A Theorem on Convex Bodies and Applications to Banach Spaces

What is known about unit cubes

“…As for the shapes of the cross sections of I n , our knowledge is very limited. According to a well-known theorem of Dvoretzky [28], for any fixed k, when n is sufficiently large there is a k-dimensional hyperplane H such that I n ∩ H is almost spherical. On the other hand, any n-dimensional centrally symmetric convex polytope with m pairs of facets can be realized as an n-dimensional cross section of an m-dimensional cube (see Ball [8]).…”

Geometrical Background of Metric Fixed Point Theory

“…Remark 20 gives us therefore another way of extending formulae (3) and (4) to the space X = L p (Ω) where 1 < p < ∞ and Ω is an arbitrary measure space such that L p (Ω) is infinite dimensional. By Dvoretzky's theorem (see [34] or [75]), l 2 is finitely representable in any infinite dimensional Banach space X. Consequently,…”

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

“…§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. …”

“…The famous Dvoretsky theorem [1] states that each multidimensional convex body admits cross-sections of small dimension that are close to a ball.…”

“…Let π be the orthogonal projection of the space R n+1 onto the hyperplane of the midsection of C. Then the ellipsoid πK 1 is inscribed in the section πP ∩ K, and this section lies in the concentric ellipsoid (1 + ε 1 (n + 1, k))πK 1 .…”