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Les géométries de Hilbert sont à géométrie locale bornée

Bruno Colbois·Constantin Vernicos
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Abstract

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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Bruno Colbois, Constantin Vernicos. 2006-04-21. Les géométries de Hilbert sont à géométrie locale bornée. https://arxiv.org/abs/math/0604461

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