Let $X$ be a locally compact metric space of integer Hausdorff dimension $n$. Let $K\subset X$ be a compact subset. Let $\{B_i\}_i$ be a finite family of balls covering $K$. One may assume that all balls have the same radius, but it might be unnecessary.

Is it true that one can choose a subcovering such that every point of $K$ is covered by at most $N$ balls, where $N$ depends on $n$ only?

This question might be rather trivial to experts; I have very little experience with the subject.


The answer is no.

Hausdorff dimension does not reflect any global geometry. Say you can construct a metric graph which approximates any compact length-metric space (as well as a finite metric space which approximates any compact metric space).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.