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Convexity: Extreme points and the Krein–Milman theorem

The next four chapters will focus on an important geometric aspect of compact sets, namely, the role of extreme points where: Definition An extreme point of a convex set, A, is a point x ∈ A, with the property that if x = θy + (1 − θ)z with y, z ∈ A and θ ∈ [0, 1], then y = x and/or z = x. E(A) will denote the set of extreme points of A. In other words, an extreme point is a point that is not an interior point of any line segment lying entirely in A. This chapter will prove a point is a limit of convex combinations of extreme points and the following chapters will refine this representation of a general point. Example 8.1 The ν-simplex, ∆ ν , is given by (5.3) as the convex hull in R ν +1 of {δ 1 ,. .. , δ ν +1 }, the coordinate vectors. It is easy to see its extreme points are precisely the ν + 1 points {δ j }