Convex sets

Convexity is a very intuitive and geometric notion, but plays a fundamental role in many (also abstract) branches of mathematics. In vector spaces, this intuitive notion can be easily defined in a formal way: a convex set is a set which completely contains any segment connecting two of its points. This is the most central concept in this book. Some remarkable aspects of this idea are discussed in the Notes section at the end of this chapter, and a historical overview of the theory is given in Chapter 12 . For now, we highlight that the simple notion of convexity gives rise to a very rich and deep theory, as it particularly becomes clear also in this chapter.