Gruber P. Convex and Discrete Geometry

286 Convex Polytopes
T ∩ Z, respectively. Let ±u be the exterior normal unit vectors of S ∩ Z at the facets
F, F − lv. Clearly, ±v are the exterior normal unit vectors of T ∩ Z at the facets
G, G − lv. Choose p ∈ F and q ∈ G arbitrarily. Then
(19) The contributions of the facets F, F − lv of S ∩ Z to V (S ∩ Z) and of the
facets G, G − lv of T ∩ Z to V (T ∩ Z) are
1 � � l
p · u v(F) + (p − lv) · (−u) v(F−lv) = u · vv(F),
d
d
1 � � l
q · vv(G) + (q − lv) · (−v) v(G − lv) =
d
d v(G).
Here, the translation invariance of v was used. Clearly, G = F ′ , where “ ′ ”is
the orthogonal projection of the hyperplane HF = aff F onto the hyperplane HG =
aff G. Since S, T are not parallel, HF, HG are also not parallel. Let w : P(HF) → R
be defined by:
w(Q) = v(Q ′ ) for Q ∈ P(HF).
Since, by induction, v is a simple translation invariant, monotone valuation on
P(HG), it is easy to see, that this also holds for w on P(HF). Thus, there are two
simple, translation invariant valuations v and w on P(HF). The uniqueness theorem
16.1 then yields
w(Q) = αv(Q) for Q ∈ P(HF),
where α ≥ 0 is a suitable constant. To determine its value, choose a (d − 1)dimensional
cube K in HF of edge-length 1 such that a (d − 2)-dimensional facet of
K is contained in HF ∩ HG. Then K ′ is a box in HG of edge-lengths 1,...,1, u · v.
Thus
w(K ) = v(K ′ ) = u · v = αv(K ) = α,
i.e. α = u · v. Here we have used property (12) for v. Hence
By (19), this shows that
v(G) = v(F ′ ) = αv(F) = u · vv(F).
(20) The contributions of the facets F, F − lv of S ∩ Z to V (S ∩ Z) and of the
facets G, G − lv of T ∩ Z to V (T ∩ Z) are the same.
Having proved (18) and (20), the proof of (17) is complete, concluding the induction
for (13).
For the proof of (14), let P ∈ P and t ∈ E d .IfP is improper, V (P) = 0 =
V (P + t) since V is simple by (11), and we are done. Assume now that P is proper.
Let L be a line through o with direction t and call it vertical. Let H be the (d − 1)dimensional
subspace of E d orthogonal to L. LetF1,...,Fm be the facets of P on
the upper side and G1,...,Gn the facets on the lower side of P. The translate P + t
of P may be obtained as follows. Dissect each facet Fi into pieces Fi1,...,Fin and
each facet G j into pieces G1 j,...,Gmj such that