k-Sets in Four Dimensions

AbstractWe 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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