Gruber P. Convex and Discrete Geometry

266 Convex Polytopes
(geometric) face of both of them, and thus is contained in C. (What we call a polytopal
complex is often called a polyhedral complex. The reason for our terminology
is that we want to stress that our complexes consist of convex polytopes and not of,
possibly, unbounded convex polyhedra.) The underlying set of C is the union of all
polytopes in C.Thedimension of C,dimC, is the maximum dimension of a polytope
in C. The complex C is pure if any polytope in C is a (geometric) face of a polytope
in C of dimension dim C. Two (abstract or geometric) complexes are isomorphic if
there is a bijection between their (abstract or geometric) vertices which preserves
their (abstract or geometric) faces in both directions.
Given a convex polytope P in E d , the family of all its (geometric) faces, including
∅ and P, is a polytopal complex, the complex C = C(P) of P. Also the family
of all its proper (geometric) faces, including ∅, is a polytopal complex, the boundary
complex C(relbd P) of P. See Theorems 14.5 and 14.6.
By a shelling of a convex polytope P or, more precisely, of its boundary complex
C(relbd P), we mean a linear ordering F1,...,Fm of the facets of P such that either
dim P = 0, 1, or which otherwise satisfies the following conditions:
(i) (The boundary complex of) F1 has a shelling.
(ii) For i = 2,...,m we have (F1 ∪···∪Fi−1) ∩ Fi = G1 ∪···∪G j,
where j ≤ k and G1 = Gi1,...,Gk = Gik is a shelling of Fi.
The Bruggesser-Mani Shelling Theorem
Before stating this result, we present a simple property of shellings.
Proposition 15.1. Let G1,...,Gk be a shelling of a convex polytope F. Then
Gk,...,G1 is also a shelling of F.
Proof (by induction on d = dim F). The assertion clearly holds for d = 0, 1.
Assume now that d > 1 and that it holds in dimension d − 1. We prove it for F.
First, each Gi and thus, in particular, Gk is shellable by (i) and (ii). Second, let
i < k. By (ii),
(2) (G1 ∪···∪Gi−1) ∩ Gi = H1 ∪···∪ H j,
where j ≤ n and H1,...,Hn is a shelling of Gi. For each facet Hl of Gi there is a
unique facet Gm of F such that Hl = Gi ∩ Gm. Thus (2) implies that
Gi ∩ (Gi+1 ∪···∪Gk) = H j+1 ∪···∪ Hn.
Now note that Hn,...,H1 is also a shelling of Gi by induction. The induction is
complete. ⊓⊔
The Bruggesser-Mani shelling theorem is as follows:
Theorem 15.4. Convex polytopes are shellable.