On approximation of a three-dimensional convex body by cylinders

Abstract:

New results on approximation of a convex body $K\subset \mathbb {R}^3$ by affine images of circular cylinders, parallelepipeds, hexagonal and octagonal regular (and some other) prisms are obtained.

Two of the theorems obtained are as follows ($V(K)$ denotes the volume of a body $K\subset \mathbb {R}^3$).

Theorem 1. Let $K$ be an arbitrary convex body in $\mathbb {R}^3$. There exists a regular octagonal prism an affine image of which is circumscribed about $K$ and has volume at most $3\sqrt {2}V(K)$, and there exists a circular cylinder an affine image of which is circumscribed about $K$ and has volume at most $\frac {3\pi }{2}V(K)$. For a tetrahedron $K$ both estimates are the best possible.

Theorem 2. Let $K$ be a centrally symmetric convex body in $\mathbb {R}^3$. There exists a regular octagonal prism, an affine image of which lies in $K$ and has volume at least $\frac {4}{9}(2\sqrt {2}-2)V(K)$.