Extreme point

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 $S$ in a real or complex vector space or affine space is a point in $S$ that does not lie in any open line segment joining two points of $S.$ 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 $S.$

Definition

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 $p$ lies between $x$ and $y$ if $x\ne y$ and there exists a $0<t<1$ such that $p=tx+(1-t)y.$

If $K$ is a subset of $X$ and $p\in K,$ then $p$ is called an extreme point of $K$ if it does not lie between any two distinct points of $K.$ That is, if there does not exist $x,y\in K$ and $0<t<1$ such that $x\ne y$ and $p=tx+(1-t)y.$ The set of all extreme points of $K$ is denoted by $\mathrm{extreme}(K).$

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 support variety 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 midpoint of two elements $x$ and $y$ in a vector space is the vector $\frac{1}{2}(x+y).$

For any elements $x$ and $y$ in a vector space, the set $[x,y]=\{tx+(1-t)y:0\le t\le 1\}$ is called the closed line segment or closed interval between $x$ and $y.$ The open line segment or open interval between $x$ and $y$ is $(x,x)=\varnothing$ when $x=y$ while it is $(x,y)=\{tx+(1-t)y:0<t<1\}$ when $x\ne y.$ The points $x$ and $y$ 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 $[x,y]$ is equal to the convex hull of $(x,y)$ if (and only if) $x\ne y.$ 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 face of $K$ if whenever a point $p\in F$ lies between two points of $K,$ then those two points necessarily belong to $F.$

Theorem—Let $K$ be a non-empty convex subset of a vector space $X$ and let $p\in K.$ Then the following statements are equivalent:

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

Examples

If $a<b$ 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 fail 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.

Krein–Milman theorem—If $S$ is convex and compact in a locally convex topological vector space, then $S$ 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 $E$ be a Banach space with the Radon–Nikodym property, let $C$ be a separable, closed, bounded, convex subset of $E,$ and let $a$ be a point in $C.$ Then there is a probability measure $p$ on the universally measurable sets in $C$ such that $a$ is the barycenter of $p,$ and the set of extreme points of $C$ has $p$-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 $S$ 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 $k\le n.$ 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

• Extreme set
• Exposed point
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• Bang–bang control

Bibliography

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