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 masspartitioning results for chessboard colourings of R d using hyperplanes with fixed directions.
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Supported by the Dynasty Foundation and the Russian Foundation for Basic Research grants 15-31-20403 (mol a ved) and 15-01-99563 (A).
Supported CONACyT project 166306.
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Karasev, R.N., Roldán-Pensado, E. & Soberón, P. Measure partitions using hyperplanes with fixed directions. Isr. J. Math. 212, 705–728 (2016). https://doi.org/10.1007/s11856-016-1303-z
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DOI: https://doi.org/10.1007/s11856-016-1303-z