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

Abstract

Let K ⊂ ℝ3 be a convex body of unit volume. It is proved that K contains an affine-regular pentagonal prism of volume $$ {{4\left( {5 - 2\sqrt 5 } \right)} \mathord{\left/{\vphantom {{4\left( {5 - 2\sqrt 5 } \right)} 9}} \right.} 9} $$ (which is greater than 0.2346) and an affine-regular pentagonal antiprism of volume $$ {{4\left( {3\sqrt 5 - 5} \right)} \mathord{\left/{\vphantom {{4\left( {3\sqrt 5 - 5} \right)} {27}}} \right.} {27}} $$ (which is greater than 0,253). Furthermore, K is contained in an affine-regular pentagonal prism of volume $$ 6\left( {3 - \sqrt 5 } \right) $$ (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.