It is proven that if
X
X
is a real strictly convex 2-dimensional space, then there exists
δ
>
0
\delta >0
such that if
K
K
and
S
S
are locally compact Hausdorff spaces and
T
T
is an isomorphism from
C
0
(
K
,
X
)
C_{0}(K, X)
onto
C
0
(
S
,
X
)
C_{0}(S, X)
satisfying
‖
T
‖
‖
T
−
1
‖
≤
λ
(
X
)
+
δ
,
\begin{equation*} \|T\| \ \|T^{-1}\| \leq \lambda (X)+ \delta , \end{equation*}
then
K
K
and
S
S
are homeomorphic. Here
λ
(
X
)
\lambda (X)
is the Schäffer constant of
X
X
given by
λ
(
X
)
=
inf
{
max
{
‖
x
−
y
‖
,
‖
x
+
y
‖
}
:
‖
x
‖
=
1
and
‖
y
‖
=
1
}
.
\begin{equation*} \lambda (X)=\inf \{\max \{\|x-y\|,\|x+y\|\}\,:\,\|x\|= 1 \text { and } \|y\|= 1\}. \end{equation*}
Even for the classical cases
X
=
ℓ
p
2
X=\ell _p^2
,
1
>
p
>
∞
1>p> \infty
, this result is the form of Banach-Stone theorem to
C
0
(
K
,
X
)
C_{0}(K, X)
spaces with the largest known distortion
‖
T
‖
‖
T
−
1
‖
\|T\| \ \|T^{-1}\|
. In particular, it shows that the Banach-Stone constant of
ℓ
p
2
\ell _p^2
is strictly greater than
2
1
−
1
/
p
2^{1-1/p}
if
1
>
p
≤
2
1>p \leq 2
and strictly greater than
2
1
/
p
2^{1/p}
if
2
≤
p
>
∞
2 \leq p >\infty
. Until then this theorem had only been proved for
p
=
2
p=2
.