Measure partitions using hyperplanes with fixed directions

Abstract

We study nested partitions of R d obtained by successive cuts using hyperplanes with fixed directions. We establish the number of measures that can be split evenly simultaneously by taking a partition of this kind and then distributing the parts among k sets. This generalises classical necklace splitting results and their more recent high-dimensional versions. With similar methods we show that in the plane, for any t measures there is a path formed only by horizontal and vertical segments using at most t−1 turns that splits them by half simultaneously, and optimal mass-partitioning results for chessboard colourings of R d using hyperplanes with fixed directions.

Thieves can make sandwiches

“…A similar problem has been considered if only vertical and horizontal segments are allowed. It is conjectured that measures can be split evenly using a path that only has turns , as opposed to the measures we can split with the result above. If the condition on the directions of the paths is completely dropped, we get an interesting problem: Problem Given a polygonal path in the plane that uses at most turns, what is the largest number of measures that we can always guarantee to be able to split simultaneously by half?…”

“…A partition of into convex parts is an iterated hyperplane partition if it can be made out of by successively partitioning each convex part, from a previous partition, with a hyperplane (that means each new hyperplane only cuts one of the existing convex parts, see Figure ). In the authors showed that for thieves and measures in , there is a fair distribution of each measure among the thieves using an iterated hyperplane partition that has hyperplane cuts, whose directions are fixed in advance, as long as is a prime power. This has the advantage that the total number of parts is .…”

Two's Company, Three's a Crowd: Consensus-Halving for a Constant Number of Agents

“…Deligkas et al [22] studied the computational complexity of the exact version of the problem, and obtained among other results its membership in a newly introduced class BU (for "Borsuk-Ulam" [12]) and its computational hardness for the well-known class FIXP of Etessami and Yannakakis [25]. Very recently, Deligkas et al [21] showed the PPA-completeness of the related Pizza Sharing problem [35], via a reduction from Consensus-Halving.…”

Lifting methods in mass partition problems

“…Kano conjectured that for any n smooth measures in R 2 there exists a path formed only by horizontal and vertical segments, that takes at most n − 1 turns, that simultaneously halves each measure. The conjecture is only known for k = 1, 2 or if the path is allowed to go through infinity [UKK09,KRPS16]. We wonder if the following way to mix Kano's conjecture with Theorem 2.1 holds.…”