Les géométries de Hilbert sont à géométrie locale bornée

Details Les géométries de Hilbert sont à géométrie locale bornée Les géométries de Hilbert sont à géométrie locale bornée

2006-04-21

arxiv.org/abs/math/0604461v1

Annales de l'Institut Fourier 57, 4 (2007) 1359-1375

Mathematics

Differential Geometry

A para\^{i}tre aux annales de l'Institut Fourier

Scientific paper

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.

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