Be review basic facts about polytopes, cell complexes, Voronoi diagrams, and duality. These topics are treated in detail in the texts [118, 335] and the collection of surveys [152]. We assume that the reader is familiar with the notion of a linear subspace V of Rd and its affine version, a flat, ie, x + V (x ∈ Rd). The affine span of a set is the lowest dimensional flat enclosing it.

Polytopes

A convex polyhedron in Rd is the intersection of a finite number of closed halfspaces, ie, sets of the form {x ∈ Rd ∣a · x ≤ b}, for a, b ∈ Rd, where a ≠ 0 and a·x denotes the inner product of a and x. A polytope is a bounded convex polyhedron. Equivalently, it is the convex hull of a finite point set. A face of a polytope P ⊂ Rd is the relative interior of the intersection of P with a supporting hyperplane. The dimension of a face is that of its affine span. A face of dimension 0 (resp. 1 or d − 1) is called a vertex (resp. edge or facet). The collection of all faces, ordered by inclusion of their closures, forms a cell complex with a lattice structure. It is convenient to represent it by a facial graph. Each node denotes a face, and an arc connects two incident faces whose dimensions differ by exactly one.