(Solution Download) If x is not an extreme point of the convex


If x is not an extreme point of the convex set S ? X, then there exists y ? X such x + y ? S and x - y ? 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 ? S, the point  = ax + (1 - a)y ? F for any a ? (0, 1), then x and y are also in F. An extreme point is a face containing a single point (figure 1.23).

If x is not an extreme point of the convex
Figure 1.23
Faces and extreme points

 







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