Editing Extreme point (section)

===''k''-extreme points===

More generally, a point in a convex set <math>S</math> is '''<math>k</math>-extreme''' if it lies in the interior of a <math>k</math>-dimensional convex set within <math>S,</math> but not a <math>k + 1</math>-dimensional convex set within <math>S.</math>  Thus, an extreme point is also a <math>0</math>-extreme point.  If <math>S</math> is a polytope, then the <math>k</math>-extreme points are exactly the interior points of the <math>k</math>-dimensional faces of <math>S.</math>  More generally, for any convex set <math>S,</math> the <math>k</math>-extreme points are partitioned into <math>k</math>-dimensional open faces.

The finite-dimensional Krein–Milman theorem, which is due to Minkowski, can be quickly proved using the concept of <math>k</math>-extreme points.  If <math>S</math> is closed, bounded, and <math>n</math>-dimensional, and if <math>p</math> is a point in <math>S,</math> then <math>p</math> is <math>k</math>-extreme for some <math>k \leq n.</math>  The theorem asserts that <math>p</math> is a convex combination of extreme points.  If <math>k = 0</math> then it is immediate.  Otherwise <math>p</math> lies on a line segment in <math>S</math> which can be maximally extended (because <math>S</math> is closed and bounded).  If the endpoints of the segment are <math>q</math> and <math>r,</math> then their extreme rank must be less than that of <math>p,</math> and the theorem follows by induction.