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 Di, 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.