Hilbert metric, beyond convexity

TL;DR

This paper extends the Hilbert metric from convex subsets of real space to complex projective spaces, providing new examples and applications in various mathematical fields.

Contribution

It introduces a generalized Hilbert metric applicable to complex projective spaces, broadening the scope beyond convex subsets.

Findings

Classical Hilbert metric coincides with hyperbolic metrics in real and complex hyperbolic spaces.

Provides examples of applications in diverse mathematical fields.

Establishes a framework for further exploration of metrics in complex projective geometry.

Abstract

The Hilbert metric on convex subsets of $\mathbb R^n$ has proven a rich notion and has been extensively studied. We propose here a generalization of this metric to subset of complex projective spaces and give examples of applications to diverse fields. Basic examples include the classical Hilbert metric which coincides with the hyperbolic metric on real hyperbolic spaces as well as the complex hyperbolic metric on complex hyperbolic spaces.