Line and Plane Arrangements with Large Average Diameter

Let ∆A(n, d) denote the largest possible average diameter of a bounded cell of a simple arrangement defined by n hyperplanes in dimension d. We have ∆A(n, 2) ≤ 2+ 2 n−1 in the plane, and ∆A(n, 3) ≤ 3+ 4 n−1 in dimension 3. In general, the average diameter of a bounded cell of a simple arrangement is conjectured to be less than the dimension; that is, ∆A(n, d) ≤ d. We propose line and plane arrangements with large average diameter yielding ∆A(n, 2) ≥ 2 − 2d n 2 e (n−1)(n−2) and ∆A(n, 3) ≥ 3− 6 n−1 + 6(bn 2 c−2) (n−1)(n−2)(n−3) .