Convexity is one of the oldest concepts in mathematics. It already appears in the works of Archimedes, around three centuries B.C. It was not until the 1950s, however, that this theme developed widely in the works of modern mathematicians. Convexity is a fundamental notion for computational geometry, at the core of many computer engineering applications, for instance in robotics, computer graphics, or optimization.

A convex set has the basic property that it contains the segment joining any two of its points. This property guarantees that a convex object has no hole or bump, is not hollow, and always contains its center of gravity. Convexity is a purely affine notion: no norm or distance is needed to express the property of being convex. Any convex set can be expressed as the convex hull of a certain point set, that is, the smallest convex set that contains those points. It can also be expressed as the intersection of a set of half-spaces. In the following chapters, we will be interested in linear convex sets. These can be defined as convex hulls of a finite number of points, or intersections of a finite number of half-spaces. Traditionally, a bounded linear convex set is called a polytope. We follow the tradition here, but we understand the word polytope as a shorthand for bounded polytope. This lets us speak of an unbounded polytope for the non-bounded intersection of a finite set of half-spaces.