On Approximation of a Three-dimensional Convex Body by Cylinders

New results on approximation of a convex body K ⊂ R3 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 ⊂ R3). Theorem 1. Let K be an arbitrary convex body in R3. There exists a regular octagonal prism an affine image of which is circumscribed about K and has volume at most 3 √ 2V (K), and there exists a circular cylinder an affine image of which is circumscribed about K and has volume at most 3π 2 V (K). For a tetrahedron K both estimates are the best possible. Theorem 2. Let K be a centrally symmetric convex body in R3. There exists a regular octagonal prism, an affine image of which lies in K and has volume at least 4 9 (2 √ 2− 2)V (K). In what follows, by a convex body K ⊂ R (a figure K ⊂ R) we mean a compact convex set with nonempty interior; by S(K) and V (K) we mean the area of K ⊂ R and the volume of K ⊂ R, respectively. By Gk(R) (respectively, G+k (R )), we mean the Grassmann manifold of k-planes (respectively, of oriented k-planes) in R passing through 0 ∈ R. Let γ k : Ek(R ) → Gk(R) (respectively, (γ k ) : E k (R ) → G+k (R )) be the tautological bundle over the Grassmann manifold, in which the fiber over a plane is the same plane regarded as a k-dimensional subspace of R. As usual, let Vk(R) be the Stiefel manifold of orthonormal k-frames in R. We say that a polyhedron M is inscribed in a convex body K ⊂ R if the vertices of M belong to the boundary of K. A polyhedron M is circumscribed about K if K ⊂ M and K intersects all faces of M . §1. A lemma on approximation of a convex body by a cylinder In the sequel, we need the following simple (and, apparently, well-known) statement. Lemma 1. For every convex body K ⊂ R and a line l in R, there exists a convex cylinder C of volume V (C) ≤ nV (K) that contains K, has ruling parallel to l, and is such that the line connecting two points where K meets the bases of C is parallel to l. Proof. Consider the union C ′ of lines that are parallel to l and intersect K. By construction, C ′ is an infinite convex cylinder. Let AB be the longest chord AB of K parallel to l. 2000 Mathematics Subject Classification. Primary 52B10.