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.
Categories: math.CA