  We prove that the Hilbert geometry of a convex domain in ${\mathbb R}^n$ has
bounded local geometry, i.e., for a given radius, all balls are bilipschitz to
a euclidean domain of ${\mathbb R}^n$. 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 which shows that
the reciprocal is not true for non plane Hilbert geometries.
