On the intersection of a pyramid and a ball

• Autores: August Florian
• Localización: Aequationes mathematicae, ISSN 0001-9054, Vol. 77, Nº. 1-2, 2009, págs. 147-170
• Idioma: inglés
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• Resumen

  • Let Q be a convex n-sided pyramid contained in the unit ball S and having its vertex at the centre of S. We denote by a corresponding n-sided pyramid, based on a regular n-gon with its vertices on the boundary of S. Let us assume that the radial projections of the bases of Q and have the same area. Let be a ball with the same centre as S and having radius ?.Main Theorem. The volume of the intersection of Q and K(?) is not greater than the volume of the intersection of and K(?), for any ? = 1. If and ? is not too small, then we have strict inequality. In a previous paper, this result was stated without proof. The theorem was used to establish a lower bound to the edge-curvature of a convex polyhedron with given numbers of faces and vertices, and given inradius. Equality is attained only for the five regular polyhedra.