Extreme point

In mathematics, an extreme point of a convex set ${\displaystyle S}${\displaystyle S} in a real or complex vector space or affine space is a point in ${\displaystyle S}${\displaystyle S} that does not lie in any open line segment joining two points of ${\displaystyle S.}${\displaystyle 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 ${\displaystyle S.}${\displaystyle S.}^{[citation needed ]}

Throughout, it is assumed that ${\displaystyle X}${\displaystyle X} is a real or complex vector space or affine space.

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

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

Generalizations

If ${\displaystyle S}${\displaystyle S} is a subset of a vector space then a linear sub-variety (that is, an affine subspace) ${\displaystyle A}${\displaystyle A} of the vector space is called a support variety if ${\displaystyle A}${\displaystyle A} meets ${\displaystyle S}${\displaystyle S} (that is, ${\displaystyle A\cap S}${\displaystyle A\cap S} is not empty) and every open segment ${\displaystyle I\subseteq S}${\displaystyle I\subseteq S} whose interior meets ${\displaystyle A}${\displaystyle A} is necessarily a subset of ${\displaystyle A.}${\displaystyle A.}^[2] A 0-dimensional support variety is called an extreme point of ${\displaystyle S.}${\displaystyle S.}^[2]

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

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

If ${\displaystyle K}${\displaystyle K} is a nonempty subset of ${\displaystyle X}${\displaystyle X} and ${\displaystyle F}${\displaystyle F} is a nonempty subset of ${\displaystyle K,}${\displaystyle K,} then ${\displaystyle F}${\displaystyle F} is called a face^[1] of ${\displaystyle K}${\displaystyle K} if whenever a point ${\displaystyle p\in F}${\displaystyle p\in F} lies between two points of ${\displaystyle K,}${\displaystyle K,} then those two points necessarily belong to ${\displaystyle F.}${\displaystyle F.}

If ${\displaystyle a<b}${\displaystyle a<b} are two real numbers then ${\displaystyle a}${\displaystyle a} and ${\displaystyle b}${\displaystyle b} are extreme points of the interval ${\displaystyle [a,b].}${\displaystyle [a,b].} However, the open interval ${\displaystyle (a,b)}${\displaystyle (a,b)} has no extreme points.^[1] Any open interval in ${\displaystyle \mathbb {R} }${\displaystyle \mathbb {R} } has no extreme points while any non-degenerate closed interval not equal to ${\displaystyle \mathbb {R} }${\displaystyle \mathbb {R} } does have extreme points (that is, the closed interval's endpoint(s)). More generally, any open subset of finite-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}${\displaystyle \mathbb {R} ^{n}} has no extreme points.

The extreme points of the closed unit disk in ${\displaystyle \mathbb {R} ^{2}}${\displaystyle \mathbb {R} ^{2}} is the unit circle.

The perimeter of any convex polygon in the plane is a face of that polygon.^[1] The vertices of any convex polygon in the plane ${\displaystyle \mathbb {R} ^{2}}${\displaystyle \mathbb {R} ^{2}} are the extreme points of that polygon.

An injective linear map ${\displaystyle F:X\to Y}${\displaystyle F:X\to Y} sends the extreme points of a convex set ${\displaystyle C\subseteq X}${\displaystyle C\subseteq X} to the extreme points of the convex set ${\displaystyle F(X).}${\displaystyle F(X).}^[1] This is also true for injective affine maps.

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 ${\displaystyle X.}${\displaystyle X.}^[1]

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

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.^[3] )

Edgar’s theorem implies Lindenstrauss’s theorem.

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.^[5] The unit ball of any Hilbert space is a strictly convex set.^[5]

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

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

• Extreme set
• Exposed point
• Choquet theory – Area of functional analysis and convex analysis
• Bang–bang control ^[3]

• Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
• Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
• Paul E. Black, ed. (2004年12月17日). "extreme point". Dictionary of algorithms and data structures . US National institute of standards and technology . Retrieved 2011年03月24日.
• Borowski, Ephraim J.; Borwein, Jonathan M. (1989). "extreme point". Dictionary of mathematics. Collins dictionary. HarperCollins. ISBN 0-00-434347-6.
• Grothendieck, Alexander (1973). Topological Vector Spaces . Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
• Halmos, Paul R. (8 November 1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90685-0. OCLC 8169781.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
• Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.

• Convex geometry
• Convex hulls
• Functional analysis
• Mathematical analysis

• Articles with short description
• Short description is different from Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from May 2026