Dividing an apple equally -- no easy job! (Q2315808)

This paper can be considered as a sequel to [Math. Semesterber. 62, No. 2, 173-194 (2015; Zbl 1331.51012)] by the same author. The pizza theorem says that if a circular disk is divided by \(k\geq4\) chords at equally spaced angles through a single point of the disk, \(k\) is even, and the slices are taken alternately in two sets, then the total area of one set is equal to the total area of the other set. An 3-dimensional analogue of the pizza theorem is the ``apple theorem'' -- dividing a ball in two sets of equal volumes. The author shows that using Cavalieri's principle, the proof of the apple theorem is an easy corollary of the pizza theorem. Moreover, in the appendix he shows that as in the case of dividing a pizza when the two persons share the same amount of pizza crust, in the case of the dividing the apple they get the same amount of apple skin (the sums of the surface areas of the ball pieces in the two sets are equal).