Les géométries de Hilbert sont à géométrie locale bornée

Bruno Colbois (1) , Constantin Vernicos (1)

1 UNINE - Institut de Mathématiques

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.

Mots clés: Géométrie de Hilbert; hyperbolicité; bas du spectre.; bas du spectre

Domaines: Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]

Bruno Colbois, Constantin Vernicos. Les géométries de Hilbert sont à géométrie locale bornée. Annales de l'Institut Fourier, 2007, 57 (4), pp.1359-1375. ⟨hal-00022975⟩