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

Let K ⊂ ℝ3 be a convex body of unit volume. It is proved that K contains an affine-regular pentagonal prism of volume 4(5-2√5)/9 (which is greater than 0.2346) and an affine-regular pentagonal antiprism of volume 4(3√5-5)/27(which is greater than 0.253). Furthermore, K is contained in an affine-regular pentagonal prism of volume 6(3-√5) (which is less than 4.5836), and in an affine-regular heptagonal prism of volume 21(2 cos π/7 - 1)/4 (which is less than 4.2102). If K is a tetrahedron, then the latter estimate is sharp. Bibliography: 8 titles. © 2009 Springer Science+Business Media, Inc.