Bi-Lipschitz maps and identity

Question

Let $n,m$ be positive integers and consider a bi-Lipschitz linear map $L:\mathbb{R}^n\rightarrow \mathbb{R}^d$ where $d\leq n$ and $\mathbb{R}^d$ is viewed as a subspace of $\mathbb{R}^n$. Is it possible to bound the distance: $$ \sup_{x\in [0,1]^n}\,\|x-L(x)\| $$ in terms of the (bi)-Lipschitz constants of $L$?

Answer 1

If $k$ is the Lipschitz constant of $L$, then for any $x$, $\|L(x)\|\leqslant k\|x\|$. The triangle inequality thus yields $$ \|x-L(x)\| \leqslant \|x\|+ \|L(x)\| \leqslant (1+k)\|x\| $$

This bound is sharp since for $L(x) = -x$, it holds that $$ \|x-L(x)\| = 2\|x\| $$