On the intersection of a pyramid and a ball

Abstract

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 $$\overline{Q}$$ 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 $$\overline{Q}$$ have the same area. Let $$K(\rho)$$ be a ball with the same centre as S and having radius ρ. Main Theorem. The volume of the intersection of Q and K(ρ) is not greater than the volume of the intersection of $$\overline{Q}$$ and K(ρ), for any ρ ≤ 1. If $$Q \neq \overline{Q}$$ and ρ 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.