论文标题
在高维
On the lower bound for packing densities of superballs in high dimensions
论文作者
论文摘要
用半径$ r $和中心$ {\ boldsymbol 0} $在$ \ mathbb {r}^n $中定义超级球，为set $$ \ left \ left \ {{\ boldsymbol X} \ in \ Mathbb {r}^n：\ sum_ {j = 1}^{m} \ left（x_ {x_j+1}^2+x__ {k_j+2}^2+\ cdots+x_ r^p \ right \}，0 = k_1 <k_2 <\ cdots <k_ {m+1} = n，$$，这是$ \ ell_p $ -balls的概括。我们给出了两个新的证据，以庆祝的结果，即$ 1 <p \ leq2 $，$ \ mathbb {r}^n $的超级球的翻译包装密度为$ω（n/2^n）$。这种结合首先是由施密特获得的，随后由罗杰斯和施密特改善了恒定因子。我们的第一个证明是基于硬超级球模型，第二个证明是基于图的独立性数。我们还研究了包装的熵，该包装可以衡量这种包装的丰富。
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 $Ω(n/2^n)$. 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.
