Splitting Multidimensional Necklaces

The well-known “splitting necklace theorem” of Alon [1] says that each necklace with k · ai beads of color i = 1, . . . , n can be fairly divided between k “thieves” by at most n(k − 1) cuts. Alon deduced this result from the fact that such a division is possible also in the case of a continuous necklace [0, 1] where beads of given color are interpreted as measurable sets Ai ⊂ [0, 1] (or more generally as continuous measures μi). We demonstrate that Alon’s result is a special case of a multidimensional, consensus division theorem of n continuous probability measures μ1, . . . , μn on a d-cube [0, 1]. The dissection is performed by m1 + . . . + md = n(k − 1) hyperplanes parallel to the sides of [0, 1] dividing the cube into m1 · . . . ·md elementary parallelepipeds where the integers mi are prescribed in advance.