Gruber P. Convex and Discrete Geometry

32 Tiling with Convex Polytopes 469
Let N be a bounded convex neighbourhood of f . Since T is locally finite, N intersects
only finitely many tiles. By choosing N even smaller, if necessary, we may
suppose that N meets only those faces of a tile which contain f .Lets, t ∈ N be
interior points of S and T , respectively, such that the line segment [s, t] intersects
the boundaries of tiles (all of which meet N) only in interior points of facets. Order
the tiles which intersect [s, t],sayS = T1,...,Tm = T , so that successive tiles have
common facets. Let F12 be the common facet of S = T1 and T2. By our choice of N,
the facet F12 contains the relative interior point f of the face F of S = T1. Hence
F ⊆ F12. Since F12 is a facet of T2 too, we deduce that F is contained in and is a
face of T2. LetF23 be the common facet of T2 and T3. By our choice of N, the facet
F23 contains the point f in the relative interior of the face F of T2. Hence F ⊆ F23.
Continuing in this way, we finally see that F is contained in and is a facet of Tm = T .
Thus F, G are both faces of T . Since f is a relative interior point of each of them,
this can hold only if F = G, concluding the proof of (10) and thus of (7).
Next, the following will be shown:
(11) Let S, T ∈ T, F a face of S and G a face of T . Then F ∩ G is a face of
both S and T .
If F = S or G = T , the statement (11) is an immediate consequence of (7). Otherwise,
each of F, G is the intersection of finitely many facets of S, respectively, T .
Since T is a facet-to-facet tiling, we then have
say, where S1,...,Tn ∈ T. Hence
F = S ∩ S1 ∩···∩Sm, G = T1 ∩···∩Tn ∩ T,
F ∩ G = (S ∩ S1) ∩···∩(S ∩ Sm) ∩ (S ∩ T1) ∩···∩(S ∩ Tn).
is an intersection of faces of S by (7), and thus itself is a face of S. Similarly, F ∩ G
is a face of T , concluding the proof of (11).
Having proved (11), the proof of the theorem is complete. ⊓⊔
An immediate consequence of Propositions 32.1, 32.2 and of Theorem 32.1, is
the following result.
Corollary 32.1. Dirichlet–Voronoĭ tilings and Delone triangulations give rise to
polyhedral complexes.
Dirichlet–Voronoĭ and Delone Tilings are Dual
Two (generalized) polyhedral complexes C, D in E d are dual polyhedral complexes
if there is a one-to one mapping of the cells of C onto the cells of D such that the
following statements hold:
(i) Each cell F ∈ C of dimension i is mapped onto a cell G ∈ D of dimension d − i
such that these cells have precisely one point in common which is in the relative
interior of both F and G. F is disjoint from any other cell H ∈ D of dimension
d − i and similarly for G. The cells F and G are then said to be dual to each
other.