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- Title
Banach's isometric subspace problem in dimension four.
- Authors
Ivanov, Sergei; Mamaev, Daniil; Nordskova, Anya
- Abstract
We prove that if all intersections of a convex body B ⊂ 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.
- Subjects
CONVEX bodies; GEOMETRIC approach; VECTOR spaces; ISOMETRICS (Mathematics); ELLIPSOIDS; NORMED rings
- Publication
Inventiones Mathematicae, 2023, Vol 233, Issue 3, p1393
- ISSN
0020-9910
- Publication type
Article
- DOI
10.1007/s00222-023-01197-2