A rigidity result for Hilbert metrics. (Un résultat de rigidité pour les métriques de Hilbert.) (French) Zbl 0979.52001

Séminaire de théorie spectrale et géométrie. Année 1999-2000. St. Martin d’Hères: Université de Grenoble I, Institut Fourier, Sémin. Théor. Spectr. Géom. 18, 171-173 (2000).
The author proves the following result: Let \( \mathcal{C} \) be an open, convex and bounded set of \( \mathbb{R} ^n \) such that its boundary \( \partial \mathcal{C} \) is a strictly convex hypersurface of \( \mathbb{R} ^n \) of class \( C ^3 \). If \( \partial \mathcal{C} \) is not an ellipsoid then each subgroup of \( \text{Iso} (\mathcal{C}, d _{\mathcal{C}}) \), where \( d _{\mathcal{C}} \) is the Hilbert metric, which does not act proper and discontinuous on \( \mathcal{C} \) is a finite one.
For the entire collection see [Zbl 0955.00015].


52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
53C22 Geodesics in global differential geometry
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