Splitting multidimensional necklaces
Combinatorics
2007-05-23 v1 Algebraic Topology
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.
Keywords
Cite
@article{arxiv.math/0610800,
title = {Splitting multidimensional necklaces},
author = {Mark de Longueville and Rade Zivaljevic},
journal= {arXiv preprint arXiv:math/0610800},
year = {2007}
}