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On Non-positive Curvature Properties of the Hilbert Metric


In this paper, we consider different types of non-positive curvature properties of the Hilbert metric of a convex domain in \({\mathbb {R}^n}\). First, we survey the relationships among the concepts and prove that in the case of Hilbert metric some of them are equivalent. Furthermore, we show some condition which implies the rigidity feature: if the Hilbert metric is Berwald, i.e., its Finslerian Chern connection reduces to a linear one, then the domain is an ellipsoid and the metric is Riemannian.

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Correspondence to László Kozma.

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Alabdulsada, L.M., Kozma, L. On Non-positive Curvature Properties of the Hilbert Metric. J Geom Anal 29, 569–576 (2019).

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