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-1 log n) , 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).
Bibliographical noteFunding Information:
Work on this paper has been supported by Grant 338/09 from the Israel Science Fund and by the Hermann Minkowski MINERVA Center for Geometry at Tel Aviv University.
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- Hyperplane arrangements
- Zone theorem