A localization of the Artstein-Avidan-Milman theorem

Abstract

Assume that C is a nonempty convex subset of Rn such that C is not a singleton, and Conv(C) is the set of all extended-real-valued lower semi-continuous proper convex functions defined on C. In this paper, we localize the “Artstein-Avidan-Milman” representation theorem for order preserving mappings in the following manner: For every full order preserving mapping φ:Conv(C)→Conv(C), there exist a linear isomorphism E:span(C)→span(C), x0∈span(C), u0,v0∈(Rn)⁎/span(C)⊥ and α,β∈R such that(i)〈u0,x〉+α>0 for all x∈C;(ii)the restriction E|VC is a bijection from VC:=⋃λ≥0λC onto itself;(iii)the mapping F:C→C defined by F(x)=Ex+x0〈u0,x〉+α for x∈C is a fractional linear isomorphism from C onto itself;(iv)for every f∈Conv(C), x∈C, we have(φf)(x)=(〈u0,x〉+α)f(Ex+x0〈u0,x〉+α)+〈v0,x〉+β. In particular, if C=Rn, our result reduces to the classical “Artstein-Avidan-Milman” theorem. We also show that there is no nontrivial such localization of the “Artstein-Avidan-Milman” theorem for order reversing mappings.