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Hilbert metric and quasiconformal mappings
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We prove a functional identity between the Hilbert metric and the visual angle metric in the unit disk. The proof utilizes the Poincar\'e hyperbolic metric in terms of which both metrics can be expressed. This identity then yields sharp distortion results for quasiregular mappings and analytic functions, expressed in terms of the Hilbert metric. We also prove that Hilbert circles are, in fact, Euclidean ellipses. The proof makes use of computer algebra methods. In particular, Gr\"obner bases are used.
Forward citations
Cited by 2 Pith papers
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On families of Finsler metrics
For Finsler metrics with bi-geodesics, the arithmetic and maximum symmetrisations of the distance are shown to be Finsler, with explicit Lagrangians, generalising known Hilbert and Funk results.
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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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