Geodesic
Planar sections of convex bodies and universal fibrations
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 7, Tome 280 (2001), pp. 219-233 
V. V. Makeev. Planar sections of convex bodies and universal fibrations. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 7, Tome 280 (2001), pp. 219-233. http://geodesic.mathdoc.fr/item/ZNSL_2001_280_a15/
@article{ZNSL_2001_280_a15,
     author = {V. V. Makeev},
     title = {Planar sections of convex bodies and universal fibrations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {219--233},
     year = {2001},
     volume = {280},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_280_a15/}
}
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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.