Splitting multidimensional necklaces
Splitting multidimensional necklaces
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分裂多维项链
DOI：
10.1016/j.aim.2008.02.003
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
R. Živaljević
中科院分区：
文献类型：
--
作者：
M. Longueville;R. Živaljević
Alon [N. Alon，Splitting necklaces，Adv. Math. 63（1987）247-253]说，每一条有k个颜色i= 1，.，n的珠子的项链，可以在k个小偷之间公平地分成至多n（k-1）个切口。阿隆从这样一个事实中推导出这个结果，即在连续项链[0，1]的情况下，这种划分也是可能的，其中给定颜色的珠子被解释为可测集合Ai <$[0，1]（或更一般地解释为连续测度μi）。证明了Alon的结果是关于n个连续概率测度μ1，…，μnon和d-立方[0，1]d的多维一致除法定理的一个特例.通过m1+ md +md=n（k−1）个平行于[0，1] d的边的超平面将立方体分割成m1个<$m个基本长方体（平行六面体），其中整数mi是预先指定的。
The well-known “splitting necklace theorem” of Alon [N. Alon, Splitting necklaces, Adv. Math. 63 (1987) 247–253] says that each necklace with k⋅aibeads 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,…,μnon a d-cube [0,1]d. The dissection is performed by m1+⋯+md=n(k−1) hyperplanes parallel to the sides of [0,1]ddividing the cube into m1⋅⋯⋅mdelementary cuboids (parallelepipeds) where the integers miare prescribed in advance.