Finite Metric Subsets of Banach Spaces
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The central idea in this thesis is the introduction of a new isometric invariant of a Banach space. This is Property AI-I. A Banach space has Property AI-I if whenever a finite metric space almost-isometrically embeds into the space, it isometrically embeds. To study this property we introduce two further properties that can be thought of as finite metric variants of Dvoretzky's Theorem and Krivine's Theorem. We say that a Banach space satisfies the Finite Isometric Dvoretzky Property (FIDP) if it contains every finite subset of