Approximation of three-dimensional convex bodies by affine-regular prisms

Let $K\subset\mathbb R^3$ be a convex body of unit volume. It is proved that $K$ contains an affine-regular pentagonal prism of volume $4(5-2\sqrt5)/9>0.2346$ and an affine-regular pentagonal antiprism of volume $4(3\sqrt5-5)/27>0.253$. Furthermore, $K$ is contained in an affine-regular pentagonal prism of volume $6(3-\sqrt5)<4.5836$, and in an affine-regular heptagonal prism of volume $21(2\cos\pi/7-1)/4<4.2102$. If $K$ is a tetrahedron, then the latter estimate is sharp. Bibl. – 8 titles.