Extreme point

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A convex set in light blue, and its extreme points in red.

In mathematics, an extreme point of a convex set in a real or complex vector space is a point in which does not lie in any open line segment joining two points of In linear programming problems, an extreme point is also called vertex or corner point of [1]

Definition[edit]

Throughout, it is assumed that is a real or complex vector space.

For any say that lies between[2] and if and there exists a such that

If is a subset of and then is called an extreme point[2] of if it does not lie between any two distinct points of That is, if there does not exist and such that and The set of all extreme points of is denoted by

Generalizations

If is a subset of a vector space then a linear sub-variety (that is, an affine subspace) of the vector space is called a support variety if meets (that is, is not empty) and every open segment whose interior meets is necessarily a subset of [3] A 0-dimensional support variety is called an extreme point of [3]

Characterizations[edit]

The midpoint[2] of two elements and in a vector space is the vector

For any elements and in a vector space, the set is called the closed line segment or closed interval between and The open line segment or open interval between and is when while it is when [2] The points and are called the endpoints of these interval. An interval is said to be a non−degenerate interval or a proper interval if its endpoints are distinct. The midpoint of an interval is the midpoint of its endpoints.

The closed interval is equal to the convex hull of if (and only if) So if is convex and then

If is a nonempty subset of and is a nonempty subset of then is called a face[2] of if whenever a point lies between two points of then those two points necessarily belong to

Theorem[2] — Let be a non-empty convex subset of a vector space and let Then the following statements are equivalent:

  1. is an extreme point of
  2. is convex.
  3. is not the midpoint of a non-degenerate line segment contained in
  4. for any if then
  5. if is such that both and belong to then
  6. is a face of

Examples[edit]

If are two real numbers then and are extreme points of the interval However, the open interval has no extreme points.[2] Any open interval in has no extreme points while any non-degenerate closed interval not equal to does have extreme points (that is, the closed interval's endpoint(s)). More generally, any open subset of finite-dimensional Euclidean space has no extreme points.

The extreme points of the closed unit disk in is the unit circle.

The perimeter of any convex polygon in the plane is a face of that polygon.[2] The vertices of any convex polygon in the plane are the extreme points of that polygon.

An injective linear map sends the extreme points of a convex set to the extreme points of the convex set [2] This is also true for injective affine maps.

Properties[edit]

The extreme points of a compact convex form a Baire space (with the subspace topology) but this set may fail to be closed in [2]

Theorems[edit]

Krein–Milman theorem[edit]

The Krein–Milman theorem is arguably one of the most well-known theorems about extreme points.

Krein–Milman theorem — If is convex and compact in a locally convex topological vector space, then is the closed convex hull of its extreme points: In particular, such a set has extreme points.

For Banach spaces[edit]

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).[4]

Theorem (Gerald Edgar) — Let be a Banach space with the Radon-Nikodym property, let be a separable, closed, bounded, convex subset of and let be a point in Then there is a probability measure on the universally measurable sets in such that is the barycenter of and the set of extreme points of has -measure 1.[5]

Edgar's theorem implies Lindenstrauss's theorem.

Related notions[edit]

A closed convex subset of a topological vector space is called strictly convex if every one of its (topological) boundary points is an extreme point.[6] The unit ball of any Hilbert space is a strictly convex set.[6]

k-extreme points[edit]

More generally, a point in a convex set is -extreme if it lies in the interior of a -dimensional convex set within but not a -dimensional convex set within Thus, an extreme point is also a -extreme point. If is a polytope, then the -extreme points are exactly the interior points of the -dimensional faces of More generally, for any convex set the -extreme points are partitioned into -dimensional open faces.

The finite-dimensional Krein-Milman theorem, which is due to Minkowski, can be quickly proved using the concept of -extreme points. If is closed, bounded, and -dimensional, and if is a point in then is -extreme for some The theorem asserts that is a convex combination of extreme points. If then it is immediate. Otherwise lies on a line segment in which can be maximally extended (because is closed and bounded). If the endpoints of the segment are and then their extreme rank must be less than that of and the theorem follows by induction.

See also[edit]

Citations[edit]

  1. ^ Saltzman, Matthew. "What is the difference between corner points and extreme points in linear programming problems?".
  2. ^ a b c d e f g h i j Narici & Beckenstein 2011, pp. 275–339.
  3. ^ a b Grothendieck 1973, p. 186.
  4. ^ 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. MR 0564562.
  5. ^ Edgar GA. A noncompact Choquet theorem. Proceedings of the American Mathematical Society. 1975;49(2):354-8.
  6. ^ a b Halmos 1982, p. 5.

Bibliography[edit]