Geodesic complexity for non-geodesic spaces

TL;DR

This paper extends the concept of geodesic complexity to non-geodesic metric spaces by introducing near geodesics, with potential applications in topological robotics, and provides explicit calculations in various cases.

Contribution

It introduces near geodesics for non-geodesic spaces and generalizes geodesic complexity, expanding its applicability beyond geodesic spaces.

Findings

Defined near geodesics in non-geodesic spaces

Extended geodesic complexity to these spaces

Computed explicit examples of near geodesics and complexity

Abstract

We define the notion of near geodesic between points of a metric space when no geodesic exists, and use this to extend Recio-Mitter's notion of geodesic complexity to non-geodesic spaces. This has potential application to topological robotics. We determine explicit near geodesics and geodesic complexity in a variety of cases.