ABSTRACT
In convex analysis, the conjugate function of f: Rn → R̄ is defined by f*(x*) = sup{<x*, x> − f(x)}. If H respresents the family of functions {hb | b ϵR̄}, with hb (t) = t − b, f*(x*) is characterized by the relation: https://www.w3.org/1998/Math/MathML"> h f * x ∗ = sup h b ∣ h b ∈ H , h b x ∗ , x ≤ f ( x ) ∀ x ⋅ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/eqn0067.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> This suggests to extend the notion of conjugation by considering other families in such a way that the essential properties of the classical theory subsist. In particular, taking the family of continuous from the left nondecreasing functions one obtains results equivalent in a sense to the theories for quasiconvex conjugation given by Crouzeix and by Greenberg and Pierskalla respectively [3]. In the same way that a closed convex function is a supremum of affine functions, quasiaffine functions are defined being in a similar relation with the quasiconvex ones.
