On Envy-Free Cake Division

The old problem introduced by Gamov and Stern [3], to divide a cake among m people so that each person thinks that he has at least as much cake as anybody else (envy-free division), has been solved by Brams and Taylor [1]. This discovery attracted further interest to the area and, a few years later, Robertson and Webb [4] found a new construction. Here we present another solution, which is similar in spirit to the latter. Let each player Pi define a measure Ai on the cake C such that Ai(C) = 1, i E [m] {1, .. ., m}. We assume that Pi can always cut off a piece B cA with Ai(B) = c for any A c C and 0 < c < ,ui(A). We want to find a procedure for constructing a partition C = W1 u U. U Wm with Ai (W) ? Ai(Wj) for all i, j E [im]. We use the following lemma from [4], for which we present an elementary proof.