Abstract.
A collection of n hyperplanes in \({\Bbb R}\) d forms a hyperplane arrangement. The depth of a point \(\theta \in {\Bbb R}^d\) is the smallest number of hyperplanes crossed by any ray emanating from θ . For d=2 we prove that there always exists a point θ with depth at least \(\lceil n/3\rceil\) . For higher dimensions we conjecture that the maximal depth is at least \(\lceil n/(d+1)\rceil\) . For arrangements in general position, an upper bound on the maximal depth is also established. Finally, we discuss algorithms to compute points with maximal depth.
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Received December 1, 1997, and in revised form June 6, 1998.
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Rousseeuw, P., Hubert, M. Depth in an Arrangement of Hyperplanes. Discrete Comput Geom 22, 167–176 (1999). https://doi.org/10.1007/PL00009452
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DOI: https://doi.org/10.1007/PL00009452