257. Given a regular tetrahedron with edge $a$. Find the radius of a sphere centered at the center of the tetrahedron, such that the combined volume of the part of the tetrahedron outside the sphere and the part of the sphere outside the tetrahedron is minimized.
257. Let $x$ be the radius of the sphere, $V(x)$ be the sum of the volume of the part of the sphere located outside the tetrahedron and the part of the tetrahedron located outside the sphere. It is easy to see that $V^{\prime \prime}(x)=$ $=S_{1}(x)-S_{2}(x)$, where $S_{1}(x)$ is the surface area of the part of the sphere located outside the tetrahedron, $S_{2}(x)$ is the surface area of the part of the sphere located inside the tetrahedron. The minimum is achieved when $S_{1}(x)=S_{2}(x)$, from which $\cdot x=a \frac{1}{3} \sqrt{\frac{2}{3}}$.