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Abstract

We study the coarse Lipschitz geometry of Banach spaces with several asymptotic properties. Specifically, we look at asymptotic uniform smoothness and convexity, and several distinct Banach–Saks-like properties. We characterize the Banach spaces which are either coarsely or uniformly homeomorphic to $T^{p_1}\oplus \cdots \oplus T^{p_n}$, where each $T^{p_j}$ denotes the $p_j$-convexification of the Tsirelson space, for $p_1,\ldots ,p_n\in (1,\ldots , \infty )$ and $2\not \in \{p_1,\ldots ,p_n\}$. We obtain applications to the coarse Lipschitz geometry of the $p$-convexifications of the Schlumprecht space, and some hereditarily indecomposable Banach spaces. We also obtain some new results in the linear theory of Banach spaces.