A simple question from someone new to the field:

In a metric space, the Hausdorff dimension of a subset is defined by covering the subset with $\epsilon$-balls and looking at how the number of required balls grows as a power or $\epsilon$ in the limit $\epsilon \to 0$.

My question is this: If my space is $\mathbb{R}^n$ and the metric $d_p$ is the metric induced by the $p$-norm, $p \in (1, \infty]$, does the Hausdorff dimension of a subset $A \subseteq \mathbb{R}^n$ depend on the choice of $p$?

(My initial thought was that every $d_p$-ball has the same $d_q$-dimension for all $q$ (namely $n$), so that I believe all the dimensions should coincide, but I'm sure I'm overlooking something.)

Thanks!