On inscribed and circumscribed polyhedra for a centrally symmetric convex body

We construct new polyhedra with the property that some of their similar or affine images can be inscribed in (or circumscribed about) every centrally symmetric convex body. One of the theorems is as follows. If a three-dimensional body $K$ is centrally symmetric and convex, then either an affine image of the regular dodecahedron is inscribed in $K$, or there are two affine images $D_1$ and $D_2$ of the regular dodecahedron such that nine pairs of opposite vertices of $D_i$, $i=1,2$, lie on the boundary of $K$. Furthermore, the two remaining vertices of $D_1$ lie outside $K$, while the two remaining vertices of $D_2$ lie inside $K$.