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

TL;DR

This paper establishes lower bounds on the packing densities of superballs in high-dimensional spaces using two novel proofs, and explores the entropy of such packings to understand their abundance.

Contribution

It provides two new proofs for the lower bound on superball packing densities and investigates the entropy of packings, enhancing understanding of high-dimensional packing behavior.

Findings

Lower bound of (n/2^n) for packing density of superballs in high dimensions.

Two distinct proofs: one using the hard superball model, another based on graph independence.

Analysis of packing entropy indicating the abundance of packings.

Abstract

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.