Un lemme de Kazhdan-Margulis-Zassenhaus pour les g\'eom\'etries de Hilbert

We prove a Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometries. More precisely, in every dimension $n$ there exists a constant ${\epsilon }_n>0$ such that, for any properly open convex set $\text{O}$ and any point $x\in \text{O}$, any discrete group generated by a finite number of automorphisms of $\text{O}$, which displace $x$ at a distance less than ${\epsilon }_n$, is virtually nilpotent.