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.

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 $X$.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 $\epsilon$.

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

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

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

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.

## 4 comments:

Real line is not a snowflake of anything (unless you really want to allows epsilon=1...). Indeed, otherwise |x-y|^p would be comparable to a metric for some p>1, which it is not (the triangle inequality, applied to fine partitions, would collapse such a metric). Laakso's appendix to the Tyson-Wu snowflake paper gives a very nice answer to the "snowflake or not" question in general.

http://www.acadsci.fi/mathematica/Vol30/tyson.html

thanks, L. it sounded like this sort of thing has been investigated before. i'll check out the reference.

Also, If it were a snowflake of something, then it would be homeomorphic to space with Hausdorff dimension less than one, which would therefore have topological dimension less than one. Same applies for R^n, or more generally, any space whose Hausdorff and topological dimensions coincide.

The above, by the way, is one way to see that a snowflake space has no rectifiable curves.

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