Authors: de Longueville, Mark
Živaljević, Rade 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Splitting multidimensional necklaces
Journal: Advances in Mathematics
Volume: 218
Issue: 3
First page: 926
Last page: 939
Issue Date: 20-Jun-2008
Rank: M21a
ISSN: 0001-8708
DOI: 10.1016/j.aim.2008.02.003
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.
Keywords: Alon's splitting necklace theorem | Consensus division | Equivariant maps | Topological shellability
Publisher: Elsevier

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