On Φ-convexity of convex functions

We construct a non-trivial set Φ of extended-real valued functions on ℝn, containing all affine functions, such that an extended-real valued function f on ℝn 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 ℝn which are simultaneously convex and concave and we show that it contains properly the above set Φ. Hence, a function f on ℝn is convex if and only if it is the (pointwise) supremum of a set of simultaneously convex and concave functions. © 1998 Elsevier Science Inc. All rights reserved.