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

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.