Gruber P. Convex and Discrete Geometry

The Inclusion–Exclusion Formula
19 Lattice Polytopes 325
R. Stein [955] and Betke [106] proved the following result but, unfortunately, did not
publish their proofs.
Theorem 19.5. Let φ : PZd → A be an integer unimodular invariant valuation,
where A is an Abelian group. Then φ satisfies the following inclusion–exclusion
formula for lattice polytopes:
φ(P1 ∪···∪Pm) = �
φ(Pi) − �
φ(Pi ∩ Pj ) +···+(−1) m−1 φ(P1 ∩···∩Pm),
i
whenever Pi, Pi ∩ Pj,...,P1 ∩···∩ Pm, P1 ∪···∪ Pm ∈ P Z d .
Algebraic Preparations
i< j
Let Gd p be the free Abelian group generated by the proper convex lattice polytopes
P ∈ PZd p and let Hd p be its subgroup generated by the following elements of Gdp :
P − UP− u : P ∈ PZd p , U ∈ U, u ∈ Zd P − �
i
Pi : P = P1 ˙∪··· ˙∪Pm, Pi ∈ P Z d p , Pi ∩ Pj, Pi ∩ Pj ∩ Pk, ···∈P Z d
Let S0 ={o} and denote by Si the simplex conv{o, b1,...,bi } for i = 1,...,d,
where {b1,...,bd} is the standard basis of E d .
Proposition 19.1. G d p /Hd p is an infinite cyclic group generated by the coset Sd +H d p .
Proof. Let P ∈ PZd p . By Theorem 14.9, P = T1 ˙∪··· ˙∪Tm, where Ti ∈ PZd p and
Ti ∩ Tj, Ti ∩ Tj ∩ Tk, ···∈P Zd are simplices. The definition of Hd p then shows that
P + H d p = T1 + H d p +···+Tm + H d p .
For the proof of the proposition it is thus sufficient to show the following:
(1) Let S ∈ P Z d p be a simplex of volume V (V is an integer multiple of 1/d!).
Then
S + H d p = (d !V )Sd + H d p .
This will be proved by induction on d ! V .Ifd ! V = 1, there are U ∈ U, and u ∈ Z d ,
such that S = USd + u, which implies (1), on noting that
S + H d p = S − USd − u + USd + u + H d p = USd + u + H d p
= USd + u − Sd + Sd + H d p = Sd + H d p
by the definition of H d p .
Assume now that d ! V > 1 and that (1) holds for all proper convex lattice simplices
of volume less than V .LetS = conv{p0,...,pd} ∈P Z d p be a simplex with