Isometries of two dimensional Hilbert Geometries

TL;DR

This paper proves that in two-dimensional Hilbert geometries, all isometries are projective transformations except when the domains are triangles, where exceptions occur.

Contribution

It establishes a complete characterization of isometries in 2D Hilbert geometries, identifying the special case of triangular domains.

Findings

All isometries are projective transformations in non-triangular domains.

Triangular domains are exceptions with non-projective isometries.

Provides a classification of isometries in 2D Hilbert geometries.

Abstract

We prove that any isometry between two dimensional Hilbert geometries is a projective transformation unless the domains are interiors of triangles.