Splitting multidimensional necklaces

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 · a i 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 A i ⊂ [0, 1] (or more generally as continuous measures μ i ). We demonstrate that Alon's result is a special case of a multidimensional consensus division theorem about n continuous probability measures μ 1 , . . . , μ n 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 · · · · · m d elementary cuboids (parallelepipeds) where the integers m i are prescribed in advance.

Measure partitions using hyperplanes with fixed directions

“…In Section 3 we generalise this result to R d , where the cuts are made with hyperplanes with fixed directions. This further generalises a d-dimensional version byŽivaljević and de Longueville of the theorem above [DLŽ08], where the partition was made using hyperplanes with fixed directions, but only choosing out of d possible directions. In the case when k is a prime power, we also optimise on the number of parts in the resulting partition, which has not been done before.…”

Illumination complexes, Δ-zonotopes, and the polyhedral curtain theorem

“…It will become clear from the proof that it is really an offspring of the multidimensional splitting necklace theorem [8], a higher dimensional generalization of the celebrated splitting necklace theorem of Alon [1,2].…”

“…As a consequence, the configuration space Ω (3,2) of all divisions of the square of the type (3,2), together with all possible allocations of rectangular pieces to two parties involved, is the union of 2 12 polyhedral cells (copies of Δ 3 × Δ 2 ). These cells are glued together, along their boundaries, whenever some of the elementary rectangles degenerate (for example if x i = x i+1 or y j = y j+1 ), see [8,Section 2] for more details.…”

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

“…One may, for instance, explore necklace splittings with the added constraint that adjacent pieces of the necklace cannot be claimed by certain pairs of thieves; for example, Asada et al [32] prove that four thieves on a circle can share the beads of the necklace, with the restriction that the two pairs of nonadjacent thieves will not receive adjacent pieces of the necklace. There are also several nice high-dimensional generalizations of (convex) splitting of booty; see [70,133] and the references therein.…”