Gruber P. Convex and Discrete Geometry

116 Convex Bodies
If P has dimension less than d, (10) holds by the induction hypothesis. Thus we may
assume that P has dimension d. Then in each of the cases (i) and (ii) we have that
Q = P1 ∪···∪ Pm−1 ∈ P and P = Q (note that P is convex) and thus, trivially,
P = Q ∪ Pm. Since φ is a valuation:
(11) φ(P) = φ(Q) + φ(Pm) − φ(Q ∩ Pm).
Since Q = P1 ∪···∪ Pm−1 and Q ∩ Pm = (P1 ∩ Pm) ∪···∪(Pm−1 ∩ Pm), and
(10) holds for d and m − 1 by induction, the equality (11) readily leads to (10).
In the second step, the general case is traced back to the cases (i) and (ii). If P1
has dimension less than d or coincides with P, (10) holds since the cases (i) and (ii)
are already settled. Thus we may suppose that P1 has dimension d and is a proper
subpolytope of P. Then we proceed as follows: Clearly, P1 has facets which meet
int P. Given such a facet, let H be the hyperplane containing it and let H − (⊇ P1) and
H + be the closed halfspaces determined by H. Then P − = H − ∩ P, P + = H + ∩ P,
P − ∪ P + = P ∈ P and since φ is a valuation, we have:
(12) φ(P) = φ(P − ) + φ(P + ) − φ(P − ∩ P + ).
Clearly, P +
i = H + ∩ Pi, P −
i = H − ∩ Pi ∈ P for i = 1,...,m. This leads to the
representations
(13) P− = P −
1 ∪···∪ P− m , P+ = P +
1 ∪···∪ P+ m ,
P− ∩ P + = (P −
1 ∩ P+
1 ) ∪···∪(P− m ∩ P+ m ) and
φ(PI ) = φ(P − I ) + φ(P+ I ) − φ(P− I ∩ P+ I ) since φ is a valuation.
dim P +
1 < d and dim(P− ∩ P + )