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.

  • $\begingroup$ 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

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.

  • $\begingroup$ 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

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