On the zone of the boundary of a convex body

We consider an arrangement $\text{A}$ of $n$ hyperplanes in ${\mathbb{R}}^d$ and the zone $\mathbb{Z}$ in $\text{A}$ of the boundary of an arbitrary convex set in ${\mathbb{R}}^d$ in such an arrangement. We show that, whereas the combinatorial complexity of $\mathbb{Z}$ is known only to be $O<n^{d-1}\log n>$ \cite{APS}, the outer part of the zone has complexity $O<n^{d-1}>$ (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 $d=2$).