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!