Gruber P. Convex and Discrete Geometry

18 Theorems of Alexandrov, Minkowski and Lindelöf 303
Next, define a graph G on bd R as follows: in each facet of R choose a relative
interior point, a knot. Two knots are connected by an edge, that is a Jordan curve
in bd R if the facets containing the knots have a common (ordinary) edge (of R).
Clearly, the edges of G may be chosen such that they meet only at their endpoints, if
at all. Mark an edge of G by +, −, or leave it unmarked, if the corresponding edge
of R is so marked. Proposition (2) now reads as follows.
If a knot of G is the endpoint of a marked edge of G, then, on a circuit of the
knot, there are at least four changes of sign of the edges of G, omitting the
unmarked edges.
Cauchy’s combinatorial lemma 17.2 then shows that there is no marked edge of G.
This, in turn, implies that none of the edges of R is marked. Hence (2) implies the
following.
For each facet H = F +G of R the faces F and G of P and Q, respectively,
are both facets and they coincide up to translation.
Since this exhausts all facets of P and Q, we see that
The facets of P and Q appear in parallel pairs and the facets of any such
pair coincide up to translation.
Finally, building up bd P and bd Q starting with a pair of parallel facets and adding
the adjacent facets, etc., we see that bd P and bd Q and thus P and Q coincide up to
translation.
(ii)⇒(i) Trivial. ⊓⊔
Remark. This result does not extend to E d , d ≥ 4 in a straightforward way. To see
this, consider a box in E d with edge-lengths 1, 1, 3,...,3 and a cube of edge-length
2 with edges parallel to those of the box.
Problem 18.1. Find a version of Alexandrov’s theorem which holds in every
dimension.
18.2 Minkowski’s Existence Theorem and Symmetry Condition
It is a natural question to ask whether a convex polytope is determined by the areas of
its facets or the curvatures at its vertices. The first pertinent result of this type seems
to be Minkowski’s [739] existence and uniqueness theorem for convex polytopes
with given exterior normal vectors and areas of the facets. This result is the first
step in the proof of the existence of a convex body with given surface area measure,
see Sect. 10.1. A different result is Alexandrov’s theorem [16] on the existence and
uniqueness of convex polytopes with vertices on fixed rays and given corresponding
curvatures.
Here, we present Minkowski’s existence and uniqueness theorem for convex
polytopes. As an application, we show Minkowski’s condition for the central symmetry
of a convex polytope which will be used in the proof of the Venkov–McMullen
theorem on tilings in Sect. 32.2.