Metric Spaces where Geodesics are Never Unique

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.

8 pages, 3 figures. This article has been accepted for publication in the American Mathematical Monthly

References in corpus (1)

• Branching geodesics in sub-Riemannian geometry