Measure Partitions Using Hyperplanes with Fixed Directions

We study nested partitions of Rd obtained by successive cuts us-ing hyperplanes with fixed directions. We establish the number of measures that can be split evenly simultaneously by taking a parti-tion 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.