Stability of almost surjective ε-isometries of Banach spaces

Abstract

We show that for every pair of Banach spaces X and Y and for every ε-isometry $f:X\to Y$ with $\underset{y\in S_Y}{\sup }\underset{\left|t\right|\to \infty }{\lim \inf }\mathrm{dist}(ty,f(X\left)\right)/\left|t\right|<1/2$ there exists an affine surjective isometry $V:Y\to X$ such that $\Vert f\left(x\right)-Vx\Vert \le 2\epsilon$ for all $x\in X$.