On inscribing a regular octahedron in a three-dimensional convex body with smooth boundary

A norm $\|\cdot\|$ and a convex body $K$ with smooth boundary in the standard Euclidean space $\mathbb R^3$ are considered. It is proved that the boundary $\partial K$ of $K$ contains the vertices $AA'BB'CC'$ of a regular octahedron with $\|AA'\|=\|BB'\|\ge\|CC'\|$ (respectively, $\|AA'\|=\|BB'\|\le\|CC'\|$).