Choquet theory

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In mathematics, Choquet theory is an area of functional analysis and convex analysis 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 topological vector space along lines similar to the finite-dimensional case. The name is for Gustave Choquet, whose main concerns 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

c = Σ w(e)e

with

Σ w(e) = 1.

In the case of a simplex the weights are unique, but in other cases such as a ball that is not so. In any case the w(e) give a probability measure with support (a finite set) within E. The idea in Choquet theory is to relax the finite support condition, but retain the probability measure, by replacing the sum by an integral and w by a more general function. Implicit in this is the need to define integrals for vector-valued functions, taking values in V. This will only make good sense if V is at least a topological vector space, and in practice a complete metric space as well so that infinite sums can be made from Cauchy sequences.

Choquet's theorem states that for a compact convex subset C in a normed space V, any c in C is the barycentre of a probability measure supported on the set E of extreme points of C. In practice V will be a Banach space. The original Krein-Milman theorem is a corollary of Choquet's result.

More generally, for V a locally convex topological vector space, the Choquet-Bishop-de Leeuw theorem gives the same formal statement.