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

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

17 pages, 2 figures