Krein–Milman theorem


In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces.
This theorem generalizes to infinite-dimensional spaces and to arbitrary compact convex sets the following basic observation: a convex triangle, including its perimeter and the area "inside of it", is equal to the convex hull of its three vertices, where these vertices are exactly the extreme points of this shape.
This observation also holds for any other convex polygon in the plane.

Statement

Throughout, we assume that is a real or complex vector space.
Note that and always contains its endpoints while and never contains its endpoints.
If and are points in the real line, then the above definition of is the same as its usual definition as a closed interval.
For example, the vertices of any convex polygon in the plane are the extreme points of that polygon.
The extreme points of the closed unit disk in is the unit circle.
Note that any open interval in has no extreme points while the extreme points of a non-degenerate closed interval are and.
For example, the convex hull of any set of three distinct points forms a solid triangle.
Also, in the plane, the unit circle is not convex but the closed unit disk is convex and furthermore, this disk is equal to the convex hull of the circle.
It is straightforward to show that the convex hull of the extreme points forms a subset of, so the main burden of the proof is to show that there are enough extreme points so that their convex hull covers all of.
As a corollary, it follows that every non-empty compact convex subset of a Hausdorff locally convex TVS as extreme points.
This corollary is also some called "the Krein-Milman theorem".
A particular case of this theorem, which can be easily visualized, states that given a convex polygon, one only needs the corners of the polygon to recover the polygon shape.
The statement of the theorem is false if the polygon is not convex, as then there can be many ways of drawing a polygon having given points as corners.

More general settings

The assumption of local convexity for the ambient space is necessary, because constructed a counter-example for the non-locally convex space Lp space| where.
Linearity is also needed, because the statement fails for weakly compact convex sets in CAT spaces, as proved by. However, proved that the Krein–Milman theorem does hold for metrically compact CAT spaces.

Related results

Under the previous assumptions on, if is a subset of and the closed convex hull of is all of, then every extreme point of belongs to the closure of.
This result is known as Milman's converse to the Krein–Milman theorem.
The Choquet–Bishop–de Leeuw theorem states that every point in is the barycenter of a probability measure supported on the set of extreme points of.

Relation to the axiom of choice

The axiom of choice, or some weaker version of it, is needed to prove this theorem in Zermelo–Fraenkel set theory.
Conversely, this theorem together with the Boolean prime ideal theorem can prove the axiom of choice.

History

The original statement proved by was somewhat less general than the form stated here.
Earlier, proved that if is 3-dimensional then equals the convex hull of the set of its extreme points. This assertion was expanded to the case of any finite dimension by.
The Krein–Milman theorem generalizes this to arbitrary locally convex ; however, to generalize from finite to infinite dimensional spaces, it is necessary to use the closure.