Surface projective convexe de volume fini

Abstract

A convex projective surface is the quotient of a properly convex open $Ω$ of $\mathbb{P}(\R)$ by a discret subgroup $Γ$ of $\mathrm{SL}_3(\R)$. We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann's measure. We deduce of this that if $Ω$ is not a triangle then $Ω$ is strictly convex, with $\Cc^1$ boundary and that a convex projective surface $S$ is of finite volume if and only if the dual surface is of finite volume.