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.
Read more about Extreme Point: k-extreme Points
Famous quotes containing the words extreme and/or point:
“I wish to speak a word for Nature, for absolute freedom and wildness, as contrasted with a freedom and culture merely civil,—to regard man as an inhabitant, or a part and parcel of Nature, rather than as a member of society. I wish to make an extreme statement, if so I may make an emphatic one, for there are enough champions of civilization: the minister and the school committee and every one of you will take care of that.”
—Henry David David (1817–1862)
“Our culture still holds mothers almost exclusively responsible when things go wrong with the kids. Sensing this ultimate accountability, women are understandably reluctant to give up control or veto power. If the finger of blame was eventually going to point in your direction, wouldn’t you be?”
—Ron Taffel (20th century)