Gruber P. Convex and Discrete Geometry

15 Combinatorial Theory of Convex Polytopes 279
where N ∈ N and the pi and the q j are real polynomials on E N with integer coefficients.
For example, {0, 1} and ]0, 1[ are primary semi-algebraic sets, but (0, 1] and
[0, 1] are not.
On the family of all primary semi-algebraic sets, there is an equivalence relation,
called stable equivalence, which preserves certain geometric properties, for example
the homotopy type, see [834].
What do Realization Spaces Look Like?
It is easy to show that the realization space of a convex polytope P is a primary
semi-algebraic set. (The polynomials pi and q j are formed with determinants of
d × d minors of the d × n matrices x = (x1,...,xn) ∈ E N .)
The following result, for d = 3, follows from a close inspection of a proof of the
Steinitz representation theorem, see Richter-Gebert [833].
Theorem 15.10. The realization space of a proper convex polytope P in E 3 is a
smooth open ball in E f1−6 , where f1 is the number of edges of P.
For d > 3 the situation is much more involved. Improving upon a deep theorem
of Mnëv [746], Richter-Gebert [833] proved the following universality theorem.
Theorem 15.11. For every primary semi-algebraic set A, there is a (not necessarily
proper) convex polytope P in E 4 , such that A is stably equivalent to R(P).
Tools for the proof are so-called Lawrence extensions and connected sums. These
are elementary geometric operations on polytopes.
Remark. The universality theorem shows that the realization spaces of convex polytopes
may be arbitrarily complicated. This has been interpreted to indicate that there
might not exist a reasonable extension of the representation theorem of Steinitz to
higher dimensions. The universality theorem has a series of consequences, for example
the following: There is a convex polytope in E 4 which admits no realization with
rational vertices, compare the discussion in Sect. 15.3.
How Many Combinatorial Types are There?
For d = 3 results of Bender and Wormald [94] and Bender and Richmond [93] yield
asymptotic formulae for the number of combinatorial types of convex polytopes with
n vertices, edges, or faces, as n →∞. For a survey, see Bender [92].
For d > 3 results of Shemer [930], Goodman and Pollack [385] and Alon [25]
together yield the following estimates.
Theorem 15.12. Let cs(d, n) and c(d, n) be the numbers of combinatorial types of
proper simplicial and general convex polytopes in E d with n vertices. Then
�
n − d
� nd
4
≤ cs(d, n) ≤ c(d, n) ≤
d
�
n
�d2n(1+o(1)) as n →∞.
d