Krein–Milman theorem and Talk:Leah Cairns: Difference between pages

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In [[mathematics]], more precisely in [[functional analysis]], the '''Krein-Milman theorem''' is a statement about [[convex set]]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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If I'm not mistaken, Leah broke her back being thrown from a horse, not a car accident.

Can anyone comfirm?

She definatley broke it in car accident.

[[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 ''B'' is ''K''.]]

Goes to show how memory plays tricks on you.

Formally, let <math>X</math> be a [[locally convex topological vector space]], 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.

Actor Noah Hathaway who played Boxey in the original Battlestar Galactica series broke his back when he was thrown from a horse and stepped on (during the filming of NeverEnding Story in which he played Atreyu). Maybe you mixed these two up?

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.

The original statement proved by [[Mark Krein]] and [[David Milman]] (1912-1982), a student of Krein's from Odessa and father to mathematicians [[Vitali Milman]] and [[Pierre Milman]], was somewhat less general than this.

== References ==

*M. Krein, D. Milman (1940) ''On the extreme points of regularly convex sets'', Studia Mathematica 9 133-138.

* H. L. Royden. ''Real Analysis''. Prentice-Hall, Englewood Cliffs, New Jersey, 1988.

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{{planetmath|id=5921|title=Krein-Milman theorem}}

[[Category:Convex geometry]]

[[Category:Discrete geometry]]

[[Category:Functional analysis]]

[[Category:Mathematical theorems]]

[[de:Satz von Krein-Milman]]

[[fr:Théorème de Krein-Milman]]