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 (TVSs).
This theorem generalizes to infinite-dimensional spaces and to arbitrary compact convex sets the following basic observation: a convex (i.e. "filled") 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
An interval is said to be non-degenerate or proper if its endpoints are distinct.
The extreme points of the closed unit disk in
It is also equal to the intersection of all closed convex subsets that contain
For example, the convex hull of any set of three distinct points forms either a closed line segment (if they are collinear) or else a solid (that is, "filled") triangle, including its perimeter.
[4] But if a Hausdorff locally convex space is not complete then it is in general not guaranteed that
is; an example can even be found in a (non-complete) pre-Hilbert vector subspace of
Every compact subset is totally bounded (also called "precompact") and the closed convex hull of a totally bounded subset of a Hausdorff locally convex space is guaranteed to be totally bounded.
is a compact subset of a Hausdorff locally convex topological vector space then the set of extreme points of
is equal to the closed convex hull of its extreme points:
so the main burden of the proof is to show that there are enough extreme points so that their convex hull covers all of
(KM) Krein–Milman theorem (Existence)[2] — Every non-empty compact convex subset of a Hausdorff locally convex topological vector space has an extreme point; that is, the set of its extreme points is not empty.
be compact can be weakened to give the following strengthened generalization version of the theorem.
[7] (SKM) Strong Krein–Milman theorem (Existence)[8] — Suppose
is a Hausdorff locally convex topological vector space and
The property above is sometimes called quasicompactness or convex compactness.
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 (rather than all closed subsets).
The assumption of local convexity for the ambient space is necessary, because James Roberts (1977) constructed a counter-example for the non-locally convex space
[9] Linearity is also needed, because the statement fails for weakly compact convex sets in CAT(0) spaces, as proved by Nicolas Monod (2016).
[10] However, Theo Buehler (2006) proved that the Krein–Milman theorem does hold for metrically compact CAT(0) spaces.
is the barycenter of a probability measure supported on the set of extreme points of
Under the Zermelo–Fraenkel set theory (ZF) axiomatic framework, the axiom of choice (AC) suffices to prove all versions of the Krein–Milman theorem given above, including statement KM and its generalization SKM.
[8] 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 (HB) are also equivalent to the axiom of choice.
The original statement proved by Mark Krein and David Milman (1940) was somewhat less general than the form stated here.
equals the convex hull of the set of its extreme points.
[15] This assertion was expanded to the case of any finite dimension by Ernst Steinitz (1916).
[16] 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.
