On parallelepipeds and centrally symmetric hexagonal prisms circumscribed about a three-dimensional centrally symmetric convex body

Let $K$ be a three-dimensional centrally symmetric compact convex set of unit volume. It is proved that $K$ is contained in a centrally symmetric hexagonal prism or a parallelepiped with volume $4/\root3\of3<2.7735$. This fact implies that $K$ admits a lattice packing in space with density at least $\root3\of3/4>0.3605$. Furthermore, $K$ is contained in a parallelepiped with volume $4(3+6/(\sqrt3(1+\operatorname{ctg}(\pi/12))))^{2/3}/3<3.2082$. Bibl. – 6 titles.