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In [[mathematics]], '''Choquet theory''' is an area of [[functional analysis]] and [[convex analysis]] created by [[Gustave Choquet]]. It is concerned with [[measure (mathematics)|measures]] with [[support (mathematics)|support]] on the [[extreme points]] of a [[convex set]] ''C''. Roughly speaking, all [[Euclidean vector|vector]]s of ''C'' should appear as 'averages' of extreme points, a concept made more precise by the idea of [[convex combination]]s replaced by [[integral]]s taken over the set ''E'' of extreme points. Here ''C'' is a subset of a [[real vector space]] ''V'', and the main thrust of the theory is to treat the cases where ''V'' is an infinite-dimensional (locally convex Hausdorff) [[topological vector space]] along lines similar to the finite-dimensional case. The main concerns of Gustave Choquet were in [[potential theory]]. Choquet theory has become a general paradigm, particularly for treating [[convex cone]]s as determined by their extreme [[Line (mathematics)#Ray|rays]], and so for many different notions of ''positivity'' in mathematics. | |||
The two ends of a [[line segment]] determine the points in between: in vector terms the segment from ''v'' to ''w'' consists of the λ''v'' + (1 − λ)''w'' with 0 ≤ λ ≤ 1. The classical result of [[Hermann Minkowski]] says that in [[Euclidean space]], a [[bounded set|bounded]], [[closed set|closed]] [[convex set]] ''C'' is the [[convex hull]] of its extreme point set ''E'', so that any ''c'' in ''C'' is a (finite) [[convex combination]] of points ''e'' of ''E''. Here ''E'' may be a finite or an [[infinite set]]. In vector terms, by assigning non-negative weights ''w''(''e'') to the ''e'' in ''E'', [[almost all]] 0, we can represent any ''c'' in ''C'' as | |||
:<math> c = \sum_{e\in E} w(e) e\ </math> | |||
with | |||
:<math> \sum_{e\in E} w(e) = 1.\ </math> | |||
In any case the ''w''(''e'') give a [[probability measure]] supported on a finite subset of ''E''. For any [[affine function]] ''f'' on ''C'', its value at the point ''c'' is | |||
:<math>f (c) = \int f(e) d w(e).</math> | |||
In the infinite dimensional setting, one would like to make a similar statement. | |||
'''Choquet's theorem''' states that for a [[compact set|compact]] convex subset ''C'' in a [[normed space]] ''V'', given ''c'' in ''C'' there exist a [[probability measure]] ''w'' supported on the set ''E'' of extreme points of ''C'' such that, for all affine function ''f'' on ''C''. | |||
:<math>f (c) = \int f(e) d w(e).</math> | |||
In practice ''V'' will be a [[Banach space]]. The original [[Krein–Milman theorem]] follows from Choquet's result. Another corollary is the [[Riesz representation theorem]] for [[state (functional analysis)|states]] on the continuous functions on a metrizable compact Hausdorff space. | |||
More generally, for ''V'' a [[locally convex topological vector space]], the '''Choquet-Bishop-de Leeuw theorem'''<ref>[[Errett Bishop]]; [[Karel deLeeuw|Karl de Leeuw]]. [http://archive.numdam.org/ARCHIVE/AIF/AIF_1959__9_/AIF_1959__9__305_0/AIF_1959__9__305_0.pdf "The representations of linear functionals by measures on sets of extreme points"]. Annales de l'institut Fourier, 9 (1959), p. 305-331.</ref> gives the same formal statement. | |||
In addition to the existence of a probability measure supported on the extreme boundary that represent a given point ''c'', one might also consider the uniqueness of such measures. It is easy to see that uniqueness does not hold even in the finite dimensional setting. One can take, for counterexamples, the convex set to be a [[cube]] or a ball in '''R'''<sup>3</sup>. Uniqueness does hold, however, when the convex set is a finite dimensional [[simplex]]. So that the weights ''w''(''e'') are unique. A finite dimensional simplex is a special case of a '''Choquet simplex'''. Any point in a Choquet simplex is represented by a unique probability measure on the extreme points. | |||
==See also== | |||
* [[Carathéodory's theorem (convex hull)|Carathéodory's theorem]] | |||
* [[Shapley–Folkman lemma]] | |||
* [[Krein–Milman theorem]] | |||
* [[Helly's theorem]] | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
* {{cite book|last1=Asimow|first1=L.|last2=Ellis|first2=A. J.|title=Convexity theory and its applications in functional analysis|series=London Mathematical Society Monographs|volume=16|publisher=Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]|location=London-New York|year=1980|pages=x+266|isbn=0-12-065340-0|MR=623459|}} | |||
* {{cite book|last=Bourgin|first=Richard D.|title=Geometric aspects of convex sets with the Radon-Nikodým property|series= Lecture Notes in Mathematics|volume=993|publisher=Springer-Verlag|location=Berlin|year=1983|pages=xii+474|isbn=3-540-12296-6|MR=704815|ref=harv}} | |||
* {{cite book|last=Phelps|first=Robert R.|authorlink=Robert R. Phelps|title=Lectures on Choquet's theorem|edition=Second edition of 1966|series=Lecture Notes in Mathematics|volume=1757|publisher=Springer-Verlag|location=Berlin|year=2001|pages=viii+124|isbn=3-540-41834-2|MR=1835574|ref=harv}} | |||
*{{Springer|id=c/c022130|title=Choquet simplex}} | |||
[[Category:Functional analysis]] | |||
[[Category:Integral representations]] | |||
[[Category:Convex hulls]] | |||
Revision as of 20:32, 9 December 2012
In mathematics, Choquet theory is an area of functional analysis and convex analysis created by Gustave Choquet. It is concerned with measures with support on the extreme points of a convex set C. Roughly speaking, all vectors of C should appear as 'averages' of extreme points, a concept made more precise by the idea of convex combinations replaced by integrals taken over the set E of extreme points. Here C is a subset of a real vector space V, and the main thrust of the theory is to treat the cases where V is an infinite-dimensional (locally convex Hausdorff) topological vector space along lines similar to the finite-dimensional case. The main concerns of Gustave Choquet were in potential theory. Choquet theory has become a general paradigm, particularly for treating convex cones as determined by their extreme rays, and so for many different notions of positivity in mathematics.
The two ends of a line segment determine the points in between: in vector terms the segment from v to w consists of the λv + (1 − λ)w with 0 ≤ λ ≤ 1. The classical result of Hermann Minkowski says that in Euclidean space, a bounded, closed convex set C is the convex hull of its extreme point set E, so that any c in C is a (finite) convex combination of points e of E. Here E may be a finite or an infinite set. In vector terms, by assigning non-negative weights w(e) to the e in E, almost all 0, we can represent any c in C as
with
In any case the w(e) give a probability measure supported on a finite subset of E. For any affine function f on C, its value at the point c is
In the infinite dimensional setting, one would like to make a similar statement.
Choquet's theorem states that for a compact convex subset C in a normed space V, given c in C there exist a probability measure w supported on the set E of extreme points of C such that, for all affine function f on C.
In practice V will be a Banach space. The original Krein–Milman theorem follows from Choquet's result. Another corollary is the Riesz representation theorem for states on the continuous functions on a metrizable compact Hausdorff space.
More generally, for V a locally convex topological vector space, the Choquet-Bishop-de Leeuw theorem[1] gives the same formal statement.
In addition to the existence of a probability measure supported on the extreme boundary that represent a given point c, one might also consider the uniqueness of such measures. It is easy to see that uniqueness does not hold even in the finite dimensional setting. One can take, for counterexamples, the convex set to be a cube or a ball in R3. Uniqueness does hold, however, when the convex set is a finite dimensional simplex. So that the weights w(e) are unique. A finite dimensional simplex is a special case of a Choquet simplex. Any point in a Choquet simplex is represented by a unique probability measure on the extreme points.
See also
Notes
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- ↑ Errett Bishop; Karl de Leeuw. "The representations of linear functionals by measures on sets of extreme points". Annales de l'institut Fourier, 9 (1959), p. 305-331.