Metric Spaces where Geodesics are Never Unique

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 $(C([0,1]),\mathrm{\mid }\mathrm{\mid }\cdot \mathrm{\mid }{\mathrm{\mid }}_1)$ 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.