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

TL;DR

This paper constructs a compact set in or rom 1 to 2 that contains spheres of all radii in that range, with a or  neighborhoods having measure that diminishes optimally as or  approaches zero, improving previous results.

Contribution

It introduces a new construction of a compact set with optimal measure decay properties for neighborhoods of hypersurfaces, extending to smooth families of curved hypersurfaces.

Findings

Optimal measure decay rate or  neighborhoods as or   approaches zero.

Construction of a set containing all spheres of radii in [1,2].

Generalization to families of smooth, curved hypersurfaces.

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.