A metric characterization of snowflakes of Euclidean spaces

TL;DR

This paper provides a precise metric-based characterization of snowflake spaces derived from Euclidean spaces, identifying key geometric properties that define such spaces.

Contribution

It establishes necessary and sufficient conditions for a metric space to be isometric to a Euclidean snowflake, linking local compactness, homogeneity, and dilation properties.

Findings

Characterization of Euclidean snowflakes via metric properties

Identification of local compactness, homogeneity, and dilations as key features

Provides a complete metric criterion for Euclidean snowflakes

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 $(d_{\rm E})^\epsilon$, for some $n\in \mathbb N_0$ and $\epsilon\in (0,1]$, where $d_{\rm E}$ is the Euclidean distance, if and only if it is locally compact, $2$-point isometrically homogeneous, and admits dilations of any factor.