2 - Geometric Preparations

Summary

In this chapter we discuss some of the basic concepts and facts regarding the geometry of convex sets. Additional definitions and notations that are of a more limited scope will be introduced when needed. In most cases no proofs are given since these are readily available in the standard textbook literature dealing with this subject area. In particular we mention the books of Bonnesen and Fenchel (1934), Hadwiger (1957), Eggleston (1958), Valentine (1964), Leichtweiss (1980), Schneider (1993b), and Webster (1995). In fact, a large portion of this material, at least in the three-dimensional case, can already be found in the original work of Minkowski (1903, 1911). If a particular result is of importance for our objectives and if it is not textbook material we include a proof.

Basic Features of Convex Sets

As before, E^d denotes the Euclidean space of dimension d (d ≥ 2) whose points are of the form x = (x₁, …, x_d) and whose origin is o = (0, …, 0). The boundary and interior of a subset X of E^d will be denoted by ∂X and int X, respectively. A nonempty compact convex subset of E^d will be called a convex body or, more specifically, a convex body in E^d, and the class of all convex bodies in E^d will be denoted by Κ^d. If it is necessary to indicate that a convex body in E^d has interior points it will be referred to as a d-dimensional convex body.