2024/05/03 by Gautam Aishwarya, Dmitry Faifman, Aishwarya, Gautam +1
Physics and Astronomy · #46C15 #52A21 #53C60 #57R15 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2405.02440
openalex publication_date 2024/05/03 · openalex created_date 2024/05/08 · openalex updated_date 2026/07/28
The Banach isometric conjecture asserts that a normed space with all of its k-dimensional subspaces isometric, where k≥ 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.