  For the semi simple and deployed Lie algebra $\mathfrak g=\mathfrak{sl}(n,
\R)$, we give an explicit construction of an overalgebra $\mathfrak
g^+=\mathfrak g\rtimes V$ of $\mathfrak g$, where $V$ is a finite dimensional
vector space. In such a setup, we prove the existence of a map $\Phi$ from the
dual $\mathfrak g^\star$ of $\mathfrak g$ into the dual $(\mathfrak g^+)^\star$
of $\mathfrak g^+$ such that the coadjoint orbits of $\Phi(\xi)$, for generic
$\xi$ in $\mathfrak g^\star$, have a distinct closed convex hulls. Therefore,
these closed convex hulls separate 'almost' the generic coadjoint orbits of
$G$.
