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 $T$.