Gruber P. Convex and Discrete Geometry

220 Convex Bodies
λC + x
νC + z
µC + y
Fig. 12.1. Choquet simplex
w ∈ C ∩ (C − λp) = (1 − λ)C, (see (1)) and w = (1 − λ)x, we have that x ∈ C.
Thus (cone C) ∩ K − ⊆ C. The proof of (2) is complete.
For each vector x parallel to the hyperplane K and such that F ∩ (F + x) �= ∅,
the set F ∩ (F + x) is a face of C ∩ (C + x), and C ∩ (C + x) is homothetic to
C by (iii). Hence F ∩ (F + x) is homothetic to F. The induction hypothesis then
shows that F is a simplex. Since o �∈ K , Proposition (2) finally implies that C is also
a simplex, concluding the proof of (i). ⊓⊔
Remark. A compact convex set C in an infinite-dimensional topological vector
space which satisfies property (iii) is called a Choquet simplex (see Fig. 12.1).
Choquet simplices are basic in Choquet theory and thus in measure theory in infinite
dimensions.
Topological Vector Lattices in Finite Dimensions
Let V be a real vector space and � a(partial) ordering of V in the following sense:
x � x for x ∈ V
x � y, y � x ⇒ x = y for x, y ∈ V
x � y, y � z ⇒ x � z for x, y, z ∈ V
〈V, �〉 is an ordered vector space if, in addition, the ordering is compatible with the
operations in V , i.e.
x � y ⇒ x + z � y + z for x, y, z ∈ V
x � y ⇒ λx � λy for x, y ∈ V, λ≥ 0
Then K ={x ∈ V : o � x} is a convex cone with apex o such that K ∩ (−K ) =
{o}, that is, K is pointed. K is called the positive cone (see Fig. 12.2) of 〈V, �〉.
Conversely, if K is a pointed convex cone in V with apex o, then the definition
(3) x � y if y − x ∈ K or, equivalently, y ∈ K + x for x, y ∈ V
makes V into an ordered vector space with positive cone K . The ordered vector space
〈V, �〉 is a vector lattice, if for any x, y ∈ V there is a greatest lower bound x ∧ y
and a smallest upper bound x ∨ y of the set {x, y} in V , that is,