  A theorem of Tits - Vinberg allows to build an action of a Coxeter group
$\Gamma$ on a properly convex open set $\Omega$ of the real projective space,
thanks to the data $P$ of a polytope and reflection across its facets. We give
sufficient conditions for such action to be of finite covolume,
convex-cocompact or geometrically finite. We describe an hypothesis that make
those conditions necessary. Under this hypothesis, we describe the Zariski
closure of $\Gamma$, find the maximal $\Gamma$-invariant convex, when there is
a unique $\Gamma$-invariant convex, when the convex $\Omega$ is strictly
convex, when we can find a $\Gamma$-invariant convex $\Omega'$ which is
strictly convex.
