This final chapter explores the history of convexity and provides comments on some of the themes discussed earlier in the book. There are varied historical roots to the study of convexity with input from both applied and pure sources and, sometimes, long delays between seminal work and its absorption into the mainstream.

One of the earliest discoverers of the wonders of multidimensional convex functions was Josiah Willard Gibbs in three remarkable papers [128, 129, 130] published from 1873 to 1878 in an obscure American journal. These papers on thermodynamics predated his later celebrated work in statistical mechanics. The content of the papers and their reception is discussed in detail in an historical overview on the use of convexity in thermal physics by Wightman [388].

To Gibbs, thermodynamic stability implied that internal energy of a system is a function of entropy and volume had to be convex, and this persisted to convexity in additional variables in multicomponent systems. For Gibbs, coexistence of phases corresponded to the convex function having a flat piece on its graph – or in modern parlance, to its Legendre transform having a multidimensional set of tangents. Gibbs also understood the role of certain Legendre transforms in thermodynamics and understood some other relations between multiple supporting hyperplanes for the Legendre transform and flat sections in the graph of the original function. Many of his deepest ideas lay dormant for about seventy-five years!

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.