On the zone of a surface in a hyperplane arrangement (Q1209841)

For a collection of \(n\) hyperplanes in \(\mathbb{R}^ d\) the complex of all open cells and their lower dimensional faces is called an arrangement. The collection of all those cells which are met by a set \(\sigma\) is called its zone and its complexity is the number of all the faces. Asymptotic estimates for the complexity are given the simplest, for the case that \(\sigma\) is the boundary of a convex set or an algebraic surface, is \(O(n^{d-1}\log n)\).