Abstract
We prove that if $X$ is a separable infinite dimensional Banach space then its isomorphism class has infinite diameter with respect to the Banach-Mazur distance. One step in the proof is to show that if $X$ is elastic then $X$ contains an isomorph of $c_0$. We call $X$ elastic if for some $K <\infty$ for every Banach space $Y$ which embeds into $X$, the space $Y$ is $K$-isomorphic to a subspace of $X$. We also prove that if $X$ is a separable Banach space such that for some $K\lt \infty$ every isomorph of $X$ is $K$-elastic then $X$ is finite dimensional.
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