Extreme point

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

For any ${\displaystyle p,x,y\in X,}$ say that ${\displaystyle p}$ lies between[2] ${\displaystyle x}$ and ${\displaystyle y}$ if ${\displaystyle x\neq y}$ and there exists a ${\displaystyle 0<t<1}$ such that ${\displaystyle p=tx+(1-t)y.}$

If ${\displaystyle K}$ is a subset of ${\displaystyle X}$ and ${\displaystyle p\in K,}$ then ${\displaystyle p}$ is called an extreme point[2] of ${\displaystyle K}$ if it does not lie between any two distinct points of ${\displaystyle K.}$ That is, if there does not exist ${\displaystyle x,y\in K}$ and ${\displaystyle 0<t<1}$ such that ${\displaystyle x\neq y}$ and ${\displaystyle p=tx+(1-t)y.}$ The set of all extreme points of ${\displaystyle K}$ is denoted by ${\displaystyle \operatorname {extreme} (K).}$

Generalizations

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

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

The extreme points of the closed unit disk in ${\displaystyle \mathbb {R} ^{2}}$ 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 ${\displaystyle \mathbb {R} ^{2}}$ are the extreme points of that polygon.

An injective linear map ${\displaystyle F:X\to Y}$ sends the extreme points of a convex set ${\displaystyle C\subseteq X}$ to the extreme points of the convex set ${\displaystyle F(X).}$[2] 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 ${\displaystyle X.}$[2]

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]

• Choquet theory – Area of functional analysis and convex analysis
• Bang–bang control[7]

• 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. (2004-12-17). "extreme point". Dictionary of algorithms and data structures. US National institute of standards and technology. Retrieved 2011-03-24.
• 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.