Gruber P. Convex and Discrete Geometry

15.2 Shelling and Euler’s Formula for General d
15 Combinatorial Theory of Convex Polytopes 265
The first attempt to extend Euler’s polytope formula to general d goes back to Schläfli
[888] in 1850 in a book which was published only in 1901. His proof that, for the
f -vector of a proper convex polytope P in E d ,wehave,
(1) f0 − f1 +−···+(−1) d−1 fd−1 = 1 − (−1) d
made use of an argument of the following type. The boundary of P can be built up
from a given facet F1 by successively adding the other facets in a suitable order, say
F2,...,Fm, such that (F1∪···∪Fi−1)∩Fi is homeomorphic to a (d−2)-dimensional
convex polytope for i = 2,...,m −1. Schläfli, and later other mathematicians, seem
to have thought that the existence of such a shelling was obvious; see the references
in Grünbaum [453]. After many unsuccessful attacks, in particular in the 1960s, this
was proved in an ingenious way by Bruggesser and Mani [172] in 1970. Before that,
the only elementary proof of the formula (1) was that of Hadwiger [467].
A topological proof of (1), by Poincaré [807, 808], also had serious gaps.
Rigorous proofs in the context of topology were given only in the 1930s when the
necessary algebraic-topological machinery was available. For this information I am
indebted to Matthias Kreck [617].
The shelling result of Bruggesser and Mani, which yields a simple proof of (1), is
also a tool to prove the equations of Dehn [251] and Sommerville [948] for f -vectors
and is used in the striking proof of the upper bound conjecture by McMullen [705].
The problem of characterizing the f -vectors of convex polytopes is settled for
d = 2 where it is trivial, and for d = 3, see the preceding section. For d > 3itis
far from a solution, but there are important contributions towards it. For simplicial
convex polytopes a characterization was proposed by McMullen [707] in the form of
his celebrated g-conjecture. The g-conjecture was proved by Billera and Lee [116]
and Stanley [951], see also McMullen [715].
In this section we present the shelling result of Bruggesser and Mani and show
how it leads to Euler’s polytope formula (1) for general d. Then other relations for
f -vectors are discussed.
Shellings are treated by Björner and Björner et al. [121, 122] and from the voluminous
literature on f -vectors we cite the books of Grünbaum [453], McMullen and
Shephard [718], Brøndsted [171] and Ziegler [1045] and the surveys of Bayer and
Lee [83] and Billera and Björner [115].
Polytopal Complexes and Shellings
A (finite) abstract complex C is a family of subsets of {1,...,v},thefaces of C, such
that ∅, {1},...,{v} ∈C and for any two faces F, G ∈ C also F ∩ G ∈ C. The faces
{1},...,{v} are called the vertices of C.
Closer to convexity, we define a (finite) polytopal complex C to be a finite family
of convex polytopes in E d , called the faces of C, such that any (geometric) face of a
polytope in C belongs also to C and the intersection of any two polytopes in C is a