It is proven that if
X
X
is a Banach space,
K
K
and
S
S
are locally compact Hausdorff spaces and there exists an
(
M
,
L
)
(M, L)
-quasi isometry
T
T
from
C
0
(
K
,
X
)
C_{0}(K,X)
onto
C
0
(
S
,
X
)
C_{0}(S, X)
, then
K
K
and
S
S
are homeomorphic whenever
1
≤
M
2
>
S
(
X
)
1 \leq M^{2}> S(X)
, where
S
(
X
)
S(X)
denotes the Schäffer constant of
X
X
, and
L
≥
0
L \geq 0
.
As a consequence, we show that the first nonlinear extension of Banach-Stone theorem for
C
0
(
K
,
X
)
C_{0}(K, X)
spaces obtained by Jarosz in 1989 can be extended to infinite-dimensional spaces
X
X
, thus reinforcing a 1991 conjecture of Jarosz himself on
ϵ
\epsilon
-bi-Lipschitz surjective maps between Banach spaces.
Our theorem is optimal when
X
X
is the classical space
ℓ
p
\ell _p
,
2
≤
p
>
∞
2 \leq p> \infty
.