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2021 An average John theorem
Assaf Naor
Geom. Topol. 25(4): 1631-1717 (2021). DOI: 10.2140/gt.2021.25.1631

Abstract

We prove that the 12–snowflake of any finite-dimensional normed space X embeds into a Hilbert space with quadratic average distortion

O(log dim(X)).

We deduce from this (optimal) statement that if an n–vertex expander embeds with average distortion D1 into X, then necessarily dim(X)nΩ(1D), which is sharp by the work of Johnson, Lindenstrauss and Schechtman (1987). This improves over the previously best-known bound dim(X)(logn)2D2 of Linial, London and Rabinovich (1995), strengthens a theorem of Matoušek (1996) which resolved questions of Johnson and Lindenstrauss (1982), Bourgain (1985) and Arias-de-Reyna and Rodríguez-Piazza (1992), and answers negatively a question that was posed (for algorithmic purposes) by Andoni, Nguyen, Nikolov, Razenshteyn and Waingarten (2016).

Citation

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Assaf Naor. "An average John theorem." Geom. Topol. 25 (4) 1631 - 1717, 2021. https://doi.org/10.2140/gt.2021.25.1631

Information

Received: 7 December 2016; Revised: 7 February 2020; Accepted: 11 May 2020; Published: 2021
First available in Project Euclid: 12 October 2021

Digital Object Identifier: 10.2140/gt.2021.25.1631

Subjects:
Primary: 30L05

Keywords: Dimension reduction , Expander graphs , Markov type , metric embeddings , nonlinear spectral gaps

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.25 • No. 4 • 2021
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