On the Geometry of Cake Division

We study partitions of a ‘‘cake’’ C among n players. Each player uses a countably additive non-atomic probability measure to evaluate the sizes of pieces of cake. If the players’ measures are m1, m2, . . . , mn, then the ‘‘Individual Pieces Set,’’ which we studied beforeŽ2000, J. Math. Econom. 33, 401 424., is the set Žm1ŽP1.,m2ŽP2., . . . ,mnŽPn..: ²P1, P2, . . . , Pn: is a partition of C4. We continue our study of this set here. Our motivating question is: What are the possible shapes of such sets? We give an exact characterization for n 2, establish some partial results for n 3, and close with open questions