Extreme point

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File:Extreme points.svg

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

In mathematics, an extreme point of a convex set <math>S</math> in a real or complex vector space is a point in <math>S</math> that does not lie in any open line segment joining two points of <math>S.</math> The extreme points of a line segment are called its endpoints. In linear programming problems, an extreme point is also called vertex or corner point of <math>S.</math><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Definition

Throughout, it is assumed that <math>X</math> is a real or complex vector space.

For any <math>p, x, y \in X,</math> say that <math>p</math> Template:Visible anchorTemplate:Sfn <math>x</math> and <math>y</math> if <math>x \neq y</math> and there exists a <math>0 < t < 1</math> such that <math>p = t x + (1-t) y.</math>

If <math>K</math> is a subset of <math>X</math> and <math>p \in K,</math> then <math>p</math> is called an Template:Visible anchorTemplate:Sfn of <math>K</math> if it does not lie between any two Template:Em points of <math>K.</math> That is, if there does Template:Em exist <math>x, y \in K</math> and <math>0 < t < 1</math> such that <math>x \neq y</math> and <math>p = t x + (1-t) y.</math> The set of all extreme points of <math>K</math> is denoted by <math>\operatorname{extreme}(K).</math>

Generalizations

If <math>S</math> is a subset of a vector space then a linear sub-variety (that is, an affine subspace) <math>A</math> of the vector space is called a Template:Em if <math>A</math> meets <math>S</math> (that is, <math>A \cap S</math> is not empty) and every open segment <math>I \subseteq S</math> whose interior meets <math>A</math> is necessarily a subset of <math>A.</math>Template:Sfn A 0-dimensional support variety is called an extreme point of <math>S.</math>Template:Sfn

Characterizations

The Template:Visible anchorTemplate:Sfn of two elements <math>x</math> and <math>y</math> in a vector space is the vector <math>\tfrac{1}{2}(x+y).</math>

For any elements <math>x</math> and <math>y</math> in a vector space, the set <math>[x, y] = \{t x + (1-t) y : 0 \leq t \leq 1\}</math> is called the Template:Visible anchor or Template:Visible anchor between <math>x</math> and <math>y.</math> The Template:Visible anchor or Template:Visible anchor between <math>x</math> and <math>y</math> is <math>(x, x) = \varnothing</math> when <math>x = y</math> while it is <math>(x, y) = \{t x + (1-t) y : 0 < t < 1\}</math> when <math>x \neq y.</math>Template:Sfn The points <math>x</math> and <math>y</math> are called the Template:Visible anchor of these interval. An interval is said to be a Template:Visible anchor or a Template:Visible anchor if its endpoints are distinct. The Template:Visible anchor is the midpoint of its endpoints.

The closed interval <math>[x, y]</math> is equal to the convex hull of <math>(x, y)</math> if (and only if) <math>x \neq y.</math> So if <math>K</math> is convex and <math>x, y \in K,</math> then <math>[x, y] \subseteq K.</math>

If <math>K</math> is a nonempty subset of <math>X</math> and <math>F</math> is a nonempty subset of <math>K,</math> then <math>F</math> is called a Template:Visible anchorTemplate:Sfn of <math>K</math> if whenever a point <math>p \in F</math> lies between two points of <math>K,</math> then those two points necessarily belong to <math>F.</math>

Examples

If <math>a < b</math> are two real numbers then <math>a</math> and <math>b</math> are extreme points of the interval <math>[a, b].</math> However, the open interval <math>(a, b)</math> has no extreme points.Template:Sfn Any open interval in <math>\R</math> has no extreme points while any non-degenerate closed interval not equal to <math>\R</math> does have extreme points (that is, the closed interval's endpoint(s)). More generally, any open subset of finite-dimensional Euclidean space <math>\R^n</math> has no extreme points.

The extreme points of the closed unit disk in <math>\R^2</math> is the unit circle.

The perimeter of any convex polygon in the plane is a face of that polygon.Template:Sfn The vertices of any convex polygon in the plane <math>\R^2</math> are the extreme points of that polygon.

An injective linear map <math>F : X \to Y</math> sends the extreme points of a convex set <math>C \subseteq X</math> to the extreme points of the convex set <math>F(X).</math>Template:Sfn 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 Template:Em to be closed in <math>X.</math>Template:Sfn

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.<ref>Template:Cite journal</ref>)

Edgar’s theorem implies Lindenstrauss’s theorem.

Related notions

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

k-extreme points

More generally, a point in a convex set <math>S</math> is <math>k</math>-extreme if it lies in the interior of a <math>k</math>-dimensional convex set within <math>S,</math> but not a <math>k + 1</math>-dimensional convex set within <math>S.</math> Thus, an extreme point is also a <math>0</math>-extreme point. If <math>S</math> is a polytope, then the <math>k</math>-extreme points are exactly the interior points of the <math>k</math>-dimensional faces of <math>S.</math> More generally, for any convex set <math>S,</math> the <math>k</math>-extreme points are partitioned into <math>k</math>-dimensional open faces.

The finite-dimensional Krein–Milman theorem, which is due to Minkowski, can be quickly proved using the concept of <math>k</math>-extreme points. If <math>S</math> is closed, bounded, and <math>n</math>-dimensional, and if <math>p</math> is a point in <math>S,</math> then <math>p</math> is <math>k</math>-extreme for some <math>k \leq n.</math> The theorem asserts that <math>p</math> is a convex combination of extreme points. If <math>k = 0</math> then it is immediate. Otherwise <math>p</math> lies on a line segment in <math>S</math> which can be maximally extended (because <math>S</math> is closed and bounded). If the endpoints of the segment are <math>q</math> and <math>r,</math> then their extreme rank must be less than that of <math>p,</math> and the theorem follows by induction.

See also

• Extreme set
• Exposed point
• Template:Annotated link
• Bang–bang control<ref>Template:Cite journal</ref>

Bibliography

• Template:Adasch Topological Vector Spaces
• Template:Bourbaki Topological Vector Spaces Part 1 Chapters 1–5
• {{#invoke:citation/CS1|citation

|CitationClass=web }}

• Template:Cite encyclopedia
• Template:Grothendieck Topological Vector Spaces
• Template:Halmos A Hilbert Space Problem Book 1982
• Template:Jarchow Locally Convex Spaces
• Template:Köthe Topological Vector Spaces I
• Template:Köthe Topological Vector Spaces II
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Robertson Topological Vector Spaces
• Template:Rudin Walter Functional Analysis
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Schechter Handbook of Analysis and Its Foundations
• Template:Trèves François Topological vector spaces, distributions and kernels
• Template:Wilansky Modern Methods in Topological Vector Spaces