Les g\'{e}om\'{e}tries de Hilbert sont \`{a} g\'{e}om\'{e}trie locale   born\'{e}e
  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. 
Bruno Colbois (UNINE), Constantin Vernicos (UNINE)