Gruber P. Convex and Discrete Geometry

466 Geometry of Numbers
F
G
p
q
H
Fig. 32.2. Dirichlet–Voronoĭ tilings are facet-to-facet
For the proof of (2) it is sufficient to show the following: if K is a cube, then only
finitely many of the Dirichlet–Voronoĭ cells D(p) meet K and these form a tiling
of K .Letq ∈ D. Since D is discrete, all but finitely many of the halfspaces {x :
�x − p� ≤�x −q�} are disjoint from K . Thus, only finitely many Dirichlet–Voronoĭ
cells D(p) meet K ,sayD(p1),...,D(pk). Any two of these cells are separated by
a hyperplane and thus have disjoint interiors. Since D is discrete, for each point of
E d there is a least one nearest point in D. Thus E d is the union of all cells. Together,
this implies that � D(p1) ∩ K,...,D(pk) ∩ K � form a tiling of K .
The third and final step is to show the following:
(3) The Dirichlet–Voronoĭ cells {D(p) : p ∈ D} form a facet-to-facet tiling
of E d .
By (1) and (2), each facet of a cell D(p), p ∈ D, is covered by facets of other such
cells. Thus, if (3) did not hold, there are points p, q ∈ D such that the tiles D(p) and
D(q) have facets F and G, respectively, which overlap, but do not coincide. Then
there is a (d − 2)-dimensional face H of G, say, which meets the relative interior of
F. Project F, G, H onto a 2-dimensional plane orthogonal to H. Then we have the
configuration of Fig. 32.2.
Since H is the intersection of two facets of D(q), one of which is G, there is
a point r ∈ D which, together with q, determines this facet. Then the hyperplane
{x :�x − p� =�x − r�} cuts off the part of F outside H. Thus this part is not
contained in D(p). This contradiction concludes the proof of (3).
Having shown (1)–(3), the proof of the proposition is complete. ⊓⊔
Delone Sets, the Empty Sphere Method and Delone Triangulations
We follow Delone [255], see also Delone and Ryshkov [259]. Let D be an (r, R)system
or Delone set in E d where r, R > 0. That is, a discrete set D in E d such that
any two distinct points have distance at least r and for any point in E d there is a point
in D at distance at most R. Lattices and periodic sets as considered by Zassenhaus,
are Delone sets but not vice versa. Delone sets still have a certain uniformity and
regularity. Now the aim is to construct a tiling of E d with convex polytopes such that
the vertices of these polytopes are precisely the points of D. Particular such tilings
r