On the lower bound for packing densities of superballs in high dimensions

Define the superball with radius $r$ and center ${\boldsymbol 0}$ in $\mathbb{R}^n$ to be the set $$\left\{{\boldsymbol x}\in\mathbb{R}^n:\sum_{j=1}^{m}\left(x_{k_j+1}^2+x_{k_j+2}^2+\cdots+x_{k_{j+1}}^2\right)^{p/2}\leq r^p\right\},0=k_1<k_2<\cdots<k_{m+1}=n,$$ which is a generalization of $\ell_p$-balls. We give two new proofs for the celebrated result that for $1<p\leq2$, the translative packing density of superballs in $\mathbb{R}^n$ is $\Omega(n/2^n)$. This bound was first obtained by Schmidt, with subsequent constant factor improvement by Rogers and Schmidt, respectively. Our first proof is based on the hard superball model, and the second proof is based on the independence number of a graph. We also investigate the entropy of packings, which measures how plentiful such packings are.