Stability of almost surjective ε-isometries of Banach spaces

We show that for every pair of Banach spaces X and Y and for every ε  -isometry f:X→Yf:X→Y with supy∈SY⁡liminf|t|→∞dist(ty,f(X))/|t|<1/2 there exists an affine surjective isometry V:Y→XV:Y→X such that ‖f(x)−Vx‖≤2ε‖f(x)−Vx‖≤2ε for all x∈Xx∈X.