On the Hausdorff dimension of the residual set of a packing by smooth curves

Abstract

Let a planar residual set be a set obtained by removing countably many disjoint topological disks from an open set in the plane. We prove that the residual set of a planar packing by curves that satisfy a certain lower curvature bound has Hausdorff dimension bounded away from 1, quantitatively, depending only on the curvature bound. As a corollary, the residual set of any circle packing has Hausdorff dimension uniformly bounded away from 1. This result generalizes the result of Larman, who obtained the same conclusion for circle packings inside a square. We also show that our theorem is optimal and does not hold in general without lower curvature bounds. In particular, we construct packings by strictly convex, smooth curves whose residual sets have dimension 1. On the other hand, we prove that any packing by strictly convex curves cannot have $\sigma$-finite Hausdorff 1-measure.

32 pages, 22 figures

References in corpus (1)

• Monotone Sobolev functions in planar domains: level sets and smooth approximation

Cited by in corpus (2)

• CNED sets: countably negligible for extremal distances
• Conformal uniformization of planar packings by disk packings