Krein–Milman theorem

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Given a convex shape K (light blue) and its set of extreme points B (red), the convex hull of B is K.

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).

Krein–Milman theorem — A compact convex subset of a Hausdorff locally convex topological vector space is equal to the closed convex hull of its extreme points.

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 2.

Statement[edit]

Throughout, we assume that X is a real or complex vector space.

For any elements x and y in a vector space, the set [x, y] := {tx + (1 - t)y : 0 ≤ t ≤ 1} is called the closed line segment or closed interval between x and y. The open line segment or open interval between x and y is (x, x) := ∅ when x = y while it is (x, y) := {tx + (1 - t)y : 0 < t < 1 } when xy.[1] We call x and y the endpoints of these interval. An interval is said to be non-degenerate or proper if its endpoints are distinct.

Note that [x, x] = { x} and [x, y] always contains its endpoints while (x, x) = ∅ and (x, y) never contains its endpoints. If x and y are points in the real line , then the above definition of [x, y] is the same as its usual definition as a closed interval.

For any p, x, yX, say that p lies between x and y if p belongs to the open line segment (x, y).[1]

If K is a subset of X and pK, then p is called an extreme point of K if it does not lie between any two distinct points of K. That is, if there does not exist x, yK and 0 < t < 1 such that xy and p = tx + (1 - t) y. The set of all extreme points of K is denoted by extreme(K).[1]

For example, the vertices of any convex polygon in the plane 2 are the extreme points of that polygon. The extreme points of the closed unit disk in 2 is the unit circle. Note that any open interval in has no extreme points while the extreme points of a non-degenerate closed interval [a, b] are a and b.

A set S is called convex if for any two points x, yS, S contains the line segment [x, y]. The smallest convex set containing S is called the convex hull of S and is denoted by co S.

For example, the convex hull of any set of three distinct points forms a solid (i.e. "filled") triangle (including the perimeter). Also, in the plane 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 closed convex hull of a set S is the smallest closed and convex set containing S. It is also equal to the closure of the convex hull of S and to the intersection of all closed convex subsets that contain S.

Krein–Milman theorem[1] — Suppose X is a Hausdorff locally convex topological vector space and K is a compact and convex subset of X. Then K is equal to the closed convex hull of its extreme points. Moreover, if BK then K is equal to the closed convex hull of B if and only if extreme K ⊆ cl B, where cl B is closure of B.

It is straightforward to show that the convex hull of the extreme points forms a subset of 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 K.

As a corollary, it follows that every non-empty compact convex subset of a Hausdorff locally convex TVS has extreme points (i.e. the set of its extreme points is not empty).[1] 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[edit]

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 Lp[0, 1] where 0 < p < 1.[2]

Linearity is also needed, because the statement fails for weakly compact convex sets in CAT(0) spaces, as proved by Nicolas Monod (2016).[3] However, Theo Buehler (2006) proved that the Krein–Milman theorem does hold for metrically compact CAT(0) spaces.[4]

Related results[edit]

Under the previous assumptions on K, if T is a subset of K and the closed convex hull of T is all of K, then every extreme point of K belongs to the closure of T. This result is known as Milman's (partial) converse to the Krein–Milman theorem.[5]

The Choquet–Bishop–de Leeuw theorem states that every point in K is the barycenter of a probability measure supported on the set of extreme points of K.

Relation to the axiom of choice[edit]

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.[6]

History[edit]

The original statement proved by Mark Krein and David Milman (1940) was somewhat less general than the form stated here.[7]

Earlier, Hermann Minkowski (1911) proved that if X is 3-dimensional then equals the convex hull of the set of its extreme points.[8] This assertion was expanded to the case of any finite dimension by Ernst Steinitz (1916).[9] The Krein–Milman theorem generalizes this to arbitrary locally convex X; however, to generalize from finite to infinite dimensional spaces, it is necessary to use the closure.

See also[edit]

Citations[edit]

  1. ^ a b c d e Narici & Beckenstein 2011, pp. 275-339.
  2. ^ Roberts, J. (1977), "A compact convex set with no extreme points", Studia Mathematica, 60: 255–266
  3. ^ Monod, Nicolas (2016), "Extreme points in non-positive curvature", Studia Mathematica, 234: 265–270, arXiv:1602.06752
  4. ^ Buehler, Theo (2006), The Krein–Mil'man theorem for metric spaces with a convex bicombing, arXiv:math/0604187
  5. ^ Milman, D. (1947), Характеристика экстремальных точек регулярно-выпуклого множества [Characteristics of extremal points of regularly convex sets], Doklady Akademii Nauk SSSR (in Russian), 57: 119–122
  6. ^ Bell, J.; Fremlin, David (1972). "A geometric form of the axiom of choice" (PDF). Fundamenta Mathematicae. 77 (2): 167–170. Retrieved 11 June 2018. Theorem 1.2. BPI [the Boolean Prime Ideal Theorem] & KM [Krein-Milman] ⇒ (*) [the unit ball of the dual of a normed vector space has an extreme point]…. Theorem 2.1. (*) ⇒ AC [the Axiom of Choice].
  7. ^ Krein, Mark; Milman, David (1940), "On extreme points of regular convex sets", Studia Mathematica, 9: 133–138
  8. ^ Minkowski, Hermann (1911), Gesammelte Abhandlungen, 2, Leipzig: Teubner, pp. 157–161
  9. ^ Steinitz, Ernst (1916), "Bedingt konvergente Reihen und konvexe Systeme VI, VII", J. Reine Angew. Math., 146: 1–52; (see p. 16)

Bibliography[edit]

This article incorporates material from Krein–Milman theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.