In mathematics, an **extreme point** of a convex set *S* in a real vector space is a point in *S* which does not lie in any open line segment joining two points of *S*. In linear programming problems, an extreme point is also called vertex or corner point of *S*.^{ [1] }

Throughout, it is assumed that S is a real or complex vector space.

For any *x*,* x _{1}*,

If K is a subset of S and *x* ∈ *K*, then x is called an **extreme point**^{ [2] } of K if it does not lie between any two *distinct* points of K. That is, if there does *not* exist *x _{1}*,

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

For any elements x and y in a vector space, the set [*x*, *y*] := {*tx* + (1 − *t*)*y* : 0 ≤ *t* ≤ 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*) := ∅ when *x* = *y* while it is (*x*, *y*) := {*tx* + (1 − *t*)*y* : 0 < *t* < 1} when *x* ≠ *y*.^{ [2] } The points x and y are called the **endpoints** of these interval. An interval is said to be **non-degenerate** or **proper** if its endpoints are distinct. The **midpoint** of an interval is the midpoint of its endpoints.

Note that [*x*, *y*] is equal to the convex hull of {*x*, *y*} so if K is convex and *x*, *y* ∈ *K*, then [*x*, *y*] ⊆ *K*.

If K is a nonempty subset of X and F is a nonempty subset of K, then F is called a **face**^{ [2] } of K if whenever a point *p* ∈ *F* lies between two points of K, then those two points necessarily belong to F.

**Theorem ^{ [2] }** — Let K be a non-empty convex subset of a vector space X and let

- p is an extreme point of K;
*K*∖ {*p*} is convex;- p is not the midpoint of a non-degenerate line segment contained in K;
- for any
*x*,*y*∈*K*, if*p*∈ [*x*,*y*] then*x*=*p*or*y*=*p*; - if
*x*∈*X*is such that both*p*+*x*and*p*−*x*belong to K, then*x*= 0; - {
*p*} is a face of K.

- 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.^{ [2] } - An injective linear map
*F*:*X*→*Y*sends the extreme points of a convex set*C*⊆*X*to the extreme points of the convex set*F*(*C*).^{ [2] }This is also true for injective affine maps. - 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 ℝ
^{2}are the extreme points of that polygon. - The extreme points of the closed unit disk in ℝ
^{2}is the unit circle. - Any open interval in ℝ has no extreme points while any non-degenerate closed interval not equal to ℝ does have extreme points (i.e. 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 a compact convex form a Baire space (with the subspace topology) but this set may *fail* to be closed in X.^{ [2] }

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 space, then *S* is the closed convex hull of its extreme points: In particular, such a set has 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] }

**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.^{ [4] }

Edgar's theorem implies Lindenstrauss's theorem.

More generally, a point in a convex set *S* is ** k-extreme** if it lies in the interior of a

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* < *n*. The theorem asserts that *p* is a convex combination of extreme points. If *k* = 0, then it's trivially true. 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.

- ↑ Saltzman, Matthew. "What is the difference between corner points and extreme points in linear programming problems?".
- 1 2 3 4 5 6 7 8 9 10 Narici & Beckenstein 2011, pp. 275-339.
- ↑ Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points".
*SIAM Review*.**22**(2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. - ↑ Edgar GA. A noncompact Choquet theorem. Proceedings of the American Mathematical Society. 1975;49(2):354-8.

- Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978).
*Topological Vector Spaces: The Theory Without Convexity Conditions*. Lecture Notes in Mathematics.**639**. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. - Bourbaki, Nicolas (1987) [1981].
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*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".
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*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).
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*Functional Analysis*. International Series in Pure and Applied Mathematics.**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.**8**(Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. - Schechter, Eric (1996).
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