A metric characterization of snowflakes of Euclidean spaces

Abstract

We give a metric characterization of snowflakes of Euclidean spaces. Namely, a metric space is isometric to ${\mathbb{R}}^n$ equipped with a distance $(d_{\mathrm{E}}{)}^{\epsilon }$, for some $n\in {\mathbb{N}}_0$ and $\epsilon \in (0,1]$, where $d_{\mathrm{E}}$ is the Euclidean distance, if and only if it is locally compact, $2$-point isometrically homogeneous, and admits dilations of any factor.