On order-preserving and order-reversing mappings defined on cones of convex functions

Abstract

In this paper, we first show that for a Banach space X there is a fully order-reversing mapping T from conv(X) (the cone of all extended real-valued lower semicontinuous proper convex functions defined on X) onto itself if and only if X is reflexive and linearly isomorphic to its dual X*. Then we further prove the following generalized Artstein-Avidan-Milman representation theorem: For every fully order-reversing mapping T: conv(X) → conv(X) there exist a linear isomorphism U: X → X*, x ^*₀ , φ₀ ∈ X*, α > 0 and r₀ ∈ ℝ so that

$$(Tf)(x) = \alpha ({\cal F}f)(Ux + x_0^ *) + \left\langle {{\varphi _0},x} \right\rangle + {r_0},\;\;\;\;\;\forall x \in X,$$

where ${\cal F}$: conv(X) → conv(X*) is the Fenchel transform. Hence, these resolve two open questions. We also show several representation theorems of fully order-preserving mappings defined on certain cones of convex functions. For example, for every fully order-preserving mapping S: semn(X) → semn(X) there is a linear isomorphism U: X → X so that

$$(Sf)(x) = f(Ux),\;\;\;\;\;\forall f \in {\rm{semn}}(X),\;\;\;\;\;x \in X,$$

where semn(X) is the cone of all lower semicontinuous seminorms on X.