On the zone of the boundary of a convex body

We consider an arrangement A of n hyperplanes in R and the zone Z in A of the boundary of an arbitrary convex set in R in such an arrangement. We show that, whereas the combinatorial complexity of Z is known only to be O ( nd−1 log n ) [3], the outer part of the zone has complexity O ( nd−1 ) (without the logarithmic factor). Whether this bound also holds for the complexity of the inner part of the zone is still an open question (even for d = 2).