A Non-Convex Analogue to Fenchel Duality

We introduce and study a new notion of conjugacy, similar to Fenchel conjugacy, in a non-convex setting. Dual versions of Šmulyanʹs classical result are established in the framework of this conjugacy, which reveal a relation between well-posed problems and the differentiability. As an application we deduce the generic Fréchet differentiability of the norm ‖·‖∞ in certain spaces of bounded continuous functions (i.e., Lipα(X) for 0<α<1).