On the zone of the boundary of a convex body

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

Covering Partial Cubes with Zones

A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the problem of covering the cells of a line arrangement with a minimum number of lines, and the problem of finding a minimum-size fibre in a bipartite poset. For several such special cases, we give upper and lower bounds on the minimum size of a covering by zones. We also consider the computational complexity of those problems, and establish some hardness results