Injectivity of quasi-isometric mappings of balls

Abstract:

Let $X$ and $Y$ be real Banach spaces. A mapping $f$ of an open subset $R$ of $X$ into $Y$ 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 $R$, where ${D^ + }f(x)$ and ${D^ - }f(x)$ are, respectively, the upper and lower limits of $|f(y) - f(x)|/|y - x|$ as $y \to x$. We show that if $R$ is a ball then all $(m,M)$isometric mappings of $R$ are injective provided that $M/m < 1.114 \ldots$ and we also give some numerical improvements of similar results of F. John for the special case that $X$ is a Hilbert space.