On the sum of squares of cell complexities in hyperplane arrangements

Let H be a collection of n hyperplanes in R^d, d≥2. For each cell c of the arrangement of H let f_i(c) denote the number of faces of c of dimension i, and let f(c) = ∑_i=0^d-1 f_i(c). We prove that ∑_c f(c)² = O(n^dlog ^{d 2-1} n), where the sum extends over all cells of the arrangement. Among other applications, we show that the total number of faces bounding any m distinct cells in an arrangement of n hyperplanes in R^d is O(m ^{1 2}n ^{d 2}log ^{( d 2-1) 2} n) and provide a lower bound on the maximum possible face count in m distinct cells, which is close to the upper bound, and for many values of m and n is Ω(m ^{1 2}n ^{d 2}).