On the zone of the boundary of a convex body

Abstract

We consider an arrangement A of n hyperplanes in Rd and the zone Z in A of the boundary of an arbitrary convex set in Rd in such an arrangement. We show that, whereas the combinatorial complexity of Z is known only to be O(nd−1log⁡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).