If $f$ is a distance preserving map from Euclidean space to itself, then I can show that it is infact a composition of linear maps and translations. If $f$ is a bijective and distance preserving between finite normed vector spaces, then
$f$ can not be a composition of linear maps and translations ?
How can we prove this ?
Example : If $V=(\mathbb{R}^2,\|\ \|)$ and $W = (\mathbb{R}^2,\|\ \|_\infty)$ has a sup norm, where $\|\ \|$-unit ball is convex hull of six points $e^{i\frac{\pi}{3} k},\ k=1,2,\cdots,6$, then there is no distance preserving map.