  We study the notion of geometrical finiteness for those Hilbert geometries
defined by strictly convex sets with $\C^1$ boundary. In Gromov-hyperbolic
spaces, geometrical finiteness is defined by a property of the group action on
the boundary of the space. We show by constructing an explicit counter-example
that this definition has to be strenghtened in order to get equivalent
characterizations in terms of the geometry of the quotient orbifold, similar to
those obtained by Brian Bowditch in hyperbolic geometry.
