Extreme point
From Wikipedia, the free encyclopedia
For other uses, see Extreme point (disambiguation).
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 "corner" 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).[1]
- 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.
[edit] See also
[edit] Notes
- ^ Artstein (1980, p. 173): Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR564562.
[edit] References
- Paul E. Black, ed. (2004–12–17). "extreme point". Dictionary of algorithms and data structures. US National institute of standards and technology. http://www.nist.gov/dads/HTML/extremepoint.html. 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.
