Gruber P. Convex and Discrete Geometry

y
o x
x ∨ y
K
12 Special Convex Bodies 221
K + y
Fig. 12.2. Positive cone, vector lattice
(K + x) ∧ (K + y) = K + x ∨ y
K + x
x ∧ y � x, y and if z � x, y, then z � x ∧ y,
and similarly for x ∨ y.
V is a topological vector space if V is Hausdorff and the mappings (x, y) →
x + y and (λ, x) → λx are continuous. Up to isomorphisms, E d is the only
d-dimensional (real) topological vector space (forget the norm, but retain the
topology).
If the vector space V is ordered and topological, it is called an ordered topological
vector space if the ordering is compatible with the topology in the sense that
o � A ⇒ o � cl A for A ⊆ V.
This is equivalent to the requirement that the positive cone is closed.
In the following we consider the finite-dimensional case. Let K be a pointed
closed convex cone in E d with apex o. Then there is a hyperplane H with o �∈ H
and such that C = H ∩ K is a convex body. Clearly, K = pos C ={λx : λ ≥ 0,
x ∈ C}. K is a simplicial cone if C is a simplex. It is easy to see that this definition
is independent of the choice of H.
Choquet’s Characterization of Topological Vector Lattices
We conclude this section with a finite-dimensional case of Choquet’s theorem on
vector lattices, see [208].
Theorem 12.2. Let K be a pointed closed convex cone in E d with apex o which
makes E d into an ordered topological vector space. Then the following statements
are equivalent:
(i) E d is a vector lattice with positive cone K .
(ii) K is a simplicial cone with dim K = d.
Proof. (i)⇒(ii) In the following the above properties and definitions will be applied
several times without explicit reference. The first step is to show the following:
(4) Let x, y ∈ E d . Then (K + x) ∩ (K + y) = K + x ∨ y.