Packings in classical Banach spaces

We obtain several new results on the simultaneous packing and covering constant $\gamma(\mathcal{X})$ of a Banach space $\mathcal{X}$, and its lattice counterpart $\gamma^*(\mathcal{X})$. These constants measure how efficient a (lattice) packing by unit balls in $\mathcal{X}$ can be, the optimal case being that $\gamma(\mathcal{X})= 1$ and the worst that $\gamma(\mathcal{X})= 2$. Our first main result is that $\gamma(\mathcal{X})> 1$ 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) $\gamma^*(\mathcal{X})= 1$ when $\mathcal{X}$ is a separable octahedral Banach space, or $\mathcal{X}= \mathcal{C}(\mathcal{K})$, where $\mathcal{K}$ is zero-dimensional; (ii) $\gamma(\ell_p(\kappa)\oplus_r \mathcal{X})= \gamma^*(\ell_p(\kappa)\oplus_r \mathcal{X})= \frac{2}{2^{1/p}}$, whenever $\rm{dens}(\mathcal{X})< \kappa$ and $1\leq r\leq p< \infty$; (iii) $\gamma(L_p(\mu))= \gamma^*(L_p(\mu))= \frac{2}{2^{1/p}}$ for $1\leq p\leq 2$ and every measure $\mu$; (iv) there exist reflexive (resp. octahedral) Banach spaces $\mathcal{X}$ with $\gamma(\mathcal{X})= 2$. We leave a large area open for further research and we indicate several possible directions.