Approximation of two-dimensional cross-sections of convex bodies by disks and ellipses

In connection with the well-known Dvoretsky theorem, the following question arises: How close to a disk or to an ellipse can a two-dimensional cross-section through an interior point $O$ of a convex body $K\subset \mathbb{R}^n$ be? In the present paper, the attention is focused on a few (close to prime) dimensions $n$ for which this problem can be solved exactly. Asymptotically, this problem was solved by the author in 1988.

Another problem treated in the paper concerns inscribing a regular polygon in a circle that belongs to a field of circles smoothly embedded into the fibers of the tautological bundle over the Grassmannian manifold $G_2(\mathbb{R}^n)$.