Convex Operators and Supports

In this paper we show that if Z is a partially ordered linear space with the property that each set with an upper bound has a least upper bound, then any convex operator F from a linear space X into Z has a support at each x ∈ X (i.e., for each x ∈ X, there exists an affine operator A such that A(x) = F(x) and A(y) ≤ F(y) for each y ∈ X).

The results in this paper are actually in the context of a more general condition than convexity, called W-convexily. W-convexity is a pointwise property of an operator and is closely related to another generalization of convexity, called order-convexity, which was introduced by Ortega and Rheinboldt in 1967.