Banach's isometric subspace problem in dimension four

• 成果类型: Article
• 署名作者: Ivanov, Sergei; Mamaev, Daniil; Nordskova, Anya
• 署名单位: Russian Academy of Sciences; St. Petersburg Scientific Centre of the Russian Academy of Sciences; Steklov Mathematical Institute of the Russian Academy of Sciences; St. Petersburg Department of the Steklov Mathematical Institute of the Russian Academy of Sciences; Hasselt University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01197-2
• 发表日期: 2023
• 页码: 1393-1425
• 关键词: convex-bodies

摘要

We prove that if all intersections of a convex body B subset of R-4 with 3-dimensional linear subspaces are linearly equivalent then B is a centered ellipsoid. This gives an affirmative answer to the case n = 3 of the following question by Banach from 1932: Is a normed vector space V whose n-dimensional linear subspaces are all isometric, for a fixed 2 <= n < dim V, necessarily Euclidean? The dimensions n = 3 and dim V = 4 is the first case where the question was unresolved. Since the 3-sphere is parallelizable, known global topological methods do not help in this case. Our proof employs a differential geometric approach.