Choquet simplex

A non-empty compact convex set $ X $ in a locally convex space $ E $ that possesses the following property: Under the imbedding of $ E $ as the hyperplane $ E \times 1 $ in the space $ E \times \mathbf R $ the projecting cone

$$ \widetilde{X} = \ \{ {\alpha x \in E \times \mathbf R } : {x \in X \subset E \times 1,\ \alpha \geq 0 } \} , $$

of $ X $ transforms the space $ E \times \mathbf R $ into a partially ordered space $ P $ for which the space generated by $ P $, which is the space of differences $ \widetilde{X} - \widetilde{X} $, is a lattice. In the case when $ E $ is finite-dimensional, a Choquet simplex is an ordinary simplex with number of vertices equal to $ \mathop{\rm dim} E+ 1 $. There exists a number of equivalent definitions of a Choquet simplex (see [1]). One of them reduces to the requirement that an intersection of $ \widetilde{X} $ with any translate of $ \widetilde{X} $ should be again a translate of $ \widetilde{X} $.

When, in addition to the above requirements, $ E $ is separable and $ X $ is metrizable, then for $ X $ to be a Choquet simplex it is necessary and sufficient that any point $ x \in X $ is the centre of gravity of the unique measure concentrated at the extreme points of $ X $. The concept of a Choquet simplex is essential when studying the uniqueness of an integral representation of a function (see [1], [2]). It was introduced by G. Choquet.

References

[1]	R.R. Phelps, "Lectures on Choquet's theorem" , v. Nostrand (1966)
[2]	E.M. Alfsen, "Compact convex sets and boundary integrals" , Springer (1971)

Comments

The Choquet unique representation theorem says that a compact convex metrizable subset of a locally convex space is a Choquet simplex if and only if for each $ x \in X $ there exists a unique measure $ \mu $ concentrated on the extremal points of $ X $ which represents $ x $( i.e. has $ x $ as "centre of gravity" ).