If x is not an extreme point of the convex set S X, then there exists y

If x is not an extreme point of the convex set S Š‚ X, then there exists y ˆˆ X such x + y ˆˆ S and x - y ˆˆ S.
A convex subset F of a convex set S is called a face of S if no point of F is an interior point of a line segment whose end points are in S but not in F. Formally, if for any x; y ˆˆ S, the point = ax + (1 - a)y ˆˆ F for any a ˆˆ (0, 1), then x and y are also in F. An extreme point is a face containing a single point (figure 1.23).

Figure 1.23
Faces and extreme points

Transcribed Image Text:

Extreme points Face