Choquet theory

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. 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