it's cold, and i'm thinking about snowflakes.

lately it seems like all i write about is teaching. i suppose that's where the recent frustration comes from, especially with this class that is new to me.

as for research frustrations, i can handle those. this line of work is so full of setbacks (and occasionally, advances) that i've developed a sense of humor about it.

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that said, before the teaching rants begin anew, here's a little problem that came to mind today. it's nothing serious (not for research or anything) but right now i can't seem to answer it.

given a metrιc space $(X,d)$ and a number $0 \leq \epsilon \leq 1$, we call $X^\epsilon := (X,d^\epsilon)$ the $\epsilon$-snowflake of $X$.

it's not hard to show that $X^\epsilon$ is also a metrιc space; usually one finds such a problem in textbooks.

question: is the real line the snowflake of some metrιc space? what about the plane, or other Euclιdean spaces? for those experts out there: if the answer is yes, then can one choose the metrιc space to be doubling?

(epilogue: the answer is easy. thanks for the comments, guys.)

as for how it came up, i was thinking about embeddings of metric spaces, including the recent result discussed in a preprint of naοr and neιman. apparently assοuad's embedding theorem now works with a fixed dimension, regardless of the snowflaking parameter $\epsilon$.

roughly speaking, the doubling condition on a metrιc space is a finite-dimensιonality condition. so the naοr-neιman result suggests (to me) that one shouldn't have to push too far from the dimension of the given space in order to embed it.