The Minkowski lower bound for packing hyperspheres of unit radii in $\mathbb{R}^n$ states that the density of hyperspheres satisfies, for all $n\geq 2$ \begin{equation} \Delta_n \geq \frac{\zeta(n)}{2^{n-1}}, \end{equation} where $\zeta(n) \rightarrow 1$ as $n$ tends to infinity.
My question is whether boundary conditions in the definition of density affect the asymptotic behavior of this bound. Any reasonable definition of density would be something like: \begin{equation} \Delta_n = \lim_{\rho \rightarrow \infty} \frac{N(\rho \mathscr{B}_n)}{\mu(\rho \mathscr{B}_n)}, \end{equation} where $\mathscr{B}_n$ is a subset of $\mathbb{R}^n$, $\rho$ is a uniform scaling factor for $\mathscr{B}_n$, $\mu(\cdot)$ indicates measure, and $N(\cdot)$ indicates the number of balls.
The Minkowski bound implies that, as $n \rightarrow \infty$ \begin{equation} \Delta_n \geq \frac{1}{2^{n-1}} + o(1). \end{equation} Does this asymptotic behavior hold independent of boundary conditions on $\mathscr{B}_n$, or should $\mathscr{B}_n$ be some "regular" shape, like a ball or a box, or some other convex shape?
What if $\rho = \rho(n)$, for example the dimension grows at a faster rate than the scaling factor used to define the density, does the same asymptotic behavior hold?