On the possibilities for partitioning a cake

Abstract

We wish to consider the following type of cake division problem: There are p p individuals. Each individual has available a measure that he or she uses to evaluate the sizes of pieces of cake. We wish to partition our cake into q q pieces in such a way that the various evaluations that the individuals make of the sizes of the pieces satisfy certain pre-assigned equalities and inequalities. Our main result yields a quite general criterion for showing that certain such partitions exist. Following the proof, we consider various applications.