Gruber P. Convex and Discrete Geometry

278 Convex Polytopes
For d = 3 it follows from Steinitz’ representation theorem that each combinatorial
type of a convex polytope is rather simple. For d > 3 the universality theorem
of Richter-Gebert says that combinatorial types may be arbitrarily complicated. For
d = 3 there are asymptotic formulae due to Bender, Richmond and Wormald for the
number of combinatorial types of convex polytopes with n vertices, edges, or facets,
respectively, as n →∞.Ford > 3 upper and lower estimates for the number of
combinatorial types of convex polytopes with n vertices are due to the joint efforts
of Shemer, Goodman and Pollack and Alon.
In the following, we describe these results but give no proofs. All polytopes considered
in this section are proper convex polytopes in E d .
For more information we refer to Grünbaum [453] and the surveys of Bender
[92], Bayer and Lee [83] and Klee and Kleinschmidt [595].
Combinatorial Types and Realization Spaces
Two (proper) convex polytopes in E d are of the same combinatorial type if they have
isomorphic boundary complexes. The equivalence classes of convex polytopes of the
same combinatorial type are the combinatorial types of convex polytopes.
A more geometric way to express the fact that two convex polytopes P and Q
are of the same combinatorial type is the following. It is possible to represent P and
Q in the form
P = conv{x1,...,xn}, Q = conv{y1,...,yn},
where the xi and the yi are the vertices of P and Q, respectively, which correspond
to each other and such that
conv{xi1 ,...,xim }∈C(P) ⇔ conv{yi1 ,...,yim }∈C(Q)
for each set {i1,...,im} ⊆{1,...,n}.
For the investigation of a combinatorial type it is sometimes more convenient
to study a proper sub-space (which easily yields the whole combinatorial type). Fix
affinely independent points x1,...,xd+1 in E d and let P = conv{x1,...,xn} be
a proper convex polytope with vertices xi such that the vertices x2,...,xd+1 are
adjacent to the vertex x1.Therealization space R(P) of P is the family of all d × n
matrices (y1,...,yn) ∈ E dn such that the convex polytopes P = conv{x1,...,xn}
and Q = conv{y1,...,yn} are of the same combinatorial type with corresponding
vertices xi and yi, where x1 = y1,...,xd+1 = yd+1. See Richter-Gebert and Ziegler
[834] and Richter-Gebert [833].
Primary Semi-Algebraic Sets
A primary semi-algebraic set A (over Z)inE N is a set of the form
A = � x ∈ E N : p1(x) =···= pk(x) = 0, q1(x),...,ql(x) >0 � ,