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On the isometric hypothesis of Banach Montejano, Luis


The following is known as the geometric hypothesis of Banach: let V be an m-dimensional Banach space with unit ball B and suppose all n-dimensional subspaces of V are isometric (all the n-sections of B are affinely equivalent). In 1932, Banach conjectured that under this hypothesis V is a Hilbert space (the boundary of B is an ellipsoid). Gromow proved in 1967 that the conjecture is true for n=even and Dvoretzky derived the same conclusion under the hypothesis n=infinity. We prove this conjecture for all positive integers of the form n=4k+1, with the possible exception of 133. The ingredients of the proof are classical homotopic theory, irreducible representations of the orthogonal group and convex geometry

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