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

Abstract

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 crosssection through an interior point O of a convex body K ⊂ 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 (R n ).Throughout this paper, by a convex body K ⊂ R n we mean a compact convex set with nonempty interior.As usual, we denote by V k (R n ) the Stiefel manifold of orthonormal k-frames in R n , andk (R n )) be the tautological vector bundle in which the fiber over a k-plane is the same plane regarded as a k-dimensional (oriented) subspace of R n . §1. Introduction and the basic conjectureThe famous Dvoretsky theorem [1] states that each multidimensional convex body admits cross-sections of small dimension that are close to a ball.We denote by ε 1 (n, k) (respectively, ε 2 (n, k)) the minimal ε > 0 such that, for each interior point O of a convex body K ⊂ R n , there is a k-plane P through O with the property that the cross-section P ∩ K contains a ball (ellipsoid) and is contained in a (1 + ε)-homothetic ball (ellipsoid) with the same center.We also consider the quantities ε 3 (n, k) and ε 4 (n, k) defined in the same way as ε 1 (n, k) and ε 2 (n, k), respectively, with the only difference that the centers of the balls (ellipsoids) in question must coincide with the distinguished point O.Similar functions ε s i (n, k) refer to the case where the convex body K ⊂ R n is symmetric relative to its interior point O ∈ K. In this case, ε sn→∞ ε s i (n, k) = 0, i = 1, 2, 3, 4, for each positive integer k. In the present paper, we focus our attention on the cases where the values of ε i (n, k) or ε s i (n, k) can be calculated explicitly.