Krein–Milman theorem

In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in [Locally Convex set|convex topological vector space|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 and definitions

Preliminaries and definitions

Throughout, will be a real or complex vector space.
For any elements and in a vector space, the set is called the ' or closed interval between and The ' or open interval between and is when while it is when it satisfies and The points and are called the endpoints of these interval. An interval is said to be ' or proper if its endpoints are distinct.
The intervals and always contain their endpoints while and never contain either of their 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 any the point is said to ' and if belongs to the open line segment
If is a subset of and then is called an extreme point of if it does not lie between any two points of That is, if there does exist and such that and In this article, the set of all extreme points of will be denoted by
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.
Every open interval and degenerate closed interval in has no extreme points while the extreme points of a non-degenerate closed interval are and
A set is called convex if for any two points contains the line segment The smallest convex set containing is called the convex hull of and it is denoted by
The closed convex hull of a set denoted by is the smallest closed and convex set containing It is also equal to the intersection of all closed convex subsets that contain and to the closure of the convex hull of ; that is,
where the right hand side denotes the closure of while the left hand side is notation.
For example, the convex hull of any set of three distinct points forms either a closed line segment or else a solid triangle, including its perimeter.
And in the plane the unit circle is convex but the closed unit disk is convex and furthermore, this disk is equal to the convex hull of the circle.
The separable Hilbert space Lp space of square-summable sequences with the usual norm has a compact subset whose convex hull is closed and thus also compact. However, like in all complete Hausdorff locally convex spaces, the convex hull of this compact subset is compact. But, if a Hausdorff locally convex space is not complete, then it is in general guaranteed that is compact whenever is; an example can even be found in a pre-Hilbert vector subspace of Every compact subset is totally bounded and the closed convex hull of a totally bounded subset of a Hausdorff locally convex space is guaranteed to be totally bounded.

Statement

In the case where the compact set is also convex, the above theorem has as a corollary the first part of the next theorem, which is also often called the Krein–Milman theorem.
The convex hull of the extreme points of forms a convex 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
For this reason, the following corollary to the above theorem is also often called the Krein–Milman theorem.
To visualized this theorem and its conclusion, consider the particular case where is a convex polygon.
In this case, the corners of the polygon are all that is needed to recover the polygon shape.
The statement of the theorem is false if the polygon is not convex, as then there are many ways of drawing a polygon having given points as corners.
The requirement that the convex set be compact can be weakened to give the following strengthened generalization version of the theorem.
The property above is sometimes called or.
Compactness implies [|convex compactness] because a topological space is compact if and only if every family of closed subsets having the intersection property has non-empty intersection.
The [|definition of convex compactness] is similar to this characterization of compact spaces in terms of the FIP, except that it only involves those closed subsets that are also convex.

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(0) spaces, as proved by. However, proved that the Krein–Milman theorem does hold for 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 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

Under the Zermelo–Fraenkel set theory axiomatic framework, the axiom of choice suffices to prove all versions of the Krein–Milman theorem given above, including statement KM and its generalization SKM.
The axiom of choice also implies, but is not equivalent to, the Boolean prime ideal theorem, which is equivalent to the Banach–Alaoglu theorem.
Conversely, the Krein–Milman theorem KM together with the Boolean prime ideal theorem imply the axiom of choice.
In summary, AC holds if and only if both KM and BPI hold.
It follows that under ZF, the axiom of choice is equivalent to the following statement:
Furthermore, SKM together with the Hahn–Banach theorem for real vector spaces are also equivalent to the axiom of choice. It is known that BPI implies HB, but that it is not equivalent to it.

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.