# 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)

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" ).