Extreme point

Throughout, it is assumed that X is a real or complex vector space or affine space. For any p, x, y \in X, say that p '''''' x and y if x \neq y and there exists a 0 such that p = t x + (1-t) y. If K is a subset of X and p \in K, then p is called an '''''' of K if it does not lie between any two points of K. That is, if there does exist x, y \in K and 0 such that x \neq y and p = t x + (1-t) y. The set of all extreme points of K is denoted by \operatorname{extreme}(K). Generalizations If S is a subset of a vector space then a linear sub-variety (that is, an affine subspace) A of the vector space is called a if A meets S (that is, A \cap S is not empty) and every open segment I \subseteq S whose interior meets A is necessarily a subset of A. A 0-dimensional support variety is called an extreme point of S. Characterizations The '''''' of two elements x and y in a vector space is the vector \tfrac{1}{2}(x+y). For any elements x and y in a vector space, the set [x, y] = \{t x + (1-t) y : 0 \leq t \leq 1\} is called the ' or between x and y. The or between x and y is (x, x) = \varnothing when x = y while it is (x, y) = \{t x + (1-t) y : 0 when x \neq y. The points x and y are called the of these interval. An interval is said to be a or a if its endpoints are distinct. The ' is the midpoint of its endpoints. The closed interval [x, y] is equal to the convex hull of (x, y) if (and only if) x \neq y. So if K is convex and x, y \in K, then [x, y] \subseteq K. If K is a nonempty subset of X and F is a nonempty subset of K, then F is called a '''''' of K if whenever a point p \in F lies between two points of K, then those two points necessarily belong to F. {{Math theorem|name=Theorem|math_statement= Let K be a non-empty convex subset of a vector space X and let p \in K. Then the following statements are equivalent: p is an extreme point of K. K \setminus \{p\} is convex. p is not the midpoint of a non-degenerate line segment contained in K. for any x, y \in K, if p \in [x, y] then x = p \text{ or } y = p. if x \in X is such that both p + x and p - x belong to K, then x = 0. \{p\} is a face of K. }} ==Examples==

If a are two real numbers then a and b are extreme points of the interval [a, b]. However, the open interval (a, b) has no extreme points. Any open interval in \R has no extreme points while any non-degenerate closed interval not equal to \R does have extreme points (that is, the closed interval's endpoint(s)). More generally, any open subset of finite-dimensional Euclidean space \R^n has no extreme points. The extreme points of the closed unit disk in \R^2 is the unit circle. The perimeter of any convex polygon in the plane is a face of that polygon. The vertices of any convex polygon in the plane \R^2 are the extreme points of that polygon. An injective linear map F : X \to Y sends the extreme points of a convex set C \subseteq X to the extreme points of the convex set F(X). This is also true for injective affine maps. ==Properties==

The extreme points of a compact convex set form a Baire space (with the subspace topology) but this set may to be closed in X. ==Theorems==

Krein–Milman theorem The Krein–Milman theorem is arguably one of the most well-known theorems about extreme points. For Banach spaces These theorems are 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 nonempty 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.) Edgar’s theorem implies Lindenstrauss’s theorem. ==Related notions==

A closed convex subset of a topological vector space is called if every one of its (topological) boundary points is an extreme point. The unit ball of any Hilbert space is a strictly convex set. 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 \leq n. The theorem asserts that p is a convex combination of extreme points. If k = 0 then it is immediate. 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. ==See also==