Extreme point

Short description: Point not between two other points

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

In mathematics, an extreme point of a convex set [math]\displaystyle{ S }[/math] in a real or complex vector space is a point in [math]\displaystyle{ S }[/math] that does not lie in any open line segment joining two points of [math]\displaystyle{ S. }[/math] In linear programming problems, an extreme point is also called vertex or corner point of [math]\displaystyle{ S. }[/math]

Definition

Throughout, it is assumed that [math]\displaystyle{ X }[/math] is a real or complex vector space.

For any [math]\displaystyle{ p, x, y \in X, }[/math] say that [math]\displaystyle{ p }[/math] lies between [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] if [math]\displaystyle{ x \neq y }[/math] and there exists a [math]\displaystyle{ 0 \lt t \lt 1 }[/math] such that [math]\displaystyle{ p = t x + (1-t) y. }[/math]

If [math]\displaystyle{ K }[/math] is a subset of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ p \in K, }[/math] then [math]\displaystyle{ p }[/math] is called an extreme point of [math]\displaystyle{ K }[/math] if it does not lie between any two distinct points of [math]\displaystyle{ K. }[/math] That is, if there does not exist [math]\displaystyle{ x, y \in K }[/math] and [math]\displaystyle{ 0 \lt t \lt 1 }[/math] such that [math]\displaystyle{ x \neq y }[/math] and [math]\displaystyle{ p = t x + (1-t) y. }[/math] The set of all extreme points of [math]\displaystyle{ K }[/math] is denoted by [math]\displaystyle{ \operatorname{extreme}(K). }[/math]

Generalizations

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

Characterizations

The midpoint of two elements [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] in a vector space is the vector [math]\displaystyle{ \tfrac{1}{2}(x+y). }[/math]

For any elements [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] in a vector space, the set [math]\displaystyle{ [x, y] = \{t x + (1-t) y : 0 \leq t \leq 1\} }[/math] is called the closed line segment or closed interval between [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y. }[/math] The open line segment or open interval between [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] is [math]\displaystyle{ (x, x) = \varnothing }[/math] when [math]\displaystyle{ x = y }[/math] while it is [math]\displaystyle{ (x, y) = \{t x + (1-t) y : 0 \lt t \lt 1\} }[/math] when [math]\displaystyle{ x \neq y. }[/math] The points [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] 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 [math]\displaystyle{ [x, y] }[/math] is equal to the convex hull of [math]\displaystyle{ (x, y) }[/math] if (and only if) [math]\displaystyle{ x \neq y. }[/math] So if [math]\displaystyle{ K }[/math] is convex and [math]\displaystyle{ x, y \in K, }[/math] then [math]\displaystyle{ [x, y] \subseteq K. }[/math]

If [math]\displaystyle{ K }[/math] is a nonempty subset of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ F }[/math] is a nonempty subset of [math]\displaystyle{ K, }[/math] then [math]\displaystyle{ F }[/math] is called a face of [math]\displaystyle{ K }[/math] if whenever a point [math]\displaystyle{ p \in F }[/math] lies between two points of [math]\displaystyle{ K, }[/math] then those two points necessarily belong to [math]\displaystyle{ F. }[/math]

Theorem — Let [math]\displaystyle{ K }[/math] be a non-empty convex subset of a vector space [math]\displaystyle{ X }[/math] and let [math]\displaystyle{ p \in K. }[/math] Then the following statements are equivalent:

1. [math]\displaystyle{ p }[/math] is an extreme point of [math]\displaystyle{ K. }[/math]
2. [math]\displaystyle{ K \setminus \{p\} }[/math] is convex.
3. [math]\displaystyle{ p }[/math] is not the midpoint of a non-degenerate line segment contained in [math]\displaystyle{ K. }[/math]
4. for any [math]\displaystyle{ x, y \in K, }[/math] if [math]\displaystyle{ p \in [x, y] }[/math] then [math]\displaystyle{ x = p \text{ or } y = p. }[/math]
5. if [math]\displaystyle{ x \in X }[/math] is such that both [math]\displaystyle{ p + x }[/math] and [math]\displaystyle{ p - x }[/math] belong to [math]\displaystyle{ K, }[/math] then [math]\displaystyle{ x = 0. }[/math]
6. [math]\displaystyle{ \{p\} }[/math] is a face of [math]\displaystyle{ K. }[/math]

Examples

If [math]\displaystyle{ a \lt b }[/math] are two real numbers then [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are extreme points of the interval [math]\displaystyle{ [a, b]. }[/math] However, the open interval [math]\displaystyle{ (a, b) }[/math] has no extreme points. Any open interval in [math]\displaystyle{ \R }[/math] has no extreme points while any non-degenerate closed interval not equal to [math]\displaystyle{ \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]\displaystyle{ \R^n }[/math] has no extreme points.

The extreme points of the closed unit disk in [math]\displaystyle{ \R^2 }[/math] 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 [math]\displaystyle{ \R^2 }[/math] are the extreme points of that polygon.

An injective linear map [math]\displaystyle{ F : X \to Y }[/math] sends the extreme points of a convex set [math]\displaystyle{ C \subseteq X }[/math] to the extreme points of the convex set [math]\displaystyle{ F(X). }[/math] 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 fail to be closed in [math]\displaystyle{ X. }[/math]

Theorems

Krein–Milman theorem

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

Krein–Milman theorem — If [math]\displaystyle{ S }[/math] is convex and compact in a locally convex topological vector space, then [math]\displaystyle{ S }[/math] is the closed convex hull of its extreme points: In particular, such a set has 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.)

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

Edgar’s theorem implies Lindenstrauss’s theorem.

Related notions

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. The unit ball of any Hilbert space is a strictly convex set.

k-extreme points

More generally, a point in a convex set [math]\displaystyle{ S }[/math] is [math]\displaystyle{ k }[/math]-extreme if it lies in the interior of a [math]\displaystyle{ k }[/math]-dimensional convex set within [math]\displaystyle{ S, }[/math] but not a [math]\displaystyle{ k + 1 }[/math]-dimensional convex set within [math]\displaystyle{ S. }[/math] Thus, an extreme point is also a [math]\displaystyle{ 0 }[/math]-extreme point. If [math]\displaystyle{ S }[/math] is a polytope, then the [math]\displaystyle{ k }[/math]-extreme points are exactly the interior points of the [math]\displaystyle{ k }[/math]-dimensional faces of [math]\displaystyle{ S. }[/math] More generally, for any convex set [math]\displaystyle{ S, }[/math] the [math]\displaystyle{ k }[/math]-extreme points are partitioned into [math]\displaystyle{ 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]\displaystyle{ k }[/math]-extreme points. If [math]\displaystyle{ S }[/math] is closed, bounded, and [math]\displaystyle{ n }[/math]-dimensional, and if [math]\displaystyle{ p }[/math] is a point in [math]\displaystyle{ S, }[/math] then [math]\displaystyle{ p }[/math] is [math]\displaystyle{ k }[/math]-extreme for some [math]\displaystyle{ k \leq n. }[/math] The theorem asserts that [math]\displaystyle{ p }[/math] is a convex combination of extreme points. If [math]\displaystyle{ k = 0 }[/math] then it is immediate. Otherwise [math]\displaystyle{ p }[/math] lies on a line segment in [math]\displaystyle{ S }[/math] which can be maximally extended (because [math]\displaystyle{ S }[/math] is closed and bounded). If the endpoints of the segment are [math]\displaystyle{ q }[/math] and [math]\displaystyle{ r, }[/math] then their extreme rank must be less than that of [math]\displaystyle{ p, }[/math] and the theorem follows by induction.

See also

• Choquet theory – Area of functional analysis and convex analysis
• Bang–bang control

Bibliography

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