A Fourier approach to pizza inequity (Q2673895)

The surprising result known as `pizza theorem' says that if a circular disk (pizza) is divided into \(2n\) slices by \(n\ge4\) chords at equally spaced angles through a single interior point \(P\) of the disk different from the centre \(C\), \(n\) is even, and the slices are taken alternately in two sets of odd and even numbered slices, then the total area of the slices in the one set is equal to the total area of the slices in the other set. When \(n\) is not even, the pizza theorem does not hold. The author considers this case and without loss of generality for a disc of radius 1 shows an upper bound for \(|E-O|\) where \(E\) is the total area of the even numbered slices and \(O\) is the total area of the odd numbered slices. The main result is \[ |E-O|\le\frac{a^n}{2(1-a^2)(1-a^{2n})}, \] where \(a\), \(0<a<1\), is the distance from \(P\) to \(C\). This is a very nice paper but regrettably not a single figure is included which could have made it even tastier for the ravenous reader.