Gruber P. Convex and Discrete Geometry

272 Convex Polytopes
A problem of Steinitz [964] asks whether each convex polytope in E d is circumscribable
that is, it is isomorphic to a convex polytope circumscribed to B d and such
that each of the facets of this polytope touches B d . A similar problem of Steiner [961]
deals with inscribable convex polytopes, i.e. there is an isomorphic convex polytope
contained in B d such that all its vertices are on the boundary of B d . Steinitz [964]
proved the existence of a non-circumscribable convex polytope in E 3 . Schulte [917]
proved the existence of non-inscribable convex polytopes in E d for d ≥ 4. For more
information, see Grünbaum [453], Grünbaum and Shephard [455] and Florian [337].
Klee asked whether each convex polytope in E d is rationally representable, that
is, it is isomorphic to a convex polytope in E d , all vertices of which have rational
coordinates. The affirmative answer for d = 3 follows from a proof of the Steinitz
representation theorem where all steps may be carried out in the rational space Q 3 ,
see Grünbaum [453]. For d ≥ 8 Perles proved that there are convex polytopes which
cannot be represented rationally, see Grünbaum [453], p. 94. Richter-Gebert [833]
could show that this holds already for d ≥ 4. For more information compare Bayer
and Lee [83] and Richter-Gebert [833].
15.4 Graphs, Complexes, and Convex Polytopes for General d
There are many connections between graphs and complexes on the one hand and
convex polytopes on the other hand. A first such result is Steinitz’ representation
theorem for d = 3, see the preceding section where we also mentioned two basic
problems. Important later contributions are due to Grünbaum, Perles, Blind, Mani,
Kalai and others.
One of the great problems in this context is the following:
Problem 15.4. Characterize, among all polytopal complexes, those which are isomorphic
to boundary complexes of convex polytopes of dimension d.
This problem remains open, except for d = 3, in which case the Steinitz representation
theorem provides an answer. More accessible is the following
Problem 15.5. Given suitable sub-complexes of the boundary complex of a convex
polytope, what can be said about the polytope?
In this section we first show that there is an algorithm to decide whether
an abstract graph can be realized as the edge graph of a convex polytope. Then
Balinski’s theorem on the connectivity of edge graphs is presented. Next, we give
the theorem of Perles, Blind and Mani together with Kalai’s proof, which says that,
for simple convex polytopes, the edge graph determines the combinatorial structure
of the polytope. For general convex polytopes, no such result can hold. Finally, we
give a short report on related problems for complexes instead of graphs.
For more information, see the books of Grünbaum [453] and Ziegler [1045].