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The geometry of Hamming-type metrics and their embeddings into Banach spaces

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Abstract

Within the class of reflexive Banach spaces, we prove a metric characterization of the class of asymptotic-c0 spaces in terms of a bi-Lipschitz invariant which involves metrics that generalize the Hamming metric on k-subsets of ℕ. We apply this characterization to show that the class of separable, reflexive, and asymptotic-c0 Banach spaces is non-Borel co-analytic. Finally, we introduce a relaxation of the asymptotic-c0 property, called the asymptotic-subsequential-c0 property, which is a partial obstruction to the equi-coarse embeddability of the sequence of Hamming graphs. We present examples of spaces that are asymptotic-subsequential-c0. In particular, T*(T*) is asymptotic-subsequential-c0 where T* is Tsirelson’s original space.

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Correspondence to Florent P. Baudier.

Additional information

The first-named author was supported by the National Science Foundation under Grant Number DMS-1800322.

The second-named author was supported by the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-03).

The fourth-named author was supported by the National Science Foundation under Grant Numbers DMS-1464713 and DMS-1711076.

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Baudier, F.P., Lancien, G., Motakis, P. et al. The geometry of Hamming-type metrics and their embeddings into Banach spaces. Isr. J. Math. 244, 681–725 (2021). https://doi.org/10.1007/s11856-021-2187-0

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  • DOI: https://doi.org/10.1007/s11856-021-2187-0

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