On uniform and coarse rigidity of $L^p([0,1])$

Abstract

If $X$ is an almost transitive Banach space with amenable isometry group (for example, if $X=L^p([0,1])$ with $1⩽p<\infty$) and $X$ admits a uniformly continuous map $X\stackrel{\varphi }{⟶}E$ into a Banach space $E$ satisfying $$\underset{\Vert x-y\Vert =r}{\inf }\Vert \varphi (x)-\varphi (y)\Vert >0$$ for some $r>0$, then $X$ admits a simultaneously uniform and coarse embedding into a Banach space $V$ that is finitely representable in $L^2(E)$.

A correction is made to a citation. The main result also applies to $L^1([0,1])$