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Splitting multidimensional necklaces

Mark de Longueville·Rade Zivaljevic
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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 m_1+...+ m_d=n(k-1) hyperplanes parallel to the sides of [0,1]^d dividing the cube into m_1 x m_2 x ... x m_d elementary parallelepipeds where the integers m_i are prescribed in advance.

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Mark de Longueville, Rade Zivaljevic. 2006-10-26. Splitting multidimensional necklaces. https://arxiv.org/abs/math/0610800

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