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:
“The melancholy of having to count souls
Where they grow fewer and fewer every year
Is extreme where they shrink to none at all.
It must be I want life to go on living.”
—Robert Frost (1874–1963)
“Some parents feel that if they introduce their children to alcohol gradually in the home environment, the children will learn to use alcohol in moderation. I’m not sure that’s such a good idea. First of all, alcohol is not healthy for the growing child. Second, introducing alcohol to a child suggests that you condone drinking—even to the point where you want to teach your child how to drink.”
—Lawrence Balter (20th century)