The $d$-dimensional Hausdorff (outer) measure of a set $A ⊆ \mathbb{R}^n$ is usually defined to be $$ \mathcal{H}^d(A) = \lim_{δ \to 0} \inf \left\{ \sum_i \operatorname{diam}(U_i)^d ~\middle|~ A ⊆ \bigcup_{i} U_i, \operatorname{diam} U_i < δ\right\} $$ However the class of sets to which the $U_i$ are supposed to belong varies: Some authors require them to be balls while some allow all open sets. Cubes seems like another natural choice.
Evidently the different definitions agree up to (dimensional) constants. They also agree for $d = n$ because then all three definitions of $\mathcal{H}^d$ coincide with the $n$-dimensional Lebesgue measure. Do the different definitions coincide in general?