Let
$X$
be a finitedimensional Banach space. We prove that if
$K$ and
$S$
are locally compact Hausdorff spaces and there exists a bijective map
$T:{C}_{0}\left(K,X\right)\to {C}_{0}\left(S,X\right)$
such that
$$\frac{1}{M}\parallel fg\parallel L\le \parallel T\left(f\right)T\left(g\right)\parallel \le M\parallel fg\parallel +L,$$
for every
$f,g\in {C}_{0}\left(K,X\right)$
then
$K$ and
$S$ are homeomorphic,
whenever
$L\ge 0$
and
$1\le {M}^{2}<S\left(X\right)$, where
$S\left(X\right)$ denotes the Schäffer
constant of $X$.
This nonlinear vectorvalued extension of the Amir–Cambern theorem via quasiisometries
$T$ with large
$M$ was previously unknown
even for the classical
${\ell}_{p}^{n}$
spaces,
$1<p<\infty $,
$p\ne 2$ and
$n\ge 2$.
