Weak Finsler Strutures and the Funk Metric

TL;DR

This paper introduces weak Finsler structures, particularly the Funk weak metric on convex domains, and explores their properties, including distances, geodesics, and convexity, expanding the understanding of Finsler geometry.

Contribution

It defines weak Finsler structures, identifies the Funk weak metric as a canonical example, and analyzes its geometric properties in convex domains.

Findings

Distances are given by the Funk weak metric.

Geodesics and convexity properties are characterized.

The structure extends classical Finsler geometry concepts.

Abstract

We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we show that these are given by the so-called Funk weak metric. We conclude the paper with a discussion of geodesics, of metric balls and of convexity properties of the Funk weak metric.