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arXiv · 1408.4830

Measure Partitions Using Hyperplanes with Fixed Directions

Roman Karasev·Edgardo Roldán-Pensado·Pablo Soberón
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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.

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Roman Karasev, Edgardo Roldán-Pensado, Pablo Soberón. 2014-08-20. Measure Partitions Using Hyperplanes with Fixed Directions. https://arxiv.org/abs/1408.4830

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