Planar sections of convex bodies and universal fibrations

A conjecture on tautological vector bundles over Grassmannians, which generalizes the well-known Dvoretskii theorem, is stated, discussed, and proved in one nontrivial case: for the Grassmannian of 2-planes. It is also proved that every three-dimensional real normed space contains a two-dimensional subspace with Banach–Mazur distance from the Euclidean plane at most $\frac12\ln(4/3)$, and the estimate is sharp.