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Title: | Banach's isometric subspace problem in dimension four | Authors: | Ivanov, Sergei Mamaev, Daniil NORDSKOVA, Anya |
Issue Date: | 2023 | Publisher: | SPRINGER HEIDELBERG | Source: | INVENTIONES MATHEMATICAE, 233 (3) , p. 1393-1425 | Abstract: | 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. | Notes: | Ivanov, S (corresponding author), Steklov Math Inst, St Petersburg Dept, Fontanka 27, St Petersburg 191023, Russia. svivanov@pdmi.ras.ru; dan.mamaev@gmail.com; anya.nordskova@uhasselt.be |
Keywords: | Ellipsoid characterization;Convex body;Cross-section | Document URI: | http://hdl.handle.net/1942/40553 | ISSN: | 0020-9910 | e-ISSN: | 1432-1297 | DOI: | 10.1007/s00222-023-01197-2 | ISI #: | 000994146100001 | Rights: | The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023 | Category: | A1 | Type: | Journal Contribution |
Appears in Collections: | Research publications |
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Banach_final_corrections-2.pdf | Peer-reviewed author version | 469.73 kB | Adobe PDF | View/Open |
Banach’s isometric subspace problem in dimension four.pdf Restricted Access | Published version | 1.31 MB | Adobe PDF | View/Open Request a copy |
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