volume 31, issue 3, P435-460 2004
DOI: 10.1007/s00454-003-2871-3
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Abstract: We prove tight and near-tight combinatorial complexity bounds for vertical decompositions of arrangements of hyperplanes and 3-simplices in four dimensions. In particular, we prove a tight upper bound of (n 4 ) for the vertical decomposition of an arrangement of n hyperplanes in four dimensions, improving the best previously known bound [8] by a logarithmic factor. We also show that the complexity of the vertical decomposition of an arrangement of n 3-simplices in four dimensions is O(n 4 α(n) log 2 n), where…

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