Cell complexities in hyperplane arrangements (Q701778)

The complexity of some cells of an hyperplane arrangement in \(R^d\) is the total number of faces of all dimensions of these cells. The authors show that the complexity of \(m\) distinct cells in an arrangement of \(n\) hyperplanes in dimension \(d\geq 4\) is \(O(m^{1/2}n^{d/2}\log^{(\lfloor d/2\rfloor-2)}n)\). They use these new bounds to reobtain a bound for the sum of squares of cell complexities in an arrangement. They also remark that in dimension 4 the bound obtained is tight for different range of values of \(m\).