Cells with many facets in arrangements of hyperplanes

Abstract For every n, d, n⩾2d+1⩾5, we prove the existence of an arrangement H of n hyperplanes in the real projective space P d, such that exactly ∑ d−2 i=0 ( n−1 i cells of H are bounded by every hyperplane of H . In the particular case d=3, this disproves aconjecture of Edelsbrunner and Haussler. We also prove that in any arrangement of n hyperplanes in P d, the average number of hyperplanes bounding the cells of H is always less than 2d+1.