Abstract Convexity and Cone-vexing Abstractions

CONVEXITY AND CONE-VEXING ABSTRACTIONS 3 X × Φ and the cone ∆2(X) × {0} × R+ constitute a nonoblate pair in X × Y × R. Cones K1 and K2 in a topological vector space X are in general position provided that (1) the algebraic span of K1 and K2 is some subspace X0 ⊂ X; i.e., X0 = K1 −K2 = K2 −K1; (2) the subspace X0 is complemented; i.e., there exists a continuous projection P : X → X such that P (X) = X0; (3) K1 and K2 constitute a nonoblate pair in X0. Let σn stand for the rearrangement of coordinates σn : ((x1, y1), . . . , (xn, yn)) 7→ ((x1, . . . , xn), (y1, . . . , yn)) which establishes an isomorphism between (X × Y ) and X × Y . Sublinear operators P1, . . . , Pn : X → E ∪ {+∞} are in general position if so are the cones ∆n(X)×E n and σn(epi(P1)×· · ·×epi(Pn)). A similar terminology applies to convex operators. Given a cone K ⊂ X, put πE(K) := {T ∈ L (X,E) : Tk ≤ 0 (k ∈ K)}. We readily see that πE(K) is a cone in L (X,E). Theorem. Let K1, . . . , Kn be cones in a topological vector space X and let E be a topological Kantorovich space. If K1, . . . , Kn are in general position then πE(K1 ∩ · · · ∩Kn) = πE(K1) + · · ·+ πE(Kn). This formula opens a way to various separation results. Sandwich Theorem. Let P,Q : X → E ∪ {+∞} be sublinear operators in general position. If P (x) + Q(x) ≥ 0 for all x ∈ X then there exists a continuous linear operator T : X → E such that −Q(x) ≤ Tx ≤ P (x) (x ∈ X). Many efforts were made to abstract these results to a more general algebraic setting and, primarily, to semigroups. The relevant separation results are collected in [10]. 3. Calculus. Consider a Kantorovich space E and an arbitrary nonempty set A. Denote by l∞(A, E) the set of all order bounded mappings from A into E; i.e., f ∈ l∞(A, E) if and only if f : A → E and the set {f(α) : α ∈ A} is order bounded in E. It is easy to verify that l∞(A, E) becomes a Kantorovich space if endowed with the coordinatewise algebraic operations and order. The operator εA,E acting from l∞(A, E) into E by the rule εA,E : f 7→ sup{f(α) : α ∈ A} (f ∈ l∞(A, E))