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Hilbert metric in the unit ball
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abstract
The Hilbert metric between two points $x,y$ in a bounded convex domain $G$ is defined as the logarithm of the cross-ratio of $x,y$ and the intersection points of the Euclidean line passing through the points $x,y$ and the boundary of the domain. Here, we study this metric in the case of the unit ball $\mathbb{B}^n$. We present an identity between the Hilbert metric and the hyperbolic metric, give several inequalities for the Hilbert metric, and results related to the inclusion properties of the balls defined in the Hilbert metric. Furthermore, we study the distortion of the Hilbert metric under conformal mappings.
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Cited by 1 Pith paper
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On the average scale-invariant Cassinian metric
The average scale-invariant Cassinian metric is sharply comparable to four standard hyperbolic-type metrics, and its balls in punctured Euclidean space are convex exactly for radius at most log 3.
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