In this note, we study some concentration properties for Lipschitz maps defined on Hamming graphs, as well as their stability under sums of Banach spaces. As an application, we extend a result of Causey on the coarse Lipschitz structure of quasi-reflexive spaces satisfying upper ℓp tree estimates to the setting of ℓp-sums of such spaces. We also give a sufficient condition for a space to be asymptotic-c0 in terms of a concentration property, as well as relevant counterexamples.2.2. Hamming graphs. Before introducing concentration properties, we need to define special metric graphs that we shall call Hamming graphs. Let M be an infinite subset of N. We denote rMs ω the set of infinite subsets of M. For M P rNs ω and k P N, we noteandThen we equip rMs k with the Hamming distance: d H pn, mq " |tj; n j ‰ m j u| for all n " pn 1 , ¨¨¨, n k q, m " pm 1 , ¨¨¨, m k q P rMs k . Let us mention that this distance can be extended to rMs ăω by letting d H pn, mq " |ti P t1, ¨¨¨, minpl, jqu; n i ‰ m i u| `maxpl, jq ´minpl, jq for all n " pn 1 , ¨¨¨, n l q, m " pm 1 , ¨¨¨, m j q P rMs ăω (with possibly l " 0 or j " 0). We also need to introduce I k pMq, the set of strictly interlaced pairs in rMs k :and, for each j P t1, ¨¨¨, ku, let us denote H j pMq " tpn, mq Ă rMs k ; @i ‰ j, n i " m i and n j ă m j u.Note that, for pn, mq P I k pMq, d H pn, mq " k and n X m " ∅.
We study four asymptotic smoothness properties of Banach spaces, denoted T p , A p , N p \mathsf {T}_p,\mathsf {A}_p, \mathsf {N}_p , and P p \mathsf {P}_p . We complete their description by proving the missing renorming characterization for A p \mathsf {A}_p . We show that asymptotic uniform flattenability (property T ∞ \mathsf {T}_\infty ) and summable Szlenk index (property A ∞ \mathsf {A}_\infty ) are three-space properties. Combined with the positive results of the first-named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three-space problem for this family of properties. We also derive from our characterizations of A p \mathsf {A}_p and N p \mathsf {N}_p in terms of equivalent renormings, new coarse Lipschitz rigidity results for these classes.
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