3
$\begingroup$

Let $M$ be a metric space and let $U \subset M$ be open. Then the Hausdorff dimension of $U$ is defined in the usual way. If there is a single dimension number $d$ that is the Hausdorff dimension of every open set $U$ in $M$ I say the metric space $M$ has uniform Hausdorff dimension $d$.

What I want to know is whether some concept like this already exists, possible under a different name.

I want to use this as a property that 'nice' metric spaces have. A simple example: Let $I$ denote the unit interval and $D$ the unit ball in $\mathbb{R}^2$. Then both $I$ and $D$ have a uniform Hausdorff dimension but the disjoint union of $I$ and $D$ does not.

$\endgroup$
  • $\begingroup$ Well, the unit interval $I$ is not open as a subset of $\mathbb{R}^2$, so it doesn't invalidate the "uniform Hausdorff dimension" of the disjoint union of $I$ and $D$ being 2 according to your definition. $\endgroup$ – N Unnikrishnan Sep 6 '16 at 7:01
4
$\begingroup$

This is not an answer, just a comment:

I have never heard a name for the property you cite.

There is a significantly stronger property that is commonly used and does have a name: A metric space is called Ahlfors $n$-regular if there is a constant $K$ such that for each closed ball $B(x,r)$ in the space ($0<r\leq\text{diam}(X)$), $$ K^{-1} r^n \leq H^n(B(x,r)) \leq K r^n $$ where $H^n$ is $n$-dimensional Hausdorff measure.

In your example, $I$ and $D$ are of course Ahlfors $1$-regular and Ahlfors $2$-regular, respectively, while the union is not Ahlfors regular for any $n$.

One could consider any number of intermediate definitions between this concept and yours.

$\endgroup$
  • $\begingroup$ Thanks, this was exactly the kind of thing I was looking for. Is there a good text book or another standard reference where this is defined/ explained? $\endgroup$ – quarague Sep 6 '16 at 7:38
  • $\begingroup$ If you google the term Ahlfors regular, you will find a lot of papers and books that use it. A standard reference on metric spaces that uses this concept is Juha Heinonen's book Lectures on Analysis on Metric Spaces. $\endgroup$ – user98074 Sep 6 '16 at 12:30

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.