Distortion and Stabilized Structure in Banach Spaces; New Geometric Phenomena for Banach and Hilbert Spaces

Abstract

Many of the fundamental research problems in the geometry of normed linear spaces can be loosely phrased as: Given a Banach space X and a class of Banach spaces Y does X contain a subspace Y ∈ Y? As a Banach space X is determined by its unit ball B x ≡ { x ∈ X :‖ x ‖ ≤ 1 } the problem can be rephrased in terms of the geometry of convex sets: Can a given unit ball B x be sliced with a subspace to obtain a set in some given class of unit balls? A result of this type is the famous theorem of Dvoretzky (see also [L], [M6], [M4], [MS], [FLM]).