Gruber P. Convex and Discrete Geometry

330 Convex Polytopes
Theorem 19.6. The integer unimodular invariant valuations φ : P Z d → R (with
ordinary addition and multiplication by real numbers) form a real vector space of
dimension d + 1. This space has a basis {L0,...,Ld} such that
Li(nP) = n i Li(P) and L(nP) = L0(P) + L1(P)n +···+Ld(P)n d
for P ∈ P Z d , n ∈ N, i = 0,...,d.
Proof. We first show that
(11) The homomorphisms ψ : G d /H d → R with ordinary addition and multiplication
with real numbers form a real vector space of dimension d + 1.
These homomorphisms clearly form a real vector space. For the proof that it is of dimension
d +1, use Proposition 19.3 to define d +1 homomorphisms ψi : G d /H d →
R, i = 0,...,d, by:
ψi(a0S0 +···+ad Sd + H d ) = ai for a0S0 +···+ad Sd + H d ∈ G d /H d .
To show that ψ0,...,ψd are linearly independent, let α0,...,αd ∈ R be such that
α0ψ0 +···+αdψd = 0. Then
0 = α0ψ0(a0S0 +···+ad Sd + H d ) +···+αdψd(a0S0 +···+ad Sd + H d )
= α0a0 +···+αdad for all a0,...,ad ∈ Z,
which implies that α0 = ··· = αd = 0 and thus shows the linear independence
of ψ0,...,ψd. To show that ψ0,...,ψd form a basis, let ψ : G d /H d → R be a
homomorphism. Then
ψ(a0S0 +···+ad Sd + H d ) = a0ψ(S0 + H d ) +···+adψ(Sd + H d )
= a0β0 +···+adβd say, where βi = ψ(Si + H d )
= β0ψ0(a0S0 +···+ad Sd + H d ) +···+βdψd(a0S0 +···+ad Sd + H d )
for all a0S0 +···+ad Sd + H d ∈ G d /H d .
Hence ψ = β0ψ0 +···+βdψd. Thus {ψ0,...,ψd} is a basis, concluding the proof
of (11).
By Proposition 19.2, the integer unimodular invariant valuations φ : P Z d → R
coincide with the restrictions of the homomorphisms ψ : G d /H d → R to a certain
subset of G d /H d . Thus (11) implies that
(12) The integer unimodular invariant valuations φ : P Z d → R form a real
vector space of dimension at most d + 1.
We next show that
(13) There are d + 1 linearly independent valuations L0,...,Ld : P Z d → R
such that
Li(nP) = n i Li(P), L(nP) = L0(P) + L1(P)n +···+Ld(P)n d
for P ∈ P Z d , n ∈ N, i = 0,...,d.