Unconventional Fenchel Duality

In the fifth chapter of this work, we give some new insights into the classical Fenchel duality. The first two sections deal with the concept of totally Fenchel unstable functions introduced by Stephen Simons in [120], while in the remaining sections we turn our attention to the study of some “unconventional” regularity conditions for Fenchel duality expressed via the quasi interior and the quasi-relative interior of the domains and epigraphs of the functions involved. Let X be a separated locally convex space, X ∗ its topological dual space and $$f, g : X \rightarrow \overline{\mathbb{R}}$$ two arbitrary proper functions. According to the terminology used in Section 5 (see also Definition 5.4), we say that f and g satisfy stable Fenchel duality if for all $$x^{\ast} \in X^{\ast}$$ , there exists $$y^{\ast} \in X^{\ast}$$ such that $$(f + g)^{\ast}(x^{\ast}) = f^{\ast}(x^{\ast} - y^{\ast}) + g^{\ast}(y^{\ast})$$ . If this property holds for x ∗ = 0, then f; g satisfy the classical Fenchel duality. Due to Stephen Simons (see [120]), the pair f; g is said to be totally Fenchel unstable if f and g satisfy Fenchel duality, but $$y^{\ast}, z^{\ast} \in X^{\ast}\ {\rm and}\ (f + g)^{\ast}(y^{\ast} + z^{\ast}) = f^{\ast}(y^{\ast}) + g^{\ast}(z^{\ast}) \Longrightarrow y^{\ast} + z^{\ast} = 0.$$