Gruber P. Convex and Discrete Geometry

32 Tiling with Convex Polytopes 465
Fig. 32.1. Dirichlet–Voronoĭ tiling and Delone triangulation
The systematic study of such tilings started with Voronoĭ [1014]. More generally, let
D be a discrete set in E d . Then the sets
{D(p) : p ∈ D}, D(p) = � x :�x − p� ≤�x − q� for all q ∈ D �
are called Dirichlet–Voronoĭ cells of D. Clearly, any cell D(p) consists of all points
x which are at least as close to p as to any other point q ∈ D. Dirichlet–Voronoĭ
cells are known under very different names, for example honeycombs, domains
of action, Brillouin, or Wigner-Seitz zones. The corresponding tilings are called
Dirichlet–Voronoĭ tilings (Fig. 32.1).
Proposition 32.1. Let D be a discrete set. Then the corresponding Dirichlet–Voronoĭ
cells are proper, generalized convex polyhedra and form a facet-to-facet tiling of E d .
Proof. First, the following will be shown:
(1) Each Dirichlet–Voronoĭ cell D(p), p ∈ D, is a proper generalized convex
polyhedron.
As the intersection of closed halfspaces, each cell is a closed convex set in E d . Since
D is discrete, each cell has non-empty interior and thus is proper. To show that it is a
generalized polyhedron, it is sufficient to prove the following: Let K be a cube. Then
the intersection D(p) ∩ K is a convex polytope for each p ∈ D. Clearly, D(p) ∩ K
is the intersection of the cube K with the halfspaces {x :�x − p� ≤�x − q�}, where
q ∈ D, q �= p. Since D is discrete, all but finitely many of these halfspaces contain
K in their interior. Thus D(p) ∩ K is the intersection of K with a finite family of
these halfspaces and thus is a convex polytope, concluding the proof of (1).
The second step is to show the following statement:
(2) The Dirichlet–Voronoĭ cells {D(p) : p ∈ D} form a locally-finite tiling
of E d .