Inscribed and Circumscribed Polyhedra for a Centrally Symmetric Convex Body

Abstract

We construct new polyhedra such that some of their similar or affine images can be inscribed in (or circumscribed about) every centrally symmetric convex body.

One of the main theorems is as follows. If K is a centrally symmetric, three-dimensional, convex body, then either an affine-regular dodecahedron is inscribed in K or there are two affine-regular dodecahedra D₁ and D₂ such that nine pairs of opposite vertices of D_i, i = 1, 2, lie on the boundary of K. Furthermore, the remaining two vertices of D₁ lie outside K, while the remaining two vertices of D₂ lie inside K.