A conjecture on tautological vector bundles over Grassmannians, which generalizes the well-known Dvoretzky theorem, is stated, discussed, and also proved in one nontrivial case: for the Grassmannian of 2-planes. It is also proved that each three-dimensional real normed space contains a two-dimensional subspace with Banach-Mazur distance from the Euclidean plane at most 1/2 log(4/3) and this estimate is sharp.
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