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 $1\leqslant p<\infty$) and $X$ admits a uniformly continuous map $X\overset\phi\longrightarrow E$ into a Banach space $E$ satisfying $$\inf_{|x-y|=r} | \phi(x)-\phi(y)|>0 $$ for some $r>0$, then $X$ admits a simultaneously uniform and coarse embedding into a Banach space $V$ that is finitely representable in $L^2(E)$.