Abstract
We show, with an elementary proof, that the number of halving simplices in a set of n points in ℝ4 in general position is O(n4-2/45). This improves the previous bound of O(n4-1/13^{4}). Our main new ingredient is a bound on the maximum number of halving simplices intersecting a fixed 2-plane.
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Matousek, J., Sharir, M., Smorodinsky, S. et al. k-Sets in Four Dimensions. Discrete Comput Geom 35, 177–191 (2006). https://doi.org/10.1007/s00454-005-1200-4
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DOI: https://doi.org/10.1007/s00454-005-1200-4