  We prove a very general theorem concerning the estimation of the expression
$\|T(\frac{a+b}{2}) - \frac{Ta+Tb}{2}\|$ for different kinds of maps $T$
satisfying some general perurbated isometry condition. It can be seen as a
quantitative generalization of the classical Mazur-Ulam theorem. The estimates
improve the existing ones for bi-Lipschitz maps. As a consequence we also
obtain a very simple proof of the result of Gevirtz which answers the
Hyers-Ulam problem and we prove a non-linear generalization of the Banach-Stone
theorem which improves the results of Jarosz and more recent results of
Dutrieux and Kalton.
