Injectivity in Banach spaces and the Mazur-Ulam theorem on isometries

A mapping $f$ of an open subset $U$ of a Banach space $X$ into another Banach space $Y$ is said to be $(m,M)$-isometric if it is a local homeomorphism for which $M \geqslant {D^ + }f(x)$ and $m \leqslant {D^ - }f(x)$ for all $x \in U$, where ${D^ + }f(x)$ and ${D^ - }f(x)$ are, respectively, the upper and lower limits of $\vert f(y) - f(x)\vert/\vert y - x\vert\;{\text{as}}\;y \to x$. For $0 < \rho \leqslant 1$ we find a number $\mu (\rho ) > 1$ which has the following property: Let $X$ and $Y$ be Banach spaces and let $U$ be an open convex subset of $X$ containing a ball of radius $r$ and contained in the concentric ball of radius $R$. Then all $(m,M)$-isometric mappings of $U$ into $Y$ are injective if $M/m < \mu (r/R)$. We also derive similar injectivity criteria for a more general class of connected open sets $U$. The basic tool used is an approximate version of the Mazur-Ulam theorem on the linearity of distance preserving transformations between normed linear spaces.