Approximation of three-dimensional convex bodies by affine-regular prisms (Q844473)

The author proves that every three-dimensional convex body \(K\) contains an affine-regular pentagonal prism of volume \({4\over 9}(5 - 2\sqrt 5) \cdot \text{Vol}(K)\) and an affine-regular pentagonal antiprism of volume \({4\over 27}(3\sqrt 5 -5) \cdot \text{Vol}(K)\). Moreover \(K\) is contained in an affine-regular pentagonal prism of volume \(6(3 - \sqrt 5) \cdot \text{Vol}(K)\). It is also shown that \(K\) is contained in an affine-regular heptagonal prism of volume \({21\over 4}(2\cos {\pi \over 7} -1) \cdot \text{Vol}(K)\). The last estimate cannot be improved for \(K\) being a tetrahedron.