Planar Sections of Convex Bodies and Universal Fibrations

Abstract

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 $$\frac{1}{2}\log (4/3)$$ and this estimate is sharp. Bibliography: 12 titles.