Let $(X,\|\cdot\|)$ be a Banach space with a Schauder basis and fix $p\in[1,\infty]$. Suppose that $X$ is asymptotic-$\ell_{p}$ with respect to this basis. It is known that the closed linear span of every (nontrivial) spreading model of $X$ is isomorphic to $\ell_{p}$ if $X$ is reflexive and at least contains an isomorphic copy of $\ell_{p}$ in general (replace $\ell_{p}$ by $c_{0}$ if $p=\infty$). In other words, the global asymptotic geometry of $X$ gives some information about the local asymptotic geometry.

Do there exist any known converse results? For example, are there general hypotheses that, in combination with the closed linear span of every spreading model containing an isomorphic copy of $\ell_{p}$, ensure that $X$ itself will be asymptotic-$\ell_{p}$?

Let $(X,\|\cdot\|)$ be a Banach space with a Schauder basis and fix $p\in[1,\infty]$. Suppose that $X$ is asymptotic-$\ell_{p}$ with respect to this basis. It is known that the closed linear span of every (nontrivial) spreading model of $X$ is isomorphic to $\ell_{p}$ if $X$ is reflexive and at least contains an isomorphic copy of $\ell_{p}$ in general (replace $\ell_{p}$ by $c_{0}$ if $p=\infty$). In other words, the global asymptotic geometry of $X$ gives some information about the local asymptotic geometry.

Do there exist any known converse results? For example, are there general hypotheses that, in combination with the closed linear span of every spreading model containing an isomorphic copy of $\ell_{p}$, ensure that $X$ itself will be asymptotic-$\ell_{p}$?

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