Geodesic paths in the finite dimensional unit sphere under sup norm

TL;DR

This paper characterizes the shortest paths, or geodesics, on the n-dimensional unit sphere when distances are measured using the supremum norm, providing a geometric understanding of these paths.

Contribution

It offers a complete characterization of geodesic paths on the finite-dimensional unit sphere under the sup norm, a problem not previously fully addressed.

Findings

Explicit description of geodesic paths under sup norm

Conditions for shortest curves on the sphere

Geometric properties of geodesics in this normed space

Abstract

We characterize geodesic paths in the $n$-dimensional unit sphere under sup norm. A geodesic path between two points is a shortest curve joining the two points.