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

Makeev, V.

doi:10.1090/S1061-0022-05-00889-7

Approximation of two-dimensional cross-sections of convex bodies by disks and ellipses
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by V. V. Makeev
Translated by: B. M. Bekker

St. Petersburg Math. J. 16 (2005), 1043-1049

DOI: https://doi.org/10.1090/S1061-0022-05-00889-7

Published electronically: November 17, 2005

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Abstract:

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 $G_2(\mathbb {R}^n)$.

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