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 ⋅ a i 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 A i ⊂ [ 0 , 1 ] (or more generally as continuous measures μ i ). We demonstrate that Alon's result is a special case of a multidimensional consensus division theorem about n continuous probability measures μ 1 , … , μ n on a d-cube [ 0 , 1 ] d . The dissection is performed by m 1 + ⋯ + m d = n ( k − 1 ) hyperplanes parallel to the sides of [ 0 , 1 ] d dividing the cube into m 1 ⋅ ⋯ ⋅ m d elementary cuboids (parallelepipeds) where the integers m i are prescribed in advance.