corners of the cube, so that they are all externally tangent to the sphere, and each is tangent to three faces of the cube. The radius of the small spheres is $\qquad$.
10. $\frac{1}{2}(2 \sqrt{3}-3)$ As shown in the figure, by symmetry, the eight small balls are all of equal size.

The diagonal plane $A C C^{\prime} A^{\prime}$ of the cube passes through the centers of five balls, 10 tangent points, and the diagonal of the cube. Using homothety, we know that points $A, O_{1}, O, O_{2}, C^{\prime}$ are collinear, and $\angle M O A=\angle C C^{\prime} A$.
Let the radius of the small balls be $x$, then $O O_{1} \cdot \cos \angle M O A=O M-O_{1} N$,
i.e., $\left(\frac{1}{2}+x\right) \cdot \frac{1}{\sqrt{3}}=\frac{1}{2}-x$.
Solving for $x$ gives $x=\frac{1}{2}(2 \sqrt{3}-3)$.