Ontology type: schema:ScholarlyArticle Open Access: True
2021-11-09
AUTHORS ABSTRACTWe discuss various aspects of a local-to-global embedding technique and the metric geometry of stable metric spaces, in particular two of its important subclasses: locally finite spaces and proper spaces. We explain how the barycentric gluing technique, which has been mostly applied to bi-Lipschitz embedding problems pertaining to locally finite spaces, can be implemented successfully in a much broader context. For instance, we show that the embeddability of an arbitrary metric space into ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _p$$\end{document} is determined by the embeddability of its balls. We also introduce the notion of upper stability. This new metric invariant lies formally between Krivine–Maurey (isometric) notion of stability and Kalton’s property Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {Q}}$$\end{document}. We show that several results of Raynaud and Kalton for stable metrics can be extended to the broader context of upper stable metrics and we point out the relevance of upper stability to a long standing embedding problem raised by Kalton. Applications to compression exponent theory are highlighted and we recall old, and state new, important open problems. This article was written in a style favoring clarity over conciseness in order to make the material appealing, accessible, and reusable to geometers from a variety of backgrounds, and not only to Banach space geometers. More... »
PAGES37
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
ISSUE1
VOLUME116
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