1 - Introduction

Summary

Convex polytopes, or simply polytopes, are geometric objects in some space $\R^{d}$; in fact, they are bounded intersections of finitely many closed halfspaces in $\R^{d}$.The space $\R^{d}$ can be regarded as a linear space or an affine space, and its linear or affine subspaces can be described by linear or affine equations. We introduce the basic concepts and results from linear algebra that allow the description and analysis of these subspaces. A polytope can alternatively be described as the convex hull of a finite set of points in $\R^{d}$, and so it is a convex set. Convex sets are therefore introduced, as well as their topological properties, with emphasis on relative notions as these are based on a more natural setting, the affine hull of the set. We then review the separation and support of convex sets by hyperplanes. A convex set is formed by fitting together other polytopes of smaller dimensions, its faces; Section 1.7 discusses them.Finally, the chapter studies convex cones and lineality spaces of convex sets; these sets are closely connected to the structure of unbounded convex sets.