Hilbert metric and quasiconformal mappings

TL;DR

This paper establishes a functional identity between the Hilbert and visual angle metrics in the unit disk, leading to sharp distortion results for quasiregular mappings and analytic functions, and proves that Hilbert circles are Euclidean ellipses using computer algebra methods.

Contribution

It introduces a novel identity linking Hilbert and visual angle metrics and applies it to derive distortion bounds, also proving the geometric shape of Hilbert circles with computer algebra.

Findings

Hilbert and visual angle metrics are functionally identical in the unit disk.

Sharp distortion bounds for quasiregular and analytic functions are derived.

Hilbert circles are Euclidean ellipses.

Abstract

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.