Fix $U$ unitary over $\mathbb{R}^{K},$ take $U_n=I_{n\times n}\otimes U$ and denote the unit ball at 0 in $\mathbb{R}^n$ as $B^n$. For $d_1,\dots,d_K>0$, fix $S_n:=U_n\left(\prod_{k=1}^K d_k B^n\right).$ How many translates of $(B^n)^K$ does it take to cover and/or pack $S_n$?
Alternatively, if $L_n$ is a random determinant-$\operatorname{vol}(B^n)$ lattice in $\mathbb{R}^{n}$ then what is $\lim _n \frac{1}{n}\log|(L_n)^K\cap (S_n+x_n)|$ for $x_n\sim \mathrm{Unif}(B^n)^K$?
I believe these are all some fairly strong function of $U$ and $(d_k)_{k=1}^K$. I have looked at some old geometry of numbers results by Rogers, but cannot find any results that bound the moments of $|(L_n)^K\cap S_n|$ strongly enough that the limit exists.
