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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
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