# metric density of a set in the plane with respect to distinct metrics

Let $A \subseteq \mathbb{R}^2$ be Borel (or even open or closed), let $\mathbf{0} = ( 0 , 0)$, and let $\lambda$ be the Lebesgue measure in $\mathbb{R}^2$. Let $\mathcal{D}^2_A ( \mathbf{0} ) = \lim_{ \varepsilon \to 0} \lambda ( A \cap B^2 ( \mathbf{0} , \varepsilon )) / \lambda ( B^2 ( \mathbf{0} , \varepsilon ) )$ be the density of the point $\mathbf{0}$ in the set $A$ computed using the euclidean metric, so that $B^2 ( \mathbf{0} , \varepsilon ) = \{ ( x , y ) \mid x^2 + y^2 < \varepsilon^2 \}$. Let $\mathcal{D}^\infty_A ( \mathbf{0} )$ be defined similarly, but using the sup metric, so that $B^\infty ( \mathbf{0} , \varepsilon ) = ( - \varepsilon , \varepsilon ) \times ( - \varepsilon , \varepsilon )$.

It is not hard to construct sets $A$ such that both densities exist and are different, and other examples where both densities exist and are equal.

Suppose $\mathcal{D}^2_A ( \mathbf{0} )$ is defined, i.e. the limit exists; must $\mathcal{D}^\infty_A ( \mathbf{0} )$ be defined as well?

What about the analogous problem in dimension $\geq 3$?

P.S.: I have posted the same question also on math-stack exchange.