Affine images of the rhombo-dodecahedron that are circumscribed about a three-dimensional convex body

Abstract

The main result of the paper is dual to the author's earlier theorem on the affine images of the cube-octahedron (the convex hull of the midpoints of edges of a cube) inscribed in a three-dimensional convex body. The rhombododecahedron is the polyhedron dual to the cube-octahedron. Let us call a convex body in Κ⊂ℝ3 exceptional if it contains a parallelogram P and is contained in a cylinder with directrix P. It is proved that any nonexceptional convex body is inscribed in an affine image of the rhombo-dodecahedron. The author does not know if the assertion is true for all three-dimensional convex bodies. Bibliography: 2 titles.