I am wondering if the metric space $(\mathbb{R}^n, d)$, where $d(x,y) = \|x-y\|$ is the standard euclidean metric, is isomorphic to itself under the squared euclidean metric: $d^2(x,y) = \|x-y\|^2$.
That is, for all pairs of points $x,y \in \mathbb{R}^n$, does there exist a bijection, $f : \mathbb{R}^n \to \mathbb{R}^n$ such that $d(x,y) = d^2(f(x),f(y))$ ?
More generally, is it true if we consider metrics derived from arbitrary p-norms, $d^p(x,y) = \|x-y\|^p$, $p\ge 1$ ?
Finally, if the answers to the above questions are "no", then can we at least get a "yes" for some values of $n$ (where $n \ge 1$)?
Or can we get a "yes" by relaxing the requirement that $f$ be a bijection from $\mathbb{R}^n$ to itself, and instead allow $f$ to embed points in some other number of dimensions, "m", i.e. $f : \mathbb{R}^n \to \mathbb{R}^m$, with $d(x,y) = d^p(f(x),f(y))$ ?