## Finite Metric Subsets of Banach Spaces

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##### Authors

Kilbane, James

##### Advisors

Zsak, Andras

##### Date

2019-01-31##### Awarding Institution

University of Cambridge

##### Author Affiliation

DPMMS

##### Qualification

Doctor of Philosophy (PhD)

##### Language

English

##### Type

Thesis

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Show full item record##### Citation

Kilbane, J. (2019). Finite Metric Subsets of Banach Spaces (Doctoral thesis). https://doi.org/10.17863/CAM.35589

##### Abstract

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 $\ell_2$ isometrically. We say that a Banach space has the Finite Isometric Krivine Property (FIKP) if whenever $\ell_p$ is finitely representable in the space then it contains every subset of $\ell_p$ isometrically. We show that every infinite-dimensional Banach space \emph{nearly} has FIDP and every Banach space nearly has FIKP. We then use convexity arguments to demonstrate that not every Banach space has FIKP, and thus we can exhibit classes of Banach spaces that fail to have Property AI-I. The methods used break down when one attempts to prove that there is a Banach space without FIDP and we conjecture that every infinite-dimensional Banach space has Property FIDP.

##### Keywords

Banach Spaces, Metric Spaces, Metric Embeddings

##### Identifiers

This record's DOI: https://doi.org/10.17863/CAM.35589

##### Rights

Attribution-ShareAlike 4.0 International (CC BY-SA 4.0)

Licence URL: https://creativecommons.org/licenses/by-sa/4.0/