Gruber P. Convex and Discrete Geometry

314 Convex Polytopes
To see this, note that
L(n int S) = L(nS) − �
L(nR) + �
L(nQ) −+··· ,
R
where R ranges over all facets of S, Q over all (d − 2)-dimensional faces, etc. Now
take into account (6) and (9) to see that L(n int S) = qS(n) where qS is a polynomial
of degree d with leading coefficient V (S) and constant term
� � � �
d + 1 d + 1
1 − + −+···+(−1)
1 2
d
� �
d + 1
= (−1)
d
d ,
concluding the proof of (10).
Analogous to the derivation of (9) from (6), the following proposition is a consequence
of (10).
(11) Let R ∈ P Z d be a simplex with c = dim R < d. Then L(n relint R) =
qR(n) for n ∈ N, where qR is a polynomial of degree < d with constant
term (−1) c .
In the fourth, and last, step of the proof of (i), we extend (6) from simplices to
convex polytopes.
(12) Let P ∈ P Z d p . Then L(nP) = pP(n) for n ∈ N, where pP is a polynomial
of degree d with leading coefficient V (P) and constant term 1.
By Theorem 14.9, the proper lattice polytope P is the union of all simplices of a
suitable simplicial complex where all simplices are lattice simplices. Hence P is the
disjoint union of the relative interiors of these simplices. See, e.g. Alexandroff and
Hopf [9], p. 128. Propositions (10) and (11) thus yield (12), except for the statement
that the constant term is 1. To see this note that the Euler characteristic of a simplicial
complex is the number of its vertices minus the number of its edges plus the number
of its 2-dimensional simplices, etc. This together with (10) and (11) shows that the
constant term in pP is just the Euler characteristic of the simplicial complex; but
this equals the Euler characteristic of P and is thus 1. The proof of (12) and thus of
Proposition (i) is complete.
(ii) Since the proof of (ii) is similar to that of (i), some details are omitted. The
first step is to show the following analogue of (1).
(13) Let S ∈ P Z d p be a simplex. Then
∞�
n=0
L o (nS)t n �
= r(t) 1 +
� d + 1
1
�
t +
Q
� d + 2
2
�
t 2 �
+··· for |t| < 1,
where r is a polynomial with integer coefficients of degree ≤ d + 1.
Instead of the fundamental parallelotope F in the proof of (1), here the fundamental
parallelotope G = � x = α1q1 +···+αd+1qd+1 : 0