An Amir–Cambern theorem for quasi-isometriesof C0(K,X) spaces

Let X be a finite-dimensional Banach space.We prove that if K and S are locally compact Hausdorff spaces and there exists a bijective mapfor every f, g ∈ C 0 (K, X) then K and S are homeomorphic, whenever L ≥ 0 and 1 ≤ M 2 < S(X), where S(X) denotes the Schäffer constant of X.This nonlinear vector-valued extension of the Amir-Cambern theorem via quasi-isometries T with large M was previously unknown even for the classical ℓ n p spaces, 1 < p < ∞, p = 2 and n ≥ 2.