Packings in classical Banach spaces

TL;DR

This paper investigates the efficiency of packings and coverings in Banach spaces, providing new bounds, methods for computation, and examples of spaces with specific packing constants, advancing understanding of geometric properties in functional analysis.

Contribution

The paper introduces new results on packing and covering constants in Banach spaces, including conditions for their values, computational methods, and examples of spaces with extremal constants.

Findings

$ ext{gamma}( ext{X}) > 1$ if $B_ ext{X}$ has a LUR point

$ ext{gamma}^*( ext{X})=1$ for certain octahedral and zero-dimensional spaces

$ ext{gamma}( ext{ell}_p( ext{kappa}) igoplus_r ext{X})= 2/2^{1/p}$ for specified spaces

Abstract

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.