On φ-convexity of convex functions

Abstract

We construct a non-trivial set φ of extended-real valued functions on R n, containing all affine functions, such that an extended-real valued function f on R n is convex if and only if it is φ-convex in the sense of Dolecki and Kurcyusz, i.e., the (pointwise) supremum of some subset of φ. Also, we prove a new sandwich theorem. Finally, we characterize the set of all extended-real valued functions on R n which are simultaneously convex and concave and we show that it contains properly the above set φ. Hence, a function f on R n is convex if and only if it is the (pointwise) supremum of a set of simultaneously convex and concave functions.