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| [[Image:Extreme points illustration.png|thumb|right|Given a convex shape ''K'' (light blue) and its set of extreme points ''B'' (red), the convex hull of ''B'' is ''K''.]]
| | Alyson is the name individuals use to call me and I think it sounds quite good when you say it. Her family lives in Ohio but her husband wants them to transfer. Office supervising is what she does for a living. To climb is something she would by no means give up.<br><br>Feel free to surf to my weblog ... spirit messages ([http://ltreme.com/index.php?do=/profile-127790/info/ ltreme.com]) |
| In the [[mathematics|mathematical theory]] of [[functional analysis]], the '''Krein–Milman theorem''' is a [[theorem|proposition]] about [[convex set]]s in [[topological vector space]]s. 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.
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| Formally, let <math>X</math> be a [[locally convex topological vector space]] (assumed to be [[Hausdorff space|Hausdorff]]), and let <math>K</math> be a [[compact set|compact]] [[convex set|convex]] [[subset]] of <math>X</math>. Then, the theorem states that <math>K</math> is the closed [[convex hull]] of its [[extreme point]]s.
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| The closed convex hull above is defined as the [[intersection (set theory)|intersection]] of all closed convex subsets of <math>X</math> that contain <math>K.</math> This turns out to be the same as the [[closure (topology)|closure]] of the [[convex hull]] in the topological vector space. One direction in the theorem is easy; the main burden is to show that there are 'enough' extreme points.
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| The original statement proved by [[Mark Krein]] and [[David Milman]] was somewhat less general than this.
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| [[Hermann Minkowski]] had already proved that if <math>X</math> is [[finite-dimensional]] then <math>K</math> equals the convex hull of the set of its extreme points. The '''Krein–Milman theorem''' generalizes this to arbitrary locally-convex <math>X</math>, with a [[:wikt:caveat|caveat]]: the closure may be needed.
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| ==Relation to the axiom of choice==
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| The [[axiom of choice]], or some weaker version of it, is needed to prove this theorem in [[Zermelo–Fraenkel set theory]]. This theorem together with the [[Boolean prime ideal theorem]], though, can prove the axiom of choice.
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| ==Related results==
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| Under the previous assumptions on <math>K,</math> if <math>T</math> is a [[subset]] of <math>K</math> and the closed convex hull of <math>T</math> is all of <math>K</math>, then every [[extreme point]] of <math>K</math> belongs to the [[closure (topology)|closure]] of <math>T.</math> This result is known as ''Milman's'' (partial) ''converse'' to the Krein–Milman theorem.
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| The [[Choquet theory|Choquet–Bishop–de Leeuw theorem]] states that every point in <math>K</math> is the barycenter of a [[probability space|probability measure]] supported on the set of [[extreme point]]s of <math>K.</math>
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| Theo Buehler proved in 2006 that the Krein–Milman theorem also holds for CAT(0) spaces. <ref>{{cite web |url=http://arxiv.org/abs/math/0604187 |title=The Krein-Mil'man Theorem for Metric Spaces with a Convex Bicombing, Theo Buehler, 2006.}}</ref>
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| ==See also==
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| * [[Carathéodory's theorem (convex hull)]]
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| * [[Shapley–Folkman lemma]]
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| * [[Helly's theorem]]
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| * [[Radon's theorem]]
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| * [[Choquet theory]]
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| ==References==
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| {{Reflist}}
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| *M. Krein, D. Milman (1940) ''On extreme points of regular convex sets'', Studia Mathematica '''9''' 133–138.
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| * {{cite journal| last=Milman | first=D. | year=1947 | title=Характеристика экстремальных точек регулярно-выпуклого множества | trans_title=Characteristics of extremal points of regularly convex sets | language=Russian | journal=[[Doklady Akademii Nauk SSSR]] | volume=57 | pages=119–122}}
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| * H. L. Royden. ''Real Analysis''. Prentice-Hall, Englewood Cliffs, New Jersey, 1988.
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| * N. K. Nikol'skij (Ed.). ''Functional Analysis I''. Springer-Verlag, 1992
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| * H. Minkowski. ''Geometrie der Zahlen''. Teubner, Leipzig, 1910
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| {{PlanetMath attribution|id=5921|title=Krein–Milman theorem}}
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| {{DEFAULTSORT:Krein-Milman theorem}}
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| [[Category:Theorems in convex geometry]]
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| [[Category:Theorems in discrete geometry]]
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| [[Category:Topological vector spaces]]
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| [[Category:Convex hulls]]
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| [[Category:Oriented matroids]]
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| [[Category:Theorems in functional analysis]]
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Alyson is the name individuals use to call me and I think it sounds quite good when you say it. Her family lives in Ohio but her husband wants them to transfer. Office supervising is what she does for a living. To climb is something she would by no means give up.
Feel free to surf to my weblog ... spirit messages (ltreme.com)