Gruber P. Convex and Discrete Geometry

256 Convex Polytopes
Then (6) implies that
(9) P = F ♦ −1 � F♦ 0 � ···� F♦ = F ♦ k � ···� F♦ d−1 � F♦ d =∅.
Propositions (4) and (6) and the fact that for faces G, H ∈ F(P) with G � H
holds dim G < dim H together imply that the sequences of inclusions (8) and (9) are
compatible only if dim F ♦
i = d − 1 − dim Fi for i =−1, 0,...,d. Proposition (7)
is the special case where i = k.
Finally, (i) and (ii) hold by (3), (4) and (7). Since by (3), (4) and (7) the mappings
♦P and ♦P∗ = ♦−1
P are onto, one-to-one and inclusion reversing, both are
anti-isomorphisms. This proves (iii). ⊓⊔
The Face Lattice of P is Atomic, Co-Atomic and Complemented
First, the necessary lattice-theoretic terminology is introduced. Given a lattice L =
〈L, ∧, ∨〉 with 0 and 1, an atom of L is an element a �= 0, such that there is no
element of L strictly between 0 and a. L is atomic if each element of L is the join of
finitely many atoms. The dual notions of co-atom and co-atomic are defined similarly
with 0, ∨ exchanged by 1, ∧. The lattice L is complemented if for each l ∈ L there
is an element m ∈ L such that l ∧ m = 0 and l ∨ m = 1.
Our aim is to show the following properties of the face lattice.
Theorem 14.8. Let P ∈ P. Then the following statements hold:
(i) � F(P), ∧, ∨ � is atomic and co-atomic.
(ii) � F(P), ∧, ∨ � is complemented.
Proof. We may assume that o ∈ int P. LetP = conv{v1,...,vn}, where the vi are
the vertices of P. LetF1,...,Fm be the facets of P.
(i) Clearly, v1,...,vn are the atoms and F1,...,Fm the co-atoms of F(P).
Given F ∈ F(P), F is the convex hull of the vertices contained in it, say v1,...,vk.
Then F is the smallest set in F(P) with respect to inclusion which contains
v1,...,vk. Hence F = v1 ∨ ··· ∨ vk. That is, F(P) is atomic. To see that it is
co-atomic, let F ∈ F(P) and consider the face F ♦ ∈ F(P ∗ ).Wehavejustproved
that the face lattice of any convex polytope is atomic. Thus, in particular, F(P ∗ ) is
atomic. This shows that F ♦ = w1 ∨···∨wl for suitable vertices w1,...,wl of P ∗ .
Now apply ♦ P ∗ and take into account Theorem 14.7 and that ♦P ♦ P ∗ is the identity
to see that
F = F ♦P♦ P ∗ = (w1 ∨···∨wk) ♦ P ∗ = w ♦ P ∗
1
∧···∧w♦ P∗ l .
Since wi is a vertex of P∗ ,thesetw ♦P∗ i is a face of P of dimension d −1 by Theorem
14.7 (i), i.e. a facet of P. Thus F(P) is co-atomic.
(ii) Let F ∈ F(P). IfF =∅or P, then G = P, respectively, ∅ is a complement
of F. Assume now that F �= ∅, P. LetG∈ F(P) be a maximal face disjoint from
F. Clearly, F ∧ G =∅. For the proof that F ∨ G = P, it is sufficient to show
that the set of vertices in F and G is not contained in a hyperplane. Assume that, on