Essential smoothness and essential strict convexity

This chapter is dedicated to the study of convex functions whose smoothness and curvature properties are preserved by Fenchel conjugation.

Definition 7.1.1. We will say a proper convex lower-semicontinuous function f : ℝN → (-∞,+∞] is:

1. (a) essentially smooth in the classical sense if it is differentiable on int dom f ≠, and ∥∇f(xn)∥ → ∞ whenever xn → x ∈ bdry dom f;

2. (b) essentially strictly convex in the classical sense, if it is strictly convex on every convex subset of dom ∂f;

3. (c) Legendre in the classical sense, if it is both essentially smooth and essentially strictly convex in the classical sense.

The duality theory for these classical functions is presented in [369, Section 26]. The qualification in the classical sense is used to distinguish the definition on ℝn from the alternate definition for general Banach spaces given below which allows the classical duality results to extend to reflexive Banach spaces as shown in [35], which we follow.

Continuity and subdifferentials

Throughout the rest of the book X denotes a normed or a Banach space. We assume familiarity with basic results and language from functional analysis, and we will often draw upon the classical spaces ℓp, c0 and others for examples. Notwithstanding, we will sketch some of the classic convexity theorems in our presentation.

We write X* for the real dual space of continuous linear functionals, and denote the unit ball and sphere by BX ≔ {x ∈ X : ∥x∥ ≤ 1} and SX ≔ {x ∈ X : ∥x∥ = 1}, respectively. We say that a subset C of X is convex if, for any x, y ∈ C and any λ ∈ [0, 1], λx + (1 - λ)y ∈ C.

This book on convex functions emerges out of 15 years of collaboration between the authors. It is far from being the first on the subject nor will it be the last. It is neither a book on convex analysis such as Rockafellar's foundational 1970 book [369] nor a book on convex programming such as Boyd and Vandenberghe's excellent recent text [128]. There are a number of fine books – both recent and less so – on both those subjects or on convexity and relatedly on variational analysis. Books such as [371, 255, 378, 256, 121, 96, 323, 332] complement or overlap in various ways with our own focus which is to explore the interplay between the structure of a normed space and the properties of convex functions which can exist thereon. In some ways, among the most similar books to ours are those of Phelps [349] and of Giles [229] in that both also straddle the fields of geometric functional analysis and convex analysis – but without the convex function itself being the central character.

We have structured this book so as to accommodate a variety of readers. This leads to some intentional repetition. Chapter 1 makes the case for the ubiquity of convexity, largely by way of examples, many but not all of which are followed up in later chapters. Chapter 2 then provides a foundation for the study of convex functions in Euclidean (finite-dimensional) space, and Chapter 3 reprises important special structures such as polyhedrality, eigenvalue optimization and semidefinite programming.

Back to the finite

We finish this book by reprising some of the ways in which finite-dimensionality has played a critical role in the previous chapters. While our list is far from complete it should help illuminate the places in which care is appropriate when ‘generalizing’. Many of our results have effectively the same proof in Banach spaces as they do in Euclidean spaces.

The general picture

For example, the equivalence of local boundedness and local Lipschitz properties for convex functions is a purely geometric argument that does not depend on the dimension of the space (compare Theorem 2.1.10 and Proposition 4.1.4). Consequently, in finite dimensions a real-valued convex functions is automatically continuous essentially because a simplex has nonempty interior (see Theorem 2.1.12). This argument clearly fails in infinite-dimensional spaces, and the result fails badly as there are discontinuous linear functionals on every infinite-dimensional Banach space (Exercise 4.1.22).

Why ‘convex’?

This introductory polemic makes the case for a study focusing on convex functions and their structural properties. We highlight the centrality of convexity and give a selection of salient examples and applications; many will be revisited in more detail later in the text – and many other examples are salted among later chapters.