Prologue: This chapter provides background material from the very classical theory of convex sets. Many of the ideas here are one hundred years old or more. You will see no analysis in this chapter—only elementary geometry. Yet there are fascinating results in this elementary context: the theorems of Helly, Kirchberger, Carathéodory, and others are nontrivial and useful. And we will make good use of them later in the book when we are doing analysis. The ideas about approximating an arbitrary convex set by a more regular convex set (like a polygon or a simplex) are also useful and nontrivial. They round out the chapter and set us up for our analytic studies that begin in Chapter 2 .