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In the
mathematical theory A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
of
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, the Krein–Milman theorem is a
proposition A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...
about
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
s in locally convex
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s (TVSs). This theorem generalizes to infinite-dimensional spaces and to arbitrary compact convex sets the following basic observation: a convex (i.e. "filled") triangle, including its perimeter and the area "inside of it", is equal to the convex hull of its three vertices, where these vertices are exactly the extreme points of this shape. This observation also holds for any other convex
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
in the plane \R^2.


Statement and definitions


Preliminaries and definitions

Throughout, X will be a real or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
. For any elements x and y in a vector space, the set
, y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
:= \ is called the or closed interval between x and y. The or open interval between x and y is (x, y) := \varnothing when x = y while it is (x, y) := \ when x \neq y; it satisfies (x, y) =
, y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
\setminus \ and
, y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
= (x, y) \cup \. The points x and y are called the endpoints of these interval. An interval is said to be or proper if its endpoints are distinct. The intervals , x= \ and
, y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> always contain their endpoints while (x, x) = \varnothing and (x, y) never contain either of their endpoints. If x and y are points in the real line \R then the above definition of
, y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is the same as its usual definition as a
closed interval In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
. For any p, x, y \in X, the point p is said to (strictly) x and y if p belongs to the open line segment (x, y). If K is a subset of X and p \in K, then p is called an
extreme point In mathematics, an extreme point of a convex set S in a Real number, real or Complex number, complex vector space is a point in S that does not lie in any open line segment joining two points of S. The extreme points of a line segment are calle ...
of K if it does not lie between any two points of K. That is, if there does exist x, y \in K and 0 < t < 1 such that x \neq y and p = tx + (1-t) y. In this article, the set of all extreme points of K will be denoted by \operatorname(K). For example, 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 In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
. Every
open interval In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
and degenerate closed interval in \R has no extreme points while the extreme points of a non-degenerate
closed interval In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
, y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> are x and y. A set S is called convex if for any two points x, y \in S, S contains the line segment
, y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
The smallest convex set containing S is called the
convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
of S and it is denoted by \operatorname S. The closed convex hull of a set S, denoted by \overline(S), is the smallest closed and convex set containing S. It is also equal to the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of all closed convex subsets that contain S and to the closure of the
convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
of S; that is, \overline(S) = \overline, where the right hand side denotes the closure of \operatorname(S) while the left hand side is notation. For example, the convex hull of any set of three distinct points forms either a closed line segment (if they are
collinear In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
) or else a solid (that is, "filled") triangle, including its perimeter. And in the plane \R^2, the unit circle is convex but the closed unit disk is convex and furthermore, this disk is equal to the convex hull of the circle. The separable Hilbert space
Lp space In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourba ...
\ell^2(\N) of square-summable sequences with the usual norm \, \cdot\, _2 has a compact subset S whose convex hull \operatorname(S) is closed and thus also compact. However, like in all complete Hausdorff locally convex spaces, the convex hull \overline S of this compact subset is compact. But, if a Hausdorff locally convex space is not complete, then it is in general guaranteed that \overline S is compact whenever S is; an example can even be found in a (non-complete) pre-Hilbert vector subspace of \ell^2(\N). Every compact subset is totally bounded (also called "precompact") and the closed convex hull of a totally bounded subset of a Hausdorff locally convex space is guaranteed to be totally bounded.


Statement

In the case where the compact set K is also convex, the above theorem has as a corollary the first part of the next theorem, which is also often called the Krein–Milman theorem. The convex hull of the extreme points of K forms a convex subset of K so the main burden of the proof is to show that there are enough extreme points so that their convex hull covers all of K. For this reason, the following corollary to the above theorem is also often called the Krein–Milman theorem. To visualized this theorem and its conclusion, consider the particular case where K is a convex
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
. In this case, the corners of the polygon (which are its extreme points) are all that is needed to recover the polygon shape. The statement of the theorem is false if the polygon is not convex, as then there are many ways of drawing a polygon having given points as corners. The requirement that the convex set K be compact can be weakened to give the following strengthened generalization version of the theorem. The property above is sometimes called or .
Compactness In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...
implies convex compactness because a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is compact if and only if every
family Family (from ) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictabili ...
of closed subsets having the finite intersection property (FIP) has non-empty intersection (that is, its kernel is not empty). The definition of convex compactness is similar to this characterization of
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...
s in terms of the FIP, except that it only involves those closed subsets that are also
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
(rather than all closed subsets).


More general settings

The assumption of local convexity for the ambient space is necessary, because constructed a counter-example for the non-locally convex space where 0 < p < 1. Linearity is also needed, because the statement fails for weakly compact convex sets in CAT(0) spaces, as proved by . However, proved that the Krein–Milman theorem does hold for compact CAT(0) spaces.


Related results

Under the previous assumptions on K, if T is a
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of K and the closed convex hull of T is all of K, then every
extreme point In mathematics, an extreme point of a convex set S in a Real number, real or Complex number, complex vector space is a point in S that does not lie in any open line segment joining two points of S. The extreme points of a line segment are calle ...
of K belongs to the closure of T. This result is known as (partial) to the Krein–Milman theorem. The Choquet–Bishop–de Leeuw theorem states that every point in K is the barycenter of a
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
supported on the set of
extreme point In mathematics, an extreme point of a convex set S in a Real number, real or Complex number, complex vector space is a point in S that does not lie in any open line segment joining two points of S. The extreme points of a line segment are calle ...
s of K.


Relation to the axiom of choice

Under the
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
(ZF) axiomatic framework, the
axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
(AC) suffices to prove all versions of the Krein–Milman theorem given above, including statement KM and its generalization SKM. The axiom of choice also implies, but is not equivalent to, the
Boolean prime ideal theorem In mathematics, the Boolean prime ideal theorem states that Ideal (order theory), ideals in a Boolean algebra (structure), Boolean algebra can be extended to Ideal (order theory)#Prime ideals , prime ideals. A variation of this statement for Filte ...
(BPI), which is equivalent to the Banach–Alaoglu theorem. Conversely, the Krein–Milman theorem KM together with the
Boolean prime ideal theorem In mathematics, the Boolean prime ideal theorem states that Ideal (order theory), ideals in a Boolean algebra (structure), Boolean algebra can be extended to Ideal (order theory)#Prime ideals , prime ideals. A variation of this statement for Filte ...
(BPI) imply the axiom of choice. In summary, AC holds if and only if both KM and BPI hold. It follows that under ZF, the axiom of choice is equivalent to the following statement: :The closed unit ball of the continuous dual space of any real normed space has an extreme point. Furthermore, SKM together with the
Hahn–Banach theorem In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...
for real vector spaces (HB) are also equivalent to the axiom of choice. It is known that BPI implies HB, but that it is not equivalent to it (said differently, BPI is strictly stronger than HB).


History

The original statement proved by was somewhat less general than the form stated here. Earlier, proved that if X is 3-dimensional then K equals the convex hull of the set of its extreme points. This assertion was expanded to the case of any finite dimension by .; (see p. 16) The Krein–Milman theorem generalizes this to arbitrary locally convex X; however, to generalize from finite to infinite dimensional spaces, it is necessary to use the closure.


See also

* * * * * * *


Citations


Bibliography

* * * * * * * * * * * * * N. K. Nikol'skij (Ed.). ''Functional Analysis I''. Springer-Verlag, 1992. * * H. L. Royden, ''Real Analysis''. Prentice-Hall, Englewood Cliffs, New Jersey, 1988. * * * * * {{DEFAULTSORT:Krein-Milman theorem Convex hulls Oriented matroids Theorems involving convexity Theorems in convex geometry Theorems in discrete geometry Theorems in functional analysis Topological vector spaces