On \(\Phi\)-convexity of convex functions (Q1307287)

The authors construct a non-trivial set \(\Phi\) of extended-real valued functions on \(R^n\) containing all affine functions, such that an extended-real valued function defined on \(R^n\) is convex if and only if it is \(\Phi\)-convex, i.e., it is the pointwise supremum of some subset of \(\Phi\). They also prove a new sandwich theorem. Finally, they characterize the set \(\widetilde{\Phi}\) (\(\widetilde{\Phi} \neq \Phi\)) of all extended-real valued functions on \(R^n\) which are simultaneously convex and concave and show that an extended-real valued function is convex if and only if it is \(\widetilde{\Phi}\)-convex.