A remark on geodesics in the Banach Mazur distance

TL;DR

This paper demonstrates the existence of uncountably many geodesics between any two non-isometric finite-dimensional normed spaces and provides explicit constructions to describe all such geodesics.

Contribution

It establishes the uncountability of geodesics in the Banach-Mazur distance and offers explicit examples to characterize all geodesics between two spaces.

Findings

Uncountably many geodesics exist between any two non-isometric n-dimensional normed spaces.

Explicit constructions of two geodesics can describe all geodesics in this setting.

The results deepen understanding of the geometric structure of the Banach-Mazur space.

Abstract

We show that there are uncountably many geodesics between any two non-isometric $n$-dimensional normed spaces. We construct two explicit geodesics that can be used to describe all the points of the other geodesics.