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.
In mathematics, Choquettheory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have...
Bifurcation theory Braid theory Brill–Noether theory Catastrophe theory Category theory Chaos theory Character theoryChoquettheory Class field theory Cobordism...
analysis, potential theory, topology and measure theory. He is known for creating the Choquettheory, the Choquet integral and the theory of capacities. He...
Approximation theory — Arakelov theory — Asymptotic theory — Bifurcation theory — Catastrophe theory — Category theory — Chaos theory — Choquettheory — Coding...
are both convex, but neither one has any extreme points. Choquettheory extends this theory from finite convex combinations of extreme points to infinite...
A Choquet integral is a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953. It was initially used in statistical...
mathematician and physicist Choquet integral, a way of measuring the expected utility of an uncertain event Choquettheory states that for a compact convex...
The theory of relativity usually encompasses two interrelated physics theories by Albert Einstein: special relativity and general relativity, proposed...
The Choquet game is a topological game named after Gustave Choquet, who was in 1969 the first to investigate such games. A closely related game is known...
operator theory, to harmonic analysis, to Choquettheory, and to variational analysis. In one field of its application, optimization theory, Ivar Ekeland...
central concept of fuzzy measure theory is the fuzzy measure (also capacity, see ), which was introduced by Choquet in 1953 and independently defined...
that of p , {\displaystyle p,} and the theorem follows by induction. Choquettheory – Area of functional analysis and convex analysis Bang–bang control...
A^{-1}(\cdot ).} Choquettheory – Area of functional analysis and convex analysis Covariance operator – Operator in probability theory Fundamental theorem...
general an invariant measure need not be ergodic, but as a consequence of Choquettheory it can always be expressed as the barycenter of a probability measure...
relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein...
Mathematical Society. p. 143. ISBN 978-1-4704-2846-4. OCLC 1020305599. Choquet-Bruhat, Yvonne (2009). General Relativity and the Einstein Equations. Oxford:...
of P with d+1 or fewer points such that x lies in its convex hull. Choquettheory - an area of functional analysis and convex analysis concerned with...
"capacitable" set was introduced by Gustave Choquet in 1950: for a detailed account, see reference (Choquet 1986). Let Σ be a closed, smooth, (n − 1)-dimensional...
of gravity is based on Albert Einstein's general theory of relativity, which incorporates his theory of special relativity and deeply modifies the understanding...
framework have been proposed. These include: Choquet expected utility: Created by French mathematician Gustave Choquet was a subadditive integral used as a way...
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert...
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