We prove that the class of reflexive asymptotic-$c_{0}$ Banach spaces is coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a reflexive asymptotic-$c_{0}$ space $Y$, then $X$ is also reflexive and asymptotic-$c_{0}$. In order to achieve this result, we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic-$c_{0}$ space, we show that this concentration inequality is not equivalent to the non-equi-coarse embeddability of the Hamming graphs.

Let $X$ be an infinite-dimensional uniformly smooth Banach space. We prove that $X$ contains an infinite equilateral set. That is, there exist a constant $\lambda \gt 0$ and an infinite sequence $\mathop{({x}_{i} )}\nolimits_{i= 1}^{\infty } \subset X$ such that $\Vert {x}_{i} - {x}_{j} \Vert = \lambda $ for all $i\not = j$.

We consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1 < p < ∞) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space $X$ . In particular we give an example of a reflexive $X$ so that all spreading models of $X$ contain ${{\ell }_{1}}$ but none of them is isomorphic to ${{\ell }_{1}}$ . We also prove that for any countable set $C$ of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of $C$ . In certain cases this ensures that $X$ admits, for each $\alpha \,<\,{{\omega }_{1}}$ , a spreading model ${{\left( \tilde{x}_{i}^{\left( \alpha \right)} \right)}_{i}}$ such that if $\alpha \,<\,\beta $ then ${{\left( \tilde{x}_{i}^{\left( \alpha \right)} \right)}_{i}}$ is dominated by (and not equivalent to) ${{\left( \tilde{x}_{i}^{\left( \beta \right)} \right)}_{i}}$ . Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.