|Authors:||de Longueville, Mark
|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||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 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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