The Characterizations Of Order Preserving And Order Reversing Mappings In Convex Analysis And Its Applications

This dissertation mainly investigates the characterizations of fully order preserv-ing and fully order reversing mappings defined on the cone Conv(X)of extended real-valued proper convex functions defined on general Banach space X.An elegant theorem of Artstein-Avidan and Milman states that every fully order reversing(resp.preserving)mappings defined from Conv(Rn)onto itself is essentially the Legendre(reps,identity)transform.The question which is apparently natural and worth consid-ering is the existence and the representation theorem of fully order reversing mappings defined from Conv(X)onto itself for general Banach space X.We mainly investigate these problems and resolve them completely.In this dissertation,we make use of sev-eral theory besides functional analysis and convex analysis,such as affine geometry,the characterization of the state space of partially ordered Abelian group with order unit,ect.The main results of this dissertation is the following.Firstly,we investigate the properties of fully order preserving mappings on the "sup-generating" class of convex functions,and then provide a necessary and sufficient condition of the ex-istence of fully order reversing mappings defined from Conv(X).Besides,we prove that if a Banach space X satisfies this condition then the "Artstein-Avidan-Milman" representation theorem holds,and hence,resolve the basic problems stated as above.Secondly,we generate the classic fundamental theorem of affine geometry to infinite dimensional setting.Combining with subdifferential operator and the funda-mental theorem we provide representation theorems of fully order preserving mapping defined on certain classes of convex functions,such as sublinear functions,Minkows-ki functions,seminorms,equivalent norms,ect.Thirdly,we make use of theory of the affine geometry,"minimax" theorem and the characterization of state spaces to investigate the "local" version of the "Artstein-Avidan-Milman" representation the-orem.We prove that in many meaningful setting the fully order preserving mappings is essentially the identity.Finally,we make use of convex analysis techniques to prove several classic results in operator theory.