Splitting multidimensional necklaces
Abstract
The well-known "splitting necklace theorem" of Noga Alon says that each "necklace" having beads of n different colors can be fairly divided between k "thieves" by at most n(k-1) cuts. We demonstrate that Alon's result is a special case of a multidimensional, consensus division theorem for n continuous probability measures 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 x m2 x ... x md elementary parallelepipeds where the integers mi are prescribed in advance.
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