Convexity of sets and functions are extremely simple notions to define, so it may be somewhat surprising the depth and breadth of ideas that these notions give rise to. It turns out that convexity is central to a vast number of applied areas, including Statistical Mechanics, Thermodynamics, Mathematical Economics, and Statistics, and that many inequalities, including Hölder's and Minkowski's inequalities, are related to convexity.

An introductory chapter (1) includes a study of regularity properties of convex functions, some inequalities (Hölder, Minkowski, and Jensen), the Hahn–Banach theorem as a statement about extending tangents to convex functions, and the introduction of two constructions that will play major roles later in this book: the Minkowski gauge of a convex set and the Legendre transform of a function.

The remainder of the book is roughly in four parts: convexity and topology on infinite-dimensional spaces (Chapters 2–5); Loewner's theorem (Chapters 6–7); extreme points of convex sets and related issues, including the Krein–Milman theorem and Choquet theory (Chapters 8–11); and a discussion of convexity and inequalities (Chapters 12–16).

The first part begins with a study of Orlicz spaces in Chapter 2, a notion that extends Lp. The most interesting new example is L1 log L but the theory also illustrates parts of Lp theory. Chapter 3 introduces the notion of locally convex spaces and includes a discussion of Lp and Hp for 0 < p < 1 to illustrate what can happen in nonlocally convex spaces.