Polyhedral Finsler spaces with locally unique geodesics

TL;DR

This paper investigates Finsler piecewise linear spaces constructed from normed simplices, focusing on conditions for local and global uniqueness of geodesics, revealing that certain convexity properties do not extend from Euclidean to Finsler settings.

Contribution

It establishes a globalization theorem for locally unique geodesics in simply-connected Finsler PL spaces, extending known results from Euclidean cases to non-Euclidean norms.

Findings

Local uniqueness of geodesics implies global uniqueness in simply-connected Finsler PL spaces.

Non-Euclidean normed spaces do not satisfy CAT(0), but share some properties with Euclidean cases.

Basic convexity properties fail to extend to the PL Finsler setting.

Abstract

We study Finsler PL spaces, that is simplicial complexes glued out of simplices cut off from some normed spaces. We are interested in the class of Finsler PL spaces featuring local uniqueness of geodesics (for complexes made of Euclidean simplices, this property is equivalent to local CAT(0)). Though non-Euclidean normed spaces never satisfy CAT(0), it turns out that they share many common features. In particular, a globalization theorem holds: in a simply-connected Finsler PL space local uniqueness of geodesics implies the global one. However the situation is more delicate here: some basic convexity properties do not extend to the PL Finsler case.