Packings in classical Banach spaces

Abstract

We obtain several new results on the simultaneous packing and covering constant ${\hat{I}}^3(\mathcal{X})$ 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 ${\hat{I}}^3(\mathcal{X})=1$ and the worst that ${\hat{I}}^3(\mathcal{X})=2$. Our first main result is that ${\hat{I}}^3(\mathcal{X})>1$ 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 $\mathrm{dens}(\mathcal{X})<\hat{\mathrm{I}}º$ and $1\le r\le p<\infty$; (iii) $γ(L_p(μ))= γ^*(L_p(μ))= \frac{2}{2^{1/p}}$ for $1\le p\le 2$ and every measure $\hat{I}¼$; (iv) there exist reflexive (resp. octahedral) Banach spaces $\mathcal{X}$ with ${\hat{I}}^3(\mathcal{X})=2$. We leave a large area open for further research and we indicate several possible directions.