Gruber P. Convex and Discrete Geometry

19 Lattice Polytopes 327
(2) conv � S ∪{p} � = S ˙∪ conv � Fi1 ∪{p}� ˙∪··· ˙∪ conv � Fik ∪{p}�
= conv � Fj1 ∪{p}� ˙∪··· ˙∪ conv � Fjl ∪{p}� .
Here, for a facet Fin , the point p is in the interior of the halfspace aff F+
in determined
by the support hyperplane aff Fin of S, and, for a facet Fjn , the point p is in the
interior of the support halfspace aff F − of S. Then
jn
(3) S + conv � Fi1 ∪{p}� +···+conv � Fik ∪{p}� + H d p
= conv � Fj1 ∪{p}� +···+conv � Fjl ∪{p}� + H d p
by the definition of Hd p . We next show that
(4) There is a point p ∈ Z d such that
V � conv � Fi ∪{p} �� < V = V (S) for i = 0,...,d.
The proof of (4) is elementary and will only be outlined. Choose m such that
maj + 1 ≤ a1 +···+a j ≤ (m + 1)a j
and put p = (0,...,0, 1,...,1, 0,...,0) ∈ Z d with j − m − 1 zeros, m + 1
ones and d − j zeros (in this order). Considering the equations of the hyperplanes
aff Fi for i = 0, i = 1,..., j − 1, i = j, i = j + 1,...,d, it turns out that
p ∈ aff Fi for i = j + 1,...,d. For all other i the point p is closer to aff Fi than pi.
This concludes the proof of (4). Propositions (4), (3) and the induction assumption
show that S satisfies (1). The induction is thus complete and (1) holds generally,
concluding the proof of the proposition. ⊓⊔
Let G d be the free Abelian group generated by the convex lattice polytopes P ∈
P Z d and let H d be the subgroup generated by the following elements of the group
G d :
P − UP− u, P ∈ PZd , U ∈ U, u ∈ Z d
P − �
Pi + �
i
i< j
Let A be an Abelian group.
Pi ∩ Pj −··· , P = P1 ∪···∪ Pm, Pi, Pi ∩ Pj ,...∈ P Z d
Proposition 19.2. Between the integer unimodular invariant valuations φ : P Z d →A
and the homomorphisms ψ : G d /H d → A there is a one-to-one correspondence
such that
(5) φ(P) = ψ(P + H d ) for P ∈ P Z d .
Proof. The first step is to show the following statement:
(6) Let φ : P Z d → A be an integer unimodular invariant valuation. Then there
is a unique homomorphism ψ : G d /H d → A such that (5) is satisfied.
First, define a mapping ψ : G d → A as follows: for P ∈ P Z d let ψ(P) = φ(P).
Then extend ψ to a mapping ψ : G d → A by linearity (over Z d ). This is possible
since G d is the free Abelian group generated by the polytopes in P Z d . Clearly ψ is a
homomorphism. Next,