Measure Partitions Using Hyperplanes with Fixed Directions

2014/08/20 by Karasev, Roman, Roldán-Pensado, Edgardo, Soberón, Pablo · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1408.4830

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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• Partitions of mass assignments with spheres and wedges

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• Bisecting masses with families of parallel hyperplanes
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