| 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( 5 - 2?5 ) \mathord | / |
\vphantom 4( 5 - 2?5 ) 9 9 {{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( 3?5 - 5 ) \mathord | / |
\vphantom 4( 3?5 - 5 ) 27 27 {{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( 3 - ?5 ) 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. |
