Packings in classical Banach spaces
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arXiv
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| Format: | Preprint |
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2026
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| author | De Bernardi, Carlo Alberto Russo, Tommaso Sezgek, Şeyda Somaglia, Jacopo |
| author_facet | De Bernardi, Carlo Alberto Russo, Tommaso Sezgek, Şeyda Somaglia, Jacopo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_12934 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Packings in classical Banach spaces De Bernardi, Carlo Alberto Russo, Tommaso Sezgek, Şeyda Somaglia, Jacopo Functional Analysis Metric Geometry 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. |
| title | Packings in classical Banach spaces |
| topic | Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2602.12934 |