Improved packing of hypersurfaces in $\mathbb R^d$
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866929663164547072 |
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| author | Chen, Xianghong Yang, Tongou Zhong, Yue |
| author_facet | Chen, Xianghong Yang, Tongou Zhong, Yue |
| contents | For $d\ge 1$, we construct a compact subset $K\subseteq \mathbb {R}^{d+1}$ containing a $d$-sphere of every radius between $1$ and $2$, such that for every $δ\in (0,1)$, the $δ$-neighbourhood of $K$ has Lebesgue measure $\lesssim |\log δ|^{-2/d}$. This is the smallest possible order when $d=2$, and improves a result of Kolasa-Wolff (Pacific J. Math., 190(1):111-154, 1999). Our construction also generalises to Holder-continuous families of $C^{2,α}$ hypersurfaces with nonzero Gaussian curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03532 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved packing of hypersurfaces in $\mathbb R^d$ Chen, Xianghong Yang, Tongou Zhong, Yue Classical Analysis and ODEs 42B99 For $d\ge 1$, we construct a compact subset $K\subseteq \mathbb {R}^{d+1}$ containing a $d$-sphere of every radius between $1$ and $2$, such that for every $δ\in (0,1)$, the $δ$-neighbourhood of $K$ has Lebesgue measure $\lesssim |\log δ|^{-2/d}$. This is the smallest possible order when $d=2$, and improves a result of Kolasa-Wolff (Pacific J. Math., 190(1):111-154, 1999). Our construction also generalises to Holder-continuous families of $C^{2,α}$ hypersurfaces with nonzero Gaussian curvature. |
| title | Improved packing of hypersurfaces in $\mathbb R^d$ |
| topic | Classical Analysis and ODEs 42B99 |
| url | https://arxiv.org/abs/2501.03532 |