Gruber P. Convex and Discrete Geometry

19 Lattice Polytopes 329
Proposition 19.3. G d /H d is the free Abelian group generated by the cosets S0 +
H d ,...,Sd + H d .
Proof. By induction on d we prove that
(9) S0 + H d ,...,Sd + H d generate G d /H d .
For d = 0, (9) is trivial. Assume now that d > 0 and that (9) holds for d − 1. Let
Ed−1 be embedded into Ed as usual (first d − 1 coordinates). Then Gd is generated
by (the elements of) Gd p and Gd−1 . According to Proposition 19.1, Gd p is generated
by Sd and Hd p and Gd−1 is generated by S0,...,Sd−1 and Hd−1 (⊆ Hd ) by the
induction assumption. Thus Gd is generated by S0,...,Sd and Hd . This concludes
the induction and thus yields (9).
It remains to show that
(10) S0 + H d ,...,Sd + H d are linearly independent (with respect to Z).
A simple induction argument shows that
� �
n + i
L(nSi) = for n ∈ N, i = 0,...,d.
i
For n ∈ N define φn : P Z d → R by
φn(P) = L(nP) for P ∈ P Z d .
Since the lattice point enumerator L : P Z d → R is a valuation, each φn is also a
valuation. Let µn : G d /H d → R be the corresponding homomorphism, see Proposition
19.2. If (10) did not hold, then
for suitable integers ai, not all 0. Then
a0S0 +···+ad Sd + H d = H d
0 = µn(a0S0 +···+ad Sd + H d )
= a0µn(S0 + H d ) +···+adµn(Sd + H d )
= a0φn(S0) +···+adφn(Sd) = a0L(nS0) +···+ad L(nSd)
� � � � � �
n + 0 n + 1
n + d
= a0 + a1 +···+ad for n ∈ N.
0
1
d
Since � n+i �
i is a polynomial in n of degree i, this can hold only if ad =···=a0 = 0.
This contradiction concludes the proof of (10).
Claims (9) and (10) together yield Proposition 19.3. ⊓⊔
The Theorem of Betke and Kneser
As a corollary of the above results, Betke and Kneser proved the following result.