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Abstract

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 perturbed 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.