View source for Choquet simplex

A non-empty compact convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c0221301.png" /> in a [[Locally convex space|locally convex space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c0221302.png" /> that possesses the following property: Under the imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c0221303.png" /> as the hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c0221304.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c0221305.png" /> the projecting cone

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c0221306.png" /></td> </tr></table>

of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c0221307.png" /> transforms the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c0221308.png" /> into a partially ordered space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c0221309.png" /> for which the space generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213010.png" />, which is the space of differences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213011.png" />, is a [[Lattice|lattice]]. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213012.png" /> is finite-dimensional, a Choquet simplex is an ordinary simplex with number of vertices equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213013.png" />. There exists a number of equivalent definitions of a Choquet simplex (see [[#References|[1]]]). One of them reduces to the requirement that an intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213014.png" /> with any translate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213015.png" /> should be again a translate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213016.png" />.

When, in addition to the above requirements, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213017.png" /> is separable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213018.png" /> is metrizable, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213019.png" /> to be a Choquet simplex it is necessary and sufficient that any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213020.png" /> is the centre of gravity of the unique measure concentrated at the extreme points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213021.png" />. The concept of a Choquet simplex is essential when studying the uniqueness of an integral representation of a function (see [[#References|[1]]], [[#References|[2]]]). It was introduced by G. Choquet.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.R. Phelps,   "Lectures on Choquet's theorem" , v. Nostrand  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.M. Alfsen,   "Compact convex sets and boundary integrals" , Springer  (1971)</TD></TR></table>



====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213022.png" /> there exists a unique measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213023.png" /> concentrated on the extremal points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213024.png" /> which represents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213025.png" /> (i.e. has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022130/c02213026.png" /> as  "centre of gravity" ).