Zones in Arrangements

Abstract

In this chapter, we investigate the complexity of the boundary of a certain collection of cells in an arrangement of hyperplanes. We formalize the problem by introducing the notion of a so-called zone which is defined relative to some chosen hyperplane in the arrangement. Intuitively, the zone of a hyperplane h contains all faces in the boundaries of those cells which are supported by h. The introduction of this concept is motivated by an algorithm that constructs an arrangement incrementally, that is, the hyperplanes are inserted one after another (see Chapter 7). A more formal definition of the zone of a hyper plane in terms of visibility is as follows.