Krein–Milman theorem

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 ${\displaystyle \mathbb {R} ^{2}.}$

Throughout, ${\displaystyle X}$ will be a real or complex vector space.

For any elements ${\displaystyle x}$ and ${\displaystyle y}$ in a vector space, the set ${\displaystyle [x,y]:=\{tx+(1-t)y:0\leq t\leq 1\}}$ is called the closed line segment or closed interval between ${\displaystyle x}$ and ${\displaystyle y.}$ The open line segment or open interval between ${\displaystyle x}$ and ${\displaystyle y}$ is ${\displaystyle (x,y):=\varnothing }$ when ${\displaystyle x=y}$ while it is ${\displaystyle (x,y):=\{tx+(1-t)y:0<t<1\}}$ when ${\displaystyle x\neq y;}$^[2] it satisfies ${\displaystyle (x,y)=[x,y]\setminus \{x,y\}}$ and ${\displaystyle [x,y]=(x,y)\cup \{x,y\}.}$ The points ${\displaystyle x}$ and ${\displaystyle y}$ are called the endpoints of these interval. An interval is said to be non-degenerate or proper if its endpoints are distinct.

The intervals ${\displaystyle [x,x]=\{x\}}$ and ${\displaystyle [x,y]}$ always contain their endpoints while ${\displaystyle (x,x)=\varnothing }$ and ${\displaystyle (x,y)}$ never contain either of their endpoints. If ${\displaystyle x}$ and ${\displaystyle y}$ are points in the real line ${\displaystyle \mathbb {R} }$ then the above definition of ${\displaystyle [x,y]}$ is the same as its usual definition as a

For any ${\displaystyle p,x,y\in X,}$ the point ${\displaystyle p}$ is said to (strictly) lie between ${\displaystyle x}$ and ${\displaystyle y}$ if ${\displaystyle p}$ belongs to the open line segment ${\displaystyle (x,y).}$^[2]

If ${\displaystyle K}$ is a subset of ${\displaystyle X}$ and ${\displaystyle p\in K,}$ then ${\displaystyle p}$ is called an extreme point of ${\displaystyle K}$ if it does not lie between any two distinct points of ${\displaystyle K.}$ That is, if there does not exist ${\displaystyle x,y\in K}$ and ${\displaystyle 0<t<1}$ such that ${\displaystyle x\neq y}$ and ${\displaystyle p=tx+(1-t)y.}$ In this article, the set of all extreme points of ${\displaystyle K}$ will be denoted by ${\displaystyle \operatorname {extreme} (K).}$^[2]

For example, the vertices of any convex polygon in the plane ${\displaystyle \mathbb {R} ^{2}}$ are the extreme points of that polygon. The extreme points of the

${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle \mathbb {R} }$

${\displaystyle [x,y]}$

${\displaystyle x}$

${\displaystyle y.}$

A set ${\displaystyle S}$ is called convex if for any two points ${\displaystyle x,y\in S,}$ ${\displaystyle S}$ contains the line segment ${\displaystyle [x,y].}$ The smallest convex set containing ${\displaystyle S}$ is called the convex hull of ${\displaystyle S}$ and it is denoted by ${\displaystyle \operatorname {co} S.}$ The closed convex hull of a set ${\displaystyle S,}$ denoted by ${\displaystyle {\overline {\operatorname {co} }}(S),}$ is the smallest closed and convex set containing ${\displaystyle S.}$ It is also equal to the intersection of all closed convex subsets that contain ${\displaystyle S}$ and to the closure of the convex hull of ${\displaystyle S}$; that is, $${\displaystyle {\overline {\operatorname {co} }}(S)={\overline {\operatorname {co} (S)}},}$$ where the right hand side denotes the closure of ${\displaystyle \operatorname {co} (S)}$ 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 (if they are collinear) or else a solid (that is, "filled") triangle, including its perimeter. And in the plane ${\displaystyle \mathbb {R} ^{2},}$ 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.

The separable Hilbert space Lp space ${\displaystyle \ell ^{2}(\mathbb {N} )}$ of square-summable sequences with the usual norm ${\displaystyle \|\cdot \|_{2}}$ has a compact subset ${\displaystyle S}$ whose convex hull ${\displaystyle \operatorname {co} (S)}$ is not closed and thus also not compact.^[3] However, like in all complete Hausdorff locally convex spaces, the closed convex hull ${\displaystyle {\overline {\operatorname {co} }}S}$ of this compact subset will be compact.^[4] But if a Hausdorff locally convex space is not complete then it is in general not guaranteed that ${\displaystyle {\overline {\operatorname {co} }}S}$ will be compact whenever ${\displaystyle S}$ is; an example can even be found in a (non-complete)

${\displaystyle \ell ^{2}(\mathbb {N} ).}$

In the case where the compact set ${\displaystyle K}$ is also convex, the above theorem has as a corollary the first part of the next theorem,^[6] which is also often called the Krein–Milman theorem.

Krein–Milman theorem^[2]—Suppose ${\displaystyle X}$ is a

normed space

) and ${\displaystyle K}$ is a compact and convex subset of ${\displaystyle X.}$ Then ${\displaystyle K}$ is equal to the closed convex hull of its extreme points: $${\displaystyle K~=~{\overline {\operatorname {co} }}(\operatorname {extreme} (K)).}$$

Moreover, if ${\displaystyle B\subseteq K}$ then ${\displaystyle K}$ is equal to the closed convex hull of ${\displaystyle B}$ if and only if ${\displaystyle \operatorname {extreme} K\subseteq \operatorname {cl} B,}$ where ${\displaystyle \operatorname {cl} B}$ is closure of ${\displaystyle B.}$

The convex hull of the extreme points of ${\displaystyle K}$ forms a convex subset of ${\displaystyle K}$ so the main burden of the proof is to show that there are enough extreme points so that their convex hull covers all of ${\displaystyle K.}$ 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 ${\displaystyle K}$ is a convex polygon. In this case, the corners of the polygon (which are its extreme points) 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 ${\displaystyle K}$ be compact can be weakened to give the following strengthened generalization version of the theorem.^[7]

The property above is sometimes called quasicompactness or convex compactness. Compactness implies convex compactness because a topological space is compact if and only if every family of closed subsets having the finite intersection property (FIP) has non-empty intersection (that is, its kernel is not empty). 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 ${\displaystyle L^{p}[0,1]}$ where ${\displaystyle 0<p<1.}$^[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.^[11]

Under the previous assumptions on ${\displaystyle K,}$ if ${\displaystyle T}$ is a subset of ${\displaystyle K}$ and the closed convex hull of ${\displaystyle T}$ is all of ${\displaystyle K,}$ then every extreme point of ${\displaystyle K}$ belongs to the closure of ${\displaystyle T.}$ This result is known as Milman's (partial) converse to the Krein–Milman theorem.^[12]

The Choquet–Bishop–de Leeuw theorem states that every point in ${\displaystyle K}$ is the

of ${\displaystyle K.}$

Relation to the axiom of choice

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. The axiom of choice also implies, but is not equivalent to, the Boolean prime ideal theorem (BPI), which is equivalent to the Banach–Alaoglu theorem. Conversely, the Krein–Milman theorem KM together with the Boolean prime ideal theorem (BPI) imply the axiom of choice.^[13] In summary, AC holds if and only if both KM and BPI hold.^[8] It follows that under ZF, the axiom of choice is equivalent to the following statement:

The closed unit ball of the continuous dual space of any real normed space has an extreme point.^[8]

Furthermore,

It is known that BPI implies HB, but that it is not equivalent to it (said differently, BPI is strictly stronger than HB).

History

The original statement proved by Mark Krein and David Milman (1940) was somewhat less general than the form stated here.^[14]

Earlier, Hermann Minkowski (1911) proved that if ${\displaystyle X}$ is

3-dimensional

then ${\displaystyle K}$ 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 ${\displaystyle X}$; however, to generalize from finite to infinite dimensional spaces, it is necessary to use the closure.

See also

• Banach–Alaoglu theorem – Theorem in functional analysis
• Carathéodory's theorem (convex hull) – Point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P
• Choquet theory – Area of functional analysis and convex analysis
• Helly's theorem – Theorem about the intersections of d-dimensional convex sets
• Radon's theorem – Says d+2 points in d dimensions can be partitioned into two subsets whose convex hulls intersect
• Shapley–Folkman lemma – Sums of sets of vectors are nearly convex
• Topological vector space – Vector space with a notion of nearness

Citations

1. ^ Rudin 1991, p. 75 Theorem 3.23.
2. ^ ^{a b c d e} Narici & Beckenstein 2011, pp. 275–339.
3. ^ Aliprantis & Border 2006, p. 185.
4. ^ Trèves 2006, p. 145.
5. ^ Trèves 2006, p. 67.
6. ^ ^{a b} Grothendieck 1973, pp. 187–188.
7. ^ Pincus 1974, pp. 204–205.
8. ^ ^{a b c d} Bell, J. L.; Jellett, F. (1971). "On the Relationship Between the Boolean Prime Ideal Theorem and Two Principles in Functional Analysis" (PDF). Bull. Acad. Polon. Sci. sciences math., astr. et phys. 19 (3): 191–194. Retrieved 23 Dec 2021.
9. doi:10.4064/sm-60-3-255-266
10. arXiv:1602.06752
11. Bibcode:2006math......4187B
12. Doklady Akademii Nauk SSSR
    (in Russian), 57: 119–122
13. doi:10.4064/fm-77-2-167-170
    . Retrieved 11 June 2018. Theorem 1.2. BPI [the Boolean Prime Ideal Theorem] & KM [Krein-Milman] ${\displaystyle \implies }$ (*) [the unit ball of the dual of a normed vector space has an extreme point].... Theorem 2.1. (*) ${\displaystyle \implies }$ AC [the Axiom of Choice].
14. doi:10.4064/sm-9-1-133-138
15. ^ Minkowski, Hermann (1911), Gesammelte Abhandlungen, vol. 2, Leipzig: Teubner, pp. 157–161
16. S2CID 122897233
    ; (see p. 16)

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