Les géométries de Hilbert sont à géométrie locale bornée

We prove that the Hilbert geometry of a convex domain in R-n has bounded local geometry, i.e., for a given radius, all balls are bilipschitz to a euclidean domain of R-n. As a consequence, if the Hilbert geometry is also Gromov hyperbolic, then the bottom of its spectrum is strictly positive. We also give a counter exemple in dimension three wich shows that the reciprocal is not true for non plane Hilbert geometries.