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

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 the 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)^(*),φ_(0)∈X^(*),α>0 and r_0∈R so that(Tf)(x)=α(Ff)(Ux+x_(0)^(*))+<φ_(0),x>+r_(0),■x∈X where T: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),■f∈semn(X),x∈X where semn(X)is the cone of all the lower semicontinuous seminorms on X.