Stability in the Banach isometric conjecture and nearly monochromatic Finsler surfaces

TL;DR

This paper proves that spaces with nearly isometric 2D subspaces are almost Euclidean and that certain Finsler surfaces are close to Riemannian, using stability analysis and the Banach-Mazur distance.

Contribution

It refines classical results on the Banach isometric conjecture by establishing stability and extends the analysis to nearly monochromatic Finsler surfaces.

Findings

Spaces with almost isometric 2D subspaces are nearly Euclidean.

Non-torus Klein bottle Finsler surfaces are approximately Riemannian.

Explicit stability bounds are provided via the Banach-Mazur distance.

Abstract

The Banach isometric conjecture asserts that a normed space with all of its $k$-dimensional subspaces isometric, where $k\geq 2$, is Euclidean. The first case of $k=2$ is classical, established by Auerbach, Mazur and Ulam using an elegant topological argument. We refine their method to arrive at a stable version of their result: if all $2$-dimensional subspaces are almost isometric, then the space is almost Euclidean. Furthermore, we show that a $2$-dimensional surface, which is not a torus or a Klein bottle, equipped with a near-monochromatic Finsler metric, is approximately Riemannian. The stability is quantified explicitly using the Banach-Mazur distance.