It is proved that there is a unique metrizable simplex $S$ whose extreme points are dense. This simplex is homogeneous in the sense that for every 2 affinely homeomorphic faces $F_1$ and $F_2$ there is an automorphism of $S$ which maps $F_1$ onto $F_2$. Every metrizable simplex is affinely homeomorphic to a face of $S$. The set of extreme points of $S$ is homeomorphic to the Hilbert space ${\ell }_2$. The matrices which represent $A\left(S\right)$ are characterized.

Standard facts about separating linear functionals will be used to determine how two cones $C$ and $D$ and their duals $C^{*}$ and $D^{*}$ may overlap. When $TV\to W$ is linear and $K\subset V$ and $D\subset W$ are cones, these results will be applied to $C=T\left(K\right)$ and $D$, giving a unified treatment of several theorems of the alternate which explain when $C$ contains an interior point of $D$. The case when $V=W$ is the space $H$ of $n\times n$ Hermitian matrices, $D$ is the $n\times n$ positive semidefinite matrices, and $T\left(X\right)=AX+X^{*}A$ yields new and known results about the existence of block diagonal...

Cet article étudie, sur l’ensemble $𝒮\left(X\right)$ des points extrémaux d’un convexe compact $X$, des topologies faciales dont les fermés sont les traces de faces $F$ “parallélisables” (il existe une plus grande face $F^{\text{'}}$ disjointe de $F$, et tout $x$ de $X$ s’écrit $x=\lambda y+(1-\lambda )y^{\text{'}},y\in F,y^{\text{'}}\in F^{\text{'}}$, avec $\lambda$ unique). Les topologies faciales uniformisables sont en bijection avec les sous-espaces réticulés fermés et contenant 1 de l’espace $A\left(X\right)$ des fonctions affines continues sur $X$. Ceci redonne des résultats classiques sur les simplexes, et permet une étude...