On the Zone of a Surface in a Hyperplane Arrangement

Let H be a collection of n hyperplanes in IR d , let A denote the arrangement of H; and let be a (d ? 1)-dimensional algebraic surface of low degree, or the boundary of a convex set in IR d. The zone of in A is the collection of cells of A crossed by. We show that the total number of faces bounding the cells of the zone of is O(n d?1 log n). More generally, if has dimension p, 0 p < d, this quantity is O(n b d+p 2 c) for d ? p even and O(n b d+p 2 c log n) for d ? p odd. These bounds are tight within a logarithmic factor.