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In mathematics, an extreme point of a convex set ''S'' in a real vector space is a point in S which does not lie in any open line segment joining two points of ''S''. Intuitively, an extreme point is a "vertex" of ''S''. * The Krein–Milman theorem states that if ''S'' is convex and compact in a locally convex space, then ''S'' is the closed convex hull of its extreme points: In particular, such a set has extreme points. The Krein–Milman theorem is stated for locally convex topological vector spaces. The next theorems are stated for Banach spaces with the Radon–Nikodym property: * A theorem of Joram Lindenstrauss states that, in a Banach space with the Radon–Nikodym property, a closed and bounded set has an extreme point. (In infinite-dimensional spaces, the property of compactness is stronger than the joint properties of being closed and being bounded).〔: 〕 * A theorem of Gerald Edgar states that, in a Banach space with the Radon–Nikodym property, a closed and bounded set is the closed convex hull of its extreme points. Edgar's theorem implies Lindenstrauss's theorem. == ''k''-extreme points == More generally, a point in a convex set ''S'' is ''k''-extreme if it lies in the interior of a ''k''-dimensional convex set within ''S'', but not a ''k+1''-dimensional convex set within ''S''. Thus, an extreme point is also a 0-extreme point. If ''S'' is a polytope, then the ''k''-extreme points are exactly the interior points of the ''k''-dimensional faces of ''S''. More generally, for any convex set ''S'', the ''k''-extreme points are partitioned into ''k''-dimensional open faces. The finite-dimensional Krein-Milman theorem, which is due to Minkowski, can be quickly proved using the concept of ''k''-extreme points. If ''S'' is closed, bounded, and ''n''-dimensional, and if ''p'' is a point in ''S'', then ''p'' is ''k''-extreme for some ''k'' < ''n''. The theorem asserts that ''p'' is a convex combination of extreme points. If ''k'' = 0, then it's trivially true. Otherwise ''p'' lies on a line segment in ''S'' which can be maximally extended (because ''S'' is closed and bounded). If the endpoints of the segment are ''q'' and ''r'', then their extreme rank must be less than that of ''p'', and the theorem follows by induction. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Extreme point」の詳細全文を読む スポンサード リンク
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