Packings in classical Banach spaces
Carlo Alberto De Bernardi, Tommaso Russo, Şeyda Sezgek, Jacopo Somaglia
TL;DR
This paper investigates the simultaneous packing and covering constants $\gamma(\mathcal{X})$ and $\gamma^*(\mathcal{X})$ in infinite-dimensional Banach spaces, revealing a nuanced geometric picture that ties unit-ball smoothness, octahedrality, and moduli of convexity to packing efficiency. It introduces two general methods—discrete subgroup constructions and $\phi$-octahedral moduli—to establish sharp upper and lower bounds and to compute exact values for many classical spaces, including $\ell_p$, $L_p$, and $\mathcal{C}(\mathcal{K})$ spaces. Notably, it proves $\gamma(\mathcal{X})>1$ whenever the unit ball has an LUR point, and that $\gamma^*(\mathcal{X})=1$ for separable octahedral spaces and for $\mathcal{C}(\mathcal{K})$ with zero-dimensional $\mathcal{K}$, while constructing explicit spaces with $\gamma(\mathcal{X})=2$. These results advance understanding of how geometric properties of the norm influence the density- and lattice-based packing/covering efficiency, and they open multiple avenues for further exploration, including renorming and extensions to broader classes of Banach spaces.
Abstract
We obtain several new results on the simultaneous packing and covering constant $γ(\mathcal{X})$ of a Banach space $\mathcal{X}$, and its lattice counterpart $γ^*(\mathcal{X})$. These constants measure how efficient a (lattice) packing by unit balls in $\mathcal{X}$ can be, the optimal case being that $γ(\mathcal{X})= 1$ and the worst that $γ(\mathcal{X})= 2$. Our first main result is that $γ(\mathcal{X})> 1$ whenever $B_\mathcal{X}$ admits a LUR point, which leads us to a negative answer to a question of Swanepoel. We also develop general methods to compute these constants for a large class of spaces. As a sample of our findings: (i) $γ^*(\mathcal{X})= 1$ when $\mathcal{X}$ is a separable octahedral Banach space, or $\mathcal{X}= \mathcal{C}(\mathcal{K})$, where $\mathcal{K}$ is zero-dimensional; (ii) $γ(\ell_p(κ)\oplus_r \mathcal{X})= γ^*(\ell_p(κ)\oplus_r \mathcal{X})= \frac{2}{2^{1/p}}$, whenever $\rm{dens}(\mathcal{X})< κ$ and $1\leq r\leq p< \infty$; (iii) $γ(L_p(μ))= γ^*(L_p(μ))= \frac{2}{2^{1/p}}$ for $1\leq p\leq 2$ and every measure $μ$; (iv) there exist reflexive (resp. octahedral) Banach spaces $\mathcal{X}$ with $γ(\mathcal{X})= 2$. We leave a large area open for further research and we indicate several possible directions.