Gruber P. Convex and Discrete Geometry

306 Convex Polytopes
where λ is a suitable constant. Since b1,...,bn > 0, the convex polytope P(b) is
proper and thus Ai(b) >0 for certain i. Hence λ �= 0 and we obtain
αi = 1
λ Ai(b) for i = 1,...,n.
1
−
The polytope P = λ d−1 P(b) then has the desired properties.
Secondly, we show that P is unique up to translation. For d = 3, this is an
immediate consequence of Alexandrov’s uniqueness theorem 18.1. For general d,
we argue as follows:
Let P, Q ∈ Pp be two convex polytopes with u1,...,un ∈ Sd−1 and α1,...,
αn > 0 as exterior normals and areas of their facets. By Lemma 6.4, we have
Thus
V (P, Q,...,Q) = 1
d
n�
h P(ui)αi = V (P,...,P) = V (P).
i=1
V (P) d = V (P, Q,...,Q) d ≥ V (P)V (Q) d−1 , or V (P) ≥ V (Q)
by Minkowski’s first inequality, see Theorem 6.11. Similarly, V (Q) ≥ V (P) and
therefore,
V (P) = V (Q) = V (P, Q,...,Q).
Hence there is equality in Minkowski’s first inequality. Together with V (P) = V (Q)
this shows that P and Q coincide up to translation. ⊓⊔
Remark. Minkowski’s theorem is an existence and uniqueness theorem. A first
algorithm to construct P, given the exterior normal vectors and the areas of the
facets, is due to Little [662]. Complexity questions are studied by Gritzmann and
Hufnagel [395].
Minkowski’s Symmetry Condition
In some contexts, it is important to know whether a convex body or a polytope is centrally
symmetric, for example in problems dealing with packing and tiling, compare
Sects. 30.1–30.3 and 32.2. Thus symmetry criteria are of interest. As an immediate
consequence of the uniqueness statement in Minkowski’s existence and uniqueness
theorem we have the following result.
Corollary 18.1. Let P ∈ Pp. Then the following statements are equivalent:
(i) P is centrally symmetric.
(ii) For any facet of P there is a (unique) facet of P with equal area and opposite
exterior unit normal vector.