Gruber P. Convex and Discrete Geometry

304 Convex Polytopes
For other pertinent results, see Alexandrov [16]. See also the report in Schneider
[907].
Minkowski’s Existence and Uniqueness Theorem for Convex Polytopes
Minkowski [739] proved the following result; the proof is taken from Alexandrov
[16]. It makes use of the Lagrange multiplier theorem from calculus.
Theorem 18.2. Let u1,...,un ∈ S d−1 and α1,...,αn > 0. Then the following
statements are equivalent:
(i) u1,...,un and α1,...,αn are the exterior normal unit vectors and the areas of
the corresponding facets of a proper convex polytope P in Ed which is unique up
to translation.
(ii) u1,...,un are not contained in a halfspace whose boundary hyperplane contains
o and
n�
αiui = o.
i=1
Proof. (i)⇒(ii) Clearly, P = � x : ui · x ≤ h P(ui), i = 1,...,n � .Letx∈ P. If
there were a vector u �= o such that ui · u ≤ 0fori = 1,...,n, then x + λu ∈ P for
all λ ≥ 0. This contradicts the boundedness of P and thus proves the first assertion
in (ii). To see the second assertion, note that
�
n�
�
· u = �
αi ui · u − �
αi ui · (−u) for each u ∈ S d−1 .
i=1
αiui
ui ·u>0
ui ·u0 and thus ui · u ≤ 0fori = 1,...,n,
in contradiction to (ii), concluding � the proof of (1).
For s1,...,sn ≥ 0, let Ai s = (s1,...,sn) � be the area of the (possibly empty)
face Fi(s) = P(s) ∩{x : ui · x = si} of P(s). (IfFi(s) =∅, put Ai(s) = 0.) Then