# zbMATH — the first resource for mathematics

On the intersection of a pyramid and a ball. (English) Zbl 1218.52009
The aim of the paper is to prove a theorem about the volume of the intersection between a pyramid and a ball, that is necessary to complete the proof of a previous result of the author referring to the edge curvature of a convex polyhedron [see Monatsh. Math. 60, 288–297 (1956; Zbl 0073.17402)]. Let $$Q$$ be a convex $$n$$-sided pyramid contained in the unit ball $$S$$ and having the apex at the center of $$S$$. Let $$\overline{Q}$$ be the corresponding $$n$$-sided pyramid based on a regular $$n$$-gon with its vertices on the boundary of $$S$$, such as the radial projections of the bases of $$Q$$ and $$\overline{Q}$$ have the same area. The ball $$K(\rho)$$ is supposed to have the same center as $$S$$. Let $$V$$ denote the volume of a set.
The main result is: “Let $$K(\rho)$$ be the ball with radius $$\rho \in (0,1]$$ and center $$O^*$$. Then $$V(Q\cap K(\rho))\leq V(\overline{Q}\cap K(\rho))$$, for any $$\rho \in (0,1]$$. Let $$d$$ be the minimum distance of $$O^*$$ from the base points of $$Q$$. If $$Q\neq \overline{Q}$$ and $$\rho >d$$, then the inequality strictly holds.”
##### MSC:
 52A40 Inequalities and extremum problems involving convexity in convex geometry 52B10 Three-dimensional polytopes
##### Keywords:
convex polyhedron; edge-curvature; inradius; volume; pyramid; ball
Full Text: