Gruber P. Convex and Discrete Geometry

120 Convex Bodies
that is, V is simply additive on L(B). Using this representation of V (A), Proposition
(1) implies the following formulae for polyboxes:
1
�
(2) V (A) = lim # A ∩
n→∞ nd 1
n Zd�
1
�
= lim # int A ∩
n→∞ nd 1
n Zd�
for A ∈ L(B).
As before, it is easy to prove the following result.
Proposition 7.2. The elementary volume on L(B) is a simple, simply additive,
homogeneous of degree d, translation invariant, non-decreasing valuation.
Note that V is not continuous on L(B) if L(B) is endowed with the topology
induced by the Hausdorff metric. To see this, let Fn, n = 1, 2,..., be finite sets
such that Fn →[0, 1] d with respect to the Hausdorff metric. Then V (Fn) = 0for
each n, butV ([0, 1] d ) = 1.
Volume, Jordan Measure or Jordan Content
AsetJ ⊆ E d is Jordan, Riemann or Peano measurable, if
sup � V (A) : A ∈ L(B), A ⊆ J � = inf � V (B) : B ∈ L(B), J ⊆ B � .
If J is Jordan measurable, then its Jordan,etc.measure, its Jordan,etc.content, or its
volume V (J) on Ed is this common value. In dimension 2 the volume is called area
and we sometimes write A(J) for V (J). LetJ = J (Ed ) be the family of Jordan
measurable sets in Ed . Each Jordan measurable set is Lebesgue measurable, but not
necessarily a Borel set. Its Jordan measure coincides with its Lebesgue measure.
Note that Jordan measure, actually, is not a measure in the sense of measure theory
since it lacks σ -additivity, and that J is a proper subfamily of the family of Lebesgue
measurable sets in Ed . As a consequence of (2) we obtain the formulae
1
�
(3) V (J) = lim # J ∩
n→∞ nd 1
n Zd�
1
�
= lim # int J ∩
n→∞ nd 1
n Zd� for J ∈ J .
These formulae will be needed in proofs of Minkowski’s fundamental theorem and
the Minkowski–Hlawka theorem. Since no further use of Jordan measure for general
sets will be made, we restrict our attention to convex and polyconvex bodies.
Volume of Convex Bodies
Convex bodies are Jordan measurable:
Theorem 7.4. C ⊆ J .
Proof. Let C ∈ C. Assume, first, that int C �= ∅. Since Jordan measure and measurability
of a set are invariant with respect to translations, we may suppose that o ∈
int C. Choose a box B ⊇ C.