Harmonic symmetrization of convex sets and of Finsler structures, with applications to Hilbert geometry

TL;DR

This paper introduces a new approach to studying the Hilbert metric on convex domains by representing it as a harmonic symmetrization of the Funk metric, avoiding smoothness assumptions.

Contribution

It develops a framework that characterizes the Hilbert metric as a harmonic symmetrization of the Funk metric using weak Finsler structures, broadening understanding without smoothness constraints.

Findings

Hilbert metric expressed as harmonic symmetrization of Funk metric

Properties derived from harmonic symmetrization of weak Finsler structures

New approach applicable without boundary smoothness assumptions

Abstract

David Hilbert discovered in 1895 an important metric that is canonically associated to any convex domain $\Omega$ in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof assumes a certain degree of smoothness of the boundary of $\Omega$ and refers to a theorem by Busemann and Mayer that produces the norm of a tangent vector from the distance function. In this paper, we develop a new approach for the study of the Hilbert metric where no differentiability is assumed. The approach exhibits the Hilbert metric on a domain as a symmetrization of a natural weak metric, known as the Funk metric. The Funk metric is described as a tautological weak Finsler metric, in which the unit ball at each tangent space is naturally identified with the domain $\Omega$ itself. The Hilbert metric is then identified with the reversible tautological weak Finsler structure on $\Omega$, and the unit ball at each point is described as the harmonic symmetrization of the unit ball of the Funk metric. Properties of the Hilbert metric then follow from general properties of harmonic symmetrizations of weak Finsler structures.