Tilings and coverings by balls in $\ell_1$
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| Format: | Preprint |
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2026
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| _version_ | 1866915940615061504 |
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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 | A famous result of Klee from 1981 is that the Banach space $\ell_1(κ)$ admits a disjoint tiling by balls of radius $1$, for all cardinals $κ$ with $κ^ω=κ$. Klee also observed that the smallest cardinal in which such a tiling might exist is $κ= 2^{\aleph_0}$, leaving open the question whether, for $κ< 2^{\aleph_0}$, $\ell_1(κ)$ might admit a tiling by balls at all. Our main result answers this question in the negative, proving in particular that $\ell_1$ does not admit any tiling by balls. We also give a companion result about star-$n$-finite coverings by balls of $\ell_1(κ)$ and we give a construction of a star-finite tiling of $\mathcal{X} \oplus_\infty c_{00}$, for each space $\mathcal{X}$ whose dimension is at most countable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_15092 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tilings and coverings by balls in $\ell_1$ De Bernardi, Carlo Alberto Russo, Tommaso Sezgek, Şeyda Somaglia, Jacopo Functional Analysis A famous result of Klee from 1981 is that the Banach space $\ell_1(κ)$ admits a disjoint tiling by balls of radius $1$, for all cardinals $κ$ with $κ^ω=κ$. Klee also observed that the smallest cardinal in which such a tiling might exist is $κ= 2^{\aleph_0}$, leaving open the question whether, for $κ< 2^{\aleph_0}$, $\ell_1(κ)$ might admit a tiling by balls at all. Our main result answers this question in the negative, proving in particular that $\ell_1$ does not admit any tiling by balls. We also give a companion result about star-$n$-finite coverings by balls of $\ell_1(κ)$ and we give a construction of a star-finite tiling of $\mathcal{X} \oplus_\infty c_{00}$, for each space $\mathcal{X}$ whose dimension is at most countable. |
| title | Tilings and coverings by balls in $\ell_1$ |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2604.15092 |