Envy-free division of discrete cakes

We address the problem of existence of an envy-free distribution of pieces among two or more players in the cake-cutting setting with the minimum number of cuts. Our cakes are discrete in the sense that the players’ valuations are concentrated on atoms. These atoms are placed on an interval and no two players give positive values to atoms placed at the same position. We prove the existence of an envy-free allocation for any discrete cake and any number of players by resorting to Sperner’s Lemma, a suitable triangulation, and moving-knife arguments. Our results also apply to pies, which are defined over circumferences instead of intervals.