Isometries of polyhedral Hilbert geometries

TL;DR

This paper characterizes the isometry groups of polyhedral Hilbert geometries, showing they match collineation groups except for n-simplices, and determines the isometry group structure for simplices, confirming several conjectures.

Contribution

It provides a complete description of isometry groups for polyhedral Hilbert geometries, especially distinguishing the case of n-simplices, and confirms related conjectures.

Findings

Isometry group equals collineation group for non-simplex polyhedra.

The isometry group of an n-simplex has collineation group as an index-two subgroup.

Confirmed several conjectures by P. de la Harpe regarding these geometries.

Abstract

We show that the isometry group of a polyhedral Hilbert geometry coincides with its group of collineations (projectivities) if and only if the polyhedron is not an n-simplex with n>=2. Moreover, we determine the isometry group of the Hilbert geometry on the n-simplex, and find that it has the collineation group as an index-two subgroup. These results confirm, for the class of polyhedral Hilbert geometries, several conjectures posed by P. de la Harpe.