Krein–Milman theorem

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 x ≠ y.[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 ℝ, then the above definition of [x, y] is the same as its usual definition as a closed interval.

For any p, x, y ∈ X, 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 p ∈ K, 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, y ∈ K and 0 < t < 1 such that x ≠ y 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 ℝ² are the extreme points of that polygon. The extreme points of the closed unit disk in ℝ² is the unit circle. Note that any open interval in ℝ 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, y ∈ S, 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 ℝ², 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.

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.

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 L^p[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]

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.

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]

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 ${\displaystyle K}$ 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.

• Banach–Alaoglu theorem – The closed unit ball in the dual of a normed vector space is compact in the weak* topology
• Carathéodory's theorem (convex hull) – A point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P
• Choquet theory
• 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
• Topological vector space – Vector space with a notion of nearness

• Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
• Bourbaki, Nicolas (1987) [1981]. Sur certains espaces vectoriels topologiques [Topological Vector Spaces: Chapters 1–5]. Annales de l'Institut Fourier. Éléments de mathématique. 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.
• Paul E. Black, ed. (2004-12-17). "extreme point". Dictionary of algorithms and data structures. US National institute of standards and technology. Retrieved 2011-03-24.
• Borowski, Ephraim J.; Borwein, Jonathan M. (1989). "extreme point". Dictionary of mathematics. Collins dictionary. Harper Collins. ISBN 0-00-434347-6.
• Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
• Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• N. K. Nikol'skij (Ed.). Functional Analysis I. Springer-Verlag, 1992.
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
• H. L. Royden, Real Analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1988.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.

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