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In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every vector of C should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an integral 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, parti

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rdfs:label	Choquet-Theorie (de) Choquet theory (en)
rdfs:comment	Die Choquet-Theorie (nach Gustave Choquet) ist eine mathematische Theorie aus dem Teilgebiet der Funktionalanalysis. Sie präzisiert die Vorstellung, dass die Punkte einer kompakten, konvexen Menge eines lokalkonvexen Raumes als „Mittelung“ über die Menge der Extremalpunkte dieser Menge dargestellt werden können. (de) In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every vector of C should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an integral 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, parti (en)
dct:subject	Integral representations Convex hulls Functional analysis
Wikipage page ID	3655598 (xsd:integer)
Wikipage revision ID	1119691129 (xsd:integer)
Link from a Wikipage to another Wikipage	Riesz representation theorem Cube Infinite set Line segment Convex analysis Convex set Mathematics Measure (mathematics) Bounded set Convex combination Convex cone Convex hull Line (mathematics) Locally convex topological vector space Integral representations Simplex State (functional analysis) Closed set Functional analysis Banach space Topological vector space Almost all Convex hulls Euclidean space Euclidean vector Normed space Gustave Choquet Hermann Minkowski Potential theory Functional analysis Support (mathematics) Real vector space Integral Krein–Milman theorem Probability measure Affine function Compact set Extreme points
sameAs	Choquet theory Choquet theory Choquet theory Choquet theory Choquet theory
dbp:wikiPageUsesTemplate	dbt:Springer dbt:Annotated_link dbt:Cite_book dbt:Reflist dbt:Analysis_in_topological_vector_spaces dbt:Convex_analysis_and_variational_analysis dbt:Functional_analysis
id	c/c022130 (en)
title	Choquet simplex (en)
has abstract	In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every vector of C should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an integral 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. (en) Die Choquet-Theorie (nach Gustave Choquet) ist eine mathematische Theorie aus dem Teilgebiet der Funktionalanalysis. Sie präzisiert die Vorstellung, dass die Punkte einer kompakten, konvexen Menge eines lokalkonvexen Raumes als „Mittelung“ über die Menge der Extremalpunkte dieser Menge dargestellt werden können. (de)
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page length (characters) of wiki page	5379 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Choquet_theory
is Link from a Wikipage to another Wikipage of	Robert Phelps De Finetti's theorem List of mathematical theories Convex set Edward George Effros Convex hull Theory Ergodicity Choquet State (functional analysis) Bunce–Deddens algebra Carathéodory's theorem (convex hull) List of convexity topics Gustave Choquet Helly's theorem Karel deLeeuw Krein–Milman theorem Séminaire Nicolas Bourbaki (1950–1959) Choquet's theorem Choquet-Bishop-de Leeuw theorem Choquet theorem Choquet–Bishop–de Leeuw theorem
is Wikipage redirect of	Choquet's theorem Choquet-Bishop-de Leeuw theorem Choquet theorem Choquet–Bishop–de Leeuw theorem
is Wikipage disambiguates of	Choquet
is known for of	Gustave Choquet

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