Let $(X,d)$ be a metric space. For any $0<\epsilon<1$, we call the metric space $(X,d^{\epsilon})$; where $d^{\epsilon}(x,y)\triangleq (d(x,y))^{\epsilon}$ the $\epsilon$-snowflake of $(X,d)$.


My question is, given $(X,d)$ and some $\epsilon\in(0,1)$, under what conditions can we deduce that there exists some other metric space $(Z,\rho)$ such that $(X,d)$ is the $\epsilon$-snowflake of $(Z,\rho)$; i.e.:
$$
(X,d)=(Z,\rho^{\epsilon})?
$$



When is a metric space a snowflake?
You may be interested in the following paper:


Jeremy T. Tyson and Jang-Mei Wu, Characterizations of Snowflake Metric Spaces. Annales Academiae Scientiarum Fennicae Mathematica. Volume 30, 2005, 313-336.
<https://www.emis.de/journals/AASF/Vol30/tyson.pdf>


The paper contains a number of equivalent characterizations of the property that a metric space $X$ is *bi-Lipschitz equivalent to* a snowflake. Some of the characterizations in the paper require that the space is embedded in a certain Banach space, and some do not.


