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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

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

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====
<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  $  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" ).