Stability in the Banach isometric conjecture and nearly monochromatic Finsler surfaces
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.02440
- arXiv:
- arXiv:2405.02440
- Bibcode:
- 2024arXiv240502440A
- Keywords:
-
- Mathematics - Metric Geometry;
- Mathematics - Differential Geometry;
- 52A21;
- 46C15;
- 57R15;
- 53C60
- E-Print:
- Minor corrections and improvements to exposition, 3 figures added