Gruber P. Convex and Discrete Geometry

19 Lattice Polytopes 323
As a consequence of Corollary 21.2 and Theorem 21.1 the following holds. There is
an integer unimodular d × d matrix which maps the lattice lin(P + Q) ∩ Z d onto
a lattice of the form Z c in E d , where c = dim lin(P + Q) and E c is embedded
into E d as usual (first c coordinates). Then P, Q are mapped onto convex lattice
polytopes with respect to Z c the sum of which is proper. Thus we may suppose that
already P + Q ∈ P Z d p . Embed E d into E d+1 as usual, denote by “ ′ ” the orthogonal
projection from E d+1 onto E d , and let u1 = (0,...,0, 1) ∈ E d+1 . When speaking
of upper side, etc. of a convex polytope in E d+1 , this is meant with respect to the last
coordinate. Let R = conv � Q ∪{u1} � ∈ P Z d+1.
After these preparations we show that
(6) L � P + (n + 1)Q � − L(P + nQ) is a polynomial in n ∈ N.
By the above, P + nQ+ R ∈ P Z d+1 p and its upper side contains the horizontal facet
F1 = P + nQ + u1 with exterior normal vector u1 and certain “non-horizontal”
facets, say F2,...,Fm. Letu2,...,um be exterior normal vectors of these. Then
(7) F1 = P + nQ + u1,
Fi = P(ui) + nQ(ui) + R(ui) for i = 2,...,m
by (4). Next, it will be shown that
(8) u2,...,um are not orthogonal to lin Q.
For, assume that ui⊥ lin Q (⊆ E d ). Then P(ui) ⊆ P, Q(ui) ⊆ Q. Since R =
conv(Q∪{u1}), ui⊥ lin Q, u1 = (0,...,0, 1) and ui has last coordinate greater than
0, we have R(ui) ={u1}. Thus Fi = (P+nQ+R)(ui) = P(ui)+nQ(ui)+R(ui) ⊆
P + nQ+{u1} =F1 by (4) and (7). This is impossible, concluding the proof of (8).
From P, Q ∈ P Z d , R ∈ P Z d+1 and (8) it follows that
(9) P(ui), Q(ui), � R(ui) ∈ P Z d+1 and thus � R(ui) ′ ∈ P Z d for i = 1,...,m,
dim Q(ui)