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

This is the authors' abstract. We don't add key points for this paper.

Two related problems concerning continuous functions on a sphere Sⁿ⁻¹⊂ Rⁿ are studied, together with the problem of finding a family of polyhedra in Rⁿ one of which is inscribed in (respectively, circumscribed about) a given smooth convex body in Rⁿ. In particular, it is proved that, in every convex body K⊂ R³, one can inscribe an eight-vertex polyhedron obtained by “equiaugmentation” of a similarity image of any given tetrahedron of class T.