Metric Spaces where Geodesics are Never Unique

TL;DR

This paper explores multigeodesic spaces where multiple shortest paths exist between points, characterizing such spaces in normed contexts and examining their properties in general metric spaces.

Contribution

It provides a simple characterization of multigeodesic normed spaces and identifies examples like (C([0,1]),||·||_1), while showing finite-dimensional spaces are not multigeodesic.

Findings

(C([0,1]),||·||_1) is multigeodesic

Finite-dimensional normed spaces are not multigeodesic

Multigeodesic spaces can have diverse additional features

Abstract

This article concerns a class of metric spaces, which we call multigeodesic spaces, where between any two distinct points there exist multiple distinct minimising geodesics. We provide a simple characterisation of multigeodesic normed spaces and deduce that $(C([0,1]),||\cdot||_1)$ is an example of such a space, but that finite-dimensional normed spaces are not. We also investigate what additional features are possible in arbitrary metric spaces which are multigeodesic.