{"history":[],"article":{"id":"03611911-c636-4607-9a11-bedbd7292f97","title":"Extreme point","slug":"extreme-point","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), an **extreme point** of a [convex set](https://bliptext.com/articles/convex-set) S in a [real](https://bliptext.com/articles/real-number) or [complex](https://bliptext.com/articles/complex-number) [vector space](https://bliptext.com/articles/vector-space) is a point in S that does not lie in any open [line segment](https://bliptext.com/articles/line-segment) joining two points of S. In [linear programming](https://bliptext.com/articles/linear-programming) problems, an **extreme point** is also called vertex or corner point of S.\n# [A convex set in light blue, and its extreme points in red. | upload.wikimedia.org/wikipedia/commons/8/8e/Extreme///points.svg]\n\n# Definition\nThroughout, it is assumed that X is a [real](https://bliptext.com/articles/real-number) or [complex](https://bliptext.com/articles/complex-number) [vector space](https://bliptext.com/articles/vector-space).\n****F****o****r**** ****a****n****y**** ****s****a****y**** ****t****h****a****t**** ****p**** **** ****x**** ****a****n****d**** ****y**** ****i****f**** ****x**** ****\\****n****e****q**** ****y**** ****a****n****d**** ****t****h****e****r****e**** ****e****x****i****s****t****s**** ****a**** ****0**** ****<**** ****t**** ****<**** ****1**** ****s****u****c****h**** ****t****h****a****t****\n****I****f**** ****K**** ****i****s**** ****a**** ****s****u****b****s****e****t**** ****o****f**** ****X**** ****a****n****d**** ****p**** ****\\****i****n**** ****K****,**** ****t****h****e****n**** ****p**** ****i****s**** ****c****a****l****l****e****d**** ****a****n**** **** ****o****f**** ****K**** ****i****f**** ****i****t**** ****d****o****e****s**** ****n****o****t**** ****l****i****e**** ****b****e****t****w****e****e****n**** ****a****n****y**** ****t****w****o**** ****p****o****i****n****t****s**** ****o****f**** ****K****.**** ****T****h****a****t**** ****i****s****,**** ****i****f**** ****t****h****e****r****e**** ****d****o****e****s**** **** ****e****x****i****s****t**** ****x****,**** ****y**** ****\\****i****n**** ****K**** ****a****n****d**** ****0**** ****<**** ****t**** ****<**** ****1**** ****s****u****c****h**** ****t****h****a****t**** ****x**** ****\\****n****e****q**** ****y**** ****a****n****d**** **** ****T****h****e**** ****s****e****t**** ****o****f**** ****a****l****l**** ****e****x****t****r****e****m****e**** ****p****o****i****n****t****s**** ****o****f**** ****K**** ****i****s**** ****d****e****n****o****t****e****d**** ****b****y****\n**Generalizations**\nIf S is a subset of a vector space then a linear sub-variety (that is, an [affine subspace](https://bliptext.com/articles/affine-subspace)) A of the vector space is called a if A meets S (that is, A \\cap S is not empty) and every open segment whose interior meets A is necessarily a subset of A. A 0-dimensional support variety is called an extreme point of S.\n\n## Characterizations\n****T****h****e**** **** ****o****f**** ****t****w****o**** ****e****l****e****m****e****n****t****s**** ****x**** ****a****n****d**** ****y**** ****i****n**** ****a**** ****v****e****c****t****o****r**** ****s****p****a****c****e**** ****i****s**** ****t****h****e**** ****v****e****c****t****o****r****\n****F****o****r**** ****a****n****y**** ****e****l****e****m****e****n****t****s**** ****x**** ****a****n****d**** ****y**** ****i****n**** ****a**** ****v****e****c****t****o****r**** ****s****p****a****c****e****,**** ****t****h****e**** ****s****e****t**** ****i****s**** ****c****a****l****l****e****d**** ****t****h****e**** ************************'**** ****o****r**** ************************'**** ****b****e****t****w****e****e****n**** ****x**** ****a****n****d**** ****y****.**** ****T****h****e**** ************************'**** ****o****r**** ************************'**** ****b****e****t****w****e****e****n**** ****x**** ****a****n****d**** ****y**** ****i****s**** **** ****w****h****e****n**** ****x**** ****=**** ****y**** ****w****h****i****l****e**** ****i****t**** ****i****s**** **** ****w****h****e****n**** ****x**** ****\\****n****e****q**** ****y****.**** ****T****h****e**** ****p****o****i****n****t****s**** ****x**** ****a****n****d**** ****y**** ****a****r****e**** ****c****a****l****l****e****d**** ****t****h****e**** **** ****o****f**** ****t****h****e****s****e**** ****i****n****t****e****r****v****a****l****.****\nAn interval is said to be a **' or a **' if its endpoints are distinct.\n****T****h****e**** **** ****i****s**** ****t****h****e**** ****m****i****d****p****o****i****n****t**** ****o****f**** ****i****t****s**** ****e****n****d****p****o****i****n****t****s****.****\nThe closed interval [x, y] is equal to the [convex hull](https://bliptext.com/articles/convex-hull) of (x, y) if (and only if) x \\neq y. So if K is convex and x, y \\in K, then\n****I****f**** ****K**** ****i****s**** ****a**** ****n****o****n****e****m****p****t****y**** ****s****u****b****s****e****t**** ****o****f**** ****X**** ****a****n****d**** ****F**** ****i****s**** ****a**** ****n****o****n****e****m****p****t****y**** ****s****u****b****s****e****t**** ****o****f**** ****K****,**** ****t****h****e****n**** ****F**** ****i****s**** ****c****a****l****l****e****d**** ****a**** **** ****o****f**** ****K**** ****i****f**** ****w****h****e****n****e****v****e****r**** ****a**** ****p****o****i****n****t**** ****p**** ****\\****i****n**** ****F**** ****l****i****e****s**** ****b****e****t****w****e****e****n**** ****t****w****o**** ****p****o****i****n****t****s**** ****o****f**** ****K****,**** ****t****h****e****n**** ****t****h****o****s****e**** ****t****w****o**** ****p****o****i****n****t****s**** ****n****e****c****e****s****s****a****r****i****l****y**** ****b****e****l****o****n****g**** ****t****o**** ****F****.****\n\n# Examples\nIf a < b are two real numbers then a and b are extreme points of the interval [a, b].\nHowever, the open interval (a, b) has no extreme points.\nAny [open interval](https://bliptext.com/articles/open-interval) in \\R has no extreme points while any non-degenerate [closed interval](https://bliptext.com/articles/closed-interval) not equal to \\R does have extreme points (that is, the [closed interval](https://bliptext.com/articles/closed-interval)'s endpoint(s)).\nMore generally, any [open subset](https://bliptext.com/articles/open-set) of finite-dimensional [Euclidean space](https://bliptext.com/articles/euclidean-space) \\R^n has no extreme points.\nThe extreme points of the [closed unit disk](https://bliptext.com/articles/closed-unit-disk) in \\R^2 is the [unit circle](https://bliptext.com/articles/unit-circle).\nThe perimeter of any convex polygon in the plane is a face of that polygon.\nThe vertices of any convex polygon in the plane \\R^2 are the extreme points of that polygon.\nAn injective linear map F : X \\to Y sends the extreme points of a convex set to the extreme points of the convex set F(X).\nThis is also true for injective affine maps.\n\n# Properties\nThe extreme points of a compact convex set form a [Baire space](https://bliptext.com/articles/baire-space) (with the subspace topology) but this set may to be closed in X.\n\n# Theorems\n\n## Krein–Milman theorem\nThe [Krein–Milman theorem](https://bliptext.com/articles/krein-milman-theorem) is arguably one of the most well-known theorems about extreme points.\n\n## For Banach spaces\nThese theorems are for [Banach spaces](https://bliptext.com/articles/banach-space) with the [Radon–Nikodym property](https://bliptext.com/articles/radon-nikodym-property).\nA theorem of [Joram Lindenstrauss](https://bliptext.com/articles/joram-lindenstrauss) states that, in a Banach space with the Radon–Nikodym property, a nonempty [closed](https://bliptext.com/articles/closed-set) and [bounded set](https://bliptext.com/articles/bounded-set) has an extreme point.\n(In infinite-dimensional spaces, the property of [compactness](https://bliptext.com/articles/compact-space) is stronger than the joint properties of being closed and being bounded. )\nEdgar’s theorem implies Lindenstrauss’s theorem.\n\n# Related notions\nA closed convex subset of a [topological vector space](https://bliptext.com/articles/topological-vector-space) is called if every one of its [(topological) boundary points](https://bliptext.com/articles/boundary-topology) is an extreme point.\nThe [unit ball](https://bliptext.com/articles/unit-ball) of any [Hilbert space](https://bliptext.com/articles/hilbert-space) is a strictly convex set.\n\n## k-extreme points\nMore 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.\nIf 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.\nThe finite-dimensional Krein–Milman theorem, which is due to Minkowski, can be quickly proved using the concept of k-extreme points.\nIf S is closed, bounded, and n-dimensional, and if p is a point in S, then p is k-extreme for some k \\leq n. The theorem asserts that p is a convex combination of extreme points.\nIf k = 0 then it is immediate.\nOtherwise p lies on a line segment in S which can be maximally extended (because S is closed and bounded).\nIf 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.\n\n# Citations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:40:47.524Z","updated_at":"2025-01-25T19:40:47.524Z","search_vector":"'extrem':1 'point':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), an **extreme point** of a [convex set](https://bliptext.com/articles/convex-set) S in a [real](https://bliptext.com/articles/real-number) or [complex](https://bliptext.com/articles/complex-number) [vector space](https://bliptext.com/articles/vector-space) is a point in S that does not lie in any open [line segment](https://bliptext.com/articles/line-segment) joining two points of S. In [linear programming](https://bliptext.com/articles/linear-programming) problems, an **extreme point** is also called vertex or corner point of S.\n# [A convex set in light blue, and its extreme points in red. | upload.wikimedia.org/wikipedia/commons/8/8e/Extreme///points.svg]\n\n# Definition\nThroughout, it is assumed that X is a [real](https://bliptext.com/articles/real-number) or [complex](https://bliptext.com/articles/complex-number) [vector space](https://bliptext.com/articles/vector-space).\n****F****o****r**** ****a****n****y**** ****s****a****y**** ****t****h****a****t**** ****p**** **** ****x**** ****a****n****d**** ****y**** ****i****f**** ****x**** ****\\****n****e****q**** ****y**** ****a****n****d**** ****t****h****e****r****e**** ****e****x****i****s****t****s**** ****a**** ****0**** ****<**** ****t**** ****<**** ****1**** ****s****u****c****h**** ****t****h****a****t****\n****I****f**** ****K**** ****i****s**** ****a**** ****s****u****b****s****e****t**** ****o****f**** ****X**** ****a****n****d**** ****p**** ****\\****i****n**** ****K****,**** ****t****h****e****n**** ****p**** ****i****s**** ****c****a****l****l****e****d**** ****a****n**** **** ****o****f**** ****K**** ****i****f**** ****i****t**** ****d****o****e****s**** ****n****o****t**** ****l****i****e**** ****b****e****t****w****e****e****n**** ****a****n****y**** ****t****w****o**** ****p****o****i****n****t****s**** ****o****f**** ****K****.**** ****T****h****a****t**** ****i****s****,**** ****i****f**** ****t****h****e****r****e**** ****d****o****e****s**** **** ****e****x****i****s****t**** ****x****,**** ****y**** ****\\****i****n**** ****K**** ****a****n****d**** ****0**** ****<**** ****t**** ****<**** ****1**** ****s****u****c****h**** ****t****h****a****t**** ****x**** ****\\****n****e****q**** ****y**** ****a****n****d**** **** ****T****h****e**** ****s****e****t**** ****o****f**** ****a****l****l**** ****e****x****t****r****e****m****e**** ****p****o****i****n****t****s**** ****o****f**** ****K**** ****i****s**** ****d****e****n****o****t****e****d**** ****b****y****\n**Generalizations**\nIf S is a subset of a vector space then a linear sub-variety (that is, an [affine subspace](https://bliptext.com/articles/affine-subspace)) A of the vector space is called a if A meets S (that is, A \\cap S is not empty) and every open segment whose interior meets A is necessarily a subset of A. A 0-dimensional support variety is called an extreme point of S.\n\n## Characterizations\n****T****h****e**** **** ****o****f**** ****t****w****o**** ****e****l****e****m****e****n****t****s**** ****x**** ****a****n****d**** ****y**** ****i****n**** ****a**** ****v****e****c****t****o****r**** ****s****p****a****c****e**** ****i****s**** ****t****h****e**** ****v****e****c****t****o****r****\n****F****o****r**** ****a****n****y**** ****e****l****e****m****e****n****t****s**** ****x**** ****a****n****d**** ****y**** ****i****n**** ****a**** ****v****e****c****t****o****r**** ****s****p****a****c****e****,**** ****t****h****e**** ****s****e****t**** ****i****s**** ****c****a****l****l****e****d**** ****t****h****e**** ************************'**** ****o****r**** ************************'**** ****b****e****t****w****e****e****n**** ****x**** ****a****n****d**** ****y****.**** ****T****h****e**** ************************'**** ****o****r**** ************************'**** ****b****e****t****w****e****e****n**** ****x**** ****a****n****d**** ****y**** ****i****s**** **** ****w****h****e****n**** ****x**** ****=**** ****y**** ****w****h****i****l****e**** ****i****t**** ****i****s**** **** ****w****h****e****n**** ****x**** ****\\****n****e****q**** ****y****.**** ****T****h****e**** ****p****o****i****n****t****s**** ****x**** ****a****n****d**** ****y**** ****a****r****e**** ****c****a****l****l****e****d**** ****t****h****e**** **** ****o****f**** ****t****h****e****s****e**** ****i****n****t****e****r****v****a****l****.****\nAn interval is said to be a **' or a **' if its endpoints are distinct.\n****T****h****e**** **** ****i****s**** ****t****h****e**** ****m****i****d****p****o****i****n****t**** ****o****f**** ****i****t****s**** ****e****n****d****p****o****i****n****t****s****.****\nThe closed interval [x, y] is equal to the [convex hull](https://bliptext.com/articles/convex-hull) of (x, y) if (and only if) x \\neq y. So if K is convex and x, y \\in K, then\n****I****f**** ****K**** ****i****s**** ****a**** ****n****o****n****e****m****p****t****y**** ****s****u****b****s****e****t**** ****o****f**** ****X**** ****a****n****d**** ****F**** ****i****s**** ****a**** ****n****o****n****e****m****p****t****y**** ****s****u****b****s****e****t**** ****o****f**** ****K****,**** ****t****h****e****n**** ****F**** ****i****s**** ****c****a****l****l****e****d**** ****a**** **** ****o****f**** ****K**** ****i****f**** ****w****h****e****n****e****v****e****r**** ****a**** ****p****o****i****n****t**** ****p**** ****\\****i****n**** ****F**** ****l****i****e****s**** ****b****e****t****w****e****e****n**** ****t****w****o**** ****p****o****i****n****t****s**** ****o****f**** ****K****,**** ****t****h****e****n**** ****t****h****o****s****e**** ****t****w****o**** ****p****o****i****n****t****s**** ****n****e****c****e****s****s****a****r****i****l****y**** ****b****e****l****o****n****g**** ****t****o**** ****F****.****\n\n# Examples\nIf a < b are two real numbers then a and b are extreme points of the interval [a, b].\nHowever, the open interval (a, b) has no extreme points.\nAny [open interval](https://bliptext.com/articles/open-interval) in \\R has no extreme points while any non-degenerate [closed interval](https://bliptext.com/articles/closed-interval) not equal to \\R does have extreme points (that is, the [closed interval](https://bliptext.com/articles/closed-interval)'s endpoint(s)).\nMore generally, any [open subset](https://bliptext.com/articles/open-set) of finite-dimensional [Euclidean space](https://bliptext.com/articles/euclidean-space) \\R^n has no extreme points.\nThe extreme points of the [closed unit disk](https://bliptext.com/articles/closed-unit-disk) in \\R^2 is the [unit circle](https://bliptext.com/articles/unit-circle).\nThe perimeter of any convex polygon in the plane is a face of that polygon.\nThe vertices of any convex polygon in the plane \\R^2 are the extreme points of that polygon.\nAn injective linear map F : X \\to Y sends the extreme points of a convex set to the extreme points of the convex set F(X).\nThis is also true for injective affine maps.\n\n# Properties\nThe extreme points of a compact convex set form a [Baire space](https://bliptext.com/articles/baire-space) (with the subspace topology) but this set may to be closed in X.\n\n# Theorems\n\n## Krein–Milman theorem\nThe [Krein–Milman theorem](https://bliptext.com/articles/krein-milman-theorem) is arguably one of the most well-known theorems about extreme points.\n\n## For Banach spaces\nThese theorems are for [Banach spaces](https://bliptext.com/articles/banach-space) with the [Radon–Nikodym property](https://bliptext.com/articles/radon-nikodym-property).\nA theorem of [Joram Lindenstrauss](https://bliptext.com/articles/joram-lindenstrauss) states that, in a Banach space with the Radon–Nikodym property, a nonempty [closed](https://bliptext.com/articles/closed-set) and [bounded set](https://bliptext.com/articles/bounded-set) has an extreme point.\n(In infinite-dimensional spaces, the property of [compactness](https://bliptext.com/articles/compact-space) is stronger than the joint properties of being closed and being bounded. )\nEdgar’s theorem implies Lindenstrauss’s theorem.\n\n# Related notions\nA closed convex subset of a [topological vector space](https://bliptext.com/articles/topological-vector-space) is called if every one of its [(topological) boundary points](https://bliptext.com/articles/boundary-topology) is an extreme point.\nThe [unit ball](https://bliptext.com/articles/unit-ball) of any [Hilbert space](https://bliptext.com/articles/hilbert-space) is a strictly convex set.\n\n## k-extreme points\nMore 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.\nIf 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.\nThe finite-dimensional Krein–Milman theorem, which is due to Minkowski, can be quickly proved using the concept of k-extreme points.\nIf S is closed, bounded, and n-dimensional, and if p is a point in S, then p is k-extreme for some k \\leq n. The theorem asserts that p is a convex combination of extreme points.\nIf k = 0 then it is immediate.\nOtherwise p lies on a line segment in S which can be maximally extended (because S is closed and bounded).\nIf 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.\n\n# Citations","revisionCount":0},"hasMore":false,"nextCursor":null}

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