For any say that lies between[2] and if and there exists a such that
If is a subset of and then is called an extreme point[2] of if it does not lie between any two distinct points of That is, if there does not exist and such that and The set of all extreme points of is denoted by
Generalizations
If is a subset of a vector space then a linear sub-variety (that is, an
affine subspace) of the vector space is called a support variety if meets (that is, is not empty) and every open segment whose interior meets is necessarily a subset of [3] A 0-dimensional support variety is called an extreme point of [3]
Characterizations
The midpoint[2] of two elements and in a vector space is the vector
For any elements and in a vector space, the set is called the closed line segment or closed interval between and The open line segment or open interval between and is when while it is when [2] The points and 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 is equal to the
convex hull of if (and only if) So if is convex and then
If is a nonempty subset of and is a nonempty subset of then is called a face[2] of if whenever a point lies between two points of then those two points necessarily belong to
Theorem[2] — Let be a non-empty convex subset of a vector space and let
Then the following statements are equivalent:
is an extreme point of
is convex.
is not the midpoint of a non-degenerate line segment contained in
for any if then
if is such that both and belong to then
is a face of
Examples
If are two real numbers then and are extreme points of the interval However, the open interval has no extreme points.[2]
Any
open interval in has no extreme points while any non-degenerate
closed interval not equal to does have extreme points (that is, the closed interval's endpoint(s)). More generally, any
open subset of finite-dimensional
Euclidean space has no extreme points.
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 are the extreme points of that polygon.
An injective linear map sends the extreme points of a convex set to the extreme points of the convex set [2] 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 [2]
Theorems
Krein–Milman theorem
The
Krein–Milman theorem is arguably one of the most well-known theorems about extreme points.
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.[4])
Theorem(
Gerald Edgar) — Let be a Banach space with the Radon–Nikodym property, let be a separable, closed, bounded, convex subset of and let be a point in Then there is a
probability measure on the universally measurable sets in such that is the
barycenter of and the set of extreme points of has -measure 1.[5]
More generally, a point in a convex set is -extreme if it lies in the interior of a -dimensional convex set within but not a -dimensional convex set within Thus, an extreme point is also a -extreme point. If is a polytope, then the -extreme points are exactly the interior points of the -dimensional faces of More generally, for any convex set the -extreme points are partitioned into -dimensional open faces.
The finite-dimensional Krein–Milman theorem, which is due to Minkowski, can be quickly proved using the concept of -extreme points. If is closed, bounded, and -dimensional, and if is a point in then is -extreme for some The theorem asserts that is a convex combination of extreme points. If then it is immediate. Otherwise lies on a line segment in which can be maximally extended (because is closed and bounded). If the endpoints of the segment are and then their extreme rank must be less than that of and the theorem follows by induction.
See also
Choquet theory – Area of functional analysis and convex analysis
^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.
JSTOR2029960.
MR0564562.
Bibliography
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.
ISBN978-3-540-08662-8.
OCLC297140003.
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.
ISBN978-3-642-64988-2.
MR0248498.
OCLC840293704.