Quasi-isometries of C0(K, E) spaces which determine K for all Euclidean spaces E

We prove that for all Euclidean spaces E and locally compact Hausdorff spaces K and S, if there exists a bijective map T : C-0(K, E) -> C-0(S, E) such that 1/M parallel to f - g parallel to - L <= parallel to T(f) - T(g)parallel to <= M parallel to f - g parallel to + L for some constants 1 <= M < (4)root 2 and L >= 0 and for all f,g is an element of C-0(K, E), then K and S are homeomorphic. In other words, by using quasi-isometrics we obtain a nonlinear extension of the classical 1976 Hilbert vector-valued Banach-Stone theorem due to Cambern. In the Lipschitz case, that is, when L = 0, our result improves Jarosz's 1989 theorem.