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.

中文翻译：

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BANACH 空间的一个新的粗刚类

我们证明了自反渐近类-$c_{0}$Banach 空间是粗略刚性的，这意味着如果 Banach 空间$X$粗略地嵌入到一个自反渐近-$c_{0}$空间$Y$， 然后$X$也是自反的和渐近的-$c_{0}$. 为了实现这一结果，我们提供了此类 Banach 空间的纯度量表征。这种度量表征采用汉明图上 Lipschitz 映射的浓度不等式的形式，在粗略嵌入下它是刚性的。使用一个准自反渐近的例子——$c_{0}$空间，我们表明这种集中不等式不等同于汉明图的非等粗可嵌入性。