The geometry of Hamming-type metrics and their embeddings into Banach spaces Academic Article uri icon

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.

published proceedings

  • ISRAEL JOURNAL OF MATHEMATICS

author list (cited authors)

  • Baudier, F. P., Lancien, G., Motakis, P., & Schlumprecht, T.

citation count

  • 0

complete list of authors

  • Baudier, Florent P||Lancien, Gilles||Motakis, Pavlos||Schlumprecht, Thomas

publication date

  • August 2021