  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 $L^2(E)$.
