Inscribed and circumscribed polyhedra for a convex body and continuous functions on a sphere in Euclidean space

Two related problems concerning continuous functions on a sphere $S^{n-1} \subset {\mathbb{R}}^n$ are studied, together with the problem of finding a family of polyhedra in ${\mathbb{R}}^n$ one of which is inscribed in (respectively, circumscribed about) a given smooth convex body in ${\mathbb{R}}^n$. In particular, it is proved that, in every convex body $K\subset{\mathbb{R}}^3$, one can inscribe an eight-vertex polyhedron obtained by ``equiaugmentation'' of a similarity image of any given tetrahedron of class $T$.