Dimension-free estimates for covering functionals of simplices and $\ell_p$ balls
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910277190025216 |
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| author | Chen, Feifei He, Chan Wu, Senlin |
| author_facet | Chen, Feifei He, Chan Wu, Senlin |
| contents | We study \(Γ_{2^n}(K)\), the least positive number \(γ>0\) such that an \(n\)-dimensional convex body \(K\) can be covered by \(2^n\) translates of \(γK\). For \(n\)-simplices \(Δ_n\), we prove that \(Γ_{2^n}(Δ_n)\), as a sequence in \(n\), tends to \(1/2\). For the cross-polytope \(B_1^n\), we show that \(Γ_{2^n}(B_1^n)\leq5/6\) holds for all \(n\geq2\), and that \(\limsup_{n\to\infty}Γ_{2^n}(B_1^n)\leq0.641\cdots\). Finally, we prove the existence of a constant \(κ_*<1\) such that \(Γ_{2^n}(B_p^n)\leqκ_*\) for all \(n\geq2\) and all \(p\in[1,\infty]\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01082 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dimension-free estimates for covering functionals of simplices and $\ell_p$ balls Chen, Feifei He, Chan Wu, Senlin Metric Geometry 52C17, 52A21, 46B20 We study \(Γ_{2^n}(K)\), the least positive number \(γ>0\) such that an \(n\)-dimensional convex body \(K\) can be covered by \(2^n\) translates of \(γK\). For \(n\)-simplices \(Δ_n\), we prove that \(Γ_{2^n}(Δ_n)\), as a sequence in \(n\), tends to \(1/2\). For the cross-polytope \(B_1^n\), we show that \(Γ_{2^n}(B_1^n)\leq5/6\) holds for all \(n\geq2\), and that \(\limsup_{n\to\infty}Γ_{2^n}(B_1^n)\leq0.641\cdots\). Finally, we prove the existence of a constant \(κ_*<1\) such that \(Γ_{2^n}(B_p^n)\leqκ_*\) for all \(n\geq2\) and all \(p\in[1,\infty]\). |
| title | Dimension-free estimates for covering functionals of simplices and $\ell_p$ balls |
| topic | Metric Geometry 52C17, 52A21, 46B20 |
| url | https://arxiv.org/abs/2606.01082 |