Editing Extreme point (section)

==Theorems==

===Krein–Milman theorem===

The [[Krein–Milman theorem]] is arguably one of the most well-known theorems about extreme points. 

{{Math theorem|name=[[Krein–Milman theorem]]|math_statement=
If <math>S</math> is convex and [[Compact space|compact]] in a [[locally convex topological vector space]], then <math>S</math> is the closed [[convex hull]] of its extreme points: In particular, such a set has extreme points.
}}

===For Banach spaces===

These theorems are for [[Banach space]]s with the [[Radon–Nikodym property]]. 

A theorem of [[Joram Lindenstrauss]] states that, in a Banach space with the Radon–Nikodym property, a nonempty [[closed set|closed]] and [[bounded set]] has an extreme point. (In infinite-dimensional spaces, the property of [[compact space|compactness]] is stronger than the joint properties of being closed and being bounded.<ref>{{cite journal|last=Artstein|first=Zvi|title=Discrete&nbsp;and&nbsp;continuous bang-bang and facial&nbsp;spaces, or: Look for the extreme points|journal=SIAM Review|volume=22|year=1980|number=2|pages=172–185|doi=10.1137/1022026|mr=564562|jstor=2029960}}</ref>)

{{Math theorem|name=Theorem|note=[[Gerald Edgar]]|math_statement=
Let <math>E</math> be a Banach space with the Radon–Nikodym property, let <math>C</math> be a separable, closed, bounded, convex subset of <math>E,</math> and let <math>a</math> be a point in <math>C.</math> Then there is a [[probability measure]] <math>p</math> on the universally measurable sets in <math>C</math> such that <math>a</math> is the [[barycenter]] of <math>p,</math> and the set of extreme points of <math>C</math> has <math>p</math>-measure 1.<ref>Edgar GA. [https://www.ams.org/journals/proc/1975-049-02/S0002-9939-1975-0372586-2/S0002-9939-1975-0372586-2.pdf A noncompact Choquet theorem.] Proceedings of the American Mathematical Society. 1975;49(2):354–8.</ref>
}}

Edgar’s theorem implies Lindenstrauss’s theorem.