Namespaces
Variants
Actions

Difference between revisions of "Choquet simplex"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
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
+
<!--
 +
c0221301.png
 +
$#A+1 = 26 n = 0
 +
$#C+1 = 26 : ~/encyclopedia/old_files/data/C022/C.0202130 Choquet simplex
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
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" />.
+
A non-empty compact convex set  $  X $
 +
in a [[Locally convex space|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
  
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.
+
$$
 +
\widetilde{X}  = \
 +
\{ {\alpha x \in E \times \mathbf R } : {x \in X \subset  E \times 1,\
 +
\alpha \geq  0 } \}
 +
,
 +
$$
 +
 
 +
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|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 [[#References|[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 [[#References|[1]]], [[#References|[2]]]). It was introduced by G. Choquet.
  
 
====References====
 
====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>
 
<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====
 
====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" ).
+
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" ).

Latest revision as of 16:44, 4 June 2020


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" ).

How to Cite This Entry:
Choquet simplex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Choquet_simplex&oldid=14569
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article