Sharp Bounds for Vertical Decompositions of Linear Arrangements in Four Dimensions

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 Θ(n) 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(nα(n) log n), where α(n) is the inverse Ackermann function, improving the best previously known bound [2] by a near-linear factor.