2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/136135We 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.A paraître aux annales de l'Institut FourierDifferential GeometryMetric Geometrytext