Almost linearity of $\epsilon$-bi-Lipschitz maps between real Banach spaces

Abstract:

Let $X$ and $Y$ be real Banach spaces. A map $f$ between $X$ and $Y$ is called an $\epsilon$-bi-Lipschitz map if $(1-\epsilon )\|x-y\| \le \|f(x) -f(y)\| \le (1+\epsilon )\|x- y\|$ for all $x, y\in X$. In this note we show that if $f$ is an $\epsilon$-bi-Lipschitz map with $f(0)=0$ from $X$ onto $Y$, then $f$ is almost linear. We also show that if $f:X\longrightarrow Y$ is a surjective $\epsilon$-bi-Lipschitz map with $f(0)=0$, then there exists a linear isomorphism $I:X\to Y$ such that \[ \|I(x)-f(x)\| \le E(\epsilon , \alpha )(\|x\|^\alpha +\|x\|^{2-\alpha })\] where $E(\epsilon ,\alpha )\to 0$ as $\epsilon \to 0$ and $0<\alpha <1$.