Splitting multidimensional necklaces

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