Choquet theory

In

, 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

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

Choquet's theorem states that for a

normed space V, given c in C there exists a probability measure

w supported on the set E of extreme points of C such that, for any affine function f on C,

$${\displaystyle f(c)=\int f(e)dw(e).}$$

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 represents 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. 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 – Point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P
• Helly's theorem – Theorem about the intersections of d-dimensional convex sets
• Krein–Milman theorem – On when a space equals the closed convex hull of its extreme points
• List of convexity topics
• Shapley–Folkman lemma – Sums of sets of vectors are nearly convex

Notes

References

• Asimow, L.; Ellis, A. J. (1980). Convexity theory and its applications in functional analysis. London Mathematical Society Monographs. Vol. 16. London-New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. pp. x+266.
  MR 0623459
  .
• Bourgin, Richard D. (1983). Geometric aspects of convex sets with the Radon-Nikodým property. Lecture Notes in Mathematics. Vol. 993. Berlin: Springer-Verlag. pp. xii+474.
  MR 0704815
  .
• MR 1835574
  .
• "Choquet simplex", Encyclopedia of Mathematics, EMS Press, 2001 [1994]