Gruber P. Convex and Discrete Geometry

Elementary Volume of Boxes
The elementary volume V on the space of boxes B is defined by
V (B) = �
(βi − αi) for B ={x : αi ≤ xi ≤ βi} ∈B.
i
7 Valuations 119
Easy arguments yield the following formulae to calculate the elementary volume of
boxes, where Zd is the integer (point) lattice in Ed , that is the set of all points in Ed with integer coordinates:
(1) V (B) = lim
for B ∈ B.
n→∞
1
#
nd �
B ∩ 1
n Zd�
1
�
= lim # int B ∩
n→∞ nd 1
n Zd�
The proof of the next result is left to the reader.
Proposition 7.1. The elementary volume on B is a simple, (positive) homogeneous
of degree d, translation invariant, non-decreasing, and continuous valuation.
When we say that V is a simple, (positive) homogeneous of degree k, translation
invariant, rigid motion invariant, non-decreasing or monotone valuation on B, the
following is meant:
V (B) = 0forB ∈ B, dim B < d
V (λB) = λ d V (B) for B ∈ B, λ≥ 0
V (B + t) = V (B) for B ∈ B, t ∈ E d
V (mB) = V (B) for B ∈ B and any rigid motion m of E d
V (B) ≤ V (C) for B, C ∈ B, B ⊆ C
V or −V is non-decreasing
Elementary Volume of Polyboxes
The elementary volume V on the space B of boxes is a valuation. Volland’s polytope
extension theorem 7.2 thus shows that it has a unique extension to a valuation V on
the space L(B) of polyboxes, the elementary volume on L(B). Each polybox A may
be represented in the form
A = B1 ∪···∪ Bm,
where the Bi are boxes with pairwise disjoint interiors. The inclusion-exclusion formula
for V on L(B) and the fact that V is simple on B, then yield
V (A) = V (B1) +···+V (Bm),