Gruber P. Convex and Discrete Geometry

324 Convex Polytopes
L � P + (n + 1)Q � = �
i
L(F ′
i
) − �
= L(P + nQ) +
= L(P + nQ) +
i< j
L(F ′
i ∩ F′ j ) +···
m�
i=2
L(F ′
i
�
) − L � (Fi ∩ Fj) ′� +···
i< j
m�
L � P(ui) + nQ(ui) + R(ui) ′�
i=2
− �
L � P � {ui, u j } � + nQ � {ui, u j } � + R � {ui, u j } � ′�
+··· .
i< j
This, combined with (9), (10) and the induction assumption, shows that L(P +
(n + 1)Q) − L(P + nQ) is a polynomial in n ∈ N, concluding the proof of (6).
Having proved (6), Lemma 19.2 shows that L(P +nQ) is a polynomial in n ∈ N.
If K is a lattice cube such that P, Q ⊆ K , then P + nQ ⊆ (n + 1)K and thus
L(P + nQ) ≤ L((n + 1)K ) = O(n d ). Hence the polynomial L(P + nQ) has degree
≤ d. Thus (5) holds for Q ∈ P Z d with dim Q = l. The induction is thus complete
and (5) holds generally.
The theorem finally follows from (5) and Lemma 19.1. ⊓⊔
19.4 The Betke–Kneser Theorem on Valuations
Many of the functions φ on the space P Z d of convex lattice polytopes which have
been studied are valuations with values in R or in some Abelian group A. This means
that
φ(P ∪ Q) + φ(P ∩ Q) = φ(P) + φ(Q)
whenever P, Q, P ∪ Q, P ∩ Q ∈ P Z d , and φ(∅) = 0.
Examples are the volume and the lattice point enumerators. Among such valuations
many are integer unimodular invariant. By this we mean that
φ(P) = φ(UP+ u) for P ∈ P Z d , U ∈ U, u ∈ Z d ,
where U is the family of all integer unimodular d × d matrices. A central result in
this context due to Betke [105] and Betke and Kneser [108] shows that the structure
of the linear space of the real, integer unimodular invariant valuations on P Z d is surprisingly
simple. It parallels Hadwiger’s functional theorem 7.9 for real, continuous
and motion invariant valuations on the space C of convex bodies. The theorem of
Betke and Kneser yields simple proofs of the results of Ehrhart [292, 293] on lattice
point enumerators.
In the following we state without proof an unpublished result of Stein and Betke
on the inclusion–exclusion principle, give a proof of the Betke–Kneser theorem and
show how it implies the results of Ehrhart. We will make use of some algebraic tools.
For more information, see the survey of McMullen [714] and the references in
the introduction of Sect. 19, to which we add Kantor [564].