Splitting multidimensional necklaces

Abstract

The well-known "splitting necklace theorem" of Alon [N. Alon, Splitting necklaces, Adv. Math. 63 (1987) 247-253] says that each necklace with k ṡ 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 about n continuous probability measures μ1, ..., μn on a d-cube [0, 1]d. The dissection is performed by m1 + ⋯ + md = n (k - 1) hyperplanes parallel to the sides of [0, 1]d dividing the cube into m1 ṡ ⋯ ṡ md elementary cuboids (parallelepipeds) where the integers mi are prescribed in advance.