Further Differentiability Results

Abstract

Further results concerning derivatives and subgradients are collected in this chapter. The Ekeland–Lebourg theorem gives conditions under which the set of points of Fréchet differentiability is a dense G_δ subset of the domain of the function. Formulas for the subdifferential of a maximum and of an infimal convolution are provided, and the basic duality between differentiability and strict convexity is presented. Another highlight of this chapter is the Baillon–Haddad theorem, which states that nonexpansiveness and firm nonexpansiveness are identical properties for gradients of convex functions. Finally, the subdifferential operator of the distance to a convex set is analyzed in detail.