Gruber P. Convex and Discrete Geometry

270 Convex Polytopes
The Lower and the Upper Bound Theorem
It is a natural question to determine the best lower and upper bounds for fi(P), i =
1,...,d − 1, for all convex polytopes P with given n = f0(P) (or fk(P) for given
k). The best lower bounds are attained by so-called stacked polytopes, as shown by
Barnette [73, 74]. In particular,
fd−1(P) ≥ (d − 1)n − (d + 1)(d − 2).
The best upper bounds are attained by so-called neighbourly polytopes. This was
shown by McMullen [705], thereby confirming Motzkin’s [758] upper bound
conjecture. In particular,
where
d 2� ∗
i=0
fd−1(P) ≤ 2
ai =
d
2�
� �
∗ n − d − 1 + i
,
i
i=0
� a0 +···+a d−1
2
a0 +···+a d−2
2
+ 1 2 a d 2
for d odd,
for d even.
15.3 Steinitz’ Polytope Representation Theorem for d = 3
The edge graph of a proper convex polytope P in E 3 is planar and 3-connected.
To see the former property, project the 1-skeleton of P from an exterior point of P
which is sufficiently close to a given relative interior point of a facet F of P onto
a plane parallel to F on the far side of P. For the latter property, see Theorem 15.8
later. The proof that, conversely, a planar, 3-connected graph can be realized as the
edge graph of a convex polytope in E 3 is due to Steinitz [963,965]. Grünbaum [453],
Sect. 13.1, stated in 1967 that this is
The most important and deepest known result on 3-polytopes ...
So far, it has resisted extension to higher dimensions. Since the determination of the
face lattices of 4-dimensional convex polytopes is NP-hard, almost surely there is no
such extension. In spite of this, the Steinitz theorem led to a collection of non-trivial
results of modern polytope theory centred around the following questions.
Problem 15.2. Characterize, among all graphs or in a given family of graphs, those
which can be realized as edge graphs of convex polytopes.
Problem 15.3. Given the edge graph of a convex polytope, what can be said about
the polytope?
In this section we give a short proof of the Steinitz theorem using the Koebe–
Brightwell–Scheinerman representation theorem 34.1 for planar graphs. Section 15.4
contains further contributions to these problems.