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arXiv · 2501.03532

Improved packing of hypersurfaces in $\mathbb R^d$

Xianghong Chen·Tongou Yang·Yue Zhong
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Abstract

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 $\delta\in (0,1)$, the $\delta$-neighbourhood of $K$ has Lebesgue measure $\lesssim |\log \delta|^{-2/d}$. 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 $C^{2,\alpha}$ hypersurfaces with nonzero Gaussian curvature.

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Xianghong Chen, Tongou Yang, Yue Zhong. 2025-01-07. Improved packing of hypersurfaces in $\mathbb R^d$. https://arxiv.org/abs/2501.03532

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