2006/10/26 by de Longueville, Mark, Zivaljevic, Rade
#05C10 #51M16 #52A40 #52B22 #55N91 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/0610800
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 m1+...+ md=n(k-1) hyperplanes parallel to the sides of [0,1]d dividing the cube into m1 x m2 x ... x md elementary parallelepipeds where the integers mi are prescribed in advance.