Measure Partitions Using Hyperplanes with Fixed Directions

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We study nested partitions of Rᵈ 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 Rᵈ using hyperplanes with fixed directions.