  The largest possible average diameter of a bounded cell of a simple
hyperplane arrangement is conjectured to be not greater than the dimension. We
prove that this conjecture holds in dimension 2, and is asymptotically tight in
fixed dimension. We give the exact value of the largest possible average
diameter for all simple arrangements in dimension 2, for arrangements having at
most the dimension plus 2 hyperplanes, and for arrangements having 6
hyperplanes in dimension 3. In dimension 3, we give lower and upper bounds
which are both asymptotically equal to the dimension.
