Fenchel’s duality theorem for nearly convex functions

We present an extension of Fenchel’s duality theorem to nearly convexity, giving weaker conditions under which it takes place. Instead of minimizing the difference between a convex and a concave function, we minimize the subtraction of a nearly concave function from a nearly convex one. The assertion in the special case of Fenchel’s duality theorem that consists in minimizing the difference between a convex function and a concave function pre-composed with a linear transformation is also proven to remain valid when one considers nearly convexity. We deliver an example where the Fenchel’s classical duality theorem is not applicable, unlike the extension we have introduced, and an application related to games theory.