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

Florent P. Baudier; Gilles Lancien; Pavlos Motakis; Thomas Schlumprecht

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arXiv · 2004.04805

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

Florent P. Baudier·Gilles Lancien·Pavlos Motakis·Thomas Schlumprecht

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Abstract

Within the class of reflexive Banach spaces, we prove a metric characterization of the class of asymptotic-$c_0$ spaces in terms of a bi-Lipschitz invariant which involves metrics that generalize the Hamming metric on $k$-subsets of $\mathbb{N}$. We apply this characterization to show that the class of separable, reflexive, and asymptotic-$c_0$ Banach spaces is non-Borel co-analytic. Finally, we introduce a relaxation of the asymptotic-$c_0$ property, called the asymptotic-subsequential-$c_0$ 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-$c_0$. In particular $T^*(T^*)$ is asymptotic-subsequential-$c_0$ where $T^*$ is Tsirelson's original space.

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Florent P. Baudier, Gilles Lancien, Pavlos Motakis, Thomas Schlumprecht. 2020-04-09. The geometry of Hamming-type metrics and their embeddings into Banach spaces. https://arxiv.org/abs/2004.04805

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